Giuseppe Ferraguto MACROECONOMICS The manual includes about one hundred questions, most in multiple parts and drawn from several years of exams at Bocconi University, on the models (IS-LM, IS-LM-PC, etc.) and topics (the macroeconomic equilibrium of a closed economy, the labor market and unemployment, inflation, the open economy, government debt, economic growth) covered by most introductory courses on Macroeconomics. The main objective of the problems is to help readers grasp the economic reasoning and intuition underlying the main conclusions of the discipline – the aspect of Macroeconomics, and more in general of Economics, that students find the most difficult to master, but that will turn out to be the most useful in their future. MACROECONOMICS 6th EDITION MyBook http://mybook.egeaonline.it tools 196-1c_2b.indd 1 MyBook Problems and questions GIUSEPPE FERRAGUTO is Associate Professor of Economics at Bocconi University, and director of the course on Macroeconomics offered at the same institution. Euro 36,00 Giuseppe Ferraguto MyBook is the gateway to access accompanying resources (both text and multimedia), the BookRoom, the EasyBook app and your purchased books. 24 mm 17/01/20 12:14 TOOLS tools 196-1f.indd 1 10/01/20 15:23 tools 196-1f.indd 2 10/01/20 15:23 Giuseppe Ferraguto MACROECONOMICS Problems and questions 6th EDITION tools 196-1f.indd 3 10/01/20 15:23 Copyright © 2014, 2020 EGEA S.p.A. Via Salasco, 5 - 20136 Milano Tel. 02/5836.5751 – Fax 02/5836.5753 egea.edizioni@unibocconi.it - www.egeaeditore.it Tutti i diritti sono riservati, compresi la traduzione, l’adattamento totale o parziale, la riproduzione, la comunicazione al pubblico e la messa a disposizione con qualsiasi mezzo e/o su qualunque supporto (ivi compresi i microfilm, i film, le fotocopie, i supporti elettronici o digitali), nonché la memorizzazione elettronica e qualsiasi sistema di immagazzinamento e recupero di informazioni. Date le caratteristiche di Internet, l’Editore non è responsabile per eventuali variazioni di indirizzi e contenuti dei siti Internet menzionati. Sesta edizione: febbraio 2020 ISBN 978-88-7534-196-1 ISBN ebook 978-88-238-1855-2 Table of Contents Introduction VII Part I. Problems and Questions 1. The goods and financial markets 2. The IS-LM model 3. The labor market, the IS-LM-PC model, and inflation 4. Expectations, financial markets, and economic policies 5. The open economy 6. Government debt and economic growth 3 37 67 105 139 187 Part II. Solutions 1. The goods and financial markets 2. The IS-LM model 3. The labor market, the IS-LM-PC model, and inflation 4. Expectations, financial markets, and economic policies 5. The open economy 6. Government debt and economic growth V 219 255 285 323 357 405 Introduction A good exam question should test a range of abilities, so that the very best students are stretched to the limit, but the weaker students can still get part of it right. (…) Good exam questions are like a scarce natural resource. There are only so many you can mine, and you can't keep on using the best ones year after year. (…) Maybe the depressing job of grading exams is still better than the stressful job of writing them. Nick Rowe, “The depressing job of grading exams”, Worthwhile Canadian Initiative - A mainly Canadian economics blog, December 31, 2012. http://worthwhile.typepad.com This exercises and solutions manual includes the exam questions for recent editions (from the Spring semester 2010 onwards) of the course ‘Introduction to Economics − Macroeconomics’ offered at Bocconi University. In addition to those questions, it also presents an updated, and sometimes drastically revised, version of some of the exercises and problems discussed during the tutorials for the same course in recent, and not so recent, years. VII Macroeconomics. Problems and Questions The questions and problems in the following pages aim not only at familiarizing readers with the subject matter of Macroeconomics, but also at deterring students from a mechanical approach to the discipline and to the test. Their main objective is to help readers grasp the economic reasoning and intuition underlying the results derived in class – the aspect of Macroeconomics, and more in general of Economics, that at first students find the most difficult to master, but that will turn out to be the most useful and rewarding in their future, academic and extra-academic, careers. The manual is divided into two parts. In the first one, readers are encouraged to work out on their own problems having exactly the same format as those included in the final exam (a great training for the test!); the second part provides students with solutions and answers, thus giving them a sense of what they ought to know. This sixth edition also includes some questions − those marked with an asterisk − that are more difficult than those typically included in a final examination. While students are encouraged to arrive on their own to an answer, even just a partial one, to those questions, they are also invited to read very carefully the solutions to these more difficult, or longer, problems which are contained in the second part of the book. The concepts discussed there, often not covered by the textbook, are in fact needed to answer some of the other questions presented in this manual, as well as possible exam questions for the course. Although this book is being published to my name, the list of people entitled to be regarded as its co-authors is very long; at the very least, it should include the many colleagues responsible for each of the twenty or so classes of Bocconi undergraduates to which the course on Macroeconomics has been taught in recent years. I am especially grateful to Angelo Porta and Donato Masciandaro, my immediate predecessors in the role of course director, who have contributed in important ways to the process that has led to the current structure and contents of the final examinations, and therefore of this manual; to Elisa Borghi, Maria Giovanna Bosco, Michela Braga, Daniela Grieco, Antonella Mori and Francesco Scervini, the instructors that, during my tenure as director of the course, have acted as referees of the exam questions I have been writing, before they were submitted to our students; and, last but not least, to Patrice De Micco, Elisa Facchetti, Marcella Nicolini and, once again, to Maria Giovanna Bosco, Marco Mantovani, Antonella Mori and Francesco Scervini, who have read preliminary drafts of the manuscript and helped me improve it VIII Introduction greatly. Needless to say, all the remaining weaknesses and errors are my own responsibility. The list of debts I have incurred would not be complete without an explicit mention of the many students who, after sitting the exam, found the time to give (in writing, during my office hours, or in other ways) their feedback – sometimes positive, sometimes negative – on the questions it included. Their suggestions have always been taken seriously, although not always followed; in any case, all of them have helped make our course better. Finally, a special thanks goes to Laura Vaini for what she has done in the recent past, and to Alessandra Startari for what she is currently doing, for the students and the instructors of the course on Macroeconomics. Giuseppe Ferraguto For additional questions and problems, go to http://mybook.egeaonline.it. IX Part I Problems and Questions Chapter 1 - The goods and financial markets Macroeconomics. Problems and Questions Question 1 In the economy there are only two firms, firm A and firm B. Their operations in a given year can be summarized as follows (all figures are in thousands of Euros): Firm A Costs Revenues Wages 170 Sales to B 300 Purchases from B 50 Sales to consumers 400 Indirect taxes 30 Firm B Costs Revenues Wages 230 Sales to A 50 Purchases from A 300 Sales to consumers 500 Indirect taxes 20 Exports 100 Compute the economy’s Gross Domestic Product (GDP) using all the definitions of this variable that it is possible to employ in this case. 4 The goods and financial markets ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 5 Macroeconomics. Problems and Questions Question 2 a. What is meant by ‘GDP deflator’? How is the GDP deflator computed, and how is it used? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 6 The goods and financial markets b. Compared to the previous year (π‘π‘ − 1), in year π‘π‘ the economy’s GDP deflator has gone down by 1% and the real GDP growth rate has been equal to −2%. b.1 Compute the growth rate of nominal GDP for this economy in year π‘π‘. b.2 What is the rate at which the economy under consideration has been growing in year π‘π‘ ? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 7 Macroeconomics. Problems and Questions Question 3 The economy of a country in which only the goods market exists is described by the following system of equations, πΆπΆ = ππ0 πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ ππ = πποΏ½ + π‘π‘π‘π‘ where, as always, the positive constant ππ0 is autonomous consumption, π‘π‘ (a constant between zero and one) is the tax rate, πποΏ½ (> 0) is the portion of net taxes that does not depend on income, and the other symbols have the usual meaning. a. Having determined graphically in the figure below the equilibrium level of production οΏ½πποΏ½οΏ½ for this economy, derive the analytical expressions of πποΏ½ and of the multiplier implied by the model specified above. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ……………………………………………………………………………………… ……………………………………………………………………………………… 8 The goods and financial markets b. Suppose that the government cuts πποΏ½ (that, as you will remember, is the fraction of net taxes independent of income), so that βπποΏ½ < 0. Show in the graph the effects of this change and, using the results derived when answering the previous point of this question, derive the analytical expressions of the changes in equilibrium production, private saving, government saving and national saving caused by this decrease in πποΏ½. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 9 Macroeconomics. Problems and Questions Question 4 The economy of a country in which only the goods market exists is described by the following system of equations: πΆπΆ = ππ0 πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ − ππ1 ππ ππ = πποΏ½ + π‘π‘π‘π‘ where, as always, the positive constant ππ0 is autonomous consumption, π‘π‘ (a constant between zero and one) is the tax rate, πΊπΊΜ and οΏ½ππ (both greater than zero) are the portions of government spending and net taxes that do not depend on income, and the parameters ππ1 and ππ1 , with 0 < ππ1 − ππ1 < 1, are both positive. a. Having determined graphically in the figure below the equilibrium level of production οΏ½πποΏ½οΏ½ for this economy, derive the analytical expressions of πποΏ½, of autonomous spending π΄π΄ and of the multiplier implied by the model specified above. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 10 The goods and financial markets ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that the government raises, at the same time and by the same amount, both πΊπΊΜ and πποΏ½, so that βπΊπΊΜ = βπποΏ½ > 0. Show in the graph the effects of these changes and, using the results derived when answering the previous point of this question, derive the analytical expressions of the changes in equilibrium production, private saving, government saving and national saving caused by the increases in πΊπΊΜ and in πποΏ½. In your answer, make sure to discuss in each case why reaching a definite conclusion about the direction in which those variables will change is possible, or not possible. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 11 Macroeconomics. Problems and Questions Question 5 The goods market of a nation is described by the following equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) ππ = πποΏ½ + π‘π‘π‘π‘ πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ, where the parameter π‘π‘ is greater than zero (so that net taxes are increasing in the level of income), but smaller than one. a. Derive the expression for the equilibrium level of income and that of the multiplier for this economy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 12 The goods and financial markets b. Two economists, Mary and Paul, debate on the effects of an increase in the autonomous component of government spending πΊπΊΜ on the government deficit, π·π·π·π·π·π· = πΊπΊΜ − ππ. According to Mary, since an increase in πΊπΊΜ leads to a higher income, and net taxes are increasing in ππ, raising πΊπΊΜ has an uncertain effect on the deficit. For sufficiently high values of π‘π‘, the increase in tax revenues could be so large that the government deficit could end up falling. Economist Paul, on the other hand, believes that an increase in πΊπΊΜ would still increase the deficit, for any π‘π‘ < 1. Derive the expression for the change in the deficit, π₯π₯π₯π₯π₯π₯π₯π₯, when πΊπΊΜ varies by π₯π₯πΊπΊΜ > 0. Which of the two economists is right? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 13 Macroeconomics. Problems and Questions Question 6 The economy of a country in which only the goods market exists is described by the following system of equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ ππ = πποΏ½ where the various symbols have the usual meaning. a. Derive the expressions of equilibrium income and of the multiplier for this economy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ……………………………………………………………………………………… ………………………………………………………………………………………. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 14 The goods and financial markets b. Suppose that, when the economy was in the equilibrium position described above, investment rises, and that the Government cuts πποΏ½ by the same amount by which πΌπΌ Μ has gone up, so that βπΌπΌ Μ = −βπποΏ½ > 0. Show in the graph the effects of these two contemporaneous changes and, using the results derived when answering the previous point, derive the analytical expressions of the changes in equilibrium production, private saving, government saving and national saving they will cause. In your answer, make sure to discuss in each case why reaching a definite conclusion about the direction in which those variables will vary is possible, or not possible. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 15 Macroeconomics. Problems and Questions Question 7 The economy of a country in which only the goods market exists is described by the following system of equations, πΆπΆ = ππ0 πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ − ππππ ππ = πποΏ½ + π‘π‘π‘π‘ where, as always, the positive constant ππ0 is autonomous consumption, π‘π‘ (a constant between zero and one) is the tax rate, ππ (> 0) is the sensitivity to income of government spending on goods and services, and the other symbols have the usual meaning. a. Write down the analytical expressions of autonomous spending, equilibrium income, and the multiplier for this economy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 16 The goods and financial markets b. Suppose that the autonomous components of government purchases of goods and services and of net taxes are cut at the same time and by the same amount, so that βπΊπΊΜ = βπποΏ½ > 0. Determine the effect of this change on the government deficit, π·π·π·π·π·π· = πΊπΊ − ππ, prevailing when the goods market is in equilibrium. Make sure to explain if, and why, the sign of the change in government deficit will, or will not, depend on the fact that, in this economy, the parameter π‘π‘ is larger or smaller than the parameter ππ. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 17 Macroeconomics. Problems and Questions Question 8 The goods market of a country is described by the following model: πΆπΆ = ππ0 + ππ1 (ππ − ππ) ππ = πποΏ½ + π‘π‘π‘π‘ πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ, where the parameter π‘π‘ (the tax rate) is positive, but smaller than one, ππ1 (> 0) is the sensitivity of investment to income, ππ1 + ππ1 < 1, and the other symbols have the usual meaning. a. Derive the expressions of equilibrium income and of the multiplier for this economy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 18 The goods and financial markets b. Suppose that autonomous consumption and the autonomous component of net taxes rise at the same time and by the same amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ > 0. By how much will equilibrium income and national saving (the sum of private and public saving) change? Derive the expressions for the changes in those two variables, and explain if and why they will rise, fall or remain unchanged. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 19 Macroeconomics. Problems and Questions Question 9 Country Macro, where only the good market exists and prices are constant, is described by the following model: οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ πΆπΆ = ππ0 + ππ1 (ππ − ππ) + ππ ππππππ οΏ½ ππ = ππ πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ. In the equations above, 0 < ππ1 < 1, ππ1 > 0, 0 < ππ1 + ππ1 < 1, the parameter ππ (> 0) is the sensitivity of consumption to the financial and housing wealth οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ), and the other ππππππ (assumed to be exogenous, so that ππππππ = ππππππ symbols have the usual meaning. a. Derive the expressions of equilibrium income and that of the multiplier for this economy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 20 The goods and financial markets b. Suppose that individuals experience an increase in their financial and οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ > 0. Write down the expression of the housing wealth, so that π₯π₯ππππππ change in equilibrium private saving caused by this change. In particular, explain if, and why, private saving will rise, fall, or remain constant. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 21 Macroeconomics. Problems and Questions Question 10 The economy of a country consisting of the goods market only is described by the following equations: πΆπΆ = ππ1 (ππ − ππ) ππ = πποΏ½ + π‘π‘π‘π‘ πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ , where the parameters ππ1 and π‘π‘ are positive but smaller than one. In this country, autonomous consumption ππ0 is therefore equal to zero, government purchases of goods and services and investment are entirely exogenous, and net taxes depend on consumption – as they include not just an exogenous component (πποΏ½), but also a portion that is increasing in consumption (π‘π‘π‘π‘). a. Plugging the expression for net taxes given by the second equation into the first one, and solving for πΆπΆ, derive the expression that the consumption function takes on in this economy. Next, using the definition of (private) saving and the consumption function you have just derived, write down the (private) saving function. In this economy, is private saving still increasing in income, ππ, as it is in the standard case? Why, or why not? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 22 The goods and financial markets b. Using the results derived when answering the previous point, write down the expression of the government deficit (π·π·π·π·π·π·) for this economy. Is the country’s government deficit increasing or decreasing in the level of income, ππ? Why? Provide the economic intuition for your answer. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 23 Macroeconomics. Problems and Questions Question 11 a. The goods market of country Zeta is described by the following equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) ππ = πποΏ½ πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ, where 0 < ππ1 < 1, ππ1 > 0 and ππ1 + ππ1 < 1. Economist Cher thinks that, in this economy, one could simultaneously change πΊπΊΜ and πποΏ½ so as to decrease the government deficit π·π·π·π·π·π· = πΊπΊΜ − πποΏ½ without changing at the same time the equilibrium level of income. Economist Sonny holds a different opinion: in Zeta, cutting the government deficit will always lead to a fall in equilibrium income. Which of the two economists is right? Why? [Hint: (i) write the expression for equilibrium income, πποΏ½; (ii) use it to find out how the change in πΊπΊΜ , π₯π₯πΊπΊΜ , and the change in net taxes, π₯π₯πποΏ½, must be related to one another for equilibrium income to remain constant (π₯π₯πποΏ½ = 0); (iii) verify whether it is possible that, for the values of π₯π₯πΊπΊΜ and π₯π₯πποΏ½ so determined, the change in the deficit π₯π₯π₯π₯π₯π₯π₯π₯ is going to be negative and, if so, under what circumstances this could be the case. Explain]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 24 The goods and financial markets b. Consider now a different country, one in which there is only a goods market described by the same set of equations specified above. Write down the goods market equilibrium condition as an equality between national (private plus public) saving and investment, and assume that the country’s government decides to raise net taxes, πποΏ½. In the new equilibrium, national and private saving will be larger or smaller than before? [Hint: you are NOT being asked to compute the expressions for the change in private and national saving, but just to provide the economic intuition that helps determine the direction in which they will change]. .......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 25 Macroeconomics. Problems and Questions * Question 12 [Equilibrium in the money market and equilibrium in the market for the monetary base] a. Define what is meant by “monetary base”, also known as “central bank money”, π»π». Assuming that individuals hold money both in the form of currency and in that of checkable deposits, show in the graph an equilibrium position in the market for the monetary base and discuss how the interest rate prevailing in such an equilibrium changes following a decrease in π»π», an increase in ππ (the fraction of their money individuals want to hold as currency) and a rise in ππ (the reserve ratio). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 26 The goods and financial markets b. Using the diagram below, show how the same changes just considered (decrease in π»π»; rise in ππ or ππ) will affect the money market equilibrium. Based on the results of your analysis, what can you conclude about the relationship between the value of the interest rate for which the market for the monetary base is in equilibrium, and the value of the interest rate for which the money market is in equilibrium? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 27 Macroeconomics. Problems and Questions * Question 13 [Endogenous money supply] a. Write down the money market equilibrium condition. Assuming that money demand, ππππ , is increasing in the general price level ππ and in real income ππ, and decreasing in the interest rate ππ, represent in the graph below a position of equilibrium in the money market, denoting by π€π€Μ the value that the interest rate takes on in that equilibrium. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 28 The goods and financial markets b. Suppose that the central bank aims at keeping the interest rate constant at the level ππ = π€π€Μ and that, starting from an initial equilibrium like the one you have just described, there is a rise in ππ, or in ππ. What should the central bank do, to prevent such a change from leading to an equilibrium interest rate different from π€π€Μ ? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 29 Macroeconomics. Problems and Questions Question 14 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “Consider a simultaneous rise in the reserve ratio, θ, and in the fraction of their money individuals want to hold in the form of currency, ππ. Since, for a given monetary base π»π», they push money supply in opposite directions, the two changes will have an ambiguous impact on ππ”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 30 The goods and financial markets b. “A new law banning cash transactions above €500 induces individuals to reduce the fraction of the money they hold as currency (that is, the parameter ππ), and to increase that held in the form of bank deposits. It follows that one effect of the new law will be a fall in money supply, ππ”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 31 Macroeconomics. Problems and Questions Question 15 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “An increase in the price of bonds in the bond market makes bonds more attractive and induces individuals to hold a smaller share of their financial wealth in the form of money – the demand for money falls”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 32 The goods and financial markets b. In country Gamma, the central bank chooses, and wants to keep constant at the chosen level, the interest rate, while in country Delta the central bank chooses the money supply (and, once again, keeps it constant at the level it has chosen). The figure below depicts the money market equilibrium in the two countries. As you can see, in this initial equilibrium position money supply is the same in both, and the same is true about the money demand curve and the equilibrium interest rate. Explain if you agree, or do not agree, with the following statement: “If, in both countries, banks decide to decrease by the same amount the fraction of deposits held as reserves, to keep the interest rate at the chosen level the central bank of Gamma will have to increase the money supply, while in Delta – whose central bank wants to prevent the money supply from changing – the interest rate will fall”. [Hint: in the figure, denote by 2γ and 2δ the new equilibria, if any, that will be reached by Gamma and Delta, respectively]. ππ 1 ππ ππππ ππ, ππππ ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 33 Macroeconomics. Problems and Questions Question 16 a. Define, briefly but rigorously, the following concepts: a.1 ‘liquidity trap’; a.2 contractionary open market operation (make sure to explain why the intervention you are describing is ‘contractionary’). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 34 The goods and financial markets b. Explain why, when the economy is in a liquidity trap, an increase in money supply does not lower the interest rate. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 35 Chapter 2 - The IS-LM model Macroeconomics. Problems and Questions * Question 1 [An analytical version of the IS-LM model − (I) The standard case (endogenous money supply)] In a closed economy, consumption, investment, government purchases of goods and services and net taxes are described by the following equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) πΌπΌ = πΌπΌ Μ + ππ1 ππ − ππ2 ππ πΊπΊ = πΊπΊΜ ππ = πποΏ½ where πΌπΌ Μ is autonomous investment, the coefficients ππ1 and ππ2 , both positive, have the interpretation of sensitivity of investment to income and to the interest rate, respectively, and the other symbols have the usual meaning. Furthermore, assume that the sum ππ1 + ππ1 (the “propensity to spend”) is positive but less than one, and that nominal money demand οΏ½ππππ οΏ½ depends positively on the general price level (ππ) and on real GDP (ππ), and negatively on the interest rate (ππ), as implied by the following functional form: ππππ = ππ(ππ1 ππ − ππ2 ππ). In the previous equation, that we have met already when answering Question 13 of Chapter 1, the coefficients ππ1 and ππ2, both positive, have the interpretation of sensitivity of (real) money demand ππππ ⁄ππ to income and to the interest rate, respectively. Finally, suppose that the central bank sets the money supply so as to make sure that the interest rate always takes on the value ππ = π€π€Μ . a. Derive the analytical expression of the IS curve for this economy, and draw the IS in the (ππ, ππ) plane. What determines the slope of the curve? And what causes parallel shifts of the IS curve in the plane? Explain, using the graphs below and providing the economic intuition underlying your answers. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 38 The IS-LM model ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 39 Macroeconomics. Problems and Questions b. Derive the analytical expression of the LM curve for this economy, and draw the LM in the (ππ, ππ) plane. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 40 The IS-LM model c. Derive the equilibrium values of ππ and ππ. By how much will equilibrium output change if autonomous demand changes by π₯π₯π₯π₯? And by how much, following a change π₯π₯π€π€Μ in the level of the interest rate chosen by the central bank? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 41 Macroeconomics. Problems and Questions * Question 2 [An analytical version of the IS-LM model − (II) Exogenous money supply] Consider an economy different from the one studied in the previous question only for the fact that, rather the interest rate, its central bank chooses a value for the nominal money supply ππ, and then lets the interest rate take on any value that turns out to be consistent with the macroeconomic equilibrium for that given ππ. οΏ½ the value of ππ chosen by the central bank, derive the anaa. Denoting by ππ lytical expressions of the IS and LM curves for this economy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 42 The IS-LM model b. Draw the portion of the LM curve entirely lying in the first quadrant of the (ππ, ππ) plane (that is, the portion of the curve corresponding to positive values of both output and the interest rate). 1 What determines its slope? Which are the causes of parallel shifts of the LM curve in that plane? Explain, using the following graphs and providing the economic intuition underlying your results. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 1 The assumption that the nominal interest rate cannot take on negative values allows us to disregard the portion of the LM curve lying in the second quadrant. We shall discuss later on the shape of the LM curve when the nominal interest rate is zero (that is, when it is at its "zero lower bound") and − as assumed in the present question − the central bank chooses the nominal money supply. 43 Macroeconomics. Problems and Questions c. Derive the equilibrium values of ππ and ππ. By how much will equilibrium output change if autonomous demand changes by π₯π₯π₯π₯? And by how much, οΏ½ in the nominal money supply? following a change π₯π₯ππ ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 44 The IS-LM model ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 45 Macroeconomics. Problems and Questions Question 3 a. Gamma is a closed economy described by a standard IS-LM model,, 1 given by the following system: π΄π΄ 1 − ππ1 − ππ1 − · ππ ππ2 ππ2 ππ = π€π€Μ . ππ = In the equations above, π€π€Μ is the value of the interest rate chosen by the central bank, and the other symbols have the usual meaning. As discussed in the answer to Question 1 of this Chapter, in this model the fiscal policy multiplier is: 1 π₯π₯πποΏ½ = . (1 − ππ1 − ππ1 ) π₯π₯π₯π₯ (∗) What does the ‘fiscal policy multiplier’ measure? Knowing that, in Gamma, it equals 2, by how much will equilibrium output change if government spending on goods and services πΊπΊΜ goes up by 200 and, at the same time, autonomous consumption ππ0 falls by 100? Explain. .......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 1 From now on, by "a standard IS-LM model" we shall mean a model like the one studied in Question 1 of this Chapter, and therefore based on the following assumptions: • consumption function of disposable income, investment function of ππ and ππ, nominal money demand function of ππ, ππ and ππ; • πΊπΊ and ππ both exogenous; • the central bank chooses a value ππ = π€π€Μ for the interest rate, and sets nominal money supply to whatever level is consistent with the attainment of this target value for ππ; • constant prices, and therefore current and future expected inflation rates equal to zero. 46 The IS-LM model b. Consider now two different countries, Delta and Epsilon, whose IS curves are drawn in the graphs below. Suppose that the different slopes of the two curves only reflect differences in the value that the parameter ππ2 , the sensitivity of investment to the interest rate, takes on in the two countries. All the remaining parameters take on identical values in Delta and in Epsilon. Explain what is meant by “monetary policy multiplier”, ππ2 π₯π₯πποΏ½ =− (1 − ππ1 − ππ1 ) π₯π₯π€π€Μ and, making explicit reference to this concept (discussed in the answer to Question 1 of this Chapter), discuss if the central bank's decision of changing the interest rate by the same π₯π₯π€π€Μ in the two countries will change equilibrium output more in Delta or in Epsilon. ππ ππ π·π·π·π·π·π·π·π·π·π· πΌπΌπΌπΌπ·π·π·π·π·π·π·π·π·π· πΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈ πΌπΌπΌπΌπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈ ππ ππ ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 47 Macroeconomics. Problems and Questions Question 4 a. In country Macro, described by the IS-LM model, government purchases of goods and services, net taxes and investment are exogenous (πΊπΊ = πΊπΊΜ , ππ = πποΏ½ and πΌπΌ = πΌπΌ )Μ , while the consumption function is πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) − β2 ππ, where the parameter β2 > 0 is the sensitivity of consumption to the interest rate. In this economy, will the slope of the IS curve in the (ππ, ππ) plane be negative, zero or positive? Explain. [Hint: no formal derivation of the IS curve is required – just provide the economic intuition underlying your answer]. Next, represent the initial equilibrium of Macro in an IS-LM diagram, denote it by ‘0’ and study the effects of a decrease in net taxes. In particular, denote by ‘1’ the new equilibrium the economy will reach and explain the reasons for the changes in equilibrium income, consumption and investment that will take place in the move from ‘0’ to ‘1’. As the economy goes from the first to the second equilibrium, how must the changes in consumption and in income be related to one another? 48 The IS-LM model ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that Macro’s central bank intends to bring back equilibrium income to its initial level, ππ0 . To achieve its goal, should it raise or lower the interest rate? Show in the graph the new equilibrium that will be reached following the monetary policy intervention that you are proposing, denoting it by ‘2’. Finally, compare the composition of aggregate demand at ‘2’ with that prevailing in the initial equilibrium ‘0’. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 49 Macroeconomics. Problems and Questions Question 5 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “In an IS-LM model that departs from the standard case only for the fact that, rather than the interest rate, the central bank chooses the nominal οΏ½ , the higher the sensimoney supply, keeping it constant at the level ππ = ππ tivity of investment to the interest rate, the larger the increase in equilibrium income caused by an expansionary fiscal policy”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 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........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 50 The IS-LM model b. “In a standard IS-LM model, in which the central bank chooses the interest rate and keeps it constant at the level deemed appropriate given the state of the economy (for instance, at ππ = π€π€Μ ), the higher the sensitivity of investment to the interest rate, the larger the increase in equilibrium income caused by an expansionary fiscal policy”. ........................................................................................................................... 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........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 51 Macroeconomics. Problems and Questions Question 6 a. Gamma is a closed economy initially in goods and in money markets equilibrium. A wave of optimism about the economic future of the country leads to an increase in autonomous consumption, ππ0 . Using a standard ISLM model, graphically represent the effects of such a change on equilibrium income. In addition, discuss the effects on equilibrium consumption, investment, private saving and national saving, explaining the reasons for the observed changes in these variables. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 52 The IS-LM model b. Consider now country Delta. The only difference between Gamma and Delta lies in the policy rule followed by their central banks. While, as we already know from the first part of this question, Gamma's central bank chooses a level for the interest rate, Delta's central bank chooses a level for the nominal supply of money ππ and, given that level, lets the interest rate take on any value turns out to be consistent with the macroeconomic equilibrium. Suppose now that, when the two countries are in an initial equilibrium with identical values of ππ and ππ, Delta experiences the same increase in ππ0 discussed in the previous point of this question. Compared with what happened in Gamma, will income change more or less in Delta? Represent graphically, and explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 53 Macroeconomics. Problems and Questions Question 7 a. Consider the economy of country Epsilon, described by an IS-LM model departing from the standard case only because the (private) saving function is ππ = −ππ0 + (1 − ππ1 )(ππ − πποΏ½) + β2 ππ, where β2 > 0 is the sensitivity of savings to the interest rate and the other symbols have the usual meaning. a.1 Write down the consumption function for this economy. a.2 Assuming that all the other behavioral functions (investment, money demand, etc.) are standard, explain if and why, following an expansionary monetary policy, when β2 > 0 equilibrium income increases more or less than in the standard case (β2 = 0) [Hint: no formal derivation of the IS and LM curves is required – just provide the economic intuition underlying your answer]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 54 The IS-LM model b. If, rather than a monetary expansion, to be implemented is going to be a fiscal expansion (consisting for instance in an increase in government purchases of goods and services, πΊπΊΜ ), how will your answer to the previous point a.2 change? In particular, compare the change in equilibrium output in Gamma (where β2 > 0) with the change that would prevail in the standard case (β2 = 0). Illustrate in the graph below, and explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 55 Macroeconomics. Problems and Questions Question 8 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “The government of an economy described by a standard IS-LM model decides to implement an expansionary fiscal policy. To keep the interest rate at its target value ππ = π€π€Μ , the central bank will have to raise money supply by an amount that is going to be larger the more sensitive is money demand to changes in income. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 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........................................................................................................................... ........................................................................................................................... 56 The IS-LM model b. “When the economy is in a liquidity trap, with the nominal interest rate at its ‘zero lower bound’, an expansionary fiscal policy cannot influence the equilibrium level of output”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 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........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 57 Macroeconomics. Problems and Questions Question 9 Consider a country described by an IS-LM model departing from the standard one only for the fact that, rather than the interest rate, the central bank chooses the nominal money supply and, given the level selected for ππ, allows the interest rate to take on any value is consistent with the macroeconomic equilibrium. We are therefore in the case considered in Question 2 of this Chapter. In particular, and as it was assumed there, real money demand is πΏπΏ(ππ, ππ) = ππ1 ππ − ππ2 ππ. a. Suppose that the country is in a ‘liquidity trap’. Represent in the graph its initial equilibrium, denoting it by ‘1’, and by ππ1 and ππ1 the values that the interest rate and production take on in that equilibrium position. Assume now that autonomous consumption ππ0 and net taxes πποΏ½ fall at the same time and by the same amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ < 0. Show in the graph, and explain, how these changes affect the levels of the country’s interest rate and output, denoting by ππ2 and ππ2 the values these variables will take on in the new equilibrium (‘2’). In this new equilibrium position, investment will be higher or lower? And what about national saving (the sum of private and public saving)? Motivate your answer. 58 The IS-LM model ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that the central bank of the country attempts to return income to the initial level, ππ1 , by implementing a conventional monetary policy − for instance, an open market operation. Show in the graph the equilibrium that will prevail after the central bank’s intervention. Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 59 Macroeconomics. Problems and Questions Question 10 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. Define the concept of real interest rate (ππ), and write the equation that shows how ππ and nominal interest rate (ππ) are related to one another. Use this equation to explain if you agree, or do not agree, with the following statements: a. “If individuals expect positive inflation, then the real interest rate will be greater than the nominal one.” ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 60 The IS-LM model b. “The zero lower bound for the nominal interest rate implies that the real interest rate cannot be greater than minus the expected inflation rate, −ππ ππ ”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 61 Macroeconomics. Problems and Questions Question 11 The economy is described by an "extended" IS-LM model − that is, by an ISLM model based on the following additional assumptions: − − the central bank can set the real rate ππ, that therefore becomes the “policy rate ” determined by monetary policy, and keep it at the chosen level, let's call it ππΜ ; spending decisions (in particular, investment decisions by firms) depend on the “real borrowing rate” ππ + π₯π₯, sum of the (real) policy rate and the risk premium (π₯π₯). a. Starting from an initial equilibrium position, to be denoted by ′0′ in the figure below, suppose that the autonomous component of investment, πΌπΌ ,Μ falls and that, at the same time, the government cuts net taxes, πποΏ½, with π₯π₯πΌπΌ Μ = π₯π₯πποΏ½ < 0. Denote by ′1′ the new equilibrium that will be reached following these two changes, and compare the composition of aggregate demand at point ′1′ with that prevailing at ′0′. 62 The IS-LM model ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. To return equilibrium output to the initial level (that is, the one prevailing at point ′0′), what kind of monetary policy intervention should the central bank implement? In the graph, denote by ′2′ the equilibrium that will be reached following the monetary policy you are suggesting, and compare the composition of aggregate demand at point ′2′ with that prevailing in the initial equilibrium (point ′0′). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 63 Macroeconomics. Problems and Questions Question 12 Consider a country described by an ‘extended’ IS-LM that differs from the one described in the previous question only for the fact that consumption takes on the following functional form: οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ . πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) + ππ ππππππ In the equation above, ππππππ is the individuals’ financial and housing wealth, assumed to be exogenous, and the parameter ππ is greater than zero. a. In this economy, will the slope of the IS curve differ from that prevailing in the standard case, where consumption is a function of disposable income only (ππ = 0)? If so, will the IS curve be steeper or flatter than in the standard case? If not, why? Explain. [Hint: you are not being asked to derive the analytical expression of the IS curve and of its slope, but just to discuss whether this slope will be different from the one prevailing in the standard case, and to provide the economic intuition underlying your answer.] ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 64 The IS-LM model b. Due to a stock market crash, individuals experience a fall in their financial and housing wealth. In addition, an increase in financial market participants’ degree of risk aversion leads to a marked rise in the risk premium π₯π₯. Assuming that it was initially in an equilibrium that you will denote by ‘1’ in the graph below, show the new equilibrium to which the economy will converge following the two changes mentioned before (fall in οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ππππππ; rise in π₯π₯), and denote it by ‘2’. In the move from ‘1’ to ‘2’, how will the composition of aggregate demand have changed? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... 65 Chapter 3 - The labor market, the IS-LM-PC model, and inflation Macroeconomics. Problems and Questions Question 1 Consider an economy that is initially in medium run equilibrium, with an unemployment rate at the natural level. a. A new law that increases unemployment benefits is passed. Show in the graph the effects of such a change on the real wage and on the natural rate of unemployment. Provide the economic intuition behind the results you get. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 68 The labor market, the IS-LM-PC model, and inflation b. Realizing that the law just passed has an impact on employment, but still determined to provide income support for those who lose their job, the government implements measures aiming at increasing the degree of competition in the goods market. Assuming that such measures succeed in returning the unemployment rate to the natural level − that is, to the level prevailing before the increase in unemployment benefits −, show the effects of this second policy intervention in the same graph used to answer the previous point of this question. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 69 Macroeconomics. Problems and Questions Question 2 a. Define briefly, but rigorously, the following concepts: a.1 expectations-augmented (sometimes also referred to as ‘modified’, or ‘accelerationist’) Phillips curve; a.2 ‘non-accelerating inflation rate of unemployment’, or NAIRU (in your answer, make sure to discuss the relationship between this rate and the natural rate of unemployment); a.3 stagflation. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 70 The labor market, the IS-LM-PC model, and inflation b. Explain if and why you agree or disagree with the following statement: “In order to increase the natural level of production ππππ , policy-makers can follow two alternative strategies: (i) they can try to increase the demand for goods permanently, for example by opting for a permanent increase in either the money supply or in government purchases of goods and services, or (ii) they can decide to implement ‘supply-side’ policies, such as those leading to an increase in the degree of competition in the goods market. The difference between the two strategies above is that the first one will lead not just to an increase in ππππ , but also to a permanently higher level of prices, while the second strategy will push the economy towards an equilibrium characterized by a higher ππππ and a lower general price level.” ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 71 Macroeconomics. Problems and Questions Question 3 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “The zero lower bound for the nominal interest rate implies that the real interest rate cannot be smaller than minus the expected inflation rate, −ππ ππ ”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 72 The labor market, the IS-LM-PC model, and inflation b. Since, in country Delta, the ‘modified’ (also known as ‘expectationsaugmented’) Phillips curve is: πππ‘π‘ − πππ‘π‘−1 = − 2 · (π’π’π‘π‘ − 0.05), then in Delta the ‘non-accelerating inflation rate of unemployment’, or NAIRU, is 5%. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 73 Macroeconomics. Problems and Questions Question 4 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. If πππ‘π‘ππ = πππ‘π‘−1 , from the expectations-augmented Phillips curve it follows that, to bring the unemployment rate below its natural level, policymakers must be willing to tolerate an increase in the inflation rate. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 74 The labor market, the IS-LM-PC model, and inflation b. The smaller the fraction of wages indexed to the inflation rate, the larger the increase in inflation associated with any given decrease in the rate of unemployment. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 75 Macroeconomics. Problems and Questions Question 5 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “A decrease in firms' market power leads to a fall in the inflation rate”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 76 The labor market, the IS-LM-PC model, and inflation b. “From the expectations-augmented Phillips curve, πππ‘π‘ = πππ‘π‘ππ − πΌπΌ(π’π’π‘π‘ − π’π’ππ ), it follows that, for a given natural rate of unemployment π’π’ππ , the current inflation rate πππ‘π‘ can fall if and only if the current rate of unemployment π’π’π‘π‘ increases". ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 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........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 77 Macroeconomics. Problems and Questions Question 6 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “In country Alpha πππ‘π‘ππ = πππ‘π‘ , while in country Beta πππ‘π‘ππ = 2%. It follows that, to keep the rate of unemployment at the natural level (π’π’π‘π‘ = π’π’ππ for each time π‘π‘), the rate of inflation must remain constant in Alpha, and increase at a rate greater than 2% in Beta”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 78 The labor market, the IS-LM-PC model, and inflation b. “In country Gamma πππ‘π‘ππ = 3%, while in country Delta πππ‘π‘ππ = πππ‘π‘−1 . It follows that, to keep the rate of unemployment at the natural level (π’π’π‘π‘ = π’π’ππ for each time π‘π‘), in both countries the rate of inflation must remain constant over time ”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 79 Macroeconomics. Problems and Questions * Question 7 [The natural rate of interest - Monetary policy and fiscal policy in the mediumrun] Consider an economy described by the following equations: πΆπΆ = πΆπΆ(ππ − πποΏ½) πΌπΌ = πΌπΌ(ππ + π₯π₯, ππ) πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ ππππ = ππ β πΏπΏ(ππ + ππ ππ , ππ) ππ⁄ππ = πΏπΏ(ππ + ππ ππ , ππ) ππ − ππ ππ = (πΌπΌ⁄πΏπΏ)(ππ − ππππ ) where ππ is nominal money supply, the term ππ ππ appearing in the Phillips curve (the last equation) is expected inflation, and the other symbols have the usual meaning. a. Define the concept of natural rate of interest, ππππ , and discuss its determinants using a graph in which the real rate ππ is measured along the vertical axis, and goods' supply and demand along the horizontal one. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 80 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 81 Macroeconomics. Problems and Questions b. Explain what is meant by 'neutrality of money'. Using the graph you have drawn to answer the previous point, and assuming that individuals expect zero inflation (ππ ππ = 0), verify that money is neutral in the model considered in this question. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 82 The labor market, the IS-LM-PC model, and inflation c. And what about fiscal policy? In the medium-run, is it neutral, too? To answer, study the medium-run effects of a restrictive fiscal policy consisting in a permanent decrease in πΊπΊΜ using a graph similar to the one employed to answer the previous point. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 83 Macroeconomics. Problems and Questions * Question 8 Consider an economy described by a standard IS-LM-PC model. The consumption function is: πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) where, as usual, the positive constant ππ0 is autonomous consumption, ππ1 − the marginal propensity to consume − is between zero and one, and the other symbols have the usual meaning. a. The economy is initially in a medium-run equilibrium, with ππ = ππππ and ππ = ππππ . Represent in the graph, and discuss, the short-run effects of a permanent increase in autonomous consumption, assuming that the central bank decides to keep the policy rate unchanged at the initial level, ππππ . In the move from the initial to the new short-run equilibrium, how will consumption and investment change? 84 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. How should the central bank intervene if, in the short-run, it intends to keep the policy rate at ππππ ? Do you think that ππ can be kept at this level indefinitely? If not, at what value should the policy rate be allowed to converge? Motivate your answer by making explicit reference both to the graph you have just drawn and to the one used to answer Question 7 of this Chapter. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 85 Macroeconomics. Problems and Questions Question 9 Country Eta, a closed economy with flexible prices, is initially in a mediumrun equilibrium. a. To reduce a budget deficit deemed too high, the government of Eta decides to raise taxes. At the same time, the central bank decides to resort to an open market operation, aimed at preventing the fiscal policy just mentioned from changing the inflation rate, π. In an IS-LM-PC diagram, denote the initial medium-run equilibrium by ‘1’ and represent the new medium-run equilibrium to which Eta will converge after the implementation of the policy-mix described above. In particular, describe the type of open market operation (purchase; sale) that the central bank should implement to reach its objective. 86 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Compare the composition of aggregate demand prevailing in the initial and in the final medium-run equilibria. How must the change in consumption and the change in investment taking place as the economy goes from the first to the second of such equilibria be related to one another? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 87 Macroeconomics. Problems and Questions Question 10 Consider a country where both investment (which, as usual, is also a function of ππ) and consumption depend on the borrowing rate ππ + π₯π₯. In particular, suppose that the consumption function is πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) − β2 (ππ + π₯π₯), where the parameter β2 (> 0) is the sensitivity of consumption to the real borrowing rate. The rest of the economy is described by an IS-LM-PC based on the usual hypotheses − among them, the exogeneity of net taxes and of government purchases of goods and services (ππ = πποΏ½ e πΊπΊ = πΊπΊΜ ), and the assumption the central bank conducts its monetary policy by choosing the real policy rate, ππ. a. Having explained if, in the (ππ, ππ) space, the slope of the IS curve is in this case negative, zero or positive, assume that the economy is initially in a medium-run equilibrium, with ππ = ππππ , ππ = ππππ and ππ − ππ−1 = 0. Show in the graph, and discuss, the short-run effects of a permanent increase in the financial markets participants’ degree of risk aversion, assuming that the central bank keeps the policy rate constant at the initial level, ππππ . In the move from the initial to the new short-run equilibrium, how will consumption and investment change? Explain. 88 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that, once the economy has reached the short-run equilibrium described in the answer to the previous point, the government decides to return output to the natural level by changing πΊπΊΜ . To achieve its goal, should the government raise or lower πΊπΊΜ ? Compare the levels of investment, consumption, private saving, public saving and national saving in the new medium-run equilibrium that will be reached after the government’s intervention to the levels of the same variables in the initial one (that is, the medium-run equilibrium prevailing before the increases in the degree of risk aversion and in πΊπΊΜ ). Explain. [Hint: write down the (private) saving function for this economy]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 89 Macroeconomics. Problems and Questions Question 11 Consider a country where investment is entirely exogenous (πΌπΌ = πΌπΌ )Μ , In addition, consumption depends not just on disposable income, but also on the real interest rate, as implied by the consumption function πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) − β2 ππ, where the parameter β2 > 0 is the sensitivity of consumption to the real interest rate. The rest of the economy is described by an IS-LM-PC model based on the usual assumptions. In particular, net taxes and government purchases of goods and services are exogenous (ππ = πποΏ½ and πΊπΊ = πΊπΊΜ ), and the central bank chooses the (real) policy interest rate, ππ. a. Discuss how the IS curve will be sloped in this economy. Furthermore, suppose that the economy was initially in a medium run equilibrium with ππ = ππππ , ππ = ππππ e ππ − ππ−1 = 0, and that, in the attempt to offset a fall in autonomous consumption by π₯π₯ππ0 < 0, the government of the country cuts net taxes by an equal amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ < 0. Show in the graph below the new short-run equilibrium that will be reached following the two, contemporaneous, changes in autonomous consumption and net taxes just described, assuming that the central bank decides to keep the policy rate at the initial level, ππππ . How will the various components of aggregate demand change, in the move from the initial medium-run equilibrium to the new short-run one? 90 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that, once the economy has reached the short run equilibrium you have described when answering the previous point, the central bank decides to bring income back to the natural level by implementing an open market operation. To achieve its aim, should it buy or sell bonds in the open market? Show in the graph the new equilibrium that will be reached following the central bank’s intervention you consider appropriate. In this new equilibrium, how will the levels of investment, consumption, private saving, public saving and national saving compare to the levels of the same variables in the initial medium-run equilibrium (that is, the equilibrium prevailing before the changes in autonomous consumption and the fiscal and monetary policy interventions described above)? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 91 Macroeconomics. Problems and Questions Question 12 Consider an economy described by an IS-LM-PC model departing from the standard one for the fact that expected inflation is not equal to yesterday's inflation, but rather to the constant value πποΏ½, so that ππ ππ = πποΏ½. For simplicity, assume this constant value equals zero, so that ππ ππ = πποΏ½ = 0. In an IS-LM-PC diagram, represent the initial medium-run equilibrium of the economy, denoting it by 1, and assuming that the associated real interest rate is positive (that is, ππππ1 > 0). a. Suppose that, due to a major, permanent fall in the autonomous components of the demand for goods, the economy ends up in a new short-run equilibrium, to be denoted by 1′ in the figure, in which output is below its natural level; furthermore, suppose that the natural interest rate associated with this lower demand, let's call it ππππ2 , is not only less than ππππ1 , but also negative (ππππ2 < 0 < ππππ1 ). In this economy, can a 'conventional' monetary policy − as the one consisting in the decision to lower the policy rate − return output to its natural level? Explain. 92 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. To return output to its natural level, which economic policies would you suggest? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 93 Macroeconomics. Problems and Questions Question 13 In country XYZ, the production function is ππ = π΄π΄ · ππ, where π΄π΄ is a positive constant. In addition, when price expectations are correct (ππππ = ππ), the pricesetting and the wage-setting relations are, respectively, ππ = (1 + ππ) · (ππ/π΄π΄) and ππ = ππ · πΉπΉ(π’π’, π§π§), where the variables have the usual meaning. a. Provide an economic interpretation of the costant π΄π΄ and, in the graph below, show how a decrease in π΄π΄ will change the equilibrium values of the real wage and of the natural rate of unemployment. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 94 The labor market, the IS-LM-PC model, and inflation b. Assuming that the economy was initially in a medium-run equilibrium, show the effects of the same change in π΄π΄ discussed above in an IS-LM-PC diagram. In particular, show the new short-run equilibrium and describe the adjustment process toward the new medium-run equilibrium to which the economy will eventually converge. Compare the levels of consumption and investment in this latter equilibrium with the levels of the same variables in the initial medium-run equilibrium, providing an explanation for any observed change in their values. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 95 Macroeconomics. Problems and Questions Question 14 Consider a country that is initially in a medium run equilibrium position and in which the expected inflation rate is the constant πποΏ½. For simplicity, assume that πποΏ½ equals zero, so that ππ ππ = 0. Aside from this assumption, the economy is described by a standard IS-LM-PC model, with a central bank that chooses the interest rate. a. Represent the initial medium run equilibrium position of the economy, to be denoted by 1 in the graph, assuming that it takes place for a positive value of the natural real rate of interest (that is, ππππ1 > 0). Suppose now that a new law leads to an increase in the minimum wage that firms must pay their workers. In the graph, denote by 1′ the new short run equilibrium. Compared to 1, how have production, consumption and investment changed? Why? Explain in detail. [Hint: assume that the position of the IS curve is not affected by the change in consideration]. 96 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. In the same graph, show the adjustment process from the new short-run equilibrium 1′ to the new medium-run equilibrium, denoting this latter by 2. Explain in detail how 2 will be reached, with special emphasis on the reason for the changes in the interest rate and output taking place during the transition. Finally, explain if and how the levels of consumption and investment will have changed in the new medium-run equilibrium (2), compared to the levels these variables were taking on in the initial medium-run equilibrium (point 1). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 97 Macroeconomics. Problems and Questions Question 15 Consider country Macro, described by an IS-LM-PC model and in which individuals expect inflation to be constant at 2%. While Macro was, till two years ago, in a medium run equilibrium with a real interest rate (ππ) equal to 3%, the past couple of years have witnessed a marked reduction in ππ, which has fallen to 1% both in last year and in the current one. In addition, this year’s inflation rate has remained constant at the same value it took on one year ago. a. To understand the cause of the fall in the real interest rate, the government of the country consults two economists, Harry and Ginny. According to Harry, the fact that the inflation rate has remained constant allows one to conclude that the economy must have been hit by an adverse demand shock, one to which the central bank has reacted by bringing ππ to the new, lower medium run equilibrium level. The reduction in the real rate therefore reflects, without any doubt, a decrease in its natural level, ππππ . Ginny instead thinks that, to be able to conclude that the economy has been experiencing a fall in ππππ , rather than a decision of the central bank to bring the real rate below an unchanged natural level, further information is needed. In particular, one needs to know whether, over the past two years, the inflation rate has remained constant at 2%, or at a level greater than 2%. Represent in the graph below the medium run equilibrium prevailing in Macro two years ago (ππππ = 3%), denoting it by ‘1’. In the same graph, denote by ‘2’ the equilibrium, associated with a real rate of 1%, that would prevail if the fall in ππ is due to a decrease in the natural, medium run, real rate, and by ‘3’ that which would instead prevail if the decrease of the real rate to 1% is the outcome of the decision of the central bank to bring ππ below an unchanged natural level. Which of the two economists is right? Why? Explain. 98 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Say you are now told that, in Macro, rather than an inflation rate constant at 2%, individuals always expect an inflation rate equal to that observed in the previous period, so that ππ ππ = ππ−1 . Which of the two economists is right, in this case? Why? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 99 Macroeconomics. Problems and Questions Question 16 A country in which expected inflation for the current year equals the inflation rate prevailing in the previous year, ππ ππ = ππ−1 , and in which ππ ≥ 0, is initially in a medium run equilibrium, with ππ = ππππ , ππππ = 0 and ππ ππ = ππ−1 = 0. a. Assuming that the central bank chooses the interest rate, represent in the two-panel IS-LM-PC diagram the initial medium run equilibrium, denoting it by 1. Having explained which value the nominal interest rate, ππ, will necessarily take on in this initial equilibrium, suppose that the government launches an ambitious program of liberalizations, leading to an increase in the degree of competition in the country’s goods markets. In the two panels of the graph below, denote by 2 the new short run equilibrium that the economy will reach. In the move from 1 to 2, how will have income changed? And what about the rate of inflation? Why? Explain. [Hint: when answering, assume that the position of the IS curve is not affected by the change described above]. ........................................................................................................................... ........................................................................................................................... 100 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. If policy-makers do not intervene, how will the real interest rate and income change next period? Why? Show in the graph, denoting by 3 the new short run equilibrium that, absent any policy intervention, the economy would attain. To make sure that, rather than from 1 to 2, from 2 to 3, etc., the economy goes immediately from the initial medium run equilibrium 1 to the new medium run equilibrium that you will denote by 4 in the graph, the implementation of the liberalization program should be combined with a fiscal policy intervention, or with a monetary policy one? And, once the kind of economic policy most appropriate to go directly from the equilibrium 1 to the equilibrium 4 has been detected, the proposed policy should be a restrictive or an expansionary one? Show in the graph the effects of the policy intervention you suggest, and comment. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 101 Macroeconomics. Problems and Questions * Question 17 a. In the IS-LM-PC model, when the economy is in a medium-run equilibrium, output and the real interest rate are at their natural levels, and expectations are correct, so that ππ = ππ ππ . Suppose that the inflation rate individuals expect for the current period is equal to last year's rate, ππ ππ = ππ−1 . From ππ = ππ ππ and ππ ππ = ππ−1 it follows that, in a medium-run equilibrium, inflation will be constant − let's say, at the level ππ ∗ (for instance, at 2%). What determines ππ ∗? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 102 The labor market, the IS-LM-PC model, and inflation ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. If, in the medium run, inflation depends on the rate of money growth chosen by the central bank, and if inflation is a 'bad', why the world's major central banks target an inflation rate that, rather than zero, is positive (albeit 'small')? Mention two factors that help explain this choice. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 103 Chapter 4 - Expectations, financial markets, and economic policies Macroeconomics. Problems and Questions Question 1 We are at time π‘π‘. In country Tau, only one-year and two-year bonds exist. The time-π‘π‘ price of two year bonds issued in that time period is €ππ2π‘π‘ = €94. In adππ − the yield that market participants expect on one-year bonds dition, ππ1π‘π‘+1 that, issued at π‘π‘ + 1, will mature at π‘π‘ + 2 − is 4%. a. Using the quantitative information provided above, compute the yield on the one-year bonds issued at π‘π‘, ππ1π‘π‘ . ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 106 Expectations, financial markets, and economic policies b. Suppose now that you do not share the market’s expectations about ππ′ = 5%, rather future short-term rates. In particular, you expect ππ1π‘π‘+1 ππ than ππ1π‘π‘+1 = 4%. If you are interested in how much you will have two years from today, should you buy two-year bonds or a sequence of one-year bonds? Or maybe you should be indifferent between the two alternatives? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 107 Macroeconomics. Problems and Questions Question 2 a. Alpha and Beta − two countries where only one, two and three-year bonds exist – differ just for the expectations held by market participants about the monetary policy that, in each of the two nations, will be implemented in the future. The graph below shows the yield curve prevailing in the two countries in the current period, time π‘π‘ [N.B.: the yield curve prevailing in Alpha (the bold, continuous line in the graph) and that observed in Beta (the dashed, bold line) overlap for maturities up to two years; afterwards, Beta’s yield curve declines faster than Alpha’s]. Which differences in individuals’ expectations about the future monetary policy that will be implemented in each of the two countries can explain the observed differences in the two yield curves? Using the data provided in the graph, compute the current and future expected one-year rates ππ1π‘π‘ , ππ1ππ π‘π‘+1 and ππ1ππ π‘π‘+2 prevailing today (time π‘π‘) in each of the two countries. Yield to maturity 7% 6% Beta 5% 1 2 Alpha 3 Maturity ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 108 Expectations, financial markets, and economic policies ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Consider now Gamma, a country where stock prices have recently soared. To deflate what it thinks is a stock market bubble, Gamma’s central bank decides to intervene to return stock prices to a level closer to fundamentals. To achieve this goal, should the central bank implement an expansionary or a restrictive monetary policy? Why? Motivate your answer by making reference to the formula for stock prices, showing the determinants of the value of these financial assets [Hint: when answering, assume that, in each time period, the economy is described by a static IS-LM model, with consumption and investment that depend on contemporaneous variables only]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 109 Macroeconomics. Problems and Questions Question 3 a. In country Zeta, only one-year and two-year bonds exist. Investors, who do care about risk, ask for a risk premium π₯π₯ to hold the two-year bond (which is risky, as they do not know the price at which they will be able to sell it in a year). Use the arbitrage equation to derive the (approximate) relation that will hold in this case among the yield to maturity of a two-year ππ bond (ππ2π‘π‘ ), the current (ππ1π‘π‘ ) and future expected (ππ1,π‘π‘+1 ) yields on one-year bonds, and the risk premium (π₯π₯). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 110 Expectations, financial markets, and economic policies b. Suppose that, in the current period (time π‘π‘), Zeta’s yield curve is the one in the figure below. Write on the axes the name of the variables measured along each of them, and use your answer to the previous point of this question to compute the numerical value that the risk premium π₯π₯ takes on in this economy, knowing that investors expect the future yield on one-year ππ bonds to be the same as the current one (ππ1π‘π‘ = ππ1,π‘π‘+1 ). 4% 1% 1 2 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 111 Macroeconomics. Problems and Questions Question 4 a. In country Zeta, only one-year and two-year bonds exist. Investors, who do care about risk, ask for a risk premium π₯π₯ to hold the two-year bond (which is risky, as they do not know the price at which they will be able to sell it in a year). Use the arbitrage equation to derive the (approximate) relation that will hold in this case among the yield to maturity of a two-year ππ bond (ππ2π‘π‘ ), the current (ππ1π‘π‘ ) and future expected (ππ1,π‘π‘+1 ) yields on one-year bonds, and the risk premium (π₯π₯). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 112 Expectations, financial markets, and economic policies b. Suppose that, in the current period (time π‘π‘), the yield curve in Zeta – a country where the lower bound on the yield on one-year bonds is not zero, but negative, equal to −1% - is the one in the figure below. Write on the axes the name of the variables measured along each of them, and use your answer to the previous point of this question to compute the numerical value that the risk premium π₯π₯ takes on in this economy, knowing that ππ = −3ππ1π‘π‘ . ππ1π‘π‘+1 0.2% 0% 1 2 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 113 Macroeconomics. Problems and Questions Question 5 a. We are at time t. Write down the expression defining the price (in nominal terms) of a stock that has already paid the current dividend, and define carefully all the variables entering it. From which principle/condition is this expression derived? [Hint: you are not expected to derive the expression for the nominal stock price formally; just describe in detail the considerations on which its derivation is based]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 114 Expectations, financial markets, and economic policies b. Explain if, and why, you agree, or do not agree, with the following statements [Hint: when answering, assume that, in each time period, the economy is described by a standard, static IS-LM model, with consumption and investment that depend on contemporaneous variables only]: b.1 the prices of stocks at time π‘π‘, €πππ‘π‘ , will unambiguously go up if individuals start expecting an increase in autonomous consumption at time π‘π‘ + 2; b.2 if, at time π‘π‘, individuals learn that from π‘π‘ + 1 onwards fiscal policy will become permanently more expansionary, and monetary policy permanently more restrictive, then time π‘π‘ stock prices, €πππ‘π‘ , will unambiguously fall. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 115 Macroeconomics. Problems and Questions Question 6 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. If, because of new information that has become available, at time π‘π‘ people begin to expect a future decrease in the autonomous component of investment, πΌπΌ ,Μ then time π‘π‘ stock prices €πππ‘π‘ will certainly rise. On the other hand, the direction in which €πππ‘π‘ will vary is uncertain if, in addition to a future decrease in πΌπΌ ,Μ individuals also expect that the government of the country will intervene to keep future income constant [Hint: when answering, assume that, in each time period, the economy is described by a static IS-LM model, with consumption and investment that depend on contemporaneous variables only]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 116 Expectations, financial markets, and economic policies b. If investment does not depend on the interest rate, the announcement (unexpected by individuals) that, from time π‘π‘ onwards, monetary policy will become permanently more expansionary does not lead to any change in stock prices at time π‘π‘ [Hint: when answering, assume as before that, in each time period, the economy is described by the same static IS-LM model considered in the previous point of this question]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 117 Macroeconomics. Problems and Questions * Question 7 [Intertemporal consumption choices – The microfoundations of the 'theory of the very farsighted consumer'] There are only two time periods, π‘π‘ (the 'present') and π‘π‘ + 1 (the 'future'). Make the following assumptions on the individuals populating the economy: • • • their preferences are described by the utility function ππ(πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 ), increasing and concave in its two arguments – present consumption, πΆπΆπ‘π‘ , and future consumption, πΆπΆπ‘π‘+1 ; they have access to a credit market, where they can borrow and lend at the real interest rate ππ; they earn a disposable (labor) income equal to (πππ‘π‘ − πππ‘π‘ ) in the first peππ ππ ) − πππ‘π‘+1 riod, and expect to earn one equal to (πππ‘π‘+1 in the second period. a. Suppose that, at time π‘π‘, the typical individual's financial wealth (stocks, bonds, etc.) and housing wealth (apartments, commercial buildings, etc.) are both zero. Write down the individual's budget constraints for each of the two periods, combine them into a single 'intertemporal budget constraint' and represent in the (πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 ) plane the problem the individual has to solve in order to maximize his utility subject to such constraint. Where is the optimal 'present consumption/future consumption' program located? What determines whether, in the first period, the individual will be a borrower, a lender, or rather consume exactly his disposable income? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 118 Expectations, financial markets, and economic policies ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Assuming that both πΆπΆπ‘π‘ and πΆπΆπ‘π‘+1 are 'normal goods', explain how the optimal choice of present consumption will be affected by the following changes: • a transitory increase in current disposable income [π₯π₯(πππ‘π‘ − πππ‘π‘ ) > 0 and ππ ππ ) − πππ‘π‘+1 = 0] π₯π₯(πππ‘π‘+1 119 Macroeconomics. Problems and Questions • • an increase in future expected disposable income [π₯π₯(πππ‘π‘ − πππ‘π‘ ) = 0 and ππ ππ ) − πππ‘π‘+1 > 0] ; π₯π₯(πππ‘π‘+1 ππ ππ ) − πππ‘π‘+1 a permanent increase in disposable income [π₯π₯(πππ‘π‘ − πππ‘π‘ ) = π₯π₯(πππ‘π‘+1 > 0]. What can you conclude about the consumption function implied by the analysis of the individual's intertemporal consumption choices? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 120 Expectations, financial markets, and economic policies ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... c. How does the consumption function you have just derived change if, at time π‘π‘, individuals have a positive − rather than zero, as assumed so far − nonhuman wealth (ππππππ), defined as the sum of financial and of housing wealth? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 121 Macroeconomics. Problems and Questions Question 8 There are only two periods, period π‘π‘ (“the present”) and period π‘π‘ + 1 (“the future”). In the attempt to boost current economic activity, the government of country Delta cuts by 100 the taxes each individual will have to pay in the current period (time π‘π‘). At the same time, the government however announces that future taxes will be increased by the same amount. In other words, denoting per-capita net taxes by ππ, the fiscal policy intervention just described can ππ = +100. be summarized as follows: βπππ‘π‘ = −100, βπππ‘π‘+1 = βπππ‘π‘+1 a. Suppose that all individuals are identical and that, at the beginning of time π‘π‘, their non-human wealth is zero. Moreover, assume that ππ = 0, where ππ is the real interest rate. Using the analysis of the intertemporal consumption choices, evaluate βπππ‘π‘ and βπππ‘π‘+1 − that is, the changes in current and future consumption levels of the typical inhabitant of country Delta caused by the intertemporal reallocation of taxes implemented by the government. Will the government be successful in its attempt to increase consumption, and hence aggregate demand and equilibrium production, at time π‘π‘? If yes, indicate the fraction of the tax cut by which time π‘π‘ consumption of the typical individual will rise. If no, why? In motivating your answer, make explicit reference to the way in which savings by the typical individual will change at time π‘π‘. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... 122 Expectations, financial markets, and economic policies ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Assume now that some individuals are subject to ‘liquidity constraints’. Typically, an individual who is facing a binding liquidity constraint consumes less than he would like to, given the present value of his disposable income. This may be due to imperfections in the financial markets, as those leading to situations in which some individuals simply have no means of getting credit, so that their consumption in each period cannot exceed their current disposable income. Does the existence of such constraints induce you to change your answer to the previous point? Why? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 123 Macroeconomics. Problems and Questions Question 9 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. Suppose that today (time π‘π‘) there is a fall in house prices. The consumption function based on the analysis of the intertemporal consumption decisions implies that time π‘π‘ consumption will rise – since houses are now less expensive than before, individuals will need to save less to purchase one, and this will allow them to consume more. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 124 Expectations, financial markets, and economic policies b. Today (time π‘π‘), stock prices fall. The consumption function based on the analysis of the intertemporal consumption decisions implies that, at time π‘π‘, households will cut their consumption expenditures. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 125 Macroeconomics. Problems and Questions Question 10 In country Macro, current and future expected inflation are equal to zero, so that nominal and real interest rates are the same. At time π‘π‘ + 1 (‘the future’), consumption depends on π‘π‘ + 1 disposable income only, while investment is enΜ . As for the current period, (π‘π‘, ‘the present’), contirely exogenous, πΌπΌπ‘π‘+1 = πΌπΌπ‘π‘+1 sumption is increasing both in the current and in the future expected levels of ππ ππ − πποΏ½π‘π‘+1 ), investment is exogenous, πΌπΌπ‘π‘ = disposable income, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 πΌπΌπ‘π‘Μ , πΊπΊ and ππ are as usual exogenous in both periods, and the central bank chooses a level for the interest rate in both periods. a. Starting from an initial equilibrium position, the central bank announces today (time π‘π‘) that it will cut, both in the current and in the future period, the policy rate. Discuss, and show in the graph below, the effects of these announcements on the current (time π‘π‘) levels of income and interest rate. How will the time π‘π‘ yield on two-year bonds, ππ2π‘π‘ , change? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 126 Expectations, financial markets, and economic policies ......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that only one-year bonds and two-year bonds exist and that, before the announcement described above, at time π‘π‘ the yield curve of Macro was flat. Draw the new yield curve that, at time π‘π‘, will prevail in each of the following three cases: (i) the central bank has announced that it will cut the future short-rate rate less than the current one; (ii) it has announced that it will proceed to an equal cut in current and future rates; (iii) it has announced that it will cut the future short-rate rate more than the current one. Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 127 Macroeconomics. Problems and Questions Question 11 a. In country Macrolandia, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), a standard IS-LM model describes the functioning of the economy. In particular, in the year π‘π‘ + 1 consumption depends only on disposable income at π‘π‘ + 1, and investment on the interest rate and output at π‘π‘ + 1 only. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income, πΆπΆπ‘π‘ = ππ ππ − πποΏ½π‘π‘+1 ), investment depend positively on current and exπΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 pected future income levels, and negatively on the current and expected future levels of short-term interest rates, the demand for money has the usual functional form, and πΊπΊ and ππ are exogenous as usual. Starting from an initial equilibrium, at time π‘π‘ the government of Macrolandia announces an Μ > 0. At increase of its purchases of goods and services at time π‘π‘ + 1, βπΊπΊπ‘π‘+1 the same time, the central bank announces that it will adjust the policy rate so as to prevent any change in equilibrium income that, at time π‘π‘ and/or at time π‘π‘ + 1, could be caused by the fiscal policy just described. Explain how these announcements, which are unexpected and credible, affect the time π‘π‘ yield to maturity on the two-year bonds circulating in Macrolandia, ππ2π‘π‘ . ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 128 Expectations, financial markets, and economic policies b. Assume for simplicity that, in Macrolandia, at time π‘π‘ the yield curve is initially horizontal. Explain how the position and the slope of this curve change after the announcement studied above. Represent both curves, the one “before” and the one “after” the announcement, in the graph below. Clarify whether it is possible to determine the position and the slope of the new yield curve unambiguously. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 129 Macroeconomics. Problems and Questions Question 12 a. In country Macrolandia, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), an IS-LM model describes the functioning of the economy. In particular, in the year π‘π‘ + 1 consumption depends only on disposable income at π‘π‘ + 1, and investment on the borrowing rate (the sum of the interest rate and the risk premium π₯π₯π‘π‘+1 ) and an output at π‘π‘ + 1 only. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current ππ ππ and expected future disposable income, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 − πποΏ½π‘π‘+1 ), investment depend positively on current and expected future income levels, and negatively on the current and expected future borrowing rates, the central bank chooses in each period the interest rate, and πΊπΊ and ππ are exogenous as usual. Starting from an initial equilibrium position in both periods, at time π‘π‘ individuals start expecting a decrease in the risk premium for time π‘π‘ + 1, π₯π₯π₯π₯π‘π‘+1 < 0. At the same time, the central bank announces that it will adjust the policy rate so as to prevent any change in equilibrium income that, at time π‘π‘ (and at time π‘π‘ only), could be caused by the change in π₯π₯π‘π‘+1 just described. Explain how these changes and announcements, which are unexpected and credible, affect the time π‘π‘ yield to maturity on the two-year bonds circulating in Macrolandia, ππ2π‘π‘ . ..................................................................................................................... ..................................................................................................................... ..................................................................................................................... ..................................................................................................................... ..................................................................................................................... ..................................................................................................................... ..................................................................................................................... ..................................................................................................................... 130 Expectations, financial markets, and economic policies b. Assume for simplicity that, in Macrolandia, at time π‘π‘ the yield curve is initially horizontal. Explain how the position and the slope of this curve change after the change and the announcement studied above. Represent both curves, the one “before” and the one “after”, in the graph below. Clarify whether it is possible to determine the position and the slope of the new yield curve unambiguously. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 131 Macroeconomics. Problems and Questions Question 13 In country Gamma, current and future expected inflation rates coincide and are equal to zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), consumption depends only on disposable income Μ + ππ1 πππ‘π‘+1 − ππ2 πππ‘π‘+1 , where symbols have at π‘π‘ + 1, and investment is πΌπΌπ‘π‘+1 = πΌπΌπ‘π‘+1 the usual meaning. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income, ππ ππ − πποΏ½π‘π‘+1 ), while investment is increasing in current and exπΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 pected future income levels, and decreasing in the current and expected future levels of short-term interest rates. Finally, πΊπΊ and ππ are exogenous in both periods. a. At time π‘π‘, individuals revise downwards their expectation about the value that the autonomous component of investment will take on in the future. Μ will drop to πΌπΌ ′Μ π‘π‘+1, with More specifically, they start expecting that πΌπΌπ‘π‘+1 Μ . At the same time, the central bank announces that, in the fuπΌπΌ ′Μ π‘π‘+1 < πΌπΌπ‘π‘+1 ture, it will implement a monetary policy aimed at preventing any change in time π‘π‘ + 1 equilibrium output that could be caused by the expected drop in autonomous investment in that period. To achieve its goal, what kind of monetary policy (expansionary? restrictive?) should the central bank implement? Describe, and represent in the graph below, the effects of the two announcements on Gamma’s current (that is to say, time π‘π‘) income and interest rate equilibrium levels. 132 Expectations, financial markets, and economic policies ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that only one-year bonds and two-year bonds exist and that, before the announcements described above, at time π‘π‘ the yield curve was flat. What will be the effects of the two announcements on the position and the slope of the yield curve of Gamma? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .................................................................................................................................... .................................................................................................................................... ................................................................................................................................... .................................................................................................................................... 133 Macroeconomics. Problems and Questions Question 14 a. In country Alpha, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), a standard IS-LM model describes the functioning of the economy. In particular, in year π‘π‘ + 1 consumption depends only on disposable income at π‘π‘ + 1, and investment on the interest rate and output at π‘π‘ + 1 only. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income levels, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − ππ ππ − πποΏ½π‘π‘+1 ), investment is exogenous, πΌπΌπ‘π‘ = πΌπΌ,Μ and the same is true πποΏ½π‘π‘ , πππ‘π‘+1 about πΊπΊ and ππ. Rather than the interest rate, Alpha's central bank chooses a value for the nominal money supply ππ and, given this level at which ππ is set, lets the interest rate free to take on any value is consistent with the macroeconomic equilibrium. It follows that, in the (ππ, ππ) plane, the LM curve of Alpha is the curve discussed in Question 2 − and, for the case of an economy in a liquidity trap, in Question 10 − of Chapter 2. Finally, in the initial equilibrium, time π‘π‘ + 1 income and interest rates levels are both positive, while at time π‘π‘ the economy is in a liquidity trap. Suppose now that, at time π‘π‘, the central bank announces that it will decrease nominal money supply at π‘π‘ + 1. Explain how this announcement, which is unexpected and credible, affects the time π‘π‘ yield to maturity on the two-year bonds circulating in Alpha, ππ2π‘π‘ . .......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 134 Expectations, financial markets, and economic policies b. Draw the yield curve prevailing at time π‘π‘ before the central bank’s announcement, and explain how this latter will affect the yield curve of Alpha. In particular, draw the new yield curve in the same graph, and explain if it is possible to determine without ambiguity how its slope and position will compare to those of the original curve. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 135 Macroeconomics. Problems and Questions Question 15 In countryABC, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), investment is entirely exogenous, while consumption depends only on disposable income at π‘π‘ + 1. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − ππ ππ − πποΏ½π‘π‘+1 ), and investment is once again exogenous. In both periods, the πποΏ½π‘π‘ , πππ‘π‘+1 demand for money has the usual functional form, and πΊπΊ and ππ are exogenous as usual. Starting from an initial equilibrium, at time π‘π‘ the government of Macrolandia announces an increase in net taxes for time π‘π‘ + 1, βπποΏ½π‘π‘+1 > 0. a. Assuming that, in both periods, ABC’s central bank chooses the interest rate, describe and represent in the two graphs below the effects of this fiscal policy announcement on income and the interest rate levels prevailing at time π‘π‘ and at time π‘π‘ + 1. How will the time π‘π‘ yield to maturity on the two-year bonds circulating in Macrolandia, ππ2π‘π‘ , change? Why? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 136 Expectations, financial markets, and economic policies ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Repeat the same analysis carried out when answering the previous point assuming that, rather than the interest rate, the central bank of the country chooses the money supply. Represent in the graphs below the effects of the same fiscal policy announcement considered before and explain how will ππ2π‘π‘ change in this case, and why. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 137 Chapter 5 - The open economy Macroeconomics. Problems and Questions Question 1 a. Country Iota’s Statistical Office announces that, between year π‘π‘ and year π‘π‘ + 1, gross investment has increased by €100bn, while public saving has fallen by €50bn. a.1 Knowing that Iota is a closed economy, by how much has private saving changed between the same two years? a.2 How would your answer change, were Iota an economy open to trade in goods, services and financial assets with the rest of the world? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 140 The open economy b. Explain, briefly but rigorously, what is meant by J-curve [in your answer, make explicit reference to the so-called ‘Marshall-Lerner condition’]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 141 Macroeconomics. Problems and Questions Question 2 Consider an economy open to foreign trade, where only the goods market exists and prices are constant. Assume that foreign trade is initially balanced. a. Suppose the government decides to cut net taxes, πποΏ½. In the two- panel diagram that allows one to analyze simultaneously the determination of the goods market equilibrium and the trade balance associated with that equilibrium position, represent the effects of this fiscal policy decision on equilibrium income and net exports. In the graph, denote by πποΏ½ the initial level of equilibrium income, and by πποΏ½′ its new level. Why do equilibrium income and net exports change? Explain. 142 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. In the same graph used above, find the equilibrium level of income that would have been reached had the same tax cut been implemented in a closed economy. Denote this new, closed-economy equilibrium income by πποΏ½′′. Will πποΏ½′′ be less than, equal to, or greater than πποΏ½′? Why? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 143 Macroeconomics. Problems and Questions Question 3 a. Consider a country that is open to trade in goods and services with the rest of the world, where prices are fixed and in which only the goods market exists. Initially, the country is in goods market equilibrium with a trade surplus. Draw in the two-panel diagram below the initial equilibrium position, and show how it changes following a reduction in the domestic price level, ππ [Hint: assume that, after this reduction, ππ remains forever constant at its new, lower level, so that the economy is still described by a model with fixed prices]. In particular, explain if and why income and net exports will change in the new equilibrium. 144 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. How would your answer to the previous point of this question change, if the quantities of goods and services exported and imported by the country did not depend on the real exchange rate, but only on foreign and domestic income levels, respectively? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 145 Macroeconomics. Problems and Questions Question 4 Consider a country that is open to trade in goods and services with the rest of the world, where the exchange rate and prices are fixed and in which only the goods market exists. In the initial equilibrium, the country has a trade surplus. a. Show in the graph the effects of an exogenous increase in investment on equilibrium income and on the trade balance. Explain. 146 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Does the increase in investment change the level of output for which trade is balanced? Why, or why not? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 147 Macroeconomics. Problems and Questions Question 5 Eta is an economy open to international trade in goods and services, and in which financial markets do not exist. Assume that prices are constant both in Eta and in the rest of the world. Eta is initially in goods market equilibrium, with πΊπΊΜ = 60, πποΏ½ = 10, ππ = 70, πΌπΌ = 20, where πΊπΊ is government spending on goods and services, ππ net taxes, ππ private saving, and πΌπΌ investment. a. Using the information provided above, represent the initial equilibrium of Eta in the two-panel diagram below, denoting it by ‘1’ [Hint: you do not need to compute the equilibrium level of income; just show clearly where the initial equilibrium position is located in each of the two panels below, and explain why]. 148 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that Eta’s policy-makers wish to raise equilibrium income, leaving net exports unchanged. Having discussed why resorting to fiscal policy only, or to an exchange rate policy only, does not allow the government to achieve both its objectives, discuss the combination of the two policies which would allow the country to reach its two goals. In the two panels of the graph, denote by ‘2’ the new equilibrium that will be reached following the policy-mix you are suggesting, and explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 149 Macroeconomics. Problems and Questions Question 6 Eta is an economy open to international trade in goods and services, and in which financial markets do not exist. Assume that prices are constant both in Eta and in the rest of the world. a. Eta is initially in goods market equilibrium, with πΊπΊΜ = 50, πποΏ½ = 50, ππ = 70, πΌπΌ = 20, where πΊπΊ is government spending on goods and services, ππ net taxes, ππ private saving, and πΌπΌ gross investment. Using this information, represent the initial equilibrium of Eta in the two-panel diagram below [Hint: you do not need to compute the equilibrium level of income; just show clearly where the initial equilibrium position is located in each of the two panels below, and explain why]. 150 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that real exchange rate ε falls – a real depreciation. Assuming that the Marshall-Lerner condition holds true, represent the effects of this change in the graph above. In particular, explain if it will affect the level of income at which Eta’s foreign trade is balanced, ππππππ , and – if your answer is positive – in what direction ππππππ will change. Finally, explain if and why the trade balance will vary. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 151 Macroeconomics. Problems and Questions Question 7 Consider a country that is open to trade in goods and services with the rest of the world, where prices are fixed and in which only the goods market exists. Initially, the country is in goods market equilibrium, and trade is balanced. a. Draw in the two-panel diagram below the initial equilibrium position and show how it changes following a reduction in foreign output, ππ ∗ . In particular, explain how and why net exports will have changed in the new equilibrium. 152 The open economy ................................................................................................................. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose the government wants to return to a situation of balanced trade, keeping however domestic output at the new level reached after the decrease in ππ ∗ . In order to achieve these objectives, economist Paul advocates the use of fiscal policy, while economist Edie thinks that one should necessarily resort to some combination of fiscal and exchange rate policies. Do you agree with Paul (in this case, explain if the appropriate fiscal policy is an expansionary or a contractionary one), or with Edie (if this is the case, describe the specific combination of fiscal policy and exchange rate policy you deem appropriate)? [Hint: just provide the economic intuition behind your answer; you are not requested to carry out any graphical analysis when answering this point.] ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 153 Macroeconomics. Problems and Questions Question 8 a. Consider a country that is open to trade in goods and services with the rest of the world, where prices are fixed and in which only the goods market exists. Initially, the country is in goods market equilibrium, and trade is balanced. Draw in the two graphs below the initial equilibrium position, and show how this equilibrium changes following an increase in πΈπΈ, the nominal exchange rate [Hint: assume that E will remain forever at the new, higher level, and that domestic and foreign prices do not change]. Finally, explain if and why, in the new equilibrium, income and net exports will have changed. 154 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Consider now a different economy, where also financial markets exist and under a flexible exchange rate regime. Show in the graph below, and discuss, how an increase in domestic money supply ππ will affect the domestic interest rate ππ, the country’s level of income and the exchange rate of its currency, assuming that investment is exogenous, πΌπΌ = πΌπΌ ,Μ and therefore not a function of ππ. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 155 Macroeconomics. Problems and Questions Question 9 a. The euro-dollar exchange rate (number of dollars needed to purchase one euro) is expected to be 1.30 next year (i.e., πΈπΈ ππ = 1.30), and the one-year interest rates are 3% in Europe and 1% in the Unites States. Assuming that the (uncovered) interest parity condition holds, compute the current eurodollar exchange rate, πΈπΈ. Do financial markets participants expect the euro to appreciate or to depreciate in one year’s time? ........................................................................................................................... .......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 156 The open economy b. Write down the approximated version of the interest parity condition, and check how good an approximation it is by repeating the computations you carried out when answering the previous point of this question. Finally, suggest two possible extensions that would likely increase the degree of realism of the interest parity condition in its simplest form (be it approximated or not) considered in this question. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 157 Macroeconomics. Problems and Questions Question 10 The one-year interest rate in Japan is 1%, and the nominal exchange rate between the yen and the euro (number of yen needed to purchase one euro) is 140. Finally, assume that financial markets participants expect the euro to appreciate by 5% against the yen in the following year. a. For a European investor, what is the expected return from holding oneyear Japanese assets? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 158 The open economy b. Suppose that financial investors care only about the expected rate of return, and therefore want to hold only the assets with the highest expected yield. Assuming that, in Europe, the one-year interest rate is 2%, would a European investor purchase European assets or Japanese assets? In this example, does the (uncovered) interest parity condition hold? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 159 Macroeconomics. Problems and Questions Question 11 Consider an economy freely trading goods, services and financial assets with the rest of the world, in a flexible exchange rate regime. Domestic and foreign prices are constant, and equal to one (ππ = ππ ∗ = 1). a. Using the pair of graphs ‘open economy IS-LM – interest parity condition’, show the initial equilibrium of the country, denoting it by ‘1’ (hint: assume that the economy is described by a standard open economy IS-LM model, with a central bank the central bank that chooses the interest rate). Suppose now that the government raises its purchases of goods and services and that, to keep equilibrium income constant, the central bank varies the interest rate by implementing an open market operation. To achieve its aim, should the central bank buy or sell bonds? In the figure, denote by 2 the new equilibrium that will prevail after the increase in πΊπΊ and the central bank intervention. In the move from 1 to 2, how will the interest rate, the exchange rate, investment and net exports change? Explain. ........................................................................................................................... ........................................................................................................................... 160 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that the country is in a fixed, rather than in a flexible, exchange rate regime. Can a central bank wishing to keep the exchange rate fixed also intervene to keep income at the initial level (the one prevailing at equilibrium 1, before the increase in πΊπΊ), as it did in the flexible exchange rate case considered before? Why, or why not? In the figure, denote by 1′ the equilibrium that will be reached in this case (increase in πΊπΊ, and a central bank that behaves in a way consistent with the constancy of the exchange rate). Finally, explain how the interest rate, consumption, investment and next exports will have changed in the move from 1 to 1′. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 161 Macroeconomics. Problems and Questions Question 12 Consider Alpha, an open economy with a flexible exchange rate and in which investment is a function of ππ and of the borrowing rate ππ + π₯π₯. Aside from this, the country is described by a standard ‘open economy IS-LM – interest parity’ model with constant domestic and foreign prices, so that ππ = ππ. a. Assuming that the central bank chooses the interest rate, represent the initial equilibrium of the economy in an ‘open economy IS-LM – interest parity’ diagram, denoting it by ‘1’, and by ππ1 , ππ1 and πΈπΈ1 the associated values of the interest rate, output and exchange rate. Suppose now that the risk premium on which the borrowing rate depends, π₯π₯, increases. In the graph, denote by ‘2’ the new equilibrium that will be reached. In the move from the initial equilibrium ‘1’ to the new one ‘2’, how will production, consumption, investment, the exchange rate, net exports and the country’s money supply and money demand change? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 162 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. How would your answer to the previous point change, should Alpha’s central bank decide to intervene in order to prevent the rise in π₯π₯ from affecting production, and therefore in order to keep ππ at the same level prevailing in the initial equilibrium (‘1’)? In the graph, denote by ‘3’ the equilibrium that would be reached in this case and explain if and why, in the move from ‘1’ to ‘3’, consumption, investment, the exchange rate, net exports and the country’s money supply and money demand will change. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 163 Macroeconomics. Problems and Questions Question 13 Consider Zeta, an open economy with a fixed exchange rate and in which investment is a function of ππ and of the borrowing rate ππ + π₯π₯. Aside from this, the country is described by a standard ‘open economy IS-LM – interest parity’ model with constant domestic and foreign prices, so that ππ = ππ, in wh ich th e foreign interest rate is, as usual, denoted by ππ ∗ , and where the value at which the central bank keep the exchange rate constant is πΈπΈοΏ½ = πΈπΈ1 , with πΈπΈοΏ½ ππ = πΈπΈοΏ½ = πΈπΈ1 . a. Assuming that the central bank chooses the interest rate, represent the initial equilibrium of the economy in an ‘open economy IS-LM – interest parity’ diagram, denoting it by ‘1’, and by ππ1 , ππ1 and πΈπΈ1 the associated values of the interest rate, output and exchange rate. Suppose now that the risk premium on which the borrowing rate depends, π₯π₯, decreases. In the graph, denote by ‘2’ the new equilibrium that will be reached. In the move from the initial equilibrium ‘1’ to the new one ‘2’, how will production, consumption, investment, net exports and the country’s money supply and money demand change? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 164 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that, to bring back income to the level prevailing before the fall in π₯π₯ (that is, to ππ1 ), Zeta’s central bank decides to change from πΈπΈ1 to πΈπΈ3 the value at which the exchange rate is kept fixed, and that individuals revise accordingly the value of the exchange rate they expect to prevail in the future, so that the new future expected exchange rate now becomes οΏ½ = πΈπΈ3 . Having explained if, to achieve the central bank’s aim, πΈπΈ3 πΈπΈοΏ½ ππ′ = πΈπΈ′ will have to be greater or smaller than πΈπΈ1 , in the pair of graphs used before denote by ‘3’ the new equilibrium the economy will reach. Comparing the initial equilibrium (‘1’) to the final one (‘3’), how will consumption, investment, net exports and the country’s money supply and money demand will change? Why? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 165 Macroeconomics. Problems and Questions Question 14 Consider an economy freely trading goods, services and financial assets with the rest of the world. Furthermore, assume that domestic and foreign prices are constant, and equal to one (ππ = ππ∗ = 1). a. Using the pair of graphs ‘open economy IS-LM – interest parity condition’, show the initial equilibrium of the country, assuming a flexible exchange rate regime and, as usual, that the domestic central bank chooses the interest rate, keeping it constant at the level it deems appropriate given the state of the economy. In the graph, denote by ‘1’ the initial equilibrium, by π€π€Μ 1 , ππ1 ed πΈπΈ1 the associated values of the interest rate, income and the exchange rate, and assume that, in this initial equilibrium, πΈπΈ ππ = πΈπΈ1 and π€π€Μ 1 = ππ ∗ (where, as usual, ππ ∗ is the foreign interest rate). Consider now an increase in πΈπΈ ππ , which now ′ becomes πΈπΈ ππ > πΈπΈ ππ = πΈπΈ1 . In other words, investors now expect the domestic currency to be ‘stronger’ than they initially thought. Show in the graph, and explain, how the economy’s income, interest rate, exchange rate, and money supply will be affected by this change. ........................................................................................................................... ........................................................................................................................... 166 The open economy ........................................................................................................................... ........................................................................................................................... ......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that the government intends to return domestic output to the level prevailing before the increase in πΈπΈ ππ by changing net taxes, πποΏ½. To achieve its aim, should the government raise or lower net taxes? Compare the composition of aggregate demand in the new equilibrium that will be attained after the fiscal policy intervention to that prevailing in the initial equilibrium (that is, the equilibrium in which the economy was before the increase in πΈπΈ ππ and the change in πποΏ½). Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 167 Macroeconomics. Problems and Questions Question 15 a. The world economy consists of just two countries, Alpha and Beta. International capital mobility is perfect, and the exchange rate between the currencies of the two countries is kept fixed at the level πΈπΈοΏ½ . Alpha has just entered a recession. To revive its economy, would you recommend Alpha the use of monetary policy or of fiscal policy? Explain your answer and illustrate graphically, using the pair of graphs 'open economy IS-LM'-'interest parity condition'. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 168 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Discuss the effects of the policy intervention you have just suggested on the trade balance of Alpha and on that of Beta, always assuming a fixed exchange rate. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 169 Macroeconomics. Problems and Questions Question 16 Consider an economy that trades goods, services and financial assets with the rest of the world, under a fixed exchange rate regime. Domestic and foreign prices are constant and, for simplicity, both equal to 1 (ππ = ππ ∗ = 1). In the pair of graphs ‘open economy IS-LM’-‘interest parity condition’, show the initial equilibrium, denoting by ππ0 , ππ0 and πΈπΈ0 the values of the interest rate, output and exchange rate in that equilibrium. Market participants expect that the exchange rate will be kept fixed at the current level πΈπΈ0 also in the future, so that πΈπΈ ππ = πΈπΈ0 . a. Suppose that there is a fall in the foreign interest rate, ππ ∗ , while foreign output ππ ∗ and all the other exogenous variables remain constant. Assuming that the country’s central bank keeps fixing the exchange rate at πΈπΈ0 , show in the graph, and explain carefully, the effects of this exogenous decrease in ππ ∗ on equilibrium output, interest rate, money supply and net exports of the country, denoting by ‘1’ the new equilibrium that will be reached. ........................................................................................................................... ........................................................................................................................... 170 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. How would your answer have changed if, when ππ ∗ falls, foreign output ππ ∗ had also gone up, rather than remaining constant? In the same graph used before, denote by ‘2’ the equilibrium that the economy would have reached in this case (lower ππ ∗ and higher ππ ∗ ) and compare it to the equilibrium (‘1’) where the economy settles when to vary is just the foreign interest rate. Finally, explain in which of the two cases the intervention of the central bank (consisting in a change in the money supply aimed at keeping the exchange rate constant) will have to be quantitatively larger. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 171 Macroeconomics. Problems and Questions Question 17 Consider an open economy under a fixed exchange rate regime. Domestic and foreign prices are constant and, for simplicity, both equal to 1 (ππ = ππ∗ = 1). In the pair of graphs ‘open economy IS-LM’-‘interest parity condition’, show the initial equilibrium, denoting it by ‘0’, and by ππ0 , ππ0 and πΈπΈ0 the associated values of the interest rate, output and the exchange rate. Market participants expect that the exchange rate will be kept fixed at the current level πΈπΈ0 also in the future, so that πΈπΈ ππ = πΈπΈ0 . a. Suppose the central bank announces a devaluation of the currency − that is, it announces that, effective immediately, the value at which the exchange rate is kept fixed is lowered to πΈπΈ1 < πΈπΈ0 . In addition, assume that individuals, who did not expect this announcement, revise accordingly the value of the exchange rate they expect to prevail in the future, so that now πΈπΈ ππ falls to πΈπΈ ππ ′ = πΈπΈ1 . Analyze in the graph, and explain, the effects of the devaluation on domestic output, interest rate and money supply, denoting by ‘1’ the new equilibrium that the economy will reach. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 172 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose that, before the devaluation, both the domestic country and the rest of the world were in a medium-run equilibrium, with income at its natural level and zero inflation. In addition, assume that the inflation rate is determined according to the following Phillips curve: ππ = (πΌπΌ/πΏπΏ)(ππ − ππππ ) Once price adjustment − as implied by the previous equation − is taken into account, do you think that a devaluation can permanently affect the level of income of a country? Why, or why not? Discuss [Hint: no formal analysis is required; just describe the likely dynamic adjustment of the economy following a devaluation]. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 173 Macroeconomics. Problems and Questions Question 18 Consider an open economy under a fixed exchange rate regime. Domestic and foreign prices are constant and, for simplicity, both equal to 1 (ππ = ππ∗ = 1). In the pair of graphs ‘open economy IS-LM’-‘interest parity condition’, show the initial equilibrium, denoting it by ‘0’, and by ππ0 , ππ0 and πΈπΈ0 the associated values of the interest rate, output and the exchange rate. Initially, market participants expect that the exchange rate will be kept fixed at the current level πΈπΈ0 also in the future, so that πΈπΈ ππ = πΈπΈ0 . a. Suppose that, in the initial equilibrium, income is below its natural level. Given that, as discussed in the answer to the previous question, a decision to devalue leads a higher equilibrium output in a country under a fixed exchange rate regime, individuals start expecting an impeding devaluation. It follows that the future expected exchange rate decreases from πΈπΈ ππ = πΈπΈ0 to πΈπΈ ππ′ = πΈπΈ1 , with πΈπΈ1 < πΈπΈ0 . Analyze in the graph the effects of this downward revision in the expectations on the exchange rate that will prevail in the future on the levels of domestic income and interest rate, assuming that the central bank intends to keep the exchange rate fixed at πΈπΈ0 . ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 174 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Which type of economic policy (monetary or fiscal; exansionary or contractionary) could prevent the expectations of an impending devaluation studied before from changing equilibrium output? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 175 Macroeconomics. Problems and Questions Question 19 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. An expansionary monetary policy rises equilibrium income under flexible exchange rates, but does not affect income under fixed exchange rates. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 176 The open economy b. If financial investors only care about the expected rate of return, and therefore want to hold only the assets with the highest expected rate of return, then from the interest parity condition it follows that, under fixed exchange rates, the domestic interest rate can only be equal to the foreign interest rate, ππ = ππ ∗ . ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 177 Macroeconomics. Problems and Questions Question 20 a. Psi is an open economy with a flexible exchange rate. Illustrate graphically, and explain, the effects of a decrease in the reserve ratio, θ, on output, the exchange rate, and the trade balance of Psi. When answering, assume that – starting from an initial interest rate that you will denote by π€π€Μ 0 in the figure − the central bank will not intervene to keep ππ equal to π€π€Μ 0, but that it will allow the interest rate to take on the new equilibrium value – to be denoted by π€π€Μ 1 in the graph − that will prevail after the fall in θ. Once ππ has become equal to π€π€Μ 1 , the central bank will however keep it at this new level from then on. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 178 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that the government of Psi implements a fiscal policy aimed at bringing equilibrium output back to the level it was taking on before the decrease in θ. What happens to the trade balance in this case? Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 179 Macroeconomics. Problems and Questions Question 21 Consider an economy freely trading goods, services and financial assets with the rest of the world, in a flexible exchange rate regime. Domestic and foreign prices are constant, and equal to one (ππ = ππ ∗ = 1). a. Using the pair of graphs ‘open economy IS-LM - interest parity condition’, show the initial equilibrium of the country, denoting it by ‘1’, assuming that the central bank chooses the interest rate, that investment is entirely exogenous (πΌπΌ = πΌπΌ )Μ , and finally that net exports depend positively on foreign income and negatively on the domestic one, but that they do not depend on the real exchange rate, ππ. Suppose now that the central bank sells bonds in the open market, in order to raise the domestic interest rate. Show in the graph how the economy’s income, interest rate and exchange rate will be affected by this change, denoting by ‘2’ the new equilibrium that the economy will reach. Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 180 The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Suppose now that, rather than being independent of the real exchange rate, net exports depend positively on ε – in other words, a real depreciation (appreciation) worsens (improves) the country’s trade balance. Assuming that the remaining assumptions (exogeneity of investment, a flexible exchange rate, a central bank that chooses the interest rate, etc.) still hold true, discuss the effects of the same monetary policy studied above on the equilibrium levels of income, interest rate and exchange rate. Show in the figure the new equilibrium that will be reached in this case, and explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 181 Macroeconomics. Problems and Questions Question 22 Gamma is an economy that trades goods, services and financial assets with the rest of the world, under a flexible exchange rate regime. In Gamma, changes in the autonomous components of the demand for domestic goods are the main cause of the observed fluctuations in real GDP. Starting from an initial equilibrium like point ‘0’ in the graph, sometimes the level of autonomous demand is high, leading to an πΌπΌπΌπΌ curve such as πΌπΌπΌπΌ1 in the figure; sometimes it is however low, so that the πΌπΌπΌπΌ curve shifts to the left in the graph (πΌπΌπΌπΌ2 ). Economists Jack and Jill are asked which monetary policy rule − the one assumed in the 'standard' version of the IS-LM model, in which the central bank chooses a value for the interest rate, or the alternative one, in which the central bank chooses the nominal money supply − would, in their opinion, help minimize the variability of output around its initial level, ππ0 , caused by such demand fluctuations. Jack thinks that the best option is for the central bank to choose the interest rate, and keep it at the chosen value; on the contrary, Jill suggests that the central bank should choose a level for the money supply, and then let the interest rate to take on any value that, given this choice of ππ, is consistent with the macroeconomic equilibrium. a. In the pair of graphs ‘open economy IS-LM - interest parity condition’ below, show the equilibrium levels of output that, under flexible exchange rates, will prevail when the central bank chooses the interest rate and, following the fluctuations in demand just discussed, the IS curve shifts to the right (πΌπΌπΌπΌ1 ) or to the left (πΌπΌπΌπΌ2 ). In the figure, make sure to denote the corresponding equilibrium levels of output by ππ1 and ππ2 , respectively. ππ πΌπΌπΌπΌ0 πΌπΌπΌπΌ2 πΌπΌπΌπΌ1 ππ 0 β ππ ππ0 182 πΈπΈ The open economy ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. In the same graph, denote by ππ1′ and ππ2′ the equilibrium levels of production that, following the same changes in the autonomous components of aggregate demand and the same shifts in the πΌπΌπΌπΌ curve discussed before, would prevail should the central bank choose the nominal money supply. On the basis of your answers to this and to the previous point, do you agree with Jack or with Jill? Provide the intuition for the lower output variability in the central banks behaves in the way suggested by the economist whose view you share. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 183 Macroeconomics. Problems and Questions Question 23 a. We are at time π‘π‘. Use the non-approximated form of the interest parity condition to write the current period exchange rate, πΈπΈπ‘π‘ , as a function of the current and future expected interest rates for each year over the next ππ ππ . years, as well as of the expected exchange rate for time π‘π‘ + ππ + 1, πΈπΈπ‘π‘+ππ+1 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 184 The open economy b. Explain how each of the following announcements – which, made at time π‘π‘, are unexpected, and believed, by financial markets participants – affects the time π‘π‘ exchange rate, πΈπΈπ‘π‘ [Hint: assume that, in each period, the functioning of the economic system is described by a static IS-LM model, with consumption and investment depending on contemporaneous variables only]: b.1 a permanently more expansionary monetary policy abroad, implemented from time π‘π‘ + 2 onwards; b.2 a permanently more expansionary fiscal policy, and a permanently more restrictive monetary policy, at home, from time π‘π‘ + 1 onwards; b.3 the emergence of expectations of a progressive, lasting worsening of the country's current account balance. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 185 Chapter 6 - Government debt and economic growth Macroeconomics. Problems and Questions * Question 1 [A graphical analysis of the evolution of the debt-to-GDP ratio] Write down the government budget constraint as an equation that, for given values of the real interest rate (ππ), of the growth rate of the economy (ππ) and of the primary deficit-to-GDP ratio (ππ), relates the debt-to-GDP ratio (π΅π΅⁄ππ) at time π‘π‘ to the value of the same ratio at time π‘π‘ − 1. Use that equation to graphically analyze the evolution of the ratio π΅π΅⁄ππ in the following four cases, providing the intuition underlying your conclusions: a. ππ < ππ and ππ > 0. 45° π΅π΅π‘π‘ πππ‘π‘ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 188 Government debt and economic growth b. ππ > ππ and ππ > 0. 45° π΅π΅π‘π‘ πππ‘π‘ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 189 Macroeconomics. Problems and Questions c. ππ > ππ and ππ < 0. 45° π΅π΅π‘π‘ πππ‘π‘ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 190 Government debt and economic growth d. ππ < ππ and ππ < 0. 45° π΅π΅π‘π‘ πππ‘π‘ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... 191 Macroeconomics. Problems and Questions Question 2 a. Consider a country that, in period π‘π‘ = 1, inherits from the previous period, π‘π‘ = 0, a debt-to-GDP ratio of 100%: ππ0 ≡ π΅π΅0 ⁄ππ0 = 1. Moreover, the real interest rate and the rate of economic growth are constant and equal to 3% and to 5%, respectively (ππ = 0.03, ππ = 0.05). Finally, the ratio of the primary deficit to GDP is 4%, assumed to be constant over time. Write down the equation that gives the dynamics of the debt ratio ππ (≡ π΅π΅⁄ππ) and use it to calculate the value of the debt-to-GDP ratio at times π‘π‘ = 1 and π‘π‘ = 2, and the steady state level of ππ, πποΏ½. Will the debt ratio diverge over time, or converge to its steady state value? Why, or why not? ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 192 Government debt and economic growth b. Suppose now that the government intends to stabilize ππ at the value observed at π‘π‘ = 0. In other words, the government wants ππ to continue to take on the value 1 both in period π‘π‘ = 1 and in all subsequent periods. To achieve this goal, the Government is considering the possibility of generating a permanent change in the ratio of the primary deficit to GDP. Compute the value that this ratio should take on to stabilize ππ at the value 1 forever, and explain whether it implies that the Government should implement a restrictive or an expansionary fiscal policy. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... 193 Macroeconomics. Problems and Questions Question 3 At time π‘π‘, a country inherits from the previous period a stock of government 0 debt greater than zero, corresponding to the debt-to-GDP ratio πππ‘π‘−1 in the graph below. Assuming that the real interest rate is smaller than the rate of growth of the economy, and that the government runs a primary surplus, a. show in the graph the steady state debt-to-GDP ratio in this economy and explain if, absent any intervention, the debt-to-GDP ratio of the country will converge or not to this stationary level; 45° π΅π΅π‘π‘ πππ‘π‘ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 194 Government debt and economic growth b. show in the graph the change in the ratio between the primary balance 0 and GDP needed to stabilize the debt-to-GDP ratio at the value πππ‘π‘−1 from time π‘π‘ onwards. Explain. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 195 Macroeconomics. Problems and Questions Question 4 a. Consider the following graph: π΅π΅π‘π‘ πππ‘π‘ debt/GDP line 45° 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 In the figure, the bold straight line, that gives the time π‘π‘ debt/GDP ratio as a function of the same ratio in the previous period, is parallel to the 45° line going through the origin. In this economy, what is the relative size of the growth rate of real GDP and of the real interest rate? Is the government running a primary deficit or a primary surplus? Explain. .......................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 196 Government debt and economic growth 0 , show in the b. Assuming that the debt-to-GDP ratio at time π‘π‘ − 1 was πππ‘π‘−1 graph the values that this ratio will take on at times π‘π‘ and π‘π‘ + 1. Will the debt-to-GDP ratio converge to a steady state value πποΏ½? If not, why? If yes, show in the figure this steady state value and explain whether it is stable (that is, if ππ will converge to it independently of the value of the debt-to0 ) or unstable (in which case, ππ will GDP ratio inherited from the past, πππ‘π‘−1 0 take on the steady state value if and only if πππ‘π‘−1 = πποΏ½). ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... 197 Macroeconomics. Problems and Questions Question 5 a. Consider a country that, in period π‘π‘ = 1, inherits from the previous period, π‘π‘ = 0, a debt-to-GDP ratio of 60%: ππ0 ≡ π΅π΅0 ⁄ππ0 = 0.6. In addition, the real interest rate and the rate of economic growth are constant and equal to 6% and to 0%, respectively (ππ = 0.06, ππ = 0). Finally, the ratio of the primary deficit to GDP is 3%. Write down the relation that describes the dynamics of the debt ratio (the government budget constraint), and use it to compute the debt-to-GDP ratio ππ for times π‘π‘ = 1 and π‘π‘ = 2. Will this ratio converge over time to a steady state value πποΏ½ > 0? Why, or why not? Represent this specific case in the graph that one uses to study the evolution of the debt ratio over time. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... .......................................................................................................................... 198 Government debt and economic growth b. Suppose now that, rather than a deficit, the same country runs a primary surplus of 6% of GDP. How would your answer to the previous point of this question change? Compute the debt ratio at times π‘π‘ = 1 and π‘π‘ = 2, and its steady state value, πποΏ½. Will the debt ratio converge to πποΏ½? Explain, using the graph below to motivate your conclusions. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 199 Macroeconomics. Problems and Questions Question 6 We are at time π‘π‘. The economy inherits from the previous period a debt-to0 GDP ratio πππ‘π‘−1 . Consider the following graph: 45° π΅π΅π‘π‘ πππ‘π‘ Debt/GDP line πποΏ½ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 a. In this economy, is the GDP growth rate higher or lower than the real interest rate? Is the government running a primary surplus or a primary deficit? Finally, is the debt ratio bοΏ½ a stable or an unstable steady state? Explain. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... 200 Government debt and economic growth b. Suppose now that, at time π‘π‘, the Government intends to stabilize the 0 debt-to-GDP ratio at the level πππ‘π‘−1 by changing the economy’s growth rate. To achieve its aim, should it attempt to increase or to decrease that growth rate? Show how your answer is going to affect the graph used to answer the previous question, and discuss. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 201 Macroeconomics. Problems and Questions * Question 7 a. Consider a country with a zero primary deficit-to-GDP ratio (ππ = 0), and in which the real interest rate and the rate of growth of the economy are constant and equal to ππ = ππΜ and ππ = ππΜ , respectively, with ππΜ < ππΜ and 1 + ππΜ − ππΜ > 0. Write down the equation that gives the dynamics of the debt-toGDP ratio for this economy (the government budget constraint) and derive the steady state value of that ratio. Assuming that at time π‘π‘ the econ0 > 0, explain omy inherits from the past a debt-to-GDP ratio equal to πππ‘π‘−1 if, and why, the economy will ever converge to that steady state. Use the graph below to motivate your answer. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 202 Government debt and economic growth b. Suppose now that the interest rate at which the government can borrow is no longer equal to ππΜ , but rather to ππΜ + π£π£πππ‘π‘−1 , where π£π£ is a positive parameter. In other words, the real interest rate is no longer constant, but increasing in the debt-to-GDP ratio (ππ) prevailing in the last period. The growth rate of the economy is however stiIl constant and, as before, ππΜ < ππΜ and 1 + ππΜ − ππΜ > 0. Repeat the analysis carried out in order to answer the previous point of this question. In particular, derive the expression of the steady state debt-to-GDP ratio, explain if and why ππ will converge to a steady state, and represent graphically. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 203 Macroeconomics. Problems and Questions Question 8 a. Consider the Solow growth model without technological progress. The 3 1 production function is ππ = πΎπΎ οΏ½4 ππ οΏ½4 and the rate of depreciation, πΏπΏ, is equal to 0.1. Calculate the propensity to save π π for which the steady state level of capital per worker is 100. Represent graphically this steady state. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 204 Government debt and economic growth ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... b. Assuming that the economy is initially in the steady state just described, explain and represent graphically what happens to capital per worker, output per worker and the growth rate of the economy following a reduction in the marginal propensity to consume. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 205 Macroeconomics. Problems and Questions Question 9 a. Consider the Solow growth model without technological progress. The 1 1 production function is ππ = πΎπΎ οΏ½2 ππ οΏ½2 and the rate of depreciation is πΏπΏ = 0.05. Calculate the propensity to save π π for which the steady state level of capital per worker is 200. Represent graphically this steady state. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 206 Government debt and economic growth b. Assuming that the economy is initially in the steady state just described, explain, and show in the graph, what happens to capital per worker, output per worker and the growth rate of the economy following an increase in the rate of depreciation, πΏπΏ. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 207 Macroeconomics. Problems and Questions Question 10 The government of a closed economy, whose budget was previously balanced, starts running a budget deficit equal to the percentage ππ of the country’s GDP, with 0 < ππ < π π , where π π is the private saving rate. a. Assuming that there is no technological progress and that both the private saving rate and the population of the country are constant, show in the graph below the impact of the emergence of a budget deficit (that is, of the increase in ππ from zero to a positive value) on output per worker and capital per worker in the Solow growth model. Explain. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 208 Government debt and economic growth ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. b. Will the budget deficit permanently affect the growth rate of the economy? Explain. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 209 Macroeconomics. Problems and Questions Question 11 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “From the Solow model without technological progress it follows that, as the saving rate s increases from its minimum value (0) to its maximum value (1), steady-state capital per worker and output per worker first increase and then decrease”. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 210 Government debt and economic growth b. Using the Solow model without technological progress and assuming a constant population (πππ΄π΄ = ππππ = 0), explain if and why you agree, or do not agree, with the following statements: b.1 b.2 an increase in the saving rate π π will always raise steady state output per worker; an increase in the saving rate π π will always raise steady state consumption per worker. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 211 Macroeconomics. Problems and Questions Question 12 a. Using the Solow model with technological progress, and assuming the economy was initially in a steady state equilibrium, study in the graph below the effects of a decrease in the saving rate on capital and output per effective worker, briefly explaining the reasons for the observed changes in these variables. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 212 Government debt and economic growth b. How would capital and output per effective worker change following an increase in the rate of technological progress? ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 213 Macroeconomics. Problems and Questions Question 13 Consider the Solow model, assuming positive rates of technological progress and population growth, so that πππ΄π΄ > 0 and ππππ > 0. a. Assuming the usual aggregate production function, and denoting by π π the saving rate and by δ the depreciation rate, represent in the graph below the steady state, or state of balanced growth, of the economy. Clearly indicate the levels of output per effective worker, investment per effective worker and consumption per effective worker in this steady state. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 214 Government debt and ieconomic growth b. Explain if and why you agree, or do not agree, with the following statements: b.1 b.2 an increase in the saving rate leads to a new balanced growth path characterized by a higher level of output per worker, but an unchanged growth rate of ππ⁄ππ; the Solow model implies that, if the growth rate of population ππππ increases, the economy will reach a new steady state where the growth rate of aggregate output ππ is unchanged, and the growth rate output per worker ππ⁄ππ is lower. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. ................................................................................................................. 215 Part II Solutions Chapter 1 - The goods and financial markets Macroeconomics. Problems and Questions Question 1 In the economy there are only two firms, firm A and firm B. Their operations in a given year can be summarized as follows (all figures are in thousands of Euros): Firm A Costs Revenues Wages 170 Sales to B 300 Purchases from B 50 Sales to consumers 400 Indirect taxes 30 Firm B Costs Revenues Wages 230 Sales to A 50 Purchases from A 300 Sales to consumers 500 Indirect taxes 20 Exports 100 Compute the economy’s Gross Domestic Product (GDP) using all the definitions of this variable that it is possible to employ in this case. 220 The goods and financial markets ........................................................................................................................... a. GDP is the value of the final goods and services produced in the economy: ........................................................................................................................... Value of final goods and services produced by firm A = 400 (sales to con........................................................................................................................... sumers); Value of final goods and services produced by firm B = 500 (sales to the ........................................................................................................................... public) + 100 (exports, which are always a final use of the goods pro........................................................................................................................... duced in a country – why?) = 600 GDP = 400 + 600 = 1000. ........................................................................................................................... ........................................................................................................................... 1. An alternative, and fully equivalent, way of computing GDP is as the sum of the value added (V.A.) by all the firms in the economy:. ........................................................................................................................... V.A. A = (300+ 400) – 50 = 650; ........................................................................................................................... V.A. B = (50 + 500+100) – 300 = 350; ........................................................................................................................... GDP = 650 + 350 = 1000. ........................................................................................................................... 2. Finally, in this economy, GDP = Incomes + Indirect taxes, where incomes ........................................................................................................................... are in this case the sum of profits (revenues – costs) and wages. Firm A’s profits = (300 + 400) – (170 + 50 + 30) = 450; ........................................................................................................................... Firm B’s profits = (50 + 500 + 100) – (230 + 300 + 20) =100; ........................................................................................................................... Wages paid by A = 170; Wages paid by B = 230. In addition, indirect taxes = 30 + 20 = 50. ........................................................................................................................... It follows that GDP = 450 +100 + 170 + 230 + 50 = 1000. ........................................................................................................................... 221 Macroeconomics. Problems and Questions Question 2 a. What is meant by ‘GDP deflator’? How is the GDP deflator computed, and how is it used? ........................................................................................................................... The GDP deflator for year π‘π‘, πππ‘π‘ , is computed by dividing nominal GDP by re........................................................................................................................... al GDP in year π‘π‘: ........................................................................................................................... €πππ‘π‘ ........................................................................................................................... πππ‘π‘ = , πππ‘π‘ ........................................................................................................................... where €πππ‘π‘ and πππ‘π‘ are year π‘π‘ nominal GDP and real GDP, respectively. ........................................................................................................................... The GDP deflator is an index number with very wide coverage (it reflects the ........................................................................................................................... prices of all goods and services taken into account when measuring GDP – that is, all the final goods and services produced in the economy). It takes on ........................................................................................................................... the value 1 in the base year, and its changes between consecutive years are ........................................................................................................................... one of the possible measures of the rate of inflation – the change over time in the general price level. ........................................................................................................................... 222 The goods and financial markets b. Compared to the previous year (π‘π‘ − 1), in year π‘π‘ the economy’s GDP deflator has gone down by 1% and the real GDP growth rate has been equal to −2%. b.1 Compute the growth rate of nominal GDP for this economy in year π‘π‘. b.2 What is the rate at which the economy under consideration has been growing in year π‘π‘ ? b.1 The definition of GDP deflator given when answering the previous point of this question implies that year π‘π‘ nominal GDP, €πππ‘π‘ , can be written as the product between the GDP deflator and real GDP for the same year: €πππ‘π‘ = πππ‘π‘ · πππ‘π‘ . It follows that the rate of growth of nominal GDP is (approximately) equal to the sum of the rate of change of the GDP deflator and the rate of change of real GDP. In the economy we have been asked to consider, the growth rate of nominal GDP in year π‘π‘ therefore amounts to (−1%) + (−2%) = −3%. b.2 The ‘rate of growth of the economy’ is the rate of change of real GDP between two consecutive years. It follows that, in year π‘π‘, the economy has been growing at a rate of −2%. 223 Macroeconomics. Problems and Questions Question 3 The economy of a country in which only the goods market exists is described by the following system of equations, πΆπΆ = ππ0 πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ ππ = πποΏ½ + π‘π‘π‘π‘ where, as always, the positive constant ππ0 is autonomous consumption, π‘π‘ (a constant between zero and one) is the tax rate, πποΏ½ (> 0) is the portion of net taxes that does not depend on income, and the other symbols have the usual meaning. a. Having determined graphically in the figure below the equilibrium level of production οΏ½πποΏ½οΏ½ for this economy, derive the analytical expressions of πποΏ½ and of the multiplier implied by the model specified above. 45° ππ, ππ ππ0 + πΌπΌ Μ + πΊπΊΜ πποΏ½ 224 ππ The goods and financial markets In this economy, the demand for goods is entirely autonomous – no component of aggregate demand depends on income. In the graph, it will therefore be represented by a horizontal line, and the equilibrium level of income will be the value of ππ for which that line crosses the 45° line going through the origin. The analytic expression of equilibrium income is πποΏ½ = ππ0 + πΌπΌ Μ + πΊπΊΜ , where ππ = π΄π΄ = ππ0 + πΌπΌ Μ + πΊπΊΜ , and the multiplier is therefore equal to 1. b. Suppose that the government cuts πποΏ½ (that, as you will remember, is the fraction of net taxes independent of income), so that βπποΏ½ < 0. Show in the graph the effects of this change and, using the results derived when answering the previous point of this question, derive the analytical expressions of the changes in equilibrium production, private saving, government saving and national saving caused by this decrease in πποΏ½. Since the demand for goods does not depend on πποΏ½, in the graph no curve will shift, and equilibrium income will therefore remain unchanged. Being βπποΏ½ = 0, private saving ππΜππππππππ = οΏ½πποΏ½ − πποΏ½ − π‘π‘πποΏ½οΏ½ − πΆπΆ = (1 − π‘π‘)πποΏ½ − πποΏ½ − ππ0 will rise by βππΜππππππππ = −βπποΏ½, government saving ππΜππππππ = ππ − πΊπΊΜ = πποΏ½ + π‘π‘πποΏ½ − πΊπΊΜ will fall by βππΜππππππ = βπποΏ½, and national saving will not change, since βππΜππππππππ + βππΜππππππ = 0. 225 Macroeconomics. Problems and Questions Question 4 The economy of a country in which only the goods market exists is described by the following system of equations: πΆπΆ = ππ0 πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ − ππ1 ππ ππ = πποΏ½ + π‘π‘π‘π‘ where, as always, the positive constant ππ0 is autonomous consumption, π‘π‘ (a constant between zero and one) is the tax rate, πΊπΊΜ and οΏ½ππ (both greater than zero) are the portions of government spending and net taxes that do not depend on income, and the parameters ππ1 and ππ1 , with 0 < ππ1 − ππ1 < 1, are both positive. a. Having determined graphically in the figure below the equilibrium level of production οΏ½πποΏ½οΏ½ for this economy, derive the analytical expressions of πποΏ½, of autonomous spending π΄π΄ and of the multiplier implied by the model specified above. ππ, ππ 2 45° 1 ππ0 + πΌπΌ Μ + πΊπΊΜ πποΏ½ In this economy, the demand for goods is πποΏ½′ ππ = ππ0 + πΌπΌ Μ + πΊπΊΜ + (ππ1 − ππ1 )ππ ππ′ ππ ππ where ππ0 + πΌπΌ Μ + πΊπΊΜ is autonomous spending, π΄π΄. In the graph on the left, ππ is therefore a straight line with slope (ππ1 − ππ1 ), by assumption a quantity which is positive but smaller than one. 226 The goods and financial markets Writing down the goods market equilibrium condition, and solving for Y, one gets the following expression for equilibrium income (the value of Y corresponding to point 1 in the figure): ππ0 + πΌπΌ Μ + πΊπΊΜ πποΏ½ = , 1 − (ππ1 − ππ1 ) from which it follows that the multiplier is 1 . 1 − (ππ1 − ππ1 ) b. Suppose that the government raises, at the same time and by the same amount, both πΊπΊΜ and πποΏ½, so that βπΊπΊΜ = βπποΏ½ > 0. Show in the graph the effects of these changes and, using the results derived when answering the previous point of this question, derive the analytical expressions of the changes in equilibrium production, private saving, government saving and national saving caused by the increases in πΊπΊΜ and in πποΏ½. In your answer, make sure to discuss in each case why reaching a definite conclusion about the direction in which those variables will change is possible, or not possible. When βπΊπΊΜ = βπποΏ½ > 0, the aggregate demand curve in the figure shifts upwards in a parallel fashion by βπΊπΊΜ , and the new equilibrium becomes point 2. From the expression for πποΏ½ just derived, it follows that equilibrium income goes up by 1 βπποΏ½ = βπΊπΊΜ (> 0). 1 − (ππ1 − ππ1 ) The change in private saving will be (ππ1 − ππ1 ) − π‘π‘ βππΜππππππππ = βοΏ½πποΏ½ − πποΏ½ − π‘π‘πποΏ½οΏ½ = (1 − π‘π‘)βπποΏ½ − βπΊπΊΜ = βπΊπΊΜ , 1 − (ππ1 − ππ1 ) a quantity whose sign is uncertain, without further information on the relative sizes of the parameters π‘π‘, ππ1 and ππ1 , while government saving will rise for sure by π‘π‘ + ππ1 βππΜππππππ = βπποΏ½ + π‘π‘βπποΏ½ − βπΊπΊΜ + ππ1 βπποΏ½ = βπΊπΊΜ (> 0). 1 − (ππ1 − ππ1 ) Finally, and although without more information it is not possible to determine the direction in which private saving will change, it is easy to conclude that national saving will go up, since ππ1 βππΜππππππ = βπΌπΌ = βπΊπΊΜ (> 0). 1 − (ππ1 − ππ1 ) 227 Macroeconomics. Problems and Questions Question 5 The goods market of a nation is described by the following equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) ππ = πποΏ½ + π‘π‘π‘π‘ πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ, where the parameter π‘π‘ is greater than zero (so that net taxes are increasing in the level of income), but smaller than one. a. Derive the expression for the equilibrium level of income and that of the multiplier for this economy. Using the first four equations into the last one, the goods market equilibrium condition, and solving for ππ, one gets the following expression for the equilibrium level of income: πποΏ½ = 1 ⋅ [ππ − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ ] , 1 − ππ1 (1 − π‘π‘) 0 where the term in square brackets is autonomous spending, and the multiplier is the ratio 1 . 1 − ππ1 (1 − π‘π‘) 228 The goods and financial markets b. Two economists, Mary and Paul, debate on the effects of an increase in the autonomous component of government spending πΊπΊΜ on the government deficit, π·π·π·π·π·π· = πΊπΊΜ − ππ. According to Mary, since an increase in πΊπΊΜ leads to a higher income, and net taxes are increasing in ππ, raising πΊπΊΜ has an uncertain effect on the deficit. For sufficiently high values of π‘π‘, the increase in tax revenues could be so large that the government deficit could end up falling. Economist Paul, on the other hand, believes that an increase in πΊπΊΜ would still increase the deficit, for any π‘π‘ < 1. Derive the expression for the change in the deficit, π₯π₯π₯π₯π₯π₯π₯π₯, when πΊπΊΜ varies by π₯π₯πΊπΊΜ > 0. Which of the two economists is right? In the initial equilibrium, π·π·π·π·π·π· = πΊπΊΜ − ππ = πΊπΊΜ − πποΏ½ − π‘π‘πποΏ½ ⇒ π₯π₯π₯π₯π₯π₯π₯π₯ = π₯π₯πΊπΊΜ − π‘π‘π‘π‘πποΏ½. From the expression for equilibrium income derived before, it follows that, following a change in πΊπΊΜ , πποΏ½ will vary by: π₯π₯πποΏ½ = 1 ⋅ π₯π₯πΊπΊΜ . 1 − ππ1 (1 − π‘π‘) Plugging this expression for π₯π₯πποΏ½ into π₯π₯π₯π₯π₯π₯π₯π₯ = π₯π₯πΊπΊΜ − π‘π‘π‘π‘πποΏ½ and collecting terms, one gets: π₯π₯π₯π₯π₯π₯π₯π₯ = οΏ½ (1 − π‘π‘)(1 − ππ1 ) οΏ½ ⋅ π₯π₯πΊπΊΜ . 1 − ππ1 (1 − π‘π‘) The term in square brackets is positive for any π‘π‘ < 1, and an increase in πΊπΊΜ will therefore raise the deficit. Paul is right. [However, it is true that the larger is π‘π‘, the smaller will be the worsening of the deficit.] 229 Macroeconomics. Problems and Questions Question 6 The economy of a country in which only the goods market exists is described by the following system of equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ ππ = πποΏ½ where the various symbols have the usual meaning. a. Derive the expressions of equilibrium income and of the multiplier for this economy. ππ, ππ 1 π΄π΄ πποΏ½ In this economy, the demand for goods is 2 πποΏ½′ 45° ππ′ ππ ππ ππ = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ + ππ1 ππ where ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ is autonomous spending, π΄π΄. In the graph on the left, ππ is therefore a straight line with slope ππ1 , positive but smaller than one. Writing down the goods market equilibrium condition, and solving for Y, one gets the following expression for equilibrium income (the value of Y corresponding to point 1 in the figure): πποΏ½ = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ , 1 − ππ1 from which it follows that the multiplier is 1 . 1 − ππ1 230 The goods and financial markets b. Suppose that, when the economy was in the equilibrium position described above, investment rises, and that the Government cuts πποΏ½ by the same amount by which πΌπΌ Μ has gone up, so that βπΌπΌ Μ = −βπποΏ½ > 0. Show in the graph the effects of these two contemporaneous changes and, using the results derived when answering the previous point, derive the analytical expressions of the changes in equilibrium production, private saving, government saving and national saving they will cause. In your answer, make sure to discuss in each case why reaching a definite conclusion about the direction in which those variables will vary is possible, or not possible. When βπΌπΌ Μ = −βπποΏ½ > 0, the aggregate demand curve in the figure shifts upwards in a parallel fashion by (1 + ππ1 )βπΌπΌ ,Μ and the new equilibrium becomes point 2. From the expression for πποΏ½ just derived, it follows that equilibrium income goes up by 1 + ππ1 πποΏ½ = βπΌπΌ Μ (> 0). 1 − ππ1 The change in equilibrium private saving ππΜππππππππ = οΏ½πποΏ½ − πποΏ½οΏ½ − πΆπΆΜ will be βππΜππππππππ = βοΏ½πποΏ½ − πποΏ½) − ππ1 β(πποΏ½ − πποΏ½οΏ½ = (1 − ππ1 ) 2βπΌπΌ Μ = 2βπΌπΌ ,Μ 1 − ππ1 a positive quantity. Since government saving, equal to the difference between ππ and πΊπΊ, varies by βπποΏ½ = −βπΌπΌ Μ (< 0), national saving (the sum of private and government saving) rises by βπΌπΌ.Μ [Alternatively: since investment goes up by βπΌπΌ,Μ in equilibrium national saving will have to rise by the same amount; given that government saving falls by βπΌπΌ,Μ it follows that private saving will increase by 2βπΌπΌ]Μ . 231 Macroeconomics. Problems and Questions Question 7 The economy of a country in which only the goods market exists is described by the following system of equations, πΆπΆ = ππ0 πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ − ππππ ππ = πποΏ½ + π‘π‘π‘π‘ where, as always, the positive constant ππππ is autonomous consumption, π‘π‘ (a constant between zero and one) is the tax rate, ππ (> 0) is the sensitivity to income of government spending on goods and services, and the other symbols have the usual meaning. a. Write down the analytical expressions of autonomous spending, equilibrium income, and the multiplier for this economy. In this economy, autonomous spending is π΄π΄ = ππππ + πΌπΌ Μ + πΊπΊΜ , while equilibrium income is, as usual, the value of ππ that solves the goods market equilibrium condition, that is, ππ = π΄π΄ − ππππ, πποΏ½ = 1 β π΄π΄, 1 + ππ where the multiplier is the ratio 1⁄(1 + ππ). 232 The goods and financial markets b. Suppose that the autonomous components of government purchases of goods and services and of net taxes are cut at the same time and by the same amount, so that βπΊπΊΜ = βπποΏ½ > 0. Determine the effect of this change on the government deficit, π·π·π·π·π·π· = πΊπΊ − ππ, prevailing when the goods market is in equilibrium. Make sure to explain if, and why, the sign of the change in government deficit will, or will not, depend on the fact that, in this economy, the parameter π‘π‘ is larger or smaller than the parameter ππ. When βπΊπΊΜ = βπποΏ½ < 0, in equilibrium the government deficit will change by οΏ½ = βπΊπΊΜ − ππβπποΏ½ − βπποΏ½ − π‘π‘βπποΏ½ = −(ππ + π‘π‘)βπποΏ½. βπ·π·π·π·π·π· Plugging in this equation the expression for the change in equilibrium income when βπΊπΊΜ = βπποΏ½ < 0, that is, βπποΏ½ = [1⁄(1 + ππ)] β βπΊπΊΜ , yields οΏ½ =− βπ·π·π·π·π·π· ππ + π‘π‘ β βπΊπΊΜ . 1 + ππ Since βπΊπΊΜ < 0, the government deficit will unambiguously rise, and this independently of the relative size of the two parameters π‘π‘ and ππ. The intuition is as follows: if πΊπΊ and ππ did not depend on ππ (in other words, if ππ = π‘π‘ = 0), cutting πΊπΊΜ and πποΏ½ by the same amount would not change the government deficit; however, since βπΊπΊΜ < 0 decreases equilibrium income, and given that when income goes down government outlays rise (remember that ππ > 0), and government revenues (net taxes) fall (since π‘π‘ > 0), in this economy βπΊπΊΜ = βπποΏ½ < 0 will unambigously increase the government deficit, and this independently of π‘π‘ being greater, rather than smaller, than ππ. 233 Macroeconomics. Problems and Questions Question 8 The goods market of a country is described by the following model: πΆπΆ = ππ0 + ππ1 (ππ − ππ) ππ = πποΏ½ + π‘π‘π‘π‘ πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ, where the parameter π‘π‘ (the tax rate) is positive, but smaller than one, ππ1 (> 0) is the sensitivity of investment to income, ππ1 + ππ1 < 1, and the other symbols have the usual meaning. a. Derive the expressions of equilibrium income and of the multiplier for this economy. Equilibrium income, πποΏ½, is the value of Y that solves the goods market equilibrium condition (the last equation of the model above). Using the first four equations in the fifth one, and solving, one gets πποΏ½ = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ . 1 − ππ1 − ππ1 + π‘π‘ππ1 It follows that the multiplier is the ratio 1 . 1 − ππ1 − ππ1 + π‘π‘ππ1 234 The goods and financial markets b. Suppose that autonomous consumption and the autonomous component of net taxes rise at the same time and by the same amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ > 0. By how much will equilibrium income and national saving (the sum of private and public saving) change? Derive the expressions for the changes in those two variables, and explain if and why they will rise, fall or remain unchanged. From the expression for πποΏ½ just derived, it follows that π₯π₯πποΏ½ = π₯π₯π₯π₯0 − ππ1 π₯π₯πποΏ½ (1 − ππ1 )π₯π₯π₯π₯0 = > 0. 1 − ππ1 − ππ1 + π‘π‘ππ1 1 − ππ1 − ππ1 + π‘π‘ππ1 Equilibrium income goes up. In fact, the increase in autonomous consumption raises the demand for goods by π₯π₯ππ0 ; the increase in taxes lowers disposable income, and therefore consumption, but this latter falls just by −ππ1 π₯π₯πποΏ½ = −ππ1 π₯π₯π₯π₯0 . The overall impact on autonomous demand of the two changes is therefore π₯π₯π₯π₯0 − ππ1 π₯π₯π₯π₯0 = (1 − ππ1 )π₯π₯π₯π₯0 , a positive quantity, since 0 < ππ1 < 1 and π₯π₯π₯π₯0 > 0. It follows that equilibrium production goes up. As for national saving, it will go up, too. In fact, in equilibrium it must be equal to investment, and investment rises by π₯π₯πΌπΌΜ = ππ1 π₯π₯πποΏ½ = ππ1 (1 − ππ1 )π₯π₯π₯π₯0 > 0. 1 − ππ1 − ππ1 + π‘π‘ππ1 The right-hand side of the above equation therefore also represents the equilibrium change in national saving that you were asked to derive. 235 Macroeconomics. Problems and Questions Question 9 Country Macro, where only the good market exists and prices are constant, is described by the following model: οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ πΆπΆ = ππ0 + ππ1 (ππ − ππ) + ππ ππππππ οΏ½ ππ = ππ πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ. In the equations above, 0 < ππ1 < 1, ππ1 > 0, 0 < ππ1 + ππ1 < 1, the parameter ππ (> 0) is the sensitivity of consumption to the financial and housing wealth οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ), and the other ππππππ (assumed to be exogenous, so that ππππππ = ππππππ symbols have the usual meaning. a. Derive the expressions of equilibrium income and that of the multiplier for this economy. Equilibrium income, πποΏ½, is the value of Y that solves the goods market equilibrium condition (the last equation of the model above). Using the first four equations in the fifth one, and solving, one gets πποΏ½ = οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ + ππ ππππππ . 1 − (ππ1 + ππ1 ) It follows that the multiplier is, as usual, 1 . 1 − (ππ1 + ππ1 ) 236 The goods and financial markets b. Suppose that individuals experience an increase in their financial and οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ > 0. Write down the expression of the housing wealth, so that π₯π₯ππππππ change in equilibrium private saving caused by this change. In particular, explain if, and why, private saving will rise, fall, or remain constant. Private saving is the portion of their disposable income individuals decide not to consume, ππ = (ππ − ππ) − πΆπΆ. Using in this expression the consumption function for Macro (first equation of the model above), it follows that, in ππππππ . If indiequilibrium, private saving is ππΜ = −ππ0 + (1 − ππ1 )οΏ½πποΏ½ − πποΏ½οΏ½ − ππ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ > 0, equilibrium income viduals’ financial and housing wealth rises by π₯π₯ππππππ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ . It follows that, in equilibrium, priwill rise by the multiplier times πππ₯π₯ππππππ vate saving will change by οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ = (1 − ππ1 ) · π₯π₯ππΜ = (1 − ππ1 )π₯π₯πποΏ½ − ππππππππππ = οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ππππππππππ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ − πππ₯π₯ππππππ 1 − (ππ1 + ππ1 ) ππ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ > 0. · ππ π₯π₯ππππππ 1 − (ππ1 + ππ1 ) Private saving will be higher. For a given disposable income, the increase in ππππππ leads individuals to save less, and consume more. However, this increase in consumption raises equilibrium income, and therefore saving. Although these two forces push saving in opposite directions, to prevail must necessarily be the second one, and private saving will go up, as made clear by the equation above. An alternative, and fully equivalent, way to arrive to the same conclusion is to recall that, in goods market equilibrium, investment and national saving must be equal. Since the rise in wealth leads to more consumption and a higher income, in the new equilibrium investment (which depends positively on Y) will be higher. It follows that, in the new equilibrium, national saving must be higher, too. Since government saving is unchanged, the increase in national saving can only follow from an increase in private saving. 237 Macroeconomics. Problems and Questions Question 10 The economy of a country consisting of the goods market only is described by the following equations: πΆπΆ = ππ1 (ππ − ππ) ππ = πποΏ½ + π‘π‘π‘π‘ πΌπΌ = πΌπΌ Μ πΊπΊ = πΊπΊΜ , where the parameters ππ1 and π‘π‘ are positive but smaller than one. In this country, autonomous consumption ππ0 is therefore equal to zero, government purchases of goods and services and investment are entirely exogenous, and net taxes depend on consumption – as they include not just an exogenous component (πποΏ½), but also a portion that is increasing in consumption (π‘π‘π‘π‘). a. Plugging the expression for net taxes given by the second equation into the first one, and solving for πΆπΆ, derive the expression that the consumption function takes on in this economy. Next, using the definition of (private) saving and the consumption function you have just derived, write down the (private) saving function. In this economy, is private saving still increasing in income, ππ, as it is in the standard case? Why, or why not? Explain. Plugging the expression for net taxes given by the second equation into the first one, yields: ........................................................................................................................... πΆπΆ = ππ1 ππ − ππ1 πποΏ½ − ππ1 π‘π‘π‘π‘. ........................................................................................................................... Solving for πΆπΆ, the consumption function for this economy can be written as ........................................................................................................................... ππ1 πΆπΆ = (ππ − πποΏ½). ........................................................................................................................... 1 + ππ1 π‘π‘ .................................... Using this expression for πΆπΆ and ππ = πποΏ½ + π‘π‘π‘π‘ in the definition of private saving, ππ = (ππ − ππ) − πΆπΆ, one gets: 1 − ππ1 ππ = οΏ½ οΏ½ (ππ − πποΏ½), 1 + ππ1 π‘π‘ a quantity that depends positively on ππ, since ππ1 , the marginal propensity to consume out of disposable income, is less than one. As usual, also in this economy private saving is therefore increasing in the level of income. 238 The goods and financial markets b. Using the results derived when answering the previous point, write down the expression of the government deficit (π·π·π·π·π·π·) for this economy. Is the country’s government deficit increasing or decreasing in the level of income, ππ? Why? Provide the economic intuition for your answer. .................. From the results above, it follows that, in this economy, π·π·π·π·π·π· = πΊπΊ − ππ = πΊπΊΜ − πποΏ½ − π‘π‘ππ1 (ππ − πποΏ½). 1 + ππ1 π‘π‘ The government deficit is therefore smaller the larger is the economy’s income. This is so because an increase in ππ does not change the government’s purchases of goods and services, which are entirely exogenous; by raising consumption, it however leads to an increase in net taxes (see the second equation of the model describing this economy), and therefore in government revenues. 239 Macroeconomics. Problems and Questions Question 11 a. The goods market of country Zeta is described by the following equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) ππ = πποΏ½ πΌπΌ = πΌπΌ Μ + ππ1 ππ πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ, where 0 < ππ1 < 1, ππ1 > 0 and ππ1 + ππ1 < 1. Economist Cher thinks that, in this economy, one could simultaneously change πΊπΊΜ and πποΏ½ so as to decrease the government deficit π·π·π·π·π·π· = πΊπΊΜ − πποΏ½ without changing at the same time the equilibrium level of income. Economist Sonny holds a different opinion: in Zeta, cutting the government deficit will always lead to a fall in equilibrium income. Which of the two economists is right? Why? [Hint: (i) write the expression for equilibrium income, πποΏ½; (ii) use it to find out how the change in πΊπΊΜ , π₯π₯πΊπΊΜ , and the change in net taxes, π₯π₯πποΏ½, must be related to one another for equilibrium income to remain constant (π₯π₯πποΏ½ = 0); (iii) verify whether it is possible that, for the values of π₯π₯πΊπΊΜ and π₯π₯πποΏ½ so determined, the change in the deficit π₯π₯π₯π₯π₯π₯π₯π₯ is going to be negative and, if so, under what circumstances this could be the case. Explain]. ........................................................................................................................... The equilibrium level of output is the value of Y that solves the goods market equilibrium condition (the last equation of the model above), that is ........................................................................................................................... ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ ........................................................................................................................... πποΏ½ = . 1 − ππ1 − ππ1 ........................................................................................................................... Following a simultaneous change in πΊπΊΜ and πποΏ½, in order to have π₯π₯πποΏ½ = 0 it must ........................................................................................................................... be the case that π₯π₯πΊπΊΜ = ππ1 π₯π₯πποΏ½. Using this result in π₯π₯π₯π₯π₯π₯π₯π₯ = π₯π₯πΊπΊΜ − π₯π₯πποΏ½, the change in the government deficit can be written as π₯π₯π₯π₯π₯π₯π₯π₯ = ππ1 π₯π₯πποΏ½ − π₯π₯πποΏ½ = ........................................................................................................................... (ππ1 − 1)π₯π₯πποΏ½, which is less than zero if π₯π₯πποΏ½ > 0 (remember that 0 < ππ1 < 1). ........................................................................................................................... It is easy to see that raising net taxes by π₯π₯πποΏ½ and πΊπΊΜ by π₯π₯πΊπΊΜ = ππ1 π₯π₯πποΏ½ (< π₯π₯πποΏ½) ........................................................................................................................... leads to a lower deficit. The reason why income does not change is that the decision to increase net taxes by π₯π₯πποΏ½ leads to an equal fall in disposable in........................................................................................................................... come and, for a given ππ, lowers consumption by ππ1 π₯π₯πποΏ½; however, this drop in ........................................................................................................................... consumption is offset by the increase in πΊπΊΜ by ππ1 π₯π₯πποΏ½ . Since aggregate demand does not change, equilibrium income will not change either. Cher is right. ........................................................................................................................... 240 The goods and financial markets b. Consider now a different country, one in which there is only a goods market described by the same set of equations specified above. Write down the goods market equilibrium condition as an equality between national (private plus public) saving and investment, and assume that the country’s government decides to raise net taxes, πποΏ½. In the new equilibrium, national and private saving will be larger or smaller than before? [Hint: you are NOT being asked to compute the expressions for the change in private and national saving, but just to provide the economic intuition that helps determine the direction in which they will change]. .......................................................................................................................... In goods market equilibrium, national saving (the sum of private saving and ........................................................................................................................... government saving) must equal investment, ππ + οΏ½ππ − πΊπΊοΏ½ = πΌπΌ. The increase in ........................................................................................................................... net taxes lowers equilibrium income by ........................................................................................................................... −ππ1 π₯π₯πποΏ½ π₯π₯πποΏ½ = ........................................................................................................................... 1 − ππ1 − ππ1 ........................................................................................................................... and thus leads to lower investment, since this latter variable is increasing in ππ. It follows that national saving will have to fall by the same amount I has gone ........................................................................................................................... down. Since public saving is greater than before (due to the increase in πποΏ½), ........................................................................................................................... this requires a lower private saving. Private saving will fall because of the decrease in disposable income caused by the decrease in πποΏ½ and the increase in πποΏ½. ........................................................................................................................... 241 Macroeconomics. Problems and Questions * Question 12 [Equilibrium in the money market and equilibrium in the market for the monetary base] a. Define what is meant by “monetary base”, also known as “central bank money”, π»π». Assuming that individuals hold money both in the form of currency and in that of checkable deposits, show in the graph an equilibrium position in the market for the monetary base and discuss how the interest rate prevailing in such an equilibrium changes following a decrease in π»π», an increase in ππ (the fraction of their money individuals want to hold as currency) and a rise in ππ (the reserve ratio). ππ 1 π€π€Μ π»π»’ 0 2 π»π» π»π» ππ π»π» ππ ′ π»π» π π , π»π» ππ ........................................................................................................................... ........................................................................................................................... The monetary base amounts to the central bank's overall liabilities − the sum of ........................................................................................................................... bank reserves (π π ) and currency (πΆπΆπΆπΆ). The supply of monetary base, that in the ........................................................................................................................... figure we assume to be initially given by π»π», is just the total amount of currency ........................................................................................................................... and reserves outstanding at a given point in time, π»π» = πΆπΆπΆπΆ + π π . Since it has full ........................................................................................................................... control over its liabilities, the supply of monetary base is under the control of ........................................................................................................................... the central bank (for instance, through open market operations). It follows ........................................................................................................................... that, in a diagram where the interest rate is measured on the vertical axis, it ........................................................................................................................... will be represented by a vertical line at π»π» − the value of the monetary base chosen by the central bank. The demand for monetary base, π»π» ππ , is the sum of the individuals' demand for currency, πΆπΆπΆπΆ ππ = ππππππ , and of the banks' demand for reserves, π π ππ = ππ(1 − ππ)ππππ [the fraction ππ of individuals' demand for deposits]. Since π»π» ππ = πΆπΆπΆπΆ ππ + π π ππ = [ππ + ππ(1 − ππ)]ππππ , and given that ππππ is decreasing in the interest rate, the demand for monetary base is a negatively sloped curve in the diagram above. 242 The goods and financial markets A fall in π»π», due for instance to the fact that the central bank has decided to sell bonds in the open market, shifts the supply curve to the left. The equilibrium position goes from point ′0′ to point ′1′, where the interest rate is higher (the sale of bonds lowers their price, thus rising their yield). Finally, an increase in ππ and/or ππ shifts the demand curve to the right, and the equilibrium from point ′0′ to a point such as ′2′ in the figure. In this case, the interest rate goes up for the reasons that will be discussed answering the next question. b. Using the diagram below, show how the same changes just considered (decrease in π»π»; rise in ππ or ππ) will affect the money market equilibrium. Based on the results of your analysis, what can you conclude about the relationship between the value of the interest rate for which the market for the monetary base is in equilibrium, and the value of the interest rate for which the money market is in equilibrium? ππ 1 π€π€Μ π»π» ππ + ππ(1 − ππ) ↓ π»π», ↑ ππ, ↑ ππ ππ’ ππ 0 ππππ ππ π π , ππππ Let's start by writing down the equilibrium condition in the market for the monetary base, π»π» = π»π» ππ = [ππ + ππ(1 − ππ)]ππππ . Dividing both sides by the term in squared brackets, the previous equation becomes: 243 Macroeconomics. Problems and Questions π»π» = ππππ . ππ + ππ(1 − ππ) On the right-hand side, we now have the demand for money. The term on the left-hand side is − as you might have guessed, being the equation above an equilibrium condition − the supply of money. When people hold both currency and checkable deposits, one can in fact write ππ as the ratio on the left-hand side of the equilibrium condition above.1 It follows that, whenever the market for the monetary base is in equilibrium, the money market, too, will be in equilibrium. One can therefore analyze the determination of the interest rate with reference to either market, reaching identical conclusions for the equilibrium value of this variable. If, for instance, π»π» falls, maybe because the central bank has decided to sell bonds in the open market, the left-hand side of the above equation – the supply 1 of money – declines, leading to the same increase in ππ we concluded will take place when answering to point a. of this question. The same will be true following an increase in ππ and/or in ππ – these changes, too, will lower the supply of money, thus leading to the same increase in the equilibrium interest rate that, by looking at the market for the monetary base, we concluded will take place. 1 By definition, the supply of money is the sum of currency and deposits, ππ = πΆπΆπΆπΆ + π·π·, while the monetary base is the sum of currency and reserves, π»π» = πΆπΆπΆπΆ + π π . Taking the ratio of the two previous equations, one gets: ππ πΆπΆπΆπΆ + π·π· ππππ + (1 − ππ)ππ 1 = = = π»π» πΆπΆπΆπΆ + π π ππππ + ππ(1 − ππ)ππ ππ + ππ(1 − ππ) ππ ππ where we have used the fact that, in equilibrium, πΆπΆπΆπΆ = πΆπΆπΆπΆππ = ππππ , π·π· = π·π·ππ = (1 − ππ)ππ , ππ π π = π π ππ = ππ(1 − ππ)ππ , and ππ = ππππ . Finally, multiplying both sides of the previous equation by π»π» yields ππ = π»π» . ππ + ππ(1 − ππ) 244 The goods and financial markets Lastly, let's try to understand why increases in ππ and/or in ππ end up lowering the money supply, thus raising the equilibrium value of the interest rate. The reason is that, whenever banks increase (decrease) the loans they extend to their customers, the money supply goes up (down). In fact, bank loans give rise to new deposits (I get a mortgage from my bank to buy a house; the seller of the house will deposit with her bank at least part of the price I paid), to further loans (the bank where the seller of the house has deposited the sum I have paid will lend out most of it), to new deposits, etc., in a process that leads to a multiple expansion of bank deposits and of the money supply − which, remember, is the sum of currency and bank deposits outstanding at a given point in time. If ππ goes up, individuals will deposit a smaller fraction of their money; experiencing a decrease in deposits, banks will be able to lend out less, deposits will further shrink, and the money supply will fall. If, on the other hand, to rise is ππ, the reserve ratio, banks will lend out a smaller fraction of any given amount of deposits received by their customers. In either case, ππ falls and the equilibrium value of the interest rate rises. 245 Macroeconomics. Problems and Questions * Question 13 [Endogenous money supply] a. Write down the money market equilibrium condition. Assuming that money demand, ππππ , is increasing in the general price level ππ and in real income ππ, and decreasing in the interest rate ππ, represent in the graph below a position of equilibrium in the money market, denoting by π€π€Μ the value that the interest rate takes on in that equilibrium. ππ π€π€Μ 1 0 The money market equilibrium condition is ππ ππ′ ππππ ππππ ′ ππ π π , ππππ ππ π π = ππππ where the term on the left-hand side is the nominal money supply (that in the figure we assume to be initially equal to ππ), and that on the right-hand side ........................................................................................................................... the nominal demand for money, increasing in ππ and ππ, and decreasing ππ, ........................................................................................................................... ππππ = ππππ (ππ, ππ, ππ) (∗) + + − ........................................................................................................................... Since €ππ = ππππ, a functional form for the demand for money slightly more spe........................................................................................................................... cific than the one above is ........................................................................................................................... ππππ = €ππ β πΏπΏ(ππ) (∗∗) − ........................................................................................................................... Alternatively, one could posit ........................................................................................................................... (∗∗∗) ππππ = ππ β πΏπΏ(ππ, ππ) + − ........................................................................................................................... The functional forms (∗∗) and (∗∗∗) are both consistent with what equation (∗) assumes about the way in which money demand should respond to changes 246 The goods and financial markets in its determinants; for this reason, they both lead to conclusions that are, at least qualitatively, identical about the way in which an equilibrium in the money market is reached, and on how this equilibrium is affected by shocks. For this reason, and the fact that this choice greatly simplifies the analysis of the analytical version of the model that will be introduced in the next Chapter, very often we shall assume that the demand for money takes on the functional form (∗∗∗), rather than the one given by equation (∗∗); sometimes, we shall also write the function πΏπΏ (with πΏπΏ = ππππ ⁄ππ = real money demand) as πΏπΏ(ππ, ππ) = ππ1 ππ − ππ2 ππ, so that ππππ = ππ(ππ1 ππ − ππ2 ππ). In the above equation, ππ1 and ππ2 are positive parameters having the interpretation of sensitivity of the demand for money to the level of income and to the interest rate, respectively. No matter the functional form − (∗∗) or (∗∗∗) – assumed for the demand for money, this latter will be a negatively sloped curve in a diagram where the interest rate is measured on the vertical axis; in the same diagram, money supply will be a line vertical at the value ππ chosen by the central bank. In the figure, the initial equilibrium is at point ′0′, with ππ = π€π€Μ . b. Suppose that the central bank aims at keeping the interest rate constant at the level ππ = π€π€Μ and that, starting from an initial equilibrium like the one you have just described, there is a rise in ππ, or in ππ. What should the central bank do, to prevent such a change from leading to an equilibrium interest rate different from π€π€Μ ? Explain. Rises in ππ or in ππ – changes causing an increase in nominal income – shift the money demand curve to the right, to a position such as ππππ′ in the figure. Absent any central bank intervention, the interest rate would increase. To keep ππ equal to π€π€Μ , the central bank will have to raise the money supply to ππ′, so as to make the new equilibrium (initially point ′0′) point ′1′. More generally, if the central bank targets a specific value for the interest rate, the supply of money becomes endogenous − the central bank will have to set ππ at whatever level is consistent with ππ = π€π€Μ , given the values that the other variables on which money demand depends are taking on. If, for instance, the money demand function is ππππ = ππ(ππ1 ππ − ππ2 ππ), the central bank will have to set ππ = ππ(ππ1 ππ − ππ2 π€π€Μ ), a choice that implies that, in money market equilibrium, ππ = π€π€Μ always (to convince yourself that this will indeed be the case, just plug this expression for M in the money market equilibrium condition, ππ = ππππ ). 247 Macroeconomics. Problems and Questions Question 14 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “Consider a simultaneous rise in the reserve ratio, θ, and in the fraction of their money individuals want to hold in the form of currency, ππ. Since, for a given monetary base π»π», they push money supply in opposite directions, the two changes will have an ambiguous impact on ππ”. False. Both changes tend to lower money supply. In fact, an increase in c decreases bank deposits, while an increase in θ leads bank to lend out a smaller fraction of the deposits received from their customers. It follows that, in the case we are asked to analyze, bank will end up lending a smaller fraction of a smaller volume of deposits. Bank loans will fall, thus leading to a further decrease in deposits, loans, deposits… and therefore in money supply. 248 The goods and financial markets b. “A new law banning cash transactions above €500 induces individuals to reduce the fraction of the money they hold as currency (that is, the parameter ππ), and to increase that held in the form of bank deposits. It follows that one effect of the new law will be a fall in money supply, ππ”. False. Other things the same, the rise in bank deposits caused by the fall in c will allow banks to expand their loans, thus leading to new deposits, further loans, new deposits, etc., in a process that will lead to an expansion of the money supply − which is the sum of the currency and the bank deposits outstanding. 249 Macroeconomics. Problems and Questions Question 15 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “An increase in the price of bonds in the bond market makes bonds more attractive and induces individuals to hold a smaller share of their financial wealth in the form of money – the demand for money falls”. False. If the price of bonds increases, their yield, the interest rate, will go down. Other things the same, this fall in the interest foregone by holding money rather than bonds will induce individuals to hold a larger fraction of their financial wealth in the form of money. It follows that money demand will rise. 250 The goods and financial markets b. In country Gamma, the central bank chooses, and wants to keep constant at the chosen level, the interest rate, while in country Delta the central bank chooses the money supply (and, once again, keeps it constant at the level it has chosen). The figure below depicts the money market equilibrium in the two countries. As you can see, in this initial equilibrium position money supply is the same in both, and the same is true about the money demand curve and the equilibrium interest rate. Explain if you agree, or do not agree, with the following statement: “If, in both countries, banks decide to decrease by the same amount the fraction of deposits held as reserves, to keep the interest rate at the chosen level the central bank of Gamma will have to increase the money supply, while in Delta – whose central bank wants to prevent the money supply from changing – the interest rate will fall”. [Hint: in the figure, denote by 2γ and 2δ the new equilibria, if any, that will be reached by Gamma and Delta, respectively]. ππ 1 ππ ππππ ππ, ππππ False. The equilibrium position will remain at point 1 in both countries. In fact, the fall in the reserve ratio θ leads to an increase in money supply (banks will lend out a larger fraction of any given amount of deposits received by their customers, thus causing a further increase in deposits, bank loans, deposits … and therefore M). - In Gamma, the central bank must bring back M to its initial level (e.g., by selling bonds in the open market), and this to prevent the interest rate from falling below the chosen level; - in Delta, the central bank must do the same, in this case because it wants keep M constant at the chosen level. While shocks to money demand affect the money market equilibrium in a way that depends on whether the central bank is choosing, and keeping constant, the money supply rather than the interest rate, the effects of shocks to the money supply do not depend on the rule of conduct followed by the central bank. 251 Macroeconomics. Problems and Questions Question 16 a. Define, briefly but rigorously, the following concepts: a.1 a.2 ‘liquidity trap’; contractionary open market operation (make sure to explain why the intervention you are describing is ‘contractionary’). ........................................................................................................................... a.1 Situation, typically associated with a zero (or, in any case, a very low) interest rate, in which individuals are willing to hold more money (the ........................................................................................................................... most ‘liquid’ asset) even if the interest rate doesn’t change. When the ........................................................................................................................... economy is in a liquidity trap, monetary policy (or, at least, conventional monetary policy) is unable to affect the interest rate and the equilib........................................................................................................................... rium position of the economy. ........................................................................................................................... a.2 Sale of bonds by the central bank in the market for these financial as........................................................................................................................... sets. It is a ‘contractionary’ policy intervention because it ends up reducing the supply of money (the central bank gets paid in currency or, more ........................................................................................................................... frequently, with bank reserves; it follows that the monetary base, sum of ........................................................................................................................... currency and reserves, decreases, and with it the money supply). .............................................................................................. 252 The goods and financial markets b. Explain why, when the economy is in a liquidity trap, an increase in money supply does not lower the interest rate. ............................... Normally, an increase in money supply lowers the interest rate because, at the initial income and interest rate levels, individuals have an incentive to use the ........................................................................................................................... extra money injected in the economy to purchase bonds, assets with a rate of return greater than the return on money. The ensuing increase in the demand ........................................................................................................................... for bonds will raise bond prices and lower their yield, the interest rate. This ........................................................................................................................... latter will keep dropping until it gets low enough to induce individuals to willingly hold all the extra money now in the economy. ........................................................................................................................... If, however, the economy is in a liquidity trap, with a zero (or, in any case, a ........................................................................................................................... very low) interest rate, this process does not take place any longer. Individu........................................................................................................................... als are holding more money, but they do not try to convert this extra money into bonds, as the return on bonds is now very close to that on money, and ........................................................................................................................... money has the extra advantage of being the most liquid asset. In this case, ........................................................................................................................... even if money supply increases, the demand for bonds does not change, and the interest rate therefore does not fall. ........................................................................................................................... 253 Chapter 2 - The IS-LM model Macroeconomics. Problems and Questions * Question 1 [An analytical version of the IS-LM model − (I) The standard case (endogenous money supply)] In a closed economy, consumption, investment, government purchases of goods and services and net taxes are described by the following equations: πΆπΆ = ππ0 + ππ1 (ππ − ππ) πΌπΌ = πΌπΌ Μ + ππ1 ππ − ππ2 ππ πΊπΊ = πΊπΊΜ ππ = πποΏ½ where πΌπΌ Μ is autonomous investment, the coefficients ππ1 and ππ2 , both positive, have the interpretation of sensitivity of investment to income and to the interest rate, respectively, and the other symbols have the usual meaning. Furthermore, assume that the sum ππ1 + ππ1 (the “propensity to spend”) is positive but less than one, and that nominal money demand οΏ½ππππ οΏ½ depends positively on the general price level (ππ) and on real GDP (ππ), and negatively on the interest rate (ππ), as implied by the following functional form: ππππ = ππ(ππ1 ππ − ππ2 ππ). In the previous equation, that we have met already when answering Question 13 of Chapter 1, the coefficients ππ1 and ππ2, both positive, have the interpretation of sensitivity of (real) money demand ππππ ⁄ππ to income and to the interest rate, respectively. Finally, suppose that the central bank sets the money supply so as to make sure that the interest rate always takes on the value ππ = π€π€Μ . a. Derive the analytical expression of the IS curve for this economy, and draw the IS in the (ππ, ππ) plane. What determines the slope of the curve? And what causes parallel shifts of the IS curve in the plane? Explain, using the graphs below and providing the economic intuition underlying your answers. The IS curve is the locus of all the pairs (ππ, ππ) for which the goods market is in equilibrium − that is, for which the following equilibrium condition holds true: ........................................................................................................................... ππ = πΆπΆ + πΌπΌ + πΊπΊ. ........................................................................................................................... Plugging into the previous equation the functional forms for consumption, investment and government spending given above, yields: ........................................................................................................................... ππ = ππ1 + ππ1 (ππ − πποΏ½) + πΌπΌ Μ + ππ1 ππ − ππ2 ππ + πΊπΊΜ . ........................................................................................................................... Since the IS curve is drawn in a plane in which the interest rate is measured along the vertical axis, it is convenient to solve this equation for ππ: 256 The IS-LM model ππ = π΄π΄ 1 − ππ1 − ππ1 − · ππ ππ2 ππ2 where A ( = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ ) is autonomous demand (or spending) – the sum of all the components of the aggregate demand for goods and services that do not depend on ππ or ππ. Since ππ2 > 0 and 0 < ππ1 + ππ1 < 1, in the (ππ, ππ) plane the IS curve for the economy under consideration is a straight line with a positive vertical intercept (equal to π΄π΄⁄ππ2 ) and a negative slope [− (1 − ππ1 − ππ1 )⁄ππ2 ]. The above results have the following implications for the slope and the position of the IS curve in the (ππ, ππ) plane: • The absolute value of the slope of the IS curve is greater (the IS is steeper) the smaller are the propensity to spend (ππ1 + ππ1 ) and the sensitivity of investment to the interest rate (ππ2 ). Suppose, for instance, that we are dealing with an economy where ππ2 is relatively small, and let's try to understand why this leads to a relatively steep IS curve (to check your understanding of the properties of this curve, make sure that you can explain why one can reach the same conclusion if to be relatively small are the values of ππ1 and/or ππ1 ). Let us assume that, initially, the economy is at a point lying on the IS, and therefore in a goods market equilibrium position. If, for some reason, the interest rate falls, investment will rise. It follows that demand and equilibrium output will increase. For a given decrease in ππ, the rise in investment, demand and output will be smaller the less sensitive is investment to changes in the interest rate (that is, the smaller is ππ2 ). But if, for any given decrease in the interest rate, the increase in ππ needed to return to a goods market equilibrium position is small, the IS curve will be relatively steep. In the limiting case in which investment does not depend on ππ (ππ2 = 0), changes in the interest rate will not lead to changes in the aggregate demand for goods and services, thus leaving equilibrium output unchanged. In this case, the IS curve is a vertical line. • Increases (decreases) in ππ0 , πΌπΌ ,Μ πΊπΊΜ and/or decreases (increases) in πποΏ½ cause a parallel rightward (leftward) shift of the IS curve. In fact, and for any given level of the interest rate prevailing in the economy, the changes mentioned above raise (lower) the demand for goods, thus leading to an increase (decrease) in equilibrium output. The next figures summarize the conclusions we have just reached about the slope and the position of the IS curve in the (ππ, ππ) plane. 257 Macroeconomics. Problems and Questions ππ ππ1 , ππ1 and/or ππ2 “small” πΌπΌπΌπΌ ππ ππ “large” πΌπΌπΌπΌ ππ ππ ↑ππ0 , ↑πΌπΌ Μ , ↑πΊπΊΜ , ↓πποΏ½ πΌπΌπΌπΌ ππ1 , ππ1 and/or ππ2 ππ ↓ππ0 , ↓πΌπΌ Μ , ↓πΊπΊΜ , ↑πποΏ½ πΌπΌπΌπΌ ππ ππ b. Derive the analytical expression of the LM curve for this economy, and draw the LM in the (ππ, ππ) plane. The LM curve is the locus of all the pairs (ππ, ππ) for which the money market is in equilibrium − that is, for which the following equilibrium condition holds true: ππ = ππππ , where the term on the left-hand side is the money supply in nominal terms, and that on the right-hand side nominal money demand. The same equilibrium condition can be written in real terms by dividing both sides by the general price level, ππ: ππ⁄ππ = ππππ ⁄ππ. Given the functional form that, by assumption, money demand takes on in this economy, the previous equation becomes: ππ/ππ = (ππ1 ππ − ππ2 ππ). To make sure that ππ = π€π€Μ always, the central bank will have to set ππ = ππ(ππ1 ππ − ππ2 π€π€Μ ), something that − as discussed in the answer to Question 14 of Chapter 1 – guarantees that the interest rate will always be equal to π€π€Μ . Given the way in which, in this economy, the central bank sets ππ, the equation of the LM curve is ππ = π€π€Μ . 258 The IS-LM model In the (ππ, ππ) plane, the LM is therefore a horizontal line drawn at the value of the interest rate chosen by the central bank, π€π€Μ . ππ πΏπΏπΏπΏ π€π€Μ ππ c. Derive the equilibrium values of ππ and ππ. By how much will equilibrium output change if autonomous demand changes by π₯π₯π₯π₯? And by how much, following a change π₯π₯π€π€Μ in the level of the interest rate chosen by the central bank? The equilibrium values of ππ and ππ are the solutions to the system of two equations given by the analytical expressions of the IS and the LM curves: π΄π΄ 1 − ππ1 − ππ1 ππ = − · ππ ππ2 ππ2 ππ = π€π€Μ . Given our assumption about the behavior of the central bank, in equilibrium the interest rate will always be equal to π€π€Μ . To find the equilibrium value of output, πποΏ½, replace ππ with π€π€Μ in the first equation, the IS curve. Solving for ππ, one gets: 1 ππ2 πποΏ½ = · π΄π΄ − · π€π€Μ . 1 − ππ1 − ππ1 1 − ππ1 − ππ1 The change in equilibrium output πποΏ½ caused by a change π₯π₯π₯π₯ in the level of autonomous demand is therefore π₯π₯πποΏ½ = [1⁄(1 − ππ1 − ππ1 )] · π₯π₯π₯π₯. The constant 1⁄(1 − ππ1 − ππ1 ) is sometimes referred to as “fiscal policy multiplier”, since changes in πΊπΊΜ and/or πποΏ½ are among the possible causes of the changes in π΄π΄ whose effects we are studying. When, like in this economy, the central bank chooses a level for the interest rate, the fiscal policy multiplier is therefore identical to the income, or keynesian, multiplier derived when studying the goods market in isolation. Finally, when the change disturbing the equilibrium of the economy is one in π€π€Μ , π₯π₯πποΏ½⁄π₯π₯ π€π€Μ = − ππ2 ⁄(1 − ππ1 − ππ1 ). Notice that this constant – the “monetary policy multiplier” – equals zero if ππ2 = 0, that is, if no component of aggregate demand depends on the interest rate. 259 Macroeconomics. Problems and Questions * Question 2 [An analytical version of the IS-LM model − (II) Exogenous money supply] Consider an economy different from the one studied in the previous question only for the fact that, rather than the interest rate, its central bank chooses a value for the nominal money supply ππ, and then lets the interest rate take on any value that turns out to be consistent with the macroeconomic equilibrium for that given ππ. οΏ½ the value of ππ chosen by the central bank, derive the anaa. Denoting by ππ lytical expressions of the IS and LM curves for this economy. The analytical expression of the IS curve is identical to the one derived in the answer to the previous question. Its slope and position in the (ππ, ππ) plane will therefore be determined by the same factors discussed before. As for the LM curve, to derive its expression let us write down the money market equilibrium condition, ππ = (ππ1 ππ − ππ2 ππ) ππ οΏ½ (the constant level at which the central bank keeps the supply and set ππ = ππ of money) in this equation. Since, as usual, the LM curve will be drawn in a plane in which the interest rate is measured on the vertical axis, it is convenient to solve for ππ. This leads to the following expression of the LM curve: οΏ½ 1 ππ ππ1 ππ = − οΏ½ οΏ½ · + οΏ½ οΏ½ · ππ. ππ2 ππ ππ2 b. Draw the portion of the LM curve entirely lying in the first quadrant of the (ππ, ππ) plane (that is, the portion of the curve corresponding to positive values of both output and the interest rate). 1 What determines its slope? Which are the causes of parallel shifts of the LM curve in that plane? Explain, using the following graphs and providing the economic intuition underlying your results. The assumption that the nominal interest rate cannot take on negative values allows us to disregard the portion of the LM curve lying in the second quadrant. We shall discuss later on the shape of the LM curve when the nominal interest rate is zero (that is, when it is at its "zero lower bound") and − as assumed in the present question − the central bank chooses the nominal money supply. 1 260 The IS-LM model Since ππ1 , ππ2 > 0, the LM curve for this economy is, in the (ππ, ππ) plane, a οΏ½ ⁄ππ, and straight line with a negative vertical intercept equal to −(1⁄ππ2 ) ππ positive slope equal to the ratio ππ1⁄ππ2 . The above results have the following implications for the slope and the position of the LM curve in the (ππ, ππ) plane: • • The slope of the LM curve is greater (the LM is steeper) the more sensitive is money demand to income (that is, the larger is ππ1 ), and the less sensitive it is to the interest rate (the smaller is ππ2 ). First of all, to understand why the LM curve is, in this economy, positively sloped, let us assume that we are initially on the LM, and therefore in a money market equilibrium position. If, for some reason, starting from this initial position income goes up, money demand will rise. In the money market, this change will lead to an excess demand for money over the supply of money (recall that, in this economy, the central bank keeps οΏ½ ). To return to a money marmoney supply constant at the chosen level ππ ket equilibrium, the initial increase in Y must be matched by a rise in the interest rate. This explains the positive slope of the LM. But by how much will have the interest rate to go up? The increase in the interest rate needed to return in equilibrium will be larger the greater is ππ1. If ππ1is very large (that is, money demand very sensitive to income), for any given increase in Y money demand will go up by a lot, the excess demand in the money market will be large, and to eliminate it the interest rate will have to rise a lot – the LM curve will therefore be relatively steep. To check your understanding of the properties of the LM curve in the case analyzed in the present question, make sure that you can explain why, for a given ππ1, the slope of the LM curve is decreasing in ππ2, the sensitivity of money demand to the interest rate. οΏ½ , and/or decreases (increases) in ππ cause parIncreases (decreases) in ππ allel rightward (leftward) shifts of the LM curve. For any given level of income ππ, the changes metioned above raise (lower) real money supply, thus leading to a fall (rise) in the level of the interest rate for which the money market is in equilibrium. The next figures summarize the conclusions we have just reached about the slope and the position of the LM curve in the (ππ, ππ) plane for an economy in which the central bank chooses the money supply. 261 Macroeconomics. Problems and Questions ππ ππ ππ1 “large”, and/or ππ2 “small” ππ πΏπΏπΏπΏ ππ πΏπΏπΏπΏ ππ οΏ½οΏ½οΏ½ ↓ππ ↑ππ, ππ1 “small”, and/or ππ2 “large” οΏ½ , ↑ππ ↓ππ ππ πΏπΏπΏπΏ ππ πΏπΏπΏπΏ ππ c. Derive the equilibrium values of ππ and ππ. By how much will equilibrium output change if autonomous demand changes by π₯π₯π₯π₯? And by how much, οΏ½ in the nominal money supply? following a change π₯π₯ππ The equilibrium values of ππ and ππ are the solutions to the system of two equations given by the analytical expressions of the IS and the LM curves. For the economy under consideration, the system is the following one: π΄π΄ 1 − ππ1 − ππ1 − ππ ππ2 ππ2 οΏ½ ππ1 1 ππ ππ = − οΏ½ οΏ½ + οΏ½ οΏ½ ππ . ππ2 ππ2 ππ ππ = Equating the right-hand sides of the two equations above, solving for ππ and then plugging the result in the first or in the second one, yields the following expressions for the equilibrium values of ππ and ππ: πποΏ½ = π€π€Μ = οΏ½ 1 ππ2 ππ · π΄π΄ + · (ππ1 ππ2 ⁄ππ2 ) + (1 − ππ1 − ππ1 ) ππ1 ππ2 + ππ2 (1 − ππ1 − ππ1 ) ππ οΏ½ ππ1⁄ππ2 (1 − ππ1 − ππ1 ) ππ · π΄π΄ − · (ππ1 ππ2 ⁄ππ2 ) + (1 − ππ1 − ππ1 ) ππ1 ππ2 + ππ2 (1 − ππ1 − ππ1 ) ππ 262 (∗) (∗∗) The IS-LM model These expressions, that may look rather intimidating (however, bear in mind that you are not supposed to memorize them, and that no exam question will ask you to derive them) are useful because they allow one to understand why, and by how much, policy interventions and other shocks hitting an economy in which the central bank chooses the money supply will affect the macroeconomic equilibrium. In particular, the question asks us to determine the change in equilibrium output caused by a change π₯π₯π₯π₯ in autonomous demand. From the expression for πποΏ½ we have just derived, it follows that π₯π₯πποΏ½ = 1 · π₯π₯π₯π₯ (ππ1 ππ2 ⁄ππ2 ) + (1 − ππ1 − ππ1 ) where the ratio multiplying π₯π₯π₯π₯ on the right-hand side is the “fiscal policy multiplier” − the constant by which one has to multiply any given change π₯π₯π₯π₯ in autonomous spending in order to get the ensuing change in equilibrium output – in an economy where the central bank chooses the level of ππ. Notice that, being ππ1 , ππ2 , ππ2 > 0 by assumption, this multiplier will be always lower than the fiscal policy multiplier prevailing when the central bank chooses a value for ππ and that, answering Question 1.c of this Chapter, we have concluded is given by 1⁄(1 − ππ1 − ππ1 ). Why this is necessarily the case will become clear when answering some of the next questions. Finally, the “monetary policy multiplier” (the constant by which one has to οΏ½ /ππ in the real money supply, or – which is the multiply any given change in π₯π₯ππ same, given the assumption of constant prices − in nominal money supply in order to get the corresponding change in equilibrium output, π₯π₯πποΏ½) is the ratio 1 οΏ½ ⁄ππ in equation (∗) above.1 that multiplies ππ 1 In Question 1 of this Chapter, where the central bank was choosing the interest rate, the monetary policy multiplier measured the impact on πποΏ½ of a change in the value chosen for ππ; in the economy studied in the present question, where the central bank chooses ππ, that multiplier measures instead οΏ½ , the value of ππ chosen by the central bank. Also in this latter case, the impact on πποΏ½ of a change in ππ however, a change in money supply affects πποΏ½ by changing ππ. Using equation (∗∗) above to compute οΏ½ needed to change π€π€Μ by the same π₯π₯π€π€Μ assumed in Question 1, one gets: the change in ππ οΏ½ π₯π₯ππ ππ1 ππ2 + ππ2 (1 − ππ1 − ππ1 ) =− · π₯π₯ππ.Μ ππ (1 − ππ1 − ππ1 ) οΏ½ so determined into the relation between π₯π₯ππ οΏ½ e π₯π₯πποΏ½ implied by equaPlugging the expression for π₯π₯ππ tion (∗), it is straightforward to conclude that the impact on income of a monetary impulse is identical οΏ½ ⁄π₯π₯ π€π€Μ = − ππ2 ⁄(1 − ππ1 − ππ1 )], once such an impulse is expressed in a in both cases [equal to π₯π₯ππ comparable fashion. 263 Macroeconomics. Problems and Questions Question 3 a. Gamma is a closed economy described by a standard IS-LM model, 1 given by the following system: π΄π΄ 1 − ππ1 − ππ1 − · ππ ππ2 ππ2 ππ = π€π€Μ . ππ = In the equations above, π€π€Μ is the value of the interest rate chosen by the central bank, and the other symbols have the usual meaning. As discussed in the answer to Question 1 of this Chapter, in this model the fiscal policy multiplier is: 1 π₯π₯πποΏ½ = . (1 − ππ1 − ππ1 ) π₯π₯π₯π₯ (∗) What does the ‘fiscal policy multiplier’ measure? Knowing that, in Gamma, it equals 2, by how much will equilibrium output change if government spending on goods and services πΊπΊΜ goes up by 200 and, at the same time, autonomous consumption ππ0 falls by 100? Explain. As it is straightforward to conclude from (∗), the fiscal policy multiplier measures how much the equilibrium level of output changes following a ........................................................................................................................... change in autonomous spending π΄π΄ = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ . [Alternatively, it is the ........................................................................................................................... coefficient that multiplies autonomous spending π΄π΄ in the expression of equilibrium output one gets solving the analytical version of the IS-LM model]. ........................................................................................................................... Since in this case the fiscal policy multiplier is 2, and π₯π₯π₯π₯ = π₯π₯ππ0 + π₯π₯πΊπΊΜ = ........................................................................................................................... −100 + 200 = 100, the change in equilibrium output will be π₯π₯π₯π₯ = 2 · π₯π₯π₯π₯ = ........................................................................................................................... 2 · 100 = 200. 1 From now on, by "a standard IS-LM model" we shall mean a model like the one studied in Question 1 of this Chapter, and therefore based on the following assumptions: • consumption function of disposable income, investment function of ππ and ππ, nominal money demand function of ππ, ππ and ππ; • πΊπΊ and ππ both exogenous; • the central bank chooses a value ππ = π€π€Μ for the interest rate, and sets nominal money supply to whatever level is consistent with the attainment of this target value for ππ; • constant prices, and therefore current and future expected inflation rates equal to zero. 264 The IS-LM model b. Consider now two different countries, Delta and Epsilon, whose IS curves are drawn in the graphs below. Suppose that the different slopes of the two curves only reflect differences in the value that the parameter ππ2 , the sensitivity of investment to the interest rate, takes on in the two countries. All the remaining parameters take on identical values in Delta and in Epsilon. Explain what is meant by “monetary policy multiplier”, ππ2 π₯π₯πποΏ½ =− (1 − ππ1 − ππ1 ) π₯π₯π€π€Μ and, making explicit reference to this concept (discussed in the answer to Question 1 of this Chapter), discuss if the central bank's decision of changing the interest rate by the same π₯π₯π€π€Μ in the two countries will change equilibrium output more in Delta or in Epsilon. ππ π·π·π·π·π·π·π·π·π·π· ππ πΌπΌπΌπΌπ·π·π·π·π·π·π·π·π·π· πΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈ πΌπΌπΌπΌπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈπΈ ππ ππ ........................................................................................................................... ........................................................................................................................... The monetary policy multiplier is the constant by which one has to multiply any given change in the level of interest rate chosen by the central bank in order to ........................................................................................................................... get the corresponding change in equilibrium output. The IS curve is flatter the ........................................................................................................................... larger is ππ2 , the sensitivity of investment to the interest rate. It follows from the graphs above that ππ2 is larger in Delta. Since monetary policy interven........................................................................................................................... tions − decisions of changing π€π€Μ taken by the central bank − affect equilibrium ........................................................................................................................... output by changing those components of aggregate demand that depend on the interest rate (in the standard model analyzed in the present question, invest........................................................................................................................... ment only), equilibrium output will rise more in Delta, where ππ2 is larger and ........................................................................................................................... where investment and aggregate demand will therefore respond more to any given change in the interest rate. More formally, the ratio on the right-hand ........................................................................................................................... side of the expression of the fiscal policy multiplier written above is increasing in ππ2 ; it follows that the same will be true about the change in equilibrium output caused by any given π₯π₯π€π€Μ decided by the central bank. 265 Macroeconomics. Problems and Questions Question 4 a. In country Macro, described by the IS-LM model, government purchases of goods and services, net taxes and investment are exogenous (πΊπΊ = πΊπΊΜ , ππ = πποΏ½ and πΌπΌ = πΌπΌ )Μ , while the consumption function is πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) − β2 ππ, where the parameter β2 > 0 is the sensitivity of consumption to the interest rate. In this economy, will the slope of the IS curve in the (ππ, ππ) plane be negative, zero or positive? Explain. [Hint: no formal derivation of the IS curve is required – just provide the economic intuition underlying your answer]. Next, represent the initial equilibrium of Macro in an IS-LM diagram, denote it by ‘0’ and study the effects of a decrease in net taxes. In particular, denote by ‘1’ the new equilibrium the economy will reach and explain the reasons for the changes in equilibrium income, consumption and investment that will take place in the move from ‘0’ to ‘1’. As the economy goes from the first to the second equilibrium, how must the changes in consumption and in income be related to one another? ππ 2 π€π€Μ 2 π€π€Μ 0 0 πποΏ½0 266 LM1 1 πποΏ½1 LM0 ISO IS1 ππ The IS-LM model Even though investment does not depend on i, aggregate demand is nevertheless decreasing in the interest rate, since an increase in this latter variable lowers consumption. The IS curve has therefore, as usual, a negative slope (if, starting from a goods market equilibrium position, i falls, in this economy C will rise, leading to an excess demand for goods; to return to equilibrium, the supply of goods, Y, will have to go up). If πποΏ½ decreases, the IS curve shifts to the right, since the increase in disposable income caused by the tax cut raises consumption, and with it aggregate demand, for any given level of ππ. The equilibrium point becomes 1, with a higher level of output. Being πΊπΊ and πΌπΌ (in this economy, both exogenous) unchanged, consumption must have gone up. Since in this new equilibrium supply and demand must be equal, and consumption is the only component of aggregate demand that has changed, in the move from ‘0’ to ‘1’ πΆπΆ must have risen by the same amount by which ππ, the supply of goods, has gone up. b. Suppose now that Macro’s central bank intends to bring back equilibrium income to its initial level, ππ0 . To achieve its goal, should it raise or lower the interest rate? Show in the graph the new equilibrium that will be reached following the monetary policy intervention that you are proposing, denoting it by ‘2’. Finally, compare the composition of aggregate demand at ‘2’ with that prevailing in the initial equilibrium ‘0’. The central bank will have to raise the interest rate. In the new equilibrium ‘2’, the interest rate is π€π€Μ 2 (> π€π€Μ 0 ), and income is back to its initial level. It follows that aggregate demand will have to take on, at ‘2’, the same value it was taking on at ‘0’. Being both G and I exogenous, this requires consumption to be, in the final equilibrium ‘2’, the same as in the initial equilibrium ‘0’ – the favorable impact of the decrease in net taxes, which tends to raise πΆπΆ, must have been exactly offset by the adverse impact that, in this economy, a higher interest rate has on consumption. 267 Macroeconomics. Problems and Questions Question 5 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “In an IS-LM model that departs from the standard case only for the fact that, rather than the interest rate, the central bank chooses the nominal οΏ½ , the higher the sensimoney supply, keeping it constant at the level ππ = ππ tivity of investment to the interest rate, the larger the increase in equilibrium income caused by an expansionary fiscal policy”. False, the opposite is true. The immediate effect of an expansionary fiscal policy is to make aggregate demand and output increase. However, this increase in production and income disturbs the money market equilibrium, leading to an excess demand for money and – since the central bank keeps the nominal money supply constant – to an increase in interest rate, ππ. In turn, this higher ππ will lower investment, by an amount which will be larger the higher the sensitivity of investment to the interest rate (or, alternatively, the flatter the IS curve). This decrease in investment mitigates the initial increase in income caused by the expansionary fiscal policy. We may conclude that an expansionary fiscal policy has its maximum effect on output when investment does not depend on the interest rate – the IS curve is vertical. 268 The IS-LM model b. “In a standard IS-LM model, in which the central bank chooses the interest rate and keeps it constant at the level deemed appropriate given the state of the economy (for instance, at ππ = π€π€Μ ), the higher the sensitivity of investment to the interest rate, the larger the increase in equilibrium income caused by an expansionary fiscal policy”. False. As it is immediate to conclude also noticing that the parameter ππ2 does not enter the expression for what has been defined ‘fiscal policy multiplier’ in the answer to point c of Question 1 of this Chapter, in this case the extent by which equilibrium income goes up following a fiscal stimulus is independent of the interest rate sensitivity of investment. In fact, and as discussed answering the previous point, the expansionary fiscal policy still leads to a higher aggregate demand and a higher production. For a given money supply, this would cause an increase in the interest rate. However, since it now wants ππ to remain at the chosen level, ππ = π€π€Μ , the central bank will in this case intervene, raising the money supply by the amount needed to keep the interest rate unchanged. And, with an unchanged ππ, the sensitivity of investment to changes in the interest rate will play no role in determining the extent by which the fiscal expansion will impact on output. 269 Macroeconomics. Problems and Questions Question 6 a. Gamma is a closed economy initially in goods and in money markets equilibrium. A wave of optimism about the economic future of the country leads to an increase in autonomous consumption, ππ0 . Using a standard ISLM model, graphically represent the effects of such a change on equilibrium income. In addition, discuss the effects on equilibrium consumption, investment, private saving and national saving, explaining the reasons for the observed changes in these variables. ππ π€π€Μ 1 πποΏ½ 2 οΏ½ ππ′ πΏπΏπΏπΏ πΌπΌπΌπΌ πΌπΌπΌπΌ′ ππ An increase in the autonomous component of consumption causes a rightward shift of IS curve. The equilibrium changes from point 1 to point 2. Income rises, as firms will meet the extra demand they now face by producing more. Since the central bank keeps the interest rate at π€π€Μ , in the new equilibrium investment will be higher (due to the increase in income). The same will be true about consumption, that goes up both for the increase in its autonomous component and for the increase in equilibrium income. Given that, when the economy goes from the initial equilibrium 1 to the new equilibrium 2, investment rises, the same must be true about national saving. Being public saving unchanged, private saving will be necessarily higher (by the same amount by which investment has gone up). 270 The IS-LM model b. Consider now country Delta. The only difference between Gamma and Delta lies in the policy rule followed by their central banks. While, as we already know from the first part of this question, Gamma's central bank chooses a level for the interest rate, Delta's central bank chooses a level for the nominal supply of money ππ and, given that level, lets the interest rate take on any value turns out to be consistent with the macroeconomic equilibrium. Suppose now that, when the two countries are in an initial equilibrium with identical values of ππ and ππ, Delta experiences the same increase in ππ0 discussed in the previous point of this question. Compared with what happened in Gamma, will income change more or less in Delta? Represent graphically, and explain. ππ πΌπΌπΌπΌ πΌπΌπΌπΌ′ 1 π€π€Μ πποΏ½ 2π·π· πΏπΏπΏπΏπ·π·π·π·π·π·π·π·π·π· 2πΊπΊ πποΏ½π·π·′ πποΏ½πΊπΊ′ πΏπΏπΏπΏπΊπΊπΊπΊπΊπΊπΊπΊπΊπΊ ππ Rather than flat as in Gamma, Delta's LM curve will be positively sloped. As the figure allows one to conclude, any given increase in ππ0 will raise ππ less in Delta than in Gamma. In fact, the increase in aggregate demand caused by the rise in the autonomous component of consumption will increase output, and income, in both countries. Remember, however, that Delta's central bank keeps ππ constant. It follows that, in that country, the increase in ππ, by raising money demand, will lead to an excess demand for money and to an increase in the interest rate for which the money market is in equilibrium. This increase in ππ (that, notice, takes place in Delta, but not in Gamma) mitigates the positive impact of the increase in ππ0 on aggregate demand. But if, in Delta, the rise in aggregate demand is smaller than that taking place in Gamma, then Delta's equilibrium output will rise less than Gamma's. 271 Macroeconomics. Problems and Questions Question 7 a. Consider the economy of country Epsilon, described by an IS-LM model departing from the standard case only because the (private) saving function is ππ = −ππ0 + (1 − ππ1 )(ππ − πποΏ½) + β2 ππ, where β2 > 0 is the sensitivity of savings to the interest rate and the other symbols have the usual meaning. a.1 Write down the consumption function for this economy. a.2 Assuming that all the other behavioral functions (investment, money demand, etc.) are standard, explain if and why, following an expansionary monetary policy, when β2 > 0 equilibrium income increases more or less than in the standard case (β2 = 0) [Hint: no formal derivation of the IS and LM curves is required – just provide the economic intuition underlying your answer]. a.1 πΆπΆ = (ππ − πποΏ½) − ππ = ππ0 + ππ1 (ππ − πποΏ½) − β2 ππ . a.2 When β2 > 0, a monetary expansion raises ππ more than in the standard case – the decrease in the interest rate decided by the central bank boosts not just investment, but also consumption, thus causing a larger increase in aggregate demand and equilibrium income. 272 The IS-LM model b. If, rather than a monetary expansion, to be implemented is going to be a fiscal expansion (consisting for instance in an increase in government purchases of goods and services, πΊπΊΜ ), how will your answer to the previous point a.2 change? In particular, compare the change in equilibrium output in Gamma (where β2 > 0) with the change that would prevail in the standard case (β2 = 0). Illustrate in the graph below, and explain. ππ π€π€Μ 1 0 πποΏ½0 πΌπΌπΌπΌβ2>0 πΌπΌπΌπΌβ2=0 πποΏ½1 πΏπΏπΏπΏ πΌπΌπΌπΌ′β2>0 πΌπΌπΌπΌ′β2=0 ππ The initial equilibrium is point 0; the new one point 1. .......................................................................................................................... When, in addition to investment, also consumption depends negatively on the interest rate (that is, when β2 > 0), the IS curve is less steep than in the ........................................................................................................................... standard case (β2 = 0), as shown in the figure [to check your understanding ........................................................................................................................... of the IS-LM model, make sure you can explain why this must necessarily be true]. In any case, no matter whether β2 is positive or zero, an increase in πΊπΊΜ ........................................................................................................................... causes the same parallel, rightward shift of the IS curve and, since the central ........................................................................................................................... bank keeps the interest rate at the chosen level π€π€Μ (or, equivalently, since the LM curve is horizontal), leads to the same increase in equilibrium output. On ........................................................................................................................... the basis of the analytical version of the IS-LM model introduced in Question ........................................................................................................................... 1 of this Chapter, it is also possible to conclude that, in both cases, the size of the horizontal shift to the right of the IS curve, and the amount by which equi.......................................................................................................................... librium output will rise, are both given by [1⁄(1 − ππ1 − ππ1 )] · π₯π₯πΊπΊΜ , an expression that does not depend on the sensitivity of aggregate demand to the interest rate. 273 Macroeconomics. Problems and Questions Question 8 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “The government of an economy described by a standard IS-LM model decides to implement an expansionary fiscal policy. To keep the interest rate at its target value ππ = π€π€Μ , the central bank will have to raise money supply by an amount that is going to be larger the more sensitive is money demand to changes in income. True. The expansionary fiscal policy will disturb the equilibrium in the goods market, raising output by the income multiplier times the change in autonomous demand caused by the fiscal stimulus. If, for instance, the government has decided to raise πΊπΊΜ , equilibrium output will go up by π₯π₯πποΏ½ = [1⁄(1 − ππ1 − ππ1 )] · π₯π₯πΊπΊΜ . In turn, this increase in income will disturb the money market equilibrium. Money demand, that depends positively on ππ, will rise and, should the central bank keep money supply constant, at the initial interest rate ππ = π€π€Μ there would be an excess demand for money on the supply of money. It follows that, to prevent this disequilibrium from pushing ππ above π€π€Μ , the central bank will have to intervene. More specifically, the central bank will have to increase money supply exactly by the amount by which the rise in equilibrium income has raised money demand − an amount that is larger the more sensitive money demand is to changes in income (that is, the larger is the parameter ππ1 we have defined when discussing the analytical version of the IS-LM model in Question 1 of this Chapter). 274 The IS-LM model b. “When the economy is in a liquidity trap, with the nominal interest rate at its ‘zero lower bound’, an expansionary fiscal policy cannot influence the equilibrium level of output”. False. When the economy is in a liquidity trap, to be powerless to affect output is monetary policy (or, at least, so are the “conventional” monetary policy tools discussed in this Chapter). In fact, a monetary expansion affects the demand for goods, and therefore production, by lowering the interest rate, thus raising the components of aggregate demand that depend on ππ. If, however, the interest rate is already at its lower bound, it cannot fall any further, and the channel through which monetary policy affects the real economy is no longer operational. In a similar situation, the only policy able to raise equilibrium output is fiscal policy. 275 Macroeconomics. Problems and Questions Question 9 Consider a country described by an IS-LM model departing from the standard one only for the fact that, rather than the interest rate, the central bank chooses the nominal money supply and, given the level selected for ππ, allows the interest rate to take on any value is consistent with the macroeconomic equilibrium. We are therefore in the case considered in Question 2 of this Chapter. In particular, and as it was assumed there, real money demand is πΏπΏ(ππ, ππ) = ππ1 ππ − ππ2 ππ. a. Suppose that the country is in a ‘liquidity trap’. Represent in the graph its initial equilibrium, denoting it by ‘1’, and by ππ1 and ππ1 the values that the interest rate and production take on in that equilibrium position. Assume now that autonomous consumption ππ0 and net taxes πποΏ½ fall at the same time and by the same amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ < 0. Show in the graph, and explain, how these changes affect the levels of the country’s interest rate and output, denoting by ππ2 and ππ2 the values these variables will take on in the new equilibrium (‘2’). In this new equilibrium position, investment will be higher or lower? And what about national saving (the sum of private and public saving)? Motivate your answer. ππ πΏπΏπΏπΏ1 πΌπΌπΌπΌ1 πΌπΌπΌπΌ2 ππ1 = ππ2 = 0 2 1 D ππ2 ππ1 −(1⁄ππ2 )(ππ⁄ππ) −(1⁄ππ2 )(ππ′⁄ππ ) 276 E πΏπΏπΏπΏ3 Y The IS-LM model If we allow for the possibility that the nominal interest rate reaches its lower bound (that is, zero), the LM for an economy in which the central bank chooses a value for the nominal money supply consists of two portions with different slopes: one that, as discussed in the answer to Question 2 of this Chapter, is positively sloped for values of the interest rate greater than zero; and a second one that − since the nominal interest rate cannot take on negative values − coincides with the x-axis for ππ = 0. Furthermore, increases in money supply cause parallel, rightward shifts of the positively sloped portion of the curve, prolonging at the same time the horizontal portion. In the figure, we are assuming that, initially, the LM is given by the broken line 0D-πΏπΏπΏπΏ1 . Since we are told that the economy is in a liquidity trap (that is, in an equilibrium with ππ = 0), the initial intersection between the IS and the LM curves must take place somewhere along the horizontal portion of the LM, at a point like ′1′ in the figure, where output is ππ1 and the interest rate is ππ1 = 0. If autonomous consumption ππ0 and net taxes πποΏ½ fall at the same time and by the same amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ < 0, autonomous demand π΄π΄ = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ changes by π₯π₯π₯π₯ = π₯π₯ππ0 − ππ1 π₯π₯πποΏ½ = (1 − ππ1 )π₯π₯ππ0 < 0. It follows that the IS curve will shift to the left, and that the equilibrium will become ′2′ in the graph. At point ′2′, the economy is still in a liquidity trap, with a zero interest rate and a lower equilibrium output. Since, in the move from ′1′ to ′2′, investment falls (because of the decrease in ππ, and the constancy of ππ), national saving, in equilibrium equal to investment, must be lower, too. b. Suppose now that the central bank of the country attempts to return income to the initial level, ππ1 , by implementing a conventional monetary policy − for instance, an open market operation. Show in the graph the equilibrium that will prevail after the central bank’s intervention. Explain. To achieve its end, the central bank could resort to an open market purchase ........................................................................................................................... of government bonds. However, this ‘conventional’ monetary policy would re........................................................................................................................... sult in a rightward shift of the positively sloped portion of the πΏπΏπΏπΏ curve only, and would make the horizontal one longer. The initial LM, 0D-πΏπΏπΏπΏ1 , would be ........................................................................................................................... replaced by a new one, such as 0E-πΏπΏπΏπΏ3 , with the intersection between this lat........................................................................................................................... ter curve and an unchanged πΌπΌπΌπΌ curve still taking place at point ‘′2′, that will therefore remain the equilibrium position for the economy. Since the interest ........................................................................................................................... rate was already at its lower bound, there is not going to be any decrease in ππ, ........................................................................................................................... and therefore no increase in ππ. In a liquidity trap, a conventional monetary policy is ineffective. 277 Macroeconomics. Problems and Questions Question 10 True or false? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. Define the concept of real interest rate (ππ), and write the equation that shows how ππ and nominal interest rate (ππ) are related to one another. Use this equation to explain if you agree, or do not agree, with the following statements: a. “If individuals expect positive inflation, then the real interest rate will be greater than the nominal one.” False. While the nominal interest rate tells us how a sum that has been borrowed/lent out grows over time in units of currency (Euros, for instance), the real interest rate tells us how that sum grows in real terms – that is, how its purchasing power changes over time [alternative definition:”the real interest rate is the interest rate in terms of goods; it tells us how many goods one has to repay in the future in exchange for borrowing the equivalent of one good today”]. The real and the nominal interest rates are related to one another as implied by the equation ππ = ππ − ππ ππ , where ππ ππ is expected inflation. From this relation it follows that ππ < ππ whenever ππ ππ > 0, so that we can conclude that the above statement is incorrect. 278 The IS-LM model b. “The zero lower bound for the nominal interest rate implies that the real interest rate cannot be greater than minus the expected inflation rate, −ππ ππ ”. False, the opposite is true. Being ππ = ππ − ππ ππ , and given expected inflation, ππ is lowest when ππ = 0, so that ππ ≥ −ππ ππ . The real interest rate cannot be less than minus the expected inflation rate. 279 Macroeconomics. Problems and Questions Question 11 The economy is described by an "extended" IS-LM model − that is, by an ISLM model based on the following additional assumptions: − − the central bank can set the real rate ππ, that therefore becomes the “policy rate ” determined by monetary policy, and keep it at the chosen level, let's call it ππΜ ; spending decisions (in particular, investment decisions by firms) depend on the “real borrowing rate” ππ + π₯π₯, sum of the (real) policy rate and the risk premium (π₯π₯). a. Starting from an initial equilibrium position, to be denoted by ′0′ in the figure below, suppose that the autonomous component of investment, πΌπΌ ,Μ falls and that, at the same time, the government cuts net taxes, πποΏ½, with π₯π₯πΌπΌ Μ = π₯π₯πποΏ½ < 0. Denote by ′1′ the new equilibrium that will be reached following these two changes, and compare the composition of aggregate demand at point ′1′ with that prevailing at ′0′. ππ ππΜ ππΜ ’ 1 πποΏ½1 280 0 LM0 2 πποΏ½0 LM1 IS1 ISO ππ The IS-LM model If the autonomous component of investment πΌπΌ Μ and net taxes πποΏ½ fall at the same time and by the same amount, autonomous spending π΄π΄ = ππ0 − ππ1 πποΏ½ + πΌπΌ Μ + πΊπΊΜ changes by π₯π₯π₯π₯ = π₯π₯πΌπΌ Μ − ππ1 π₯π₯πποΏ½ = (1 − ππ1 )π₯π₯πΌπΌ Μ < 0. It follows that the IS will shift to the left, and that the new equilibrium will be at point ′1′ in the figure, with an unchanged interest rate and a lower equilibrium income. In the move from equilibrium ′0′ to equilibrium ′1′, investment has gone down (due to the decrease in its autonomous component and because of the fall in ππ), while the sign of the change in consumption is uncertain (consumption tends to rise due to the cut in taxes, and to fall because of the decrease in income). Even in the case in which consumption ends up rising, the amount by which it increases will however be less than that by which investment falls – in fact, at point ′1′ production has gone down, and therefore aggregate demand must necessarily be lower than at point ′0′; being πΊπΊ unchanged, it follows that the sum of the private components of demand, πΆπΆ and πΌπΌ, must have gone down in the move from the first to the second equilibrium. b. To return equilibrium output to the initial level (that is, the one prevailing at point ′0′), what kind of monetary policy intervention should the central bank implement? In the graph, denote by ′2′ the equilibrium that will be reached following the monetary policy you are suggesting, and compare the composition of aggregate demand at point ′2′ with that prevailing in the initial equilibrium (point ′0′). To return output to the initial level, πποΏ½0 , the central bank will have to lower the real rate to ππΜ ′. Compared to ′0′, in the equilibrium point ′2′ that the economy will reach the supply of goods is unchanged; the same must therefore be true about the aggregate demand for goods. Since, at ′2′, πΊπΊΜ is unchanged and consumption is higher (income is the same as at ′0′, but net taxes are lower), in the move from ′0′ to ′2′ investment must necessarily have gone down (the impact of the decrease in the real interest rate, that tends to raise πΌπΌ, must have been more than offset by that stemming from the fall in its autonomous component, πΌπΌ )Μ . 281 Macroeconomics. Problems and Questions Question 12 Consider a country described by an ‘extended’ IS-LM that differs from the one described in the previous question only for the fact that consumption takes on the following functional form: οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ . πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) + ππ ππππππ In the equation above, ππππππ is the individuals’ financial and housing wealth, assumed to be exogenous, and the parameter ππ is greater than zero. a. In this economy, will the slope of the IS curve differ from that prevailing in the standard case, where consumption is a function of disposable income only (ππ = 0)? If so, will the IS curve be steeper or flatter than in the standard case? If not, why? Explain. [Hint: you are not being asked to derive the analytical expression of the IS curve and of its slope, but just to discuss whether this slope will be different from the one prevailing in the standard case, and to provide the economic intuition underlying your answer.] . In the (ππ, ππ) plane in which the IS-LM model is represented, the slope of the IS depends on the extent by which a given change in the interest rate (the variable measured along the vertical axis) changes the aggregate demand in the economy, and therefore the level of production for which the goods market is in equilibrium, given the values that the other variable on which aggregate demand depends and not measured along the axes of the graph. If, for inοΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½, etc.), a one percentage point decrease stance (and for given ππ0 , πΌπΌ ,Μ πΊπΊΜ , πποΏ½, ππππππ in ππ raises aggregate demand by a lot, the increase in production needed to return to goods market equilibrium will be large, and the IS curve therefore relatively ‘flat’. It follows that the fact that demand also depends on the individuals’ financial and housing wealth has nothing to do with the slope of the IS – it just affects the position of this curve in the (ππ, ππ) space. When, rather than zero, the parameter ππ is positive, an increase (a decrease) in ππππππ raises (lowers) the demand for goods for any given value of the real interest rate, thus causing a rightward (leftward) parallel shift of the IS curve. 282 The IS-LM model b. Due to a stock market crash, individuals experience a fall in their financial and housing wealth. In addition, an increase in financial market participants’ degree of risk aversion leads to a marked rise in the risk premium π₯π₯. Assuming that it was initially in an equilibrium that you will denote by ‘1’ in the graph below, show the new equilibrium to which the economy will converge following the two changes mentioned before (fall in οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ππππππ; rise in π₯π₯), and denote it by ‘2’. In the move from ‘1’ to ‘2’, how will the composition of aggregate demand change? Explain. ππ ππΜ 2 πποΏ½2 1 πποΏ½1 LM1 IS2 IS1 ππ Both changes will shift the IS curve to the left. In the new equilibrium (point’2’ in the figure), the interest rate will remain at the level chosen by the central bank, while output will be lower. As for the composition of aggregate demand, in the move from the old to the new equilibrium both investment (due to the increase in the borrowing rate caused by the rise in π₯π₯, and to the decrease in ππ) and consumption (due to the increase in π₯π₯ and to the decreases in ππ and in ππππππ) will fall. Finally, being exogenous, government purchases of goods and service will of course remain unchanged. 283 Chapter 3 - The labor market, the IS-LM-PC model, and inflation 285 Macroeconomics. Problems and Questions Question 1 Consider an economy that is initially in medium run equilibrium, with an unemployment rate at the natural level. a. A new law that increases unemployment benefits is passed. Show in the graph the effects of such a change on the real wage and on the natural rate of unemployment. Provide the economic intuition behind the results you get. ππ ππ 1 1 + ππ 1 0 ππππ ππππ′ ππππ π’π’ππ π’π’′ππ π’π’ By making the prospects of unemployment less distressing, more generous unemployment benefits lead to an increase in the wage requested by workers for any given level of the unemployment rate. The WS curve shifts upwards to the right and π’π’ππ increases to π’π’ππ′ . A higher rate of unemployment is now necessary in order to make workers willing to accept the real wage that firms are willing to pay, which remains equal to 1⁄(1 + ππ). 286 The labor market, the IS-LM-PC model, and inflation b. Realizing that the law just passed has an impact on employment, but still determined to provide income support for those who lose their job, the government implements measures aiming at increasing the degree of competition in the goods market. Assuming that such measures succeed in returning the unemployment rate to the natural level − that is, to the level prevailing before the increase in unemployment benefits −, show the effects of this second policy intervention in the same graph used to answer the previous point of this question. ππ ππ 1 1 + ππ′ 1 1 + ππ 2 1 0 ππππ′ ππππ ππππ π’π’ππ π’π’′ππ ππππ′ π’π’ The economy is initially at point 1 in the graph, reached after the increase in unemployment benefits. If the government succeeds in his attempt to increase the degree of competition in the goods market, firms' market power is reduced, the mark-up ππ falls, and the PS curve shifts upwards. Since we are told that, with this intervention, the unemployment rate is returned to the initial level, the one prevailing before the rise in benefits, the new value of ππ, ππ′, will have to be such that the new PS (the curve PS' in the figure) crosses the new WS (WS') for π’π’ = π’π’ππ − that is, at point 2 in the graph. 287 Macroeconomics. Problems and Questions Question 2 a. Define briefly, but rigorously, the following concepts: a.1 expectations-augmented (sometimes also referred to as ‘modified’, or ‘accelerationist’) Phillips curve; a.2 ‘non-accelerating inflation rate of unemployment’, or NAIRU (in your answer, make sure to discuss the relationship between this rate and the natural rate of unemployment); a.3 stagflation. ........................................................................................................................... a.1 πππ‘π‘ − πππ‘π‘−1 = − πΌπΌ(π’π’π‘π‘ − π’π’ππ ) or, equivalently, πππ‘π‘ − πππ‘π‘−1 = (ππ + π§π§) − πΌπΌπ’π’π‘π‘ ........................................................................................................................... ο relationship between the unemployment rate and the change in the inflation rate: low unemployment is associated with rising inflation, and ........................................................................................................................... high unemployment leads to decreasing inflation. a.2 Rate of unemployment at which inflation neither decreases nor increases. As clear from the equation for the Phillips curve written when answering the previous point of this question, if πππ‘π‘ππ = πππ‘π‘−1 the NAIRU (“NonAccelerating Inflation Rate of Unemployment”) is equal to the natural rate of unemployment π’π’ππ (or, if you want, it gives another way of thinking about π’π’ππ ). a.3 Situation characterized by the coexistence of stagnation and inflation, typically associated with a negative supply shock. 288 The labor market, the IS-LM-PC model, and inflation b. Explain if and why you agree or disagree with the following statement: “In order to increase the natural level of production ππππ , policy-makers can follow two alternative strategies: (i) they can try to increase the demand for goods permanently, for example by opting for a permanent increase in either the money supply or in government purchases of goods and services, or (ii) they can decide to implement ‘supply-side’ policies, such as those leading to an increase in the degree of competition in the goods market. The difference between the two strategies above is that the first one will lead not just to an increase in ππππ , but also to a permanently higher level of prices, while the second strategy will push the economy towards an equilibrium characterized by a higher ππππ and a lower general price level.” The statement is incorrect. Monetary and fiscal policies have no effect whatsoever on the natural level of production (in the medium run, monetary policy only affects the price level, while fiscal policy causes – in addition to a change in the price level – a change in the composition of aggregate demand, but not in its overall level). The only policies that can change the natural rate of unemployment (and therefore the natural level of production) are supply-side policies − for example, policies that affect the degree of competition in the goods market, or the ‘flexibility’ of the labour market. 289 Macroeconomics. Problems and Questions Question 3 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “The zero lower bound for the nominal interest rate implies that the real interest rate cannot be smaller than minus the expected inflation rate, −ππ ππ ”. True. Being ππ = ππ − ππ ππ , and given expected inflation, ππ is lowest when ππ = 0, so that ππ ≥ −ππ ππ . The real interest rate cannot be less than expected inflation. 290 The labor market, the IS-LM-PC model, and inflation b. Since, in country Delta, the ‘modified’ (also known as ‘expectationsaugmented’) Phillips curve is: πππ‘π‘ − πππ‘π‘−1 = − 2 · (π’π’π‘π‘ − 0.05), then in Delta the ‘non-accelerating inflation rate of unemployment’, or NAIRU, is 5%. True. The NAIRU (“Non Accelerating Inflation Rate of Unemployment”) is the rate of unemployment for which inflation is constant. Setting πππ‘π‘ = πππ‘π‘−1 in the Phillips curve given above, it is straightforward to conclude that, in Delta, the NAIRU is 0.05 (5%). 291 Macroeconomics. Problems and Questions Question 4 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. If πππ‘π‘ππ = πππ‘π‘−1 , from the expectations-augmented Phillips curve it follows that, to bring the unemployment rate below its natural level, policymakers must be willing to tolerate an increase in the inflation rate. ........................................................................................................................... True. With πππ‘π‘ππ = πππ‘π‘−1 , the equation for the expectations-augmented Phillips ........................................................................................................................... curve is πππ‘π‘ − πππ‘π‘−1 = −πΌπΌ(π’π’π‘π‘ − π’π’ππ ). From this expression it follows that ........................................................................................................................... π’π’π‘π‘ < π’π’ππ if and only if πππ‘π‘ > πππ‘π‘−1 − inflation must increase over time. To understand why this must be the case, recall that πππ‘π‘ππ = πππ‘π‘−1 . From πππ‘π‘ > πππ‘π‘−1 ........................................................................................................................... it then follows that, for given πππ‘π‘−1 , πππ‘π‘ > πππ‘π‘ππ − workers are underestimating ........................................................................................................................... the general price level, and therefore are overestimating the real wage they are receiving. This overestimation is what is needed to bring unemployment ........................................................................................................................... below the natural rate. 292 The labor market, the IS-LM-PC model, and inflation b. The smaller the fraction of wages indexed to the inflation rate, the larger the increase in inflation associated with any given decrease in the rate of unemployment. False. Assume that a fraction ππ (with 0 < ππ < 1) of wages is indexed, and that πππ‘π‘ππ = πππ‘π‘−1 . The expectations-augmented Phillips curve in this case becomes: πΌπΌ (π’π’ − π’π’ππ ). πππ‘π‘ − πππ‘π‘−1 = − 1 − ππ π‘π‘ The sensitivity of inflation to unemployment is πΌπΌ/(1 − ππ), which is increasing in ππ . To understand why this is the case, notice that, when wages are not indexed (ππ = 0), if this year unemployment falls, wages will increase, leading to an increase in prices and inflation. With πππ‘π‘ππ = πππ‘π‘−1 , even though current inflation is now higher, there will be no further increase in current year's wages and prices. If, however, at least some wages are indexed, the increase in prices that takes place this year leads to a further increase in those wages which are indexed, and this will in turn lead to a new increase in this year's prices. The effect of a change in unemployment on the inflation rate is therefore larger in the presence of indexation. The larger is ππ, the stronger this effect is going to be. 293 Macroeconomics. Problems and Questions Question 5 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “A decrease in firms' market power leads to a fall in the inflation rate”. True. One of the ways in which the Phillips curve can be written is: πππ‘π‘ − πππ‘π‘ππ = (ππ + π§π§) − πΌπΌπ’π’π‘π‘ . A decrease in firms' market power amounts to a reduction in the mark-up ππ and therefore, other things the same, to a fall in current inflation, πππ‘π‘ . The reason is that today, time π‘π‘, firms with less market power will set lower prices for the goods they produce. For given prices at time π‘π‘ − 1, lower prices at π‘π‘ imply a lower inflation rate. [One could reach the same conclusion also using the following, alternative version of the Phillips curve: πππ‘π‘ = πππ‘π‘ππ − πΌπΌ(π’π’π‘π‘ − π’π’ππ ). The WS-PS model implies that the natural rate of unemployment depends positively on the mark-up ππ. A decrease in ππ will therefore lower π’π’ππ , and − for any given πππ‘π‘ππ and π’π’π‘π‘ − from the previous equation it follows that πππ‘π‘ will fall]. 294 The labor market, the IS-LM-PC model, and inflation b. “From the expectations-augmented Phillips curve, πππ‘π‘ = πππ‘π‘ππ − πΌπΌ(π’π’π‘π‘ − π’π’ππ ), it follows that, for a given natural rate of unemployment π’π’ππ , the current inflation rate πππ‘π‘ can fall if and only if the current rate of unemployment π’π’π‘π‘ increases". ........................................................................................................................... False. Current inflation can fall also if, for given π’π’π‘π‘ and π’π’ππ , individuals start ........................................................................................................................... expecting a lower inflation rate − that is, if πππ‘π‘ππ falls. The reason is that, since they now expect lower inflation, workers will be willing to accept a lower nominal wage. This decrease in labor costs will enable firms to set prices at a lower level. For given prices in the previous period, a lower current rate of inflation will result. 295 Macroeconomics. Problems and Questions Question 6 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “In country Alpha πππ‘π‘ππ = πππ‘π‘ , while in country Beta πππ‘π‘ππ = 2%. It follows that, to keep the rate of unemployment at the natural level (π’π’π‘π‘ = π’π’ππ for each time π‘π‘), the rate of inflation must remain constant in Alpha, and increase at a rate greater than 2% in Beta”. To conclude that the statement is incorrect, it suffices to write the equation πππ‘π‘ − πππ‘π‘ππ = − πΌπΌ(π’π’π‘π‘ − π’π’ππ ) and to notice that in Alpha – where πππ‘π‘ππ = πππ‘π‘ and in which expectations are therefore always correct – unemployment will always be at its natural level, independently of the way the inflation rate evolves over time. Furthermore, to have π’π’π‘π‘ = π’π’ππ at each time π‘π‘, in country Beta one needs πππ‘π‘ = 2% – inflation will have to be constant at 2%, rather than increasing over time. 296 The labor market, the IS-LM-PC model, and inflation b. “In country Gamma πππ‘π‘ππ = 3%, while in country Delta πππ‘π‘ππ = πππ‘π‘−1 . It follows that, to keep the rate of unemployment at the natural level (π’π’π‘π‘ = π’π’ππ for each time π‘π‘), in both countries the rate of inflation must remain constant over time ”. To conclude that the statement is true it suffices to notice that, given the relationship between the deviation of the inflation rate from its expected value and the deviation of the unemployment rate from its natural level used to answer the previous point, to have π’π’π‘π‘ = π’π’ππ for each time π‘π‘ one needs an inflation rate always equal to 3%, and therefore constant over time, in Gamma; furthermore, in Delta the rate of inflation will have to be always equal to the level prevailing in the previous period, and therefore once again constant (even if not necessarily at the 3% level as in country Gamma). 297 Macroeconomics. Problems and Questions * Question 7 [The natural rate of interest - Monetary policy and fiscal policy in the mediumrun] Consider an economy described by the following equations: πΆπΆ = πΆπΆ(ππ − πποΏ½) πΌπΌ = πΌπΌ(ππ + π₯π₯, ππ) πΊπΊ = πΊπΊΜ ππ = πΆπΆ + πΌπΌ + πΊπΊ ππππ = ππ β πΏπΏ(ππ + ππ ππ , ππ) ππ⁄ππ = πΏπΏ(ππ + ππ ππ , ππ) ππ − ππ ππ = (πΌπΌ⁄πΏπΏ)(ππ − ππππ ) where ππ is nominal money supply, the term ππ ππ appearing in the Phillips curve (the last equation) is expected inflation, and the other symbols have the usual meaning. a. Define the concept of natural rate of interest, ππππ , and discuss its determinants using a graph in which the real rate ππ is measured along the vertical axis, and goods' supply and demand along the horizontal one. ππ 1 ππππ 0 ππππ 298 πΆπΆ + πΌπΌ + πΊπΊ supply of goods, demand for goods The labor market, the IS-LM-PC model, and inflation The natural rate of interest, ππππ , is the value of the real rate ππ associated with the natural level of output, ππππ − that is, with the level of output that the economy produces when unemployment is at its natural rate, π’π’ππ . More specifically, ππππ is the value of the real interest rate for which the goods market is in a medium-run equilibrium. Since, in this equilibrium, ππ = ππππ , the natural rate is implicitly defined by the following equation: ππππ = πΆπΆ(ππππ − πποΏ½) + πΌπΌ(ππππ + π₯π₯, ππππ ) + πΊπΊΜ . In the figure, the left-hand side (the supply of goods) is the line vertical at the natural level of output; in fact, in the medium-run the supply of goods does not depend on ππ, but rather on the factors on which the WS-PS model focuses (technology, institutional characteristics of the labour market, the degree of competition in goods and factor markets, etc.) and that jointly determine the natural rate of unemployment. The right-hand side (the demand for goods) is, in the same graph, a curve with a negative slope, as investment is decreasing in ππ. The natural rate of interest is found at the intersection of the two curves. Since, as you will remember, in goods market equilibrium national saving equals investment, an alternative (and fully equivalent) way of defining the natural rate of interest is to think of ππππ as that value of ππ for which − in medium-run equilibrium, and therefore when output is at its natural level − the sum of private and public saving equals investment. b. Explain what is meant by 'neutrality of money'. Using the graph you have drawn to answer the previous point, and assuming that individuals expect zero inflation (ππ ππ = 0), verify that money is neutral in the model considered in this question. Money is neutral if, in the medium run, monetary policy can only affect the general price level and the variables measured in nominal terms, leaving unchanged 'real' variables such as output, unemployment, consumption, investment, the real interest rate, etc. It is important to realize that the fact that money is neutral in the medium run does not mean that it never affects the economy, but just that its effects are going to be transitory, bound to disappear over time. 299 Macroeconomics. Problems and Questions To verify that, in our model, money is indeed neutral, suppose that − starting from a medium-run equilibrium like point 1 in the figure − the central bank implements a monetary expansion. In particular, assume that it decides to lower the policy rate and that, to this end, it raises the money supply (for instance, through an open market purchase of bonds). We know from the IS-LM-PC model that, in the short run, the cut in the policy rate will increase aggregate demand. Output will rise above its natural level, and inflation will become positive (that is, it will rise above its expected value, that for simplicity we have taken to be equal to zero). Prices will therefore start rising. Let us ask ourselves, however, not how the central bank intervention will affect the short-run equilibrium, but rather how it will affect the economy in the medium run, the relevant time horizon for both the previous figure and the 'neutrality of money' proposition. The monetary expansion will not change ππππ (the natural level of output does not depend on ππ, but on those 'supply-side' factors mentioned before), or the level of aggregate demand prevailing in medium-run equilibrium, πΆπΆ(ππππ − πποΏ½) + πΌπΌ(ππππ , ππππ ) + πΊπΊΜ (as ππ is not among the variables on which πΆπΆ, πΌπΌ or πΊπΊ depend). Therefore, following a monetary expansion, none of the two curves in the graph above will shift, and the natural rate of interest will not change. In the new medium-run equilibrium, output and the real interest rate will be the same as before, and the composition of aggregate demand will be unchanged, too. Compared to the initial equilibrium, the only variable that will take on a different value is the price level, which will be higher. c. And what about fiscal policy? In the medium-run, is it neutral, too? To answer, study the medium-run effects of a restrictive fiscal policy consisting in a permanent decrease in πΊπΊΜ using a graph similar to the one employed to answer the previous point. 300 The labor market, the IS-LM-PC model, and inflation ππ 1 ππππ 2 ππππ′ 0 ππππ πΆπΆ + πΌπΌ + πΊπΊΜ πΆπΆ + πΌπΌ + πΊπΊΜ ′ supply of goods, demand for goods Suppose that government spending on goods and services is cut from πΊπΊΜ to the new level πΊπΊΜ ′ < πΊπΊΜ . This change does not affect the supply of goods prevailing in the medium run, since ππππ depends on supply-side factors that we have assumed to be independent of πΊπΊ. It will however lead to a lower aggregate demand for goods. In the figure, the curve representing the aggregate demand prevailing in the medium run shifts down and to the left, and the medium-run equilibrium becomes point 2, where output is unchanged and the natural interest rate is lower. In the move from the old to the new medium-run equilibrium, the overall level of aggregate demand is unchanged (as it must be, being the supply of goods unchanged); its composition is however different − πΊπΊ is lower, consumption is unchanged (because disposable income is unchanged), and investment is higher (due to the fall in the real interest rate). Since the overall level of aggregate demand has not been affected by the fiscal policy in consideration, one can also conclude that investment must have gone up by the same amount by which πΊπΊ has been cut − the fall in the natural rate (from ππππ to ππππ′ in the figure) will take care of delivering the required rise in investment. In conclusion, and even though it does not affect output in the medium run, fiscal policy affects the natural interest rate and the composition of aggregate demand over the same time horizon. It follows that, contrary to what we have concluded about monetary policy, fiscal policy is not neutral, not even in the medium run. 301 Macroeconomics. Problems and Questions * Question 8 Consider an economy described by a standard IS-LM-PC model. The consumption function is: πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) where, as usual, the positive constant ππ0 is autonomous consumption, ππ1 − the marginal propensity to consume − is between zero and one, and the other symbols have the usual meaning. a. The economy is initially in a medium-run equilibrium, with ππ = ππππ and ππ = ππππ . Represent in the graph, and discuss, the short-run effects of a permanent increase in autonomous consumption, assuming that the central bank decides to keep the policy rate unchanged at the initial level, ππππ . In the move from the initial to the new short-run equilibrium, how will consumption and investment change? ππ ππππ′ ππππ πΌπΌπΌπΌ1 πΌπΌπΌπΌ2 2 1 ππππ ππ − ππ−1 0 ππππ 302 πΏπΏπΏπΏ2 1' πΏπΏπΏπΏ1 ππππ 1' ππ ππ The labor market, the IS-LM-PC model, and inflation The increase in ππ0 raises the demand for goods and shifts the IS curve to the right. Since the central bank keeps the policy rate at the level ππππ , the economy goes from the initial medium-run equilibrium to the new short-run equilibrium 1′. Output rises above ππππ , since firms meet the higher demand they face by producing more, and unemployment falls below π’π’ππ . This drop in the unemployment rate leads to an increase in wages, which firms will pass onto higher prices for the goods they produce. Inflation will rise, as shown in the lower panel of the graph. Finally, the increase in ππ will raise the equilibrium values of both consumption and investment. b. How should the central bank intervene if, in the short-run, it intends to keep the policy rate at ππππ ? Do you think that ππ can be kept at this level indefinitely? If not, at what value should the policy rate be allowed to converge? Motivate your answer by making explicit reference both to the graph you have just drawn and to the one used to answer Question 7 of this Chapter. Since ππ = ππ − ππ ππ , and given expected inflation, to keep ππ constant the central bank will have to rely on the control it exerts on i (for instance, through open market operations). More specifically, the central bank will have to prevent any change in the difference between ππ and ππ ππ , and therefore in the real rate, resulting from the following two forces: (i) the increase in ππ caused by the rise in ππ0 raises the price level and inflation, therefore leading, sooner or later, to an increase in ππ ππ ; (ii) both a higher ππ and a higher ππ cause an excess demand for money in the money market, and therefore tend to raise ππ. If, as a result of the two forces above, the difference between ππ and ππ ππ , and therefore ππ, tends to rise (fall) by 1%, the central bank will have to implement a monetary expansion (contraction) that lowers (raises) the nominal interest rate by the same 1%. However, this policy cannot be pursued indefinitely. In fact, the increase in ππ0 raises the natural interest rate (in the graph introduced in Question 7, the curve representing the demand for goods shifts to the right; in the IS-LM-PC graph above, the value of ππ for which the new IS curve and the dashed, vertical line going through ππ = ππππ − an intersection that, in that graph, always takes place for the natural rate of interest − is now ππππ′ > ππππ ). If the central bank persisted in keeping r at its initial level, ππ would keep rising, and the economy would never return to a medium-run equilibrium. Once it has become aware of the nature and the size of the shock, and with the lags with which it typically acquires adequate knowledge on the actual state of the economy, it is therefore plausible that − as we shall always assume in similar situations − the central bank will bring ππ to the new equilibrium level ππππ′ . 303 Macroeconomics. Problems and Questions Question 9 Country Eta, a closed economy with flexible prices, is initially in a mediumrun equilibrium. a. To reduce a budget deficit deemed too high, the government of Eta decides to raise taxes. At the same time, the central bank decides to resort to an open market operation, aimed at preventing the fiscal policy just mentioned from changing the inflation rate, π. In an IS-LM-PC diagram, denote the initial medium-run equilibrium by ‘1’ and represent the new medium-run equilibrium to which Eta will converge after the implementation of the policy-mix described above. In particular, describe the type of open market operation (purchase; sale) that the central bank should implement to reach its objective. ππ ππππ1 ππππ2 πΌπΌπΌπΌ2 πΌπΌπΌπΌ1 1' ππππ ππ − ππ−1 0 2 1 1' 304 ππππ πΏπΏπΏπΏ1 πΏπΏπΏπΏ2 ππππ ππ ππ The labor market, the IS-LM-PC model, and inflation The tax hike lowers consumption and shifts the IS curve to the left. If the central bank did not intervene, the fall in aggregate demand caused by the decrease in consumption would lower output, and the economy would go from the initial medium-run equilibrium 1 to the new short-run equilibrium 1′. Output would fall below its natural level, and the unemployment rate rise above π’π’ππ . These changes would lower wages, and therefore prices. To prevent the ensuing fall in inflation, the central bank will have to lower the policy rate to ππππ2 , thus shifting the πΏπΏπΏπΏ curve downwards to πΏπΏπΏπΏ2 . What is needed is therefore a monetary expansion, that − since it has decided to resort to an open market operation − the central bank will have to implement by purchasing bonds in the secondary market for these assets. Notice that the new LM that will result from this intervention will have to intersect the new IS curve for ππ = ππππ − in fact, from the Phillips curve it follows that an intersection for any level of output different from the natural one would lead, rather than to an unchanged π, to an inflation rate rising or falling over time, depending on whether, at the intersection point, output is greater than ππππ , or less than ππππ . b. Compare the composition of aggregate demand prevailing in the initial and in the final medium-run equilibria. How must the change in consumption and the change in investment taking place as the economy goes from the first to the second of such equilibria be related to one another? As the....................................................................................................................... supply of goods is at its natural level both at point 1 and at point 2, aggregate .... demand will have to be the same in those two equilibria. In the move from the first to the second one, government purchases are unchanged, ........................................................................................................................... consumption has gone down (income has not changed, but taxes are higher), ........................................................................................................................... and investment up (due to the fall in the real interest rate). In addition, and since aggregate demand is unchanged, one can also conclude that investment ........................................................................................................................... will have risen exactly by the amount by which consumption has fallen. 305 Macroeconomics. Problems and Questions Question 10 Consider a country where both investment (which, as usual, is also a function of ππ) and consumption depend on the borrowing rate ππ + π₯π₯. In particular, suppose that the consumption function is πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) − β2 (ππ + π₯π₯), where the parameter β2 (> 0) is the sensitivity of consumption to the real borrowing rate. The rest of the economy is described by an IS-LM-PC based on the usual hypotheses − among them, the exogeneity of net taxes and of government purchases of goods and services (ππ = πποΏ½ e πΊπΊ = πΊπΊΜ ), and the assumption the central bank conducts its monetary policy by choosing the real policy rate, ππ. a. Having explained if, in the (ππ, ππ) space, the slope of the IS curve is in this case negative, zero or positive, assume that the economy is initially in a medium-run equilibrium, with ππ = ππππ , ππ = ππππ and ππ − ππ−1 = 0. Show in the graph, and discuss, the short-run effects of a permanent increase in the financial markets participants’ degree of risk aversion, assuming that the central bank keeps the policy rate constant at the initial level, ππππ . In the move from the initial to the new short-run equilibrium, how will consumption and investment change? Explain. ππ ππππ πΌπΌπΌπΌ2 0 2 πΌπΌπΌπΌ1 1 ππππ ππ2 ππ − ππ−1 1 0 2 306 ππππ πΏπΏπΏπΏ1 ππππ ππ ππ The labor market, the IS-LM-PC model, and inflation In this economy, a decrease in ππ raises not just investment, but also consumption. As in the standard case, the demand for goods will depend negatively on the real rate, and the IS curve will therefore be negatively sloped in the (ππ, ππ) plane. When, starting from the initial medium-run equilibrium denoted by 1 in the figure, the degree of risk aversion rises, the IS shifts to the left, as the increase in the risk premium π₯π₯ leads to an increase in the borrowing rate that lowers consumption and investment. Since, by assumption, the central bank does not change the policy rate, the new short-run equilibrium becomes 2, where both consumption and investment have gone down, due to the rise in π₯π₯ and the ensuing fall in ππ. b. Suppose that, once the economy has reached the short-run equilibrium described in the answer to the previous point, the government decides to return output to the natural level by changing πΊπΊΜ . To achieve its goal, should the government raise or lower πΊπΊΜ ? Compare the levels of investment, consumption, private saving, public saving and national saving in the new medium-run equilibrium that will be reached after the government’s intervention to the levels of the same variables in the initial one (that is, the medium-run equilibrium prevailing before the increases in the degree of risk aversion and in πΊπΊΜ ). Explain. [Hint: write down the (private) saving function for this economy]. To return output to its natural level, the government will have to bring the IS curve back to its initial position. This requires an increase in πΊπΊΜ . After this fiscal policy intervention, the medium run equilibrium will be once again at point 1 in the figure, with lower consumption and investment (ππ and Y are unchanged, but the risk premium is now higher, and with it the borrowing rate). Government saving will be lower, too (πΊπΊΜ has gone up, and net taxes πποΏ½ have not changed), while private saving ππ = (ππ − πποΏ½) − πΆπΆ = −ππ0 + (1 − ππ1 )(ππ − πποΏ½) + β2 (ππ + π₯π₯) will be higher, due to the increase in π₯π₯. Since investment has gone down, the same must be true about national saving (the amount by which private saving has risen will be smaller than that by which public saving has fallen). 307 Macroeconomics. Problems and Questions Question 11 Consider a country where investment is entirely exogenous (πΌπΌ = πΌπΌ )Μ , In addition, consumption depends not just on disposable income, but also on the real interest rate, as implied by the consumption function πΆπΆ = ππ0 + ππ1 (ππ − πποΏ½) − β2 ππ, where the parameter β2 > 0 is the sensitivity of consumption to the real interest rate. The rest of the economy is described by an IS-LM-PC model based on the usual assumptions. In particular, net taxes and government purchases of goods and services are exogenous (ππ = πποΏ½ and πΊπΊ = πΊπΊΜ ), and the central bank chooses the (real) policy interest rate, ππ. a. Discuss how the IS curve will be sloped in this economy. Furthermore, suppose that the economy was initially in a medium run equilibrium with ππ = ππππ , ππ = ππππ e ππ − ππ−1 = 0, and that, in the attempt to offset a fall in autonomous consumption by π₯π₯ππ0 < 0, the government of the country cuts net taxes by an equal amount, so that π₯π₯ππ0 = π₯π₯πποΏ½ < 0. Show in the graph below the new short-run equilibrium that will be reached following the two, contemporaneous, changes in autonomous consumption and net taxes just described, assuming that the central bank decides to keep the policy rate at the initial level, ππππ . How will the various components of aggregate demand change, in the move from the initial medium-run equilibrium to the new short-run one? ππ ππππ 0 ππ − ππ−1 0 πΌπΌπΌπΌ2 2 πΌπΌπΌπΌ1 1 πΏπΏπΏπΏ1 3 ππ2 πΏπΏπΏπΏ3 ππππ 1 2 308 ππππ ππππ ππ ππ The labor market, the IS-LM-PC model, and inflation Although, in this economy, a fall in ππ will not affect investment, it will raise consumption. It follows that, as in the standard case, goods demand depends negatively on the real interest rate, so that the IS curve will still be negatively sloped in the (ππ, ππ) plane. When, starting from the initial medium-run equilibrium 1 in the figure, autonomous consumption and net taxes change by π₯π₯ππ0 = π₯π₯πποΏ½ < 0, autonomous demand will change by π₯π₯ππ0 − ππ1 π₯π₯πποΏ½ = (1 − ππ1 )π₯π₯ππ0 < 0, and the IS curve will therefore shift to the left. Since the central bank does not change the policy rate, the new short-run equilibrium becomes 2, where income is lower. Given that, being exogenous, both investment and πΊπΊ have remained constant, consumption will have to be lower in this new short-run equilibrium. b. Suppose that, once the economy has reached the short run equilibrium you have described when answering the previous point, the central bank decides to bring income back to the natural level by implementing an open market operation. To achieve its aim, should it buy or sell bonds in the open market? Show in the graph the new equilibrium that will be reached following the central bank’s intervention you consider appropriate. In this new equilibrium, how will the levels of investment, consumption, private saving, public saving and national saving compare to the levels of the same variables in the initial medium-run equilibrium (that is, the equilibrium prevailing before the changes in autonomous consumption and the fiscal and monetary policy interventions described above)? To return income to the natural level, the central bank will have to lower the policy rate. This calls for a purchase of bonds in the open market, something that will cause a downward shift in the LM curve. In the figure, the new equilibrium becomes point 3. In this new medium-run equilibrium, ππ is the same as at point 1. It follows that aggregate demand, too, will have to be the same in the two equilibria. And since πΊπΊ and πΌπΌ are unchanged, in the move from 1 to 3 consumption, too, must have remained unchanged. Finally, since investment is exogenous, and therefore constant, also national saving will have to be unchanged. It follows that, since public savings has gone down (net taxes have been cut), private saving must have gone up (thanks to the tax cut and the decrease in ππ0 , and the fall in the real interest rate notwithstanding) by the same amount. 309 Macroeconomics. Problems and Questions Question 12 Consider an economy described by an IS-LM-PC model departing from the standard one for the fact that expected inflation is not equal to yesterday's inflation, but rather to the constant value πποΏ½, so that ππ ππ = πποΏ½. For simplicity, assume this constant value equals zero, so that ππ ππ = πποΏ½ = 0. In an IS-LM-PC diagram, represent the initial medium-run equilibrium of the economy, denoting it by 1, and assuming that the associated real interest rate is positive (that is, ππππ1 > 0). a. Suppose that, due to a major, permanent fall in the autonomous components of the demand for goods, the economy ends up in a new short-run equilibrium, to be denoted by 1′ in the figure, in which output is below its natural level; furthermore, suppose that the natural interest rate associated with this lower demand, let's call it ππππ2 , is not only less than ππππ1 , but also negative (ππππ2 < 0 < ππππ1 ). In this economy, can a 'conventional' monetary policy − as the one consisting in the decision to lower the policy rate − return output to its natural level? Explain. In the figure, the IS curve shift leftwards to πΌπΌπΌπΌ2 , and output falls to ππ1′ < ππππ . To return it to its natural level, the central bank should cut the policy rate to ππππ2 , thus shifting the LM curve downwards till it becomes the dashed, horizontal line drawn for this latter value of the real rate. Since ππ = ππ − ππ ππ , a central bank wishing to reduce ππ will usually do so by lowering the nominal interest rate, ππ. In this case, however, the central bank will not be able to bring the real rate to the target level. In fact, being ππ ππ = πποΏ½ = 0, in this economy ππ = ππ. The Zero Lower Bound − that is, the fact that the nominal interest rate cannot fall below zero − implies that the real rate cannot take on negative values, and therefore that ππππ2 cannot be reached. The most the central bank can do in a similar situation is to set to zero both ππ and ππ, shifting downwards the LM to πΏπΏπΏπΏ0 and moving the economy to point 2′, where output is greater than ππ1′ , but still below its natural level. 310 The labor market, the IS-LM-PC model, and inflation ππ ππππ1 0 ππππ2 ππ 0 πΌπΌπΌπΌ2 1' πΌπΌπΌπΌ1 ππ1′ 1' 2′ 1 2 ππππ ππππ πΏπΏπΏπΏ1 πΏπΏπΏπΏ0 ππππ ππ ππ b. To return output to its natural level, which economic policies would you suggest? An expansionary fiscal policy of appropriate size could return the IS curve to its initial position. Alternatively, the country's policy-makers could implement a combination of monetary expansion (for instance, one that shifts the LM to πΏπΏπΏπΏ0 − something that, as we know, rises output somewhat) and fiscal expansion (one that shifts the IS to the right by the amount needed to fill the residual gap between ππ and ππππ ). Finally, one could also consider 'unconventional' monetary policy measures. For instance, let us suppose that, maybe by raising its medium-run target for the inflation rate, the central bank manages to induce individuals to expect that inflation will be 2%, rather than zero, in the future. Now it becomes possible for ππ to take on negative values (up to −2%, in our example); it follows that the real rate can now be brought, or in any case moved closer to, the new natural level ππππ2 . 311 Macroeconomics. Problems and Questions Question 13 In country XYZ, the production function is ππ = π΄π΄ · ππ, where π΄π΄ is a positive constant. In addition, when price expectations are correct (ππππ = ππ), the pricesetting and the wage-setting relations are, respectively, ππ = (1 + ππ) · (ππ/π΄π΄) and ππ = ππ · πΉπΉ(π’π’, π§π§), where the variables have the usual meaning. a. Provide an economic interpretation of the costant π΄π΄ and, in the graph below, show how a decrease in π΄π΄ will change the equilibrium values of the real wage and of the natural rate of unemployment. ππ/ππ π΄π΄/(1+ππ) ππππ ππππ′ π΄π΄′/(1+ππ) ππππ π’π’ππ π’π’′ππ π’π’ The constant A is labor productivity (average and marginal productivity, equal under the hypothesis of a linear production function). If A decreases, labor costs per unit of output ππ ⁄π΄π΄ (= ππππ⁄π΄π΄π΄π΄ = ππππ/ππ) rise. The PS curve shifts downwards and the equilibrium real wage decreases, while the natural rate of unemployment increases. Firms will set higher prices and will pay a lower real wage to workers who are now less productive. Employment will decrease and unemployment will increase. 312 The labor market, the IS-LM-PC model, and inflation b. Assuming that the economy was initially in a medium-run equilibrium, show the effects of the same change in π΄π΄ discussed above in an IS-LM-PC diagram. In particular, show the new short-run equilibrium and describe the adjustment process toward the new medium-run equilibrium to which the economy will eventually converge. Compare the levels of consumption and investment in this latter equilibrium with the levels of the same variables in the initial medium-run equilibrium, providing an explanation for any observed change in their values. ππ 2 ππππ2 ππππ1 ππ − ππ−1 ππππ′ ππππ′ 0 1' 1' 1 ππππ ππππ πΏπΏπΏπΏ2 πΏπΏπΏπΏ1 πΌπΌπΌπΌ1 ππππ' ππππ ππ ππ In the IS-LM-PC diagram, the PC curve shifts upwards. If the central bank does not change the policy rate, the new short-run equilibrium is point 1′, where output is unchanged and inflation higher. The rise in inflation is due to the fact that, as discussed above, the fall in A leads firms to supply each unit of output at a higher price. Given last period's prices, a higher current price level implies more inflation. To prevent inflation from rising period after period, sooner or later the central bank will have to raise the policy rate. When this latter will be set to ππππ2 , the economy will once again be in a medium-run equilibrium position, with constant inflation and output at its (new) natural level. Compared to the initial equilibrium (1), in this new medium-run equilibrium (2) both consumption (due to the fall in ππ) and investment (due to the fall in ππ and the rise in ππ) will be lower. 313 Macroeconomics. Problems and Questions Question 14 Consider a country that is initially in a medium run equilibrium position and in which the expected inflation rate is the constant πποΏ½. For simplicity, assume that πποΏ½ equals zero, so that ππ ππ = 0. Aside from this assumption, the economy is described by a standard IS-LM-PC model, with a central bank that chooses the interest rate. a. Represent the initial medium run equilibrium position of the economy, to be denoted by 1 in the graph, assuming that it takes place for a positive value of the natural real rate of interest (that is, ππππ1 > 0). Suppose now that a new law leads to an increase in the minimum wage that firms must pay their workers. In the graph, denote by 1′ the new short run equilibrium. Compared to 1, how have production, consumption and investment changed? Why? Explain in detail. [Hint: assume that the position of the IS curve is not affected by the change in consideration]. ππ 2 ππππ2 ππππ1 ππ − ππ−1 0 ππππ′ ππππ′ 1' 1' 1 ππππ ππππ πΏπΏπΏπΏ2 πΏπΏπΏπΏ1 πΌπΌπΌπΌ1 ππππ' ππππ ππ ππ An increase in the minimum wage leads to a higher wage set by wage setters for any given unemployment rate. In the WS-PS model, it is represented by an increase in π§π§ which shifts the WS curve up and to the right, thus causing an increase in the natural rate of unemployment and a decrease in the natural level of output. 314 The labor market, the IS-LM-PC model, and inflation In the figure, the new natural level of output is ππππ′ (< ππππ ). In the IS-LM-PC diagram, the PC curve shifts up and to the left. Since we are told that the position of the IS curve is not affected by the change in consideration, if the central bank does not vary the policy rate the new short-run equilibrium becomes point 1′, where output, consumption and investment are unchanged, but inflation is higher. The inflation rate rises because, due to the increase in the minimum wage and to the ensuing rise in wages, now firms charge higher unit prices for the goods they produce. For given prices prevailing in the previous period, a higher general price level today implies a higher inflation rate. b. Suppose that, once the economy has reached the short-run equilibrium described in the answer to the previous point, the central bank decides to bring back income to its natural level by implementing an open market operation. Explain if, to achieve its goal, the central bank should purchase or sell bonds, and show in the figure the new policy rate consistent with the new medium run equilibrium, denoting it by ππππ2 . Finally, discuss how the central bank should act in order to return not just output to the natural level, but also the general price level to the value it was taking on in the initial medium run equilibrium [Hint: you are not asked to show in the graph the strategy that, in this latter case, the central bank should follow; just illustrate verbally its main features]. ........................................................................................................................... To bring income to the new natural level, the central bank should raise the ........................................................................................................................... policy rate till ππππ2 , thus shifting the πΏπΏπΏπΏ curve upwards to πΏπΏπΏπΏ2 . This of course requires an open market sale of bonds. As long as the economy remains in the ........................................................................................................................... short-run equilibrium 1′, with income above the new natural level, the infla........................................................................................................................... tion rate is positive, and the price level will therefore keep rising. If the central bank aims at bringing back not just output to the natural level, but also the ........................................................................................................................... general price level to the value it was taking on in the initial medium run equilibrium, it should temporarily bring the policy rate above ππππ2 , and therefore production below the natural level. Being ππ < ππππ′ , the inflation rate will be negative, and the general price level falling. Once it is back to its initial value, the central bank should lower the real rate to ππππ2 , thus achieving its two goals. 315 Macroeconomics. Problems and Questions Question 15 Consider country Macro, described by an IS-LM-PC model and in which individuals expect inflation to be constant at 2%. While Macro was, till two years ago, in a medium run equilibrium with a real interest rate (ππ) equal to 3%, the past couple of years have witnessed a marked reduction in ππ, which has fallen to 1% both in last year and in the current one. In addition, this year’s inflation rate has remained constant at the same value it took on one year ago. a. To understand the cause of the fall in the real interest rate, the government of the country consults two economists, Harry and Ginny. According to Harry, the fact that the inflation rate has remained constant allows one to conclude that the economy must have been hit by an adverse demand shock, one to which the central bank has reacted by bringing ππ to the new, lower medium run equilibrium level. The reduction in the real rate therefore reflects, without any doubt, a decrease in its natural level, ππππ . Ginny instead thinks that, to be able to conclude that the economy has been experiencing a fall in ππππ , rather than a decision of the central bank to bring the real rate below an unchanged natural level, further information is needed. In particular, one needs to know whether, over the past two years, the inflation rate has remained constant at 2%, or at a level greater than 2%. Represent in the graph below the medium run equilibrium prevailing in Macro two years ago (ππππ = 3%), denoting it by ‘1’. In the same graph, denote by ‘2’ the equilibrium, associated with a real rate of 1%, that would prevail if the fall in ππ is due to a decrease in the natural, medium run, real rate, and by ‘3’ that which would instead prevail if the decrease of the real rate to 1% is the outcome of the decision of the central bank to bring ππ below an unchanged natural level. Which of the two economists is right? Why? Explain. ππ ππππ1 = 3% πΌπΌπΌπΌ2 πΌπΌπΌπΌ1 1% 2 0 ππ − 2% 0 1 1 πΏπΏπΏπΏ1 3 ππππ 2 ππππ 316 ππ3 3 ππ3 πΏπΏπΏπΏ2,3 ππππ ππ ππ The labor market, the IS-LM-PC model, and inflation The medium run equilibrium prevailing two years ago is 1 in the two graphs. Harry thinks that the current 1% real rate is the by-product of a leftward shift of the IS curve, and of the central bank’s decision to allow the economy to reach the new medium run equilibrium 2 by decreasing the policy rate till the new, lower natural level of 1%. According to Ginny, however, one should also consider the possibility that the economy is at point 3 – no demand shock has hit the economy; rather, the central bank is insisting on keeping income above, and the real rate below, their natural levels, both unchanged compared to two years ago. To be able to conclude which of the two economists is right, one must look at the lower panel of the figure. From the Phillips curve it follows that, if Harry if right, then over the past two years inflation should have remained constant at 2%; if Ginny is right, inflation should have been constant, but at a level greater than 2%. Since we are just told that inflation has remained constant, but not at which level, Ginny is right – to be able to assess the cause of the low real interest rate, more information is needed. b. Say you are now told that, in Macro, rather than an inflation rate constant at 2%, individuals always expect an inflation rate equal to that observed in the previous period, so that ππ ππ = ππ−1 . Which of the two economists is right, in this case? Why? Explain. In this case, the upper panel of the figure remains unchanged; however, to be measured along the vertical axis of the lower one is now the difference between π and ππ−1, rather than that between the current inflation rate and 2%. It follows that inflation will remain constant over time if, and only if, the economy is in a medium run equilibrium. The constancy of the inflation rate over the past two years allows one to conclude that, in this case, the economy must have experienced a fall in the natural real rate of interest, just like suggested by Harry. 317 Macroeconomics. Problems and Questions Question 16 A country in which expected inflation for the current year equals the inflation rate prevailing in the previous year, ππ ππ = ππ−1 , and in which ππ ≥ 0, is initially in a medium run equilibrium, with ππ = ππππ , ππππ = 0 and ππ ππ = ππ−1 = 0. a. Assuming that the central bank chooses the interest rate, represent in the two-panel IS-LM-PC diagram the initial medium run equilibrium, denoting it by 1. Having explained which value the nominal interest rate, ππ, will necessarily take on in this initial equilibrium, suppose that the government launches an ambitious program of liberalizations, leading to an increase in the degree of competition in the country’s goods markets. In the two panels of the graph below, denote by 2 the new short run equilibrium that the economy will reach. In the move from 1 to 2, how will have income changed? And what about the rate of inflation? Why? Explain. [Hint: when answering, assume that the position of the IS curve is not affected by the change described above]. ππ ππ3 ππππ1 = 0 3 ππ − ππ−1 0 1 2 ππππ ππππ 1 3 2 πΏπΏπΏπΏ3 πΏπΏπΏπΏ1 4 ππππ′ πΌπΌπΌπΌ1 4 ππππ′ ππππ πΌπΌπΌπΌ4 ππ ππππ' ππ Being ππ = ππ + ππ ππ , and since in this economy ππππ = 0 and ππ ππ = ππ−1 = 0, in the initial medium run equilibrium (points 1 in the two panels) the nominal interest rate is equal to zero. In the WS-PS model, an increase in the degree of competition in the goods markets amounts to a decrease in the mark-up ππ. It follows that the PS curve shifts upwards, the natural rate of unemployment falls, and the natural level of output becomes ππππ′ > ππππ . 318 The labor market, the IS-LM-PC model, and inflation In the IS-LM-PC diagram, the PC curve shifts to the right, crossing the xaxis for the new, higher natural level of production. If the central bank does not change its choice of the real policy rate, the new short run equilibrium becomes point 2 in the two panels, where production is unchanged and the inflation rate is lower. In particular, from zero that it was in initial the equilibrium 1, the inflation rate becomes negative – in the current period, prices are lower than in the previous one. In fact, given that their market power has fallen, firms will now be willing to supply each unit of output at a price lower than that prevailing before the launch of the liberalization program. b. If policy-makers do not intervene, how will the real interest rate and income change next period? Why? Show in the graph, denoting by 3 the new short run equilibrium that, absent any policy intervention, the economy would attain. To make sure that, rather than from 1 to 2, from 2 to 3, etc., the economy goes immediately from the initial medium run equilibrium 1 to the new medium run equilibrium that you will denote by 4 in the graph, the implementation of the liberalization program should be combined with a fiscal policy intervention, or with a monetary policy one? And, once the kind of economic policy most appropriate to go directly from the equilibrium 1 to the equilibrium 4 has been detected, the proposed policy should be a restrictive or an expansionary one? Show in the graph the effects of the policy intervention you suggest, and comment. Since ππ ππ = ππ−1, next period individuals will start expecting a negative inflation rate, and this will lead to a higher ππ. The real interest rate, which was initially zero, will become positive, rising for instance to the level denoted by ππ3 in the graph. Notice that the central bank will be unable to keep the real rate at ππππ1 ; in fact, to do so it should lower the nominal interest rate by the amount needed to offset the impact on ππ of the decrease in expected inflation. However, we know that the nominal interest rate is already at its lower bound (zero), and therefore cannot be lowered any further. We may conclude that, absent policy interventions − not just by the central bank, but also by the government −, the economy will therefore reach the new short run equilibrium 3, with a level of output which is lower, due to the rise in the real interest rate. Finally, to make sure that, from the initial equilibrium 1, the economy directly goes to the new medium run equilibrium 4, the implementation of the liberalization program should be combined with an expansionary fiscal policy, one that shifts to the right the πΌπΌπΌπΌ curve from πΌπΌπΌπΌ1 to πΌπΌπΌπΌ4 , thus immediately bringing output to the new natural level ππππ′ . In the new medium run equilibrium, the natural real rate will still be ππππ1 , and the inflation rate constant and equal to zero. 319 Macroeconomics. Problems and Questions * Question 17 a. In the IS-LM-PC model, when the economy is in a medium-run equilibrium, output and the real interest rate are at their natural levels, and expectations are correct, so that ππ = ππ ππ . Suppose that the inflation rate individuals expect for the current period is equal to last year's rate, ππ ππ = ππ−1 . From ππ = ππ ππ and ππ ππ = ππ−1 it follows that, in a medium-run equilibrium, inflation will be constant − let's say, at the level ππ ∗ (for instance, at 2%). What determines ππ ∗? In a medium-run equilibrium, all markets must be in equilibrium. In particular, in the money market the supply of money must be equal to the demand for money. Written in real terms, the equilibrium condition in this market is: ππ = πΏπΏ(ππ + ππ ππ , ππ), ππ where πΏπΏ is real money demand. In medium-run equilibrium, ππ = ππππ , a constant; the real interest rate is ππ = ππππ , and therefore constant, too; finally, the inflation rate takes on the constant value ππ ∗. The equation above therefore becomes: ππ = πΏπΏ(ππππ + ππ ∗ , ππππ ). ππ Since in a medium-run equilibrium all its determinants take on constant values, real money demand − the right-hand side of the above equation − is therefore constant. For the money market to be in equilibrium, the ratio on the left-hand side − real money supply − must be constant, too. For this to be the case, the rate at which the denominator changes over time, ππ ∗, must be equal to the rate at which the numerator, nominal money supply, varies. If ππ grows at a rate equal to zero (that is, if the central bank keeps nominal money supply constant), in equilibrium inflation, too, will be zero; if however ππ is allowed to grow at a rate of 2%, in the medium run inflation will be 2%; and if the central bank lets money supply grow at a yearly rate of 4%, inflation will double, too. It follows that, in the medium run, the rate of inflation is equal to the rate of growth of the money supply decided by the central bank. Let us sum up what, as discussed in this Chapter, the IS-LM-PC model implies for the way in which monetary policy affects the economy: 320 The labor market, the IS-LM-PC model, and inflation • in the medium run, monetary policy does not affect the natural level of output, the natural interest rate, or the composition of aggregate demand; changes in the rate of growth of nominal money supply only cause proportional changes in inflation − money is neutral; • in the short run, the policy rate can deviate from the natural one, and monetary policy does have real effects. For instance, to soften a recession and accelerate the recovery of the economy, the central bank could cut the policy rate, allowing ππ to grow at a rate temporarily higher than the one it has chosen to implement in the medium run. However, such deviations cannot be permanent, if the economy has to return to a medium-run equilibrium. It follows that the real effects of monetary policy are necessarily transitory. b. If, in the medium run, inflation depends on the rate of money growth chosen by the central bank, and if inflation is a 'bad', why the world's major central banks target an inflation rate that, rather than zero, is positive (albeit 'small')? Mention two factors that help explain this choice. 1. Let's suppose that, maybe because they suffer from money illusion, workers tend to resist cuts in their nominal wage, and that the economy is hit by an adverse supply shock. In this case, a positive inflation allows for those downward adjustments in the real wage needed to return to a medium-run equilibrium more easily than a zero rate of inflation. 2. A positive, and sufficiently high, inflation rate makes less likely that the Zero Lower Bound will be hit, thus giving central banks more room to resort to conventional monetary policy tools when the economy is in a recession. 321 Chapter 4 - Expectations, financial markets, and economic policies Macroeconomics. Problems and Questions Question 1 We are at time π‘π‘. In country Tau, only one-year and two-year bonds exist. The time-π‘π‘ price of two year bonds issued in that time period is €ππ2π‘π‘ = €94. In adππ − the yield that market participants expect on one-year bonds dition, ππ1π‘π‘+1 that, issued at π‘π‘ + 1, will mature at π‘π‘ + 2 − is 4%. a. Using the quantitative information provided above, compute the yield on the one-year bonds issued at π‘π‘, ππ1π‘π‘ . Plugging €ππ2π‘π‘ = €94 into €ππ2π‘π‘ = €100/(1 + ππ2π‘π‘ )2 , and solving, one gets 1 ππ ππ ππ2π‘π‘ = 3.14%. Since ππ2π‘π‘ = 2 (ππ1π‘π‘ + ππ1π‘π‘+1 ) and ππ1π‘π‘+1 = 4%, it is straightfor........................................................................................................................... ward to compute ππ1π‘π‘ = 2 · 3.14% − 4% = 2.28%. 324 Expectation, financial markets, and economic policies b. Suppose now that you do not share the market’s expectations about ππ′ = 5%, rather future short-term rates. In particular, you expect ππ1π‘π‘+1 ππ than ππ1π‘π‘+1 = 4%. If you are interested in how much you will have two years from today, should you buy two-year bonds or a sequence of one-year bonds? Or maybe you should be indifferent between the two alternatives? Explain. The purchase of two-year bonds at the price €ππ2π‘π‘ = €94, one that reflects the market participants’ average opinion about the future value of short-term rates, delivers an average yearly return of 3.14% (remember that ππ2π‘π‘ = 3.14%). However, since your expectations about the future level of short-term rates differ from those held by the market, by purchasing one-year bonds you will get a 2.28% return on the first year, and one that you expect to be 5% the next one, corresponding to an expected average yearly return of 3.64% (> 3.14%). You should purchase one-year bonds. 325 Macroeconomics. Problems and Questions Question 2 a. Alpha and Beta − two countries where only one, two and three-year bonds exist – differ just for the expectations held by market participants about the monetary policy that, in each of the two nations, will be implemented in the future. The graph below shows the yield curve prevailing in the two countries in the current period, time π‘π‘ [N.B.: the yield curve prevailing in Alpha (the bold, continuous line in the graph) and that observed in Beta (the dashed, bold line) overlap for maturities up to two years; afterwards, Beta’s yield curve declines faster than Alpha’s]. Which differences in individuals’ expectations about the future monetary policy that will be implemented in each of the two countries can explain the observed differences in the two yield curves? Using the data provided in the graph, compute the current and future expected one-year rates ππ1π‘π‘ , ππ1ππ π‘π‘+1 and ππ1ππ π‘π‘+2 prevailing today (time π‘π‘) in each of the two countries. Yield to maturity 7% 6% Beta 5% 1 2 Alpha 3 Maturity ........................................................................................................................... A flat (negatively sloped) yield curve signals expectations of constant (decreasing) short-term rates. In both countries short-term rates are therefore ........................................................................................................................... expected to remain next year at today’s level, and then to decrease two years ........................................................................................................................... from now. This expected future decrease is larger in country Beta. Since, by assumption, these different expectations on future interest rates only reflect ........................................................................................................................... differences in the monetary policy that is expected to be implemented in the ........................................................................................................................... future, we may conclude that market participants think that next year the stance of monetary policy will be the same as today’s both in Beta and in Al........................................................................................................................... pha; however, they also expect that, in two years, monetary policy will become more expansionary in both countries, and more so in Beta. 326 Expectation, financial markets, and economic policies From the graph it also follows that ππ1π‘π‘ = 7%, ππ2π‘π‘ = 7% and ππ3π‘π‘ = 6% in Alpha,, while ππ1π‘π‘ = 7%, ππ2π‘π‘ = 7% e ππ3π‘π‘ = 5% in Beta. Since ππ2π‘π‘ = 12(ππ1π‘π‘ + ππ ππ ππ1π‘π‘+1 ), plugging the values of ππ1π‘π‘ and ππ2π‘π‘ , and solving, in Alpha ππ1π‘π‘+1 = 7%. ππ ππ 1 Furthermore, since ππ3π‘π‘ = 3(ππ1π‘π‘ + ππ1π‘π‘+1 + ππ1π‘π‘+2 ), plugging the values of ππ3π‘π‘ , ππ1π‘π‘ ππ ππ and ππ1π‘π‘+1 , and solving, for Alpha one computes ππ1π‘π‘+2 = 4%. Following the same steps, in the case of Beta it is straightforward to conclude ππ ππ = 7% and ππ1π‘π‘+2 = 1%. that, in that country, ππ1π‘π‘ = 7%, ππ1π‘π‘+1 b. Consider now Gamma, a country where stock prices have recently soared. To deflate what it thinks is a stock market bubble, Gamma’s central bank decides to intervene to return stock prices to a level closer to fundamentals. To achieve this goal, should the central bank implement an expansionary or a restrictive monetary policy? Why? Motivate your answer by making reference to the formula for stock prices, showing the determinants of the value of these financial assets [Hint: when answering, assume that, in each time period, the economy is described by a static IS-LM model, with consumption and investment that depend on contemporaneous variables only]. Since the stock price at time π‘π‘ can be written as: €πππ‘π‘ = ππ ππ €π·π·π‘π‘+1 €π·π·π‘π‘+2 + + … , ππ 1 + ππ1π‘π‘ + π₯π₯ (1 + ππ1π‘π‘ + π₯π₯)(1 + ππ1π‘π‘+1 + π₯π₯) ππ where π₯π₯ is the equity premium and €π·π·π‘π‘+ππ denotes expected dividends at time π‘π‘ + ππ, Gamma’s central bank will have to announce and implement a restrictive monetary policy, one that will lead to higher current and future expected short-term interest rates and to lower future expected levels of firms’ production, revenues and dividends. For both reasons, today’s stock prices will fall. 327 Macroeconomics. Problems and Questions Question 3 a. In country Zeta, only one-year and two-year bonds exist. Investors, who do care about risk, ask for a risk premium π₯π₯ to hold the two-year bond (which is risky, as they do not know the price at which they will be able to sell it in a year). Use the arbitrage equation to derive the (approximate) relation that will hold in this case among the yield to maturity of a two-year ππ bond (ππ2π‘π‘ ), the current (ππ1π‘π‘ ) and future expected (ππ1,π‘π‘+1 ) yields on one-year bonds, and the risk premium (π₯π₯). In this case, the arbitrage equation is 1 + ππ1π‘π‘ + π₯π₯ = ππ €ππ1,π‘π‘+1 , €ππ2π‘π‘ where the term on the right-hand side is the expected yield on a two-year bond. Solving for €ππ2π‘π‘ , and equating the expression thus obtained for €ππ2π‘π‘ to the one that follows from the definition of the yield to maturity on the same bond, one gets ππ (1 + ππ2π‘π‘ )2 = (1 + ππ1π‘π‘ + π₯π₯)οΏ½1 + ππ1,π‘π‘+1 οΏ½. ππ For ππ2π‘π‘ , ππ1π‘π‘ e ππ1,,π‘π‘+1 ‘small enough’ (close to zero), the previous equation implies that one can write 1 ππ ππ2π‘π‘ = 2οΏ½ππ1π‘π‘ + ππ1,π‘π‘+1 + π₯π₯οΏ½. 328 (∗) Expectation, financial markets, and economic policies b. Suppose that, in the current period (time π‘π‘), Zeta’s yield curve is the one in the figure below. Write on the axes the name of the variables measured along each of them, and use your answer to the previous point of this question to compute the numerical value that the risk premium π₯π₯ takes on in this economy, knowing that investors expect the future yield on one-year ππ bonds to be the same as the current one (ππ1π‘π‘ = ππ1,π‘π‘+1 ). Yield to maturity 4% 1% 1 2 Maturity ππ In this economy, ππ2π‘π‘ = 4% and ππ1π‘π‘ = ππ1,π‘π‘+1 = 1%. Plugging these values into (∗), 1 Solving, 4% = 2(1% + 1% + π₯π₯) . π₯π₯ = 6% . 329 Macroeconomics. Problems and Questions Question 4 a. In country Zeta, only one-year and two-year bonds exist. Investors, who do care about risk, ask for a risk premium π₯π₯ to hold the two-year bond (which is risky, as they do not know the price at which they will be able to sell it in a year). Use the arbitrage equation to derive the (approximate) relation that will hold in this case among the yield to maturity of a two-year bond (ππ2π‘π‘ ), the current (ππ1π‘π‘ ) and future exππ ) yields on one-year bonds, and the risk premium (π₯π₯). pected (ππ1,π‘π‘+1 In this case, the arbitrage equation is 1 + ππ1π‘π‘ + π₯π₯ = ππ €ππ1,π‘π‘+1 , €ππ2π‘π‘ where the term on the right-hand side is the expected yield on a two-year bond. Solving for €ππ2π‘π‘ , and equating the expression thus obtained for €ππ2π‘π‘ to the one that follows from the definition of the yield to maturity on the same bond, one gets ππ (1 + ππ2π‘π‘ )2 = (1 + ππ1π‘π‘ + π₯π₯)οΏ½1 + ππ1,π‘π‘+1 οΏ½. ππ For ππ2π‘π‘ , ππ1π‘π‘ e ππ1,,π‘π‘+1 ‘small enough’ (close to zero), the previous equation implies that one can write 1 ππ ππ2π‘π‘ = 2οΏ½ππ1π‘π‘ + ππ1,π‘π‘+1 + π₯π₯οΏ½. 330 (∗) Expectation, financial markets, and economic policies b. Suppose that, in the current period (time π‘π‘), the yield curve in Zeta – a country where the lower bound on the yield on one-year bonds is not zero, but negative, equal to −1% - is the one in the figure below. Write on the axes the name of the variables measured along each of them, and use your answer to the previous point of this question to compute the numerical value that the risk premium π₯π₯ takes on in this economy, knowing that ππ = −3ππ1π‘π‘ . ππ1π‘π‘+1 Yield to maturity 0.2% 0% 1 2 Maturity ππ In this economy, ππ2π‘π‘ = 0%, ππ1π‘π‘ = 0.2% ππππππ ππ 1,π‘π‘+1 = −3 β 0.2% = −0.6%. Plugging these values into (∗), Solving, 1 0% = 2(0.2% − 0.6% + π₯π₯) . π₯π₯ = 0.4% . 331 Macroeconomics. Problems and Questions Question 5 a. We are at time t. Write down the expression defining the price (in nominal terms) of a stock that has already paid the current dividend, and define carefully all the variables entering it. From which principle/condition is this expression derived? [Hint: you are not expected to derive the expression for the nominal stock price formally; just describe in detail the considerations on which its derivation is based]. The price (in nominal terms) of a stock that has already paid the current dividend can be written as: €πππ‘π‘ = ππ ππ €π·π·π‘π‘+1 €π·π·π‘π‘+2 + + … , ππ 1 + ππ1π‘π‘ + π₯π₯ (1 + ππ1π‘π‘ + π₯π₯)(1 + ππ1π‘π‘+1 + π₯π₯) where €π·π· ππ denotes expected dividends at various future time periods, π₯π₯ is the equity premium, and the discount rates appearing at the denominator of the terms on the right-hand side of the equation are the current and the future expected nominal yields on one-year bonds. The expression for the price of a stock can be derived from the arbitrage condition, which requires the expected return from holding stocks for one year to be the same as the expected return from holding alternative assets, for instance one-year bonds. 332 Expectation, financial markets, and economic policies b. Explain if, and why, you agree, or do not agree, with the following statements [Hint: when answering, assume that, in each time period, the economy is described by a standard, static IS-LM model, with consumption and investment that depend on contemporaneous variables only]: b.1 the prices of stocks at time π‘π‘, €πππ‘π‘ , will unambiguously go up if individuals start expecting an increase in autonomous consumption at time π‘π‘ + 2; b.2 if, at time π‘π‘, individuals learn that from π‘π‘ + 1 onwards fiscal policy will become permanently more expansionary, and monetary policy permanently more restrictive, then time π‘π‘ stock prices, €πππ‘π‘ , will unambiguously fall. b.1 The statement is correct. Since the question is silent about possible changes in the monetary policy stance, we may assume that at π‘π‘ + 2 the central bank will keep the interest rate at the level prevailing before the increase in ππ0 . The only effect of the change in autonomous consumption will therefore be an increase in income at time π‘π‘ + 2, and in the dividends expected for that period. It follows that €πππ‘π‘ will rise. b.2 False. Without additional information on the size of the two policy interventions and on the characteristics of the economy, is not possible to conclude anything definite about the direction in which €πππ‘π‘ will change. In fact, the monetary contraction will lead to higher future interest rates, something that tends to lower €πππ‘π‘ (in fact, higher future expected interest rates decrease the present value of any given flow of dividends expected in the future). However, one must also take into account how the announced policy-mix will affect future income levels and dividends. The monetary contraction will lower future incomes, while the fiscal expansion will raise them. If to prevail is this latter effect, and if the increase in future incomes (and therefore dividends) is large enough, stock prices today could even increase; if, on the other hand, future incomes do not increase enough, or if they fall because the adverse impact on economic activity of the monetary contraction prevails on that of the fiscal expansion, €πππ‘π‘ would fall. 333 Macroeconomics. Problems and Questions Question 6 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. If, because of new information that has become available, at time π‘π‘ people begin to expect a future decrease in the autonomous component of investment, πΌπΌ ,Μ then time π‘π‘ stock prices €πππ‘π‘ will certainly rise. On the other hand, the direction in which €πππ‘π‘ will vary is uncertain if, in addition to a future decrease in πΌπΌ ,Μ individuals also expect that the government of the country will intervene to keep future income constant [Hint: when answering, assume that, in each time period, the economy is described by a static IS-LM model, with consumption and investment that depend on contemporaneous variables only]. False. Remember that stock prices, €πππ‘π‘ , can be written as the present value of the flow of future dividends, computed using the current and future expected short-term interest rates as discount rates. Taking this into account, it follows that the statement is incorrect. In fact, if πΌπΌ Μ falls, future incomes will be lower. Individuals will therefore expect lower future dividens, and this will tend to decrease €πππ‘π‘ . Since the queston does not mention anything about future changes in the monetary policy stance, we may assume that the central bank will keep future interest rates at the level at which they were set before the change in πΌπΌ .Μ Therefore, in the formula of the fundamental value of a stock, the only terms that will change are the numerators of the ratios appearing on the right-hand side, which will fall. It follows that, rather than rising, as stated in the question, stock prices will fall, too. 334 Expectation, financial markets, and economic policies b. If investment does not depend on the interest rate, the announcement (unexpected by individuals) that, from time π‘π‘ onwards, monetary policy will become permanently more expansionary does not lead to any change in stock prices at time π‘π‘ [Hint: when answering, assume as before that, in each time period, the economy is described by the same static IS-LM model considered in the previous point of this question]. False. Stock prices at time π‘π‘, €πππ‘π‘ , are equal to the present value of expected future dividends. If demand for investment does not depend on the interest rate, the IS curve is vertical in every period, and an expansionary monetary policy will not change either current income, nor expected future income levels. Although, for this reason, expected future dividends will remain constant, the policy we are considering will nevertheless lead to a reduction in interest rates at time π‘π‘ and in all future periods, and therefore to a decrease in the rate at which those (constant) expected future dividends are discounted. It follows that stock prices at time π‘π‘ will rise. 335 Macroeconomics. Problems and Questions * Question 7 [Intertemporal consumption choices – The microfoundations of the 'theory of the very farsighted consumer'] There are only two time periods, π‘π‘ (the 'present') and π‘π‘ + 1 (the 'future'). Make the following assumptions on the individuals populating the economy: • their preferences are described by the utility function ππ(πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 ), increasing and concave in its two arguments – present consumption, πΆπΆπ‘π‘ , and future consumption, πΆπΆπ‘π‘+1 ; • they have access to a credit market, where they can borrow and lend at the real interest rate ππ; • they earn a disposable (labor) income equal to (πππ‘π‘ − πππ‘π‘ ) in the first peππ ππ ) − πππ‘π‘+1 riod, and expect to earn one equal to (πππ‘π‘+1 in the second period. a. Suppose that, at time π‘π‘, the typical individual's financial wealth (stocks, bonds, etc.) and housing wealth (apartments, commercial buildings, etc.) are both zero. Write down the individual's budget constraints for each of the two periods, combine them into a single 'intertemporal budget constraint' and represent in the (πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 ) plane the problem the individual has to solve in order to maximize his utility subject to such constraint. Where is the optimal 'present consumption/future consumption' program located? What determines whether, in the first period, the individual will be a borrower, a lender, or rather consume exactly his disposable income? The budget constraints we are asked to write are: (πππ‘π‘ − πππ‘π‘ ) = πΆπΆπ‘π‘ + ππ ππ ππ ) (πππ‘π‘+1 − πππ‘π‘+1 + (1 + ππ)ππ = πΆπΆπ‘π‘+1 The first one refers to the first period, and tells us that, at time π‘π‘, there are two possible uses of the individual's disposable income – consumption (πΆπΆπ‘π‘ ), and saving (ππ). Notice that, depending on whether in the first period the individual consumes less than (πππ‘π‘ − πππ‘π‘ ), more than (πππ‘π‘ − πππ‘π‘ ), or exactly (πππ‘π‘ − πππ‘π‘ ), his saving ππ will be positive, negative, or equal to zero. The second constraint states that, at π‘π‘ + 1, disposable income in that period, augmented by what the individual will receive for the fact that he was a lender at π‘π‘ (that is, if ππ > 0, augmented by the repayment of the sum he has lent out, 336 Expectation, financial markets, and economic policies 1 ππ, plus interest ππππ on that lending), or reduced by the same amount, if at π‘π‘ he was a borrower (ππ < 0), is going to be entirely consumed.1 Solving the first constraint for ππ, plugging the result into the second one, and rearranging terms, it is possible to combine the two previous constraints into the following "intertemporal budget constraint": ππ ππ ) (πππ‘π‘+1 πΆπΆπ‘π‘+1 − πππ‘π‘+1 = (πππ‘π‘ − πππ‘π‘ ) + . 1 + ππ 1 + ππ The right-hand side − the present value of the individual's disposable labor income − is sometimes referred to as "human wealth". Since we shall draw the above constraint in the (πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 ) plane, it is convenient to solve it for πΆπΆπ‘π‘+1: πΆπΆπ‘π‘ + ππ ππ )] − (1 + ππ)πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 = [(1 + ππ)(πππ‘π‘ − πππ‘π‘ ) + (πππ‘π‘+1 − πππ‘π‘+1 which is the equation of the straight line in the figure, having a positive vertical intercept (given by the term in squared brackets) and a negative slope −(1 + ππ). Notice that the horizontal intercept of the constraint (the value it implies for πΆπΆπ‘π‘ when πΆπΆπ‘π‘+1 = 0) is given by what we have just termed 'human wealth'. πΆπΆπ‘π‘+1 ππ ππ ) (πππ‘π‘+1 − πππ‘π‘+1 π΄π΄ • πΈπΈ • ∗ πΆπΆπ‘π‘+1 • π΅π΅ • 1 (πππ‘π‘ − πππ‘π‘ ) πΆπΆπ‘π‘∗ (πππ‘π‘ − πππ‘π‘ ) + ππ ππ ) (πππ‘π‘+1 − πππ‘π‘+1 1 + ππ πΆπΆπ‘π‘ Why, on the right-hand side of the budget constraint for time t + 1, there is no term analogous to the term ππ appearing in the constraint for time π‘π‘? The reason is that t + 1 is the second, and last, period of the model (or, if you want, of the individual's lifetime). At t + 1, the individual would like to borrow as much as possible, since this loan will come due only at an inexistent time t + 2, and therefore never. But, exactly because of this, he will find no one willing to lend. In the last period, the individual will therefore consume all the resources that, also as a result of the choices made in the first one, he has available, nothing more and nothing less. 337 Macroeconomics. Problems and Questions ππ ππ )] The budget constraint goes always through the point [(πππ‘π‘ − πππ‘π‘ ), (πππ‘π‘+1 − πππ‘π‘+1 − a point that, in the figure, we initially assume to be π΄π΄. In fact, to consume in each period exactly his contemporaneous disposable income is a strategy always at the individual's reach (albeit certainly not the only, nor necessarily the optimal, one). In the same graph, we have drawn some of the indifference curves corresponding to the utility function ππ(πΆπΆπ‘π‘ , πΆπΆπ‘π‘+1 ). As Microeconomics allows us to conclude, the optimal choice is given by point πΈπΈ in the figure, where the budget constraint is tangent to an indifference curve, and by the as∗ . Nosociated levels of present consumption πΆπΆπ‘π‘∗ and future consumption πΆπΆπ‘π‘+1 tice that, given the way we have drawn the figure, the individual will find it optimal to borrow today [πΆπΆπ‘π‘∗ > (πππ‘π‘ − πππ‘π‘ )], and then repay this loan tomorrow ππ ππ )]. ∗ < (πππ‘π‘+1 − πππ‘π‘+1 If, however, for given preferences and slope of the [πΆπΆπ‘π‘+1 budget constraint, the point representing the sequence of disposable incomes had been π΅π΅, rather than A, to maximize his utility the individual would have chosen to be a lender at time π‘π‘. Similarly, and this time for given disposable incomes in the two periods, the fact that the individual will be a lender rather than a borrower also depends on the value of ππ, as well as on the individual's preferences (for instance, on how steep are his indifference curves). Having said that, it is important to realize that the fact that the individual will choose a positive, a negative, or a zero value for ππ in the first period does not change any of the conclusions that will be reached when answering the next point of this question, or the properties of the consumption function we are about to derive. b. Assuming that both πΆπΆπ‘π‘ and πΆπΆπ‘π‘+1 are 'normal goods', explain how the optimal choice of present consumption will be affected by the following changes: • • • a transitory increase in current disposable income [π₯π₯(πππ‘π‘ − πππ‘π‘ ) > 0 and ππ ππ ) − πππ‘π‘+1 = 0] π₯π₯(πππ‘π‘+1 an increase in future expected disposable income [π₯π₯(πππ‘π‘ − πππ‘π‘ ) = 0 and ππ ππ ) − πππ‘π‘+1 > 0] ; π₯π₯(πππ‘π‘+1 ππ ππ ) − πππ‘π‘+1 > a permanent increase in disposable income [π₯π₯(πππ‘π‘ − πππ‘π‘ ) = π₯π₯(πππ‘π‘+1 0]. What can you conclude about the consumption function implied by the analysis of the individual's intertemporal consumption choices? All the changes we are asked to investigate raise the individual's human wealth (the present value of his disposable labor incomes), causing a rightward shift in his intertemporal budget constraint. Being πΆπΆπ‘π‘ and πΆπΆπ‘π‘+1 normal goods, this will induce the consumer to raise both present and future consumption levels. 338 Expectation, financial markets, and economic policies We can therefore conclude that time π‘π‘ consumption depends positively on human wealth, πΆπΆπ‘π‘ = πΆπΆ(βπ’π’π’π’π’π’π’π’ π€π€π€π€π€π€π€π€π€π€βπ‘π‘ ). (+) Needless to say, human wealth, and therefore current consumption, will rise more if, rather than transitory (limited to the current period), the increase in current disposable income is permanent (that is, expected to last indefinitely, thus leading to an upward revision of expected future income, too). Furthermore, πΆπΆπ‘π‘ rises also if, for a given disposable income today, individuals start expecting a higher disposable income tomorrow (in fact, today's consumption depends on the present value of disposable income, which rises also when, for a given current income, future income goes up). Finally, and even though the real interest rate enters the expression of human wealth, in what follows we shall not include ππ among the variables on which πΆπΆπ‘π‘ depends, and write current consumption as a function of current and future expected disposabe income levels only, ππ ππ ). πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 − πποΏ½π‘π‘+1 (+) (+) The reason for this choice is that changes in ππ affect πΆπΆπ‘π‘ through several channels, often pushing current consumption in opposite directions (as evident from the fact that, in the figure, changes in the real interest rate change not only the intercept, but also the slope of the budget constraint). We shall therefore capture the impact of changes in ππ on aggregate demand through the effect these changes have on investment, rather than through the ambiguous impact they have on consumption. c. How does the consumption function you have just derived change if, at time π‘π‘, individuals have a positive − rather than zero, as assumed so far − nonhuman wealth (ππππππ), defined as the sum of financial and of housing wealth? In this case, the term πππππππ‘π‘ must be added to the left-hand side (resources) of the individual's budget constraint for time π‘π‘, and – as it is straightforward to conclude – to human wealth in the expression of the horizontal intercept of the intertemporal budget constraint. This latter intercept now equals the sum of human and nonhuman wealth, a sum also called 'total wealth'. If total wealth rises, because of an increase in human and/or in financial and housing wealth, the budget constraint will shift to the right, and today's consumption will go up. It follows that the consumption function now becomes: πΆπΆπ‘π‘ = πΆπΆ(π‘π‘π‘π‘π‘π‘π‘π‘π‘π‘ π€π€π€π€π€π€π€π€π€π€βπ‘π‘ ). (+) 339 Macroeconomics. Problems and Questions Question 8 There are only two periods, period π‘π‘ (“the present”) and period π‘π‘ + 1 (“the future”). In the attempt to boost current economic activity, the government of country Delta cuts by 100 the taxes each individual will have to pay in the current period (time π‘π‘). At the same time, the government however announces that future taxes will be increased by the same amount. In other words, denoting per-capita net taxes by ππ, the fiscal policy intervention just described can ππ = +100. be summarized as follows: βπππ‘π‘ = −100, βπππ‘π‘+1 = βπππ‘π‘+1 a. Suppose that all individuals are identical and that, at the beginning of time π‘π‘, their non-human wealth is zero. Moreover, assume that ππ = 0, where ππ is the real interest rate. Using the analysis of the intertemporal consumption choices, evaluate βπππ‘π‘ and βπππ‘π‘+1 − that is, the changes in current and future consumption levels of the typical inhabitant of country Delta caused by the intertemporal reallocation of taxes implemented by the government. Will the government be successful in its attempt to increase consumption, and hence aggregate demand and equilibrium production, at time π‘π‘? If yes, indicate the fraction of the tax cut by which time π‘π‘ consumption of the typical individual will rise. If no, why? In motivating your answer, make explicit reference to the way in which savings by the typical individual will change at time π‘π‘. Since individuals have zero non-human wealth at the beginning of period π‘π‘, their total wealth equals their human wealth, which − being ππ = 0 − is just the sum of their current and future expected disposable income levels, (πππ‘π‘ − ππ ππ ). − πππ‘π‘+1 πππ‘π‘ ) + (πππ‘π‘+1 ππ It follows that, since βπππ‘π‘ = −βπππ‘π‘+1 , the fiscal policy under consideration will not change the present value of disposable income. The individuals' budget constraints will remain in their initial position, and their intertemporal consumption choices will not change. But, if consumption at time π‘π‘ does not change, aggregate demand will not change either, and the policy under consideration will not affect economic activity. 340 Expectation, financial markets, and economic policies Notice that, at time π‘π‘, the tax cut raises disposable income. Since individuals do not change their consumption choices, this means that they will save the whole increase in disposable income brought about by the tax cut at time π‘π‘. Next period, individuals will use these extra savings to pay the higher taxes due at time π‘π‘ + 1, and this will allow them to keep time π‘π‘ + 1 consumption unchanged. b. Assume now that some individuals are subject to ‘liquidity constraints’. Typically, an individual who is facing a binding liquidity constraint consumes less than he would like to, given the present value of his disposable income. This may be due to imperfections in the financial markets, as those leading to situations in which some individuals simply have no means of getting credit, so that their consumption in each period cannot exceed their current disposable income. Does the existence of such constraints induce you to change your answer to the previous point? Why? Explain. The tax cut implemented by the government raises individuals’ disposable in........................................................................................................................... come at time π‘π‘, thus allowing those who are facing liquidity constraints to increase their consumption at π‘π‘, bringing it closer to their preferred level of con........................................................................................................................... sumption for that period. In this case, and even if the present value of their ........................................................................................................................... disposable income does not change, time π‘π‘ consumption by liquidityconstrained individuals will rise. It follows that aggregate demand and the ........................................................................................................................... equilibrium level of income will increase, too. 341 Macroeconomics. Problems and Questions Question 9 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. Suppose that today (time π‘π‘) there is a fall in house prices. The consumption function based on the analysis of the intertemporal consumption decisions implies that time π‘π‘ consumption will rise – since houses are now less expensive than before, individuals will need to save less to purchase one, and this will allow them to consume more. False. The analysis of the intertemporal consumption decisions implies that consumption is increasing in individuals’ total wealth. Since total wealth is the sum of human wealth and of financial and housing wealth, and since a fall in house prices amounts to a decrease in individuals’ housing wealth, current consumption will fall. 342 Expectation, financial markets, and economic policies b. Today (time π‘π‘), stock prices fall. The consumption function based on the analysis of the intertemporal consumption decisions implies that, at time π‘π‘, households will cut their consumption expenditures. True. A fall in stock prices reduces individuals’ financial wealth and therefore, as it is clear from the answer to the previous point, households’ current consumption expenditures. Other things the same, it follows that a drop in stock prices leads to lower aggregate demand and equilibrium income. 343 Macroeconomics. Problems and Questions Question 10 In country Macro, current and future expected inflation are equal to zero, so that nominal and real interest rates are the same. At time π‘π‘ + 1 (‘the future’), consumption depends on π‘π‘ + 1 disposable income only, while investment is enΜ . As for the current period, (π‘π‘, ‘the present’), contirely exogenous, πΌπΌπ‘π‘+1 = πΌπΌπ‘π‘+1 sumption is increasing both in the current and in the future expected levels of ππ ππ − πποΏ½π‘π‘+1 ), investment is exogenous, πΌπΌπ‘π‘ = disposable income, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 πΌπΌπ‘π‘Μ , πΊπΊ and ππ are as usual exogenous in both periods, and the central bank chooses a level for the interest rate in both periods. a. Starting from an initial equilibrium position, the central bank announces today (time π‘π‘) that it will cut, both in the current and in the future period, the policy rate. Discuss, and show in the graph below, the effects of these announcements on the current (time π‘π‘) levels of income and interest rate. How will the time π‘π‘ yield on two-year bonds, ππ2π‘π‘ , change? Explain. ππ ISO 0 π€π€Μ 0 π€π€Μ 1 1 πποΏ½0 344 LM0 LM1 ππ Expectation, financial markets, and economic policies Since no component of aggregate demand depends on the interest rate, implying a vertical IS curve in both periods, the permanent monetary expansion implemented by the central bank will only lead to lower current and future interest rates. Today, the IS curve will stay put, the LM curve will shift down, and the equilibrium, initially point 0 in the graph, will become point 1. Being the average of the current and of the future expected short rates, and given that both rates go down, ππ2π‘π‘ will fall. b. Suppose that only one-year bonds and two-year bonds exist and that, before the announcement described above, at time π‘π‘ the yield curve of Macro was flat. Draw the new yield curve that, at time π‘π‘, will prevail in each of the following three cases: (i) the central bank has announced that it will cut the future short-rate rate less than the current one; (ii) it has announced that it will proceed to an equal cut in current and future rates; (iii) it has announced that it will cut the future short-rate rate more than the current one. Explain. Yield to maturity “before” (i) (ii) “after” (iii) 1 2 Maturity Given that both the current one-year rate and the current two-year one (ππ2π‘π‘ ) go down, the new yield curve will lie below the initial one. Since π₯π₯π₯π₯2π‘π‘ = 1 ππ οΏ½π₯π₯ππ1π‘π‘ + π₯π₯ππ1,π‘π‘+1 οΏ½ − or, if you prefer, recalling that a positively (negatively) 2 sloped yield curve signals expectations of short-term rates rising (falling) over time, and a flat one expectations of short rates constant over time −, it is straightforward to conclude that, in each of the three cases considered in the question, the position in the plane and the slope of the yield curve will be those in the figure. 345 Macroeconomics. Problems and Questions Question 11 a. In country Macrolandia, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), a standard IS-LM model describes the functioning of the economy. In particular, in the year π‘π‘ + 1 consumption depends only on disposable income at π‘π‘ + 1, and investment on the interest rate and output at π‘π‘ + 1 only. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income, πΆπΆπ‘π‘ = ππ ππ πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 − πποΏ½π‘π‘+1 ), investment depend positively on current and expected future income levels, and negatively on the current and expected future levels of short-term interest rates, the demand for money has the usual functional form, and πΊπΊ and ππ are exogenous as usual. Starting from an initial equilibrium, at time π‘π‘ the government of Macrolandia announces an increase of its purchases of goods and services at time π‘π‘ + 1, βπΊπΊΜ π‘π‘+1 > 0. At the same time, the central bank announces that it will adjust the policy rate so as to prevent any change in equilibrium income that, at time π‘π‘ and/or at time π‘π‘ + 1, could be caused by the fiscal policy just described. Explain how these announcements, which are unexpected and credible, affect the time π‘π‘ yield to maturity on the two-year bonds circulating in Macrolandia, ππ2π‘π‘ . ........................................................................................................................... The increase in government purchases of goods and services that will be implemented at π‘π‘ + 1 will shift to the right the IS curve for that period. To keep ........................................................................................................................... πππ‘π‘+1 constant, the central bank will have to raise the short-term interest rate, ........................................................................................................................... ππ1π‘π‘+1 . At time π‘π‘, this will lead individuals to expect a higher short-term future interest rate. These expectations will lower time π‘π‘ investment, and will there........................................................................................................................... fore cause a leftward shift of the IS curve for the current period. It follows that, to prevent πππ‘π‘ from falling, the central bank will have to cut the current short-term rate, ππ1π‘π‘ . Since ππ2π‘π‘ is (approximately) equal to the average of the current and the future expected short term interest rates, and since these rates move in opposite directions, one cannot conclude anything definite about the way in which yield to maturity on the two-year bonds at time t will change − it could rise, fall, or remain constant. 346 Expectation, financial markets, and economic policies b. Assume for simplicity that, in Macrolandia, at time π‘π‘ the yield curve is initially horizontal. Explain how the position and the slope of this curve change after the announcement studied above. Represent both curves, the one “before” and the one “after” the announcement, in the graph below. Clarify whether it is possible to determine the position and the slope of the new yield curve unambiguously. Yield to maturity “before” “after” 1 2 Maturity Since ππ1π‘π‘ falls, while the sign of the change in ππ2π‘π‘ is uncertain, the dashed segments in the figure show three new, equally plausible yield curves for the economy in consideration. In particular, and no matter the direction in which ........................................................................................................................... ππ2π‘π‘ will change, the new curve will have a positive slope. Since we know that ........................................................................................................................... ππ1π‘π‘ will be lower, it is easy to understand why the new yield curve will be positively sloped if ππ2π‘π‘ will rise, or remain constant. But what if ππ2π‘π‘ ends up fall........................................................................................................................... 1 ππ ππ > 0 it follows that ing? From π₯π₯π₯π₯2π‘π‘ = οΏ½π₯π₯ππ1π‘π‘ + π₯π₯ππ1,π‘π‘+1 οΏ½, π₯π₯ππ1π‘π‘ < 0 and π₯π₯ππ1,π‘π‘+1 2 ........................................................................................................................... π₯π₯π₯π₯2π‘π‘ − π₯π₯ππ1π‘π‘ > 0. The slope of the yield curve, whose sign is equal to that of the difference (ππ2π‘π‘ − ππ1π‘π‘ ), will therefore rise − going from zero to a positive value − also in this case. [To put it differently, even if it should end up falling, being the average of the current short-term yield, which falls, and the future expected one, which rises, ππ2π‘π‘ will necessarily fall less than ππ1π‘π‘ ]. As usual, the positive slope of the new yield curve signals expectations of short term rates increasing over time (in this case, due to the fall in the current short rate, and the expectation of a rise in the future one). 347 Macroeconomics. Problems and Questions Question 12 a. In country Macrolandia, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), an IS-LM model describes the functioning of the economy. In particular, in the year π‘π‘ + 1 consumption depends only on disposable income at π‘π‘ + 1, and investment on the borrowing rate (the sum of the interest rate and the risk premium π₯π₯π‘π‘+1 ) and an output at π‘π‘ + 1 only. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current ππ ππ and expected future disposable income, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 − πποΏ½π‘π‘+1 ), investment depend positively on current and expected future income levels, and negatively on the current and expected future borrowing rates, the central bank chooses in each period the interest rate, and πΊπΊ and ππ are exogenous as usual. Starting from an initial equilibrium position in both periods, at time π‘π‘ individuals start expecting a decrease in the risk premium for time π‘π‘ + 1, π₯π₯π₯π₯π‘π‘+1 < 0. At the same time, the central bank announces that it will adjust the policy rate so as to prevent any change in equilibrium income that, at time π‘π‘ (and at time π‘π‘ only), could be caused by the change in π₯π₯π‘π‘+1 just described. Explain how these changes and announcements, which are unexpected and credible, affect the time π‘π‘ yield to maturity on the two-year bonds circulating in Macrolandia, ππ2π‘π‘ . At time π‘π‘ + 1, the decrease in the risk premium lowers the borrowing rate, raises investment and shifts to the right the IS curve for that period. It follows that income rises at π‘π‘ + 1. At time π‘π‘, the IS curve shifts to the right both for the increase in the future expected equilibrium income and for the decrease in the borrowing rate for time π‘π‘ + 1. To prevent this from increasing time π‘π‘ equilibrium income, the central bank will have to raise the current policy rate. Since the future expected short rate remains constant, while the current one goes up, the current long rate ππ2π‘π‘ increases. 348 Expectation, financial markets, and economic policies b. Assume for simplicity that, in Macrolandia, at time π‘π‘ the yield curve is initially horizontal. Explain how the position and the slope of this curve change after the change and the announcement studied above. Represent both curves, the one “before” and the one “after”, in the graph below. Clarify whether it is possible to determine the position and the slope of the new yield curve unambiguously. Yield to maturity “after” “before” 2 1 Maturity Given that both the current short rate (ππ1π‘π‘ ) and the current long one (ππ2π‘π‘ ) go up, the new yield curve will lie above the initial one. Since π₯π₯π₯π₯2π‘π‘ = 1 ππ ππ = 0, the increase in the current long rate οΏ½π₯π₯ππ1π‘π‘ + π₯π₯ππ1,π‘π‘+1 οΏ½, and being π₯π₯ππ1,π‘π‘+1 2 will be half that in short one, so that the new curve will be negatively sloped. 349 Macroeconomics. Problems and Questions Question 13 In country Gamma, current and future expected inflation rates coincide and are equal to zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), consumption depends only on disposable income Μ + ππ1 πππ‘π‘+1 − ππ2 πππ‘π‘+1 , where symbols have at π‘π‘ + 1, and investment is πΌπΌπ‘π‘+1 = πΌπΌπ‘π‘+1 the usual meaning. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income, ππ ππ − πποΏ½π‘π‘+1 ), while investment is increasing in current and exπΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − πποΏ½π‘π‘ , πππ‘π‘+1 pected future income levels, and decreasing in the current and expected future levels of short-term interest rates. Finally, πΊπΊ and ππ are exogenous in both periods. a. At time π‘π‘, individuals revise downwards their expectation about the value that the autonomous component of investment will take on in the future. Μ will drop to πΌπΌ ′Μ π‘π‘+1, with More specifically, they start expecting that πΌπΌπ‘π‘+1 Μ . At the same time, the central bank announces that, in the fuπΌπΌ ′Μ π‘π‘+1 < πΌπΌπ‘π‘+1 ture, it will implement a monetary policy aimed at preventing any change in time π‘π‘ + 1 equilibrium output that could be caused by the expected drop in autonomous investment in that period. To achieve its goal, what kind of monetary policy (expansionary? restrictive?) should the central bank implement? Describe, and represent in the graph below, the effects of the two announcements on Gamma’s current (that is to say, time π‘π‘) income and interest rate equilibrium levels. πππ‘π‘ 0 1 πΏπΏπΏπΏπ‘π‘ πΌπΌπΌπΌπ‘π‘0 πΌπΌπΌπΌπ‘π‘1 πππ‘π‘ 350 Expectation, financial markets, and economic policies To offset the impact of the fall in autonomous investment on time π‘π‘ + 1 equilibrium output, the central bank will have to implement an expansionary monetary policy in that period − for instance, a purchase of bonds in the open market, that will lead to a drop in the interest rate at time π‘π‘ + 1. In the future, the IS curve will therefore shift to the left, and the LM curve downwards, with the two curve crossing for the same level of output as the one prevailing before the fall investment, but for a lower interest rate. At time π‘π‘, the expectation of a lower interest rate in the future will boost investment, thus shifting the IS curve to the right. In the new equilibrium, if the central bank keeps the current interest rate at the level it had initially chosen, current output will be higher, and the interest rate unchanged. b. Suppose that only one-year bonds and two-year bonds exist and that, before the announcements described above, at time π‘π‘ the yield curve was flat. What will be the effects of the two announcements on the position and the slope of the yield curve of Gamma? The current one-year yield does not change, while the future expected one 1 ππ falls. Since ππ2π‘π‘ = 2 οΏ½ππ1π‘π‘ + ππ1,π‘π‘+1 οΏ½, the time π‘π‘ yield to maturity on two-year 1 ππ ). The changes bonds will drop, too (more specifically, it will fall by π₯π₯ππ1,π‘π‘+1 2 analyzed in the previous point of this question will therefore lead to a negatively sloped yield curve. 351 Macroeconomics. Problems and Questions Question 14 a. In country Alpha, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), a standard IS-LM model describes the functioning of the economy. In particular, in year π‘π‘ + 1 consumption depends only on disposable income at π‘π‘ + 1, and investment on the interest rate and output at π‘π‘ + 1 only. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income levels, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − ππ ππ − πποΏ½π‘π‘+1 ), investment is exogenous, πΌπΌπ‘π‘ = πΌπΌ,Μ and the same is true πποΏ½π‘π‘ , πππ‘π‘+1 about πΊπΊ and ππ. Rather than the interest rate, Alpha's central bank chooses a value for the nominal money supply ππ and, given this level at which ππ is set, lets the interest rate free to take on any value is consistent with the macroeconomic equilibrium. It follows that, in the (ππ, ππ) plane, the LM curve of Alpha is the curve discussed in Question 2 − and, for the case of an economy in a liquidity trap, in Question 10 − of Chapter 2. Finally, in the initial equilibrium, time π‘π‘ + 1 income and interest rates levels are both positive, while at time π‘π‘ the economy is in a liquidity trap. Suppose now that, at time π‘π‘, the central bank announces that it will decrease nominal money supply at π‘π‘ + 1. Explain how this announcement, which is unexpected and credible, affects the time π‘π‘ yield to maturity on the two-year bonds circulating in Alpha, ππ2π‘π‘ . ......................................................................................................................... The decrease in money supply announced for π‘π‘ + 1 will shift the LM for that year to the left. This will lead to a new equilibrium in which income πππ‘π‘+1 is ........................................................................................................................... lower, and the short-term interest rate ππ1π‘π‘+1 higher. At time π‘π‘, the fall in ex........................................................................................................................... pected future income will decrease consumption, the IS will shift to the left, ........................................................................................................................... and the economy will converge to an equilibrium in which πππ‘π‘ is lower and ππ1π‘π‘ unchanged (since at time π‘π‘ the economy is in a liquidity trap, the intersection ........................................................................................................................... between the IS and the LM curves will still take place in the horizontal por........................................................................................................................... tion of the latter curve, even after the leftward shift in the IS). Since ππ2π‘π‘ is (approximately) equal to the average of the current and the future expected ........................................................................................................................... ππ rises and ππ1π‘π‘ does not change, the short term interest rates, and since ππ1π‘π‘+1 yield to maturity of the two-year bonds at time t will definitely rise. 352 Expectation, financial markets, and economic policies b. Draw the yield curve prevailing at time π‘π‘ before the central bank’s announcement, and explain how this latter will affect the yield curve of Alpha. In particular, draw the new yield curve in the same graph, and explain if it is possible to determine without ambiguity how its slope and position will compare to those of the original curve. Yield to maturity “after” “before” 1 2 Maturity ππ Since we know that, before the announcement, ππ1π‘π‘ = 0 and ππ1π‘π‘+1 > 0, initially the slope of the yield curve is positive. After the announcement, ππ1π‘π‘ remains equal to zero, while ππ2π‘π‘ rises. It follows that the new curve will still be positively sloped, but steeper than the initial one. 353 Macroeconomics. Problems and Questions Question 15 In country ABC, there are only one-year and two-year bonds. The current and the expected future inflation rates are both zero, so that the real and the nominal interest rates are equal. In period π‘π‘ + 1 (“the future”), investment is entirely exogenous, while consumption depends only on disposable income at π‘π‘ + 1. Turning now to the current period (π‘π‘, “the present”), consumption is increasing in both current and expected future disposable income, πΆπΆπ‘π‘ = πΆπΆ(πππ‘π‘ − ππ ππ − πποΏ½π‘π‘+1 ), and investment is once again exogenous. In both periods, the πποΏ½π‘π‘ , πππ‘π‘+1 demand for money has the usual functional form, and πΊπΊ and ππ are exogenous as usual. Starting from an initial equilibrium, at time π‘π‘ the government of Macrolandia announces an increase in net taxes for time π‘π‘ + 1, βπποΏ½π‘π‘+1 > 0. a. Assuming that, in both periods, ABC’s central bank chooses the interest rate, describe and represent in the two graphs below the effects of this fiscal policy announcement on income and the interest rate levels prevailing at time π‘π‘ and at time π‘π‘ + 1. How will the time π‘π‘ yield to maturity on the two-year bonds circulating in Macrolandia, ππ2π‘π‘ , change? Why? ππ1π‘π‘ π€π€1π‘π‘ οΏ½οΏ½οΏ½ πΌπΌπΌπΌπ‘π‘2 πΌπΌπΌπΌπ‘π‘1 1 2 t ππ1π‘π‘+1 πΏπΏπΏπΏπ‘π‘ π€π€1π‘π‘+1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ πππ‘π‘ 2 πΌπΌπΌπΌπ‘π‘+1 1 πΌπΌπΌπΌπ‘π‘+1 1 2 t+1 πΏπΏπΏπΏπ‘π‘+1 πππ‘π‘+1 At time π‘π‘ + 1, the rise in πποΏ½ will shift the πΌπΌπΌπΌ – a curve that is vertical in both periods, as no component of aggregate demand depends on the interest rate − to the left, thus leading to a lower equilibrium income. At time π‘π‘, the ensuing decrease in expected future income and the expectation of higher future taxes will lower consumption, causing a leftward shift in the IS curve. It follows that production will fall in both periods, and that – given the assumption about the way the central bank behaves – both the current and the future expected short-term rates will remain unchanged. Being (approximately) equal to the ππ average of ππ1π‘π‘ and ππ1π‘π‘+1 , and that these rates remain constant, ππ2π‘π‘ will remain unchanged, too. 354 Expectation, financial markets, and economic policies b. Repeat the same analysis carried out when answering the previous point assuming that, rather than the interest rate, the central bank of the country chooses the money supply. Represent in the graphs below the effects of the same fiscal policy announcement considered before and explain how will ππ2π‘π‘ change in this case, and why. ππ1π‘π‘ 2 πΌπΌπΌπΌπ‘π‘2 1 πΌπΌπΌπΌπ‘π‘1 πΏπΏπΏπΏπ‘π‘ ππ1π‘π‘+1 2 1 2 1 πΌπΌπΌπΌπ‘π‘+1 πΌπΌπΌπΌπ‘π‘+1 πππ‘π‘ t+1 πΏπΏπΏπΏπ‘π‘+1 πππ‘π‘+1 The πΏπΏπΏπΏ curve is now positively sloped in both periods. The increase in πποΏ½ announced for time π‘π‘ + 1 will lead to a leftward shift of the πΌπΌπΌπΌ for that period that will now cause a fall not just in income, but also in the future interest rate (the decrease in income will lower money demand; since now the central bank keeps ππ constant at the chosen level, the interest rate for which the money market is in equilibrium goes down). At time π‘π‘, the expectation of a lower future income and higher future taxes decreases consumption, shifting the IS to the left. With lower current and future expected short rates, ππ2π‘π‘ (which is just ππ ) will fall. the average of ππ1π‘π‘ and ππ1π‘π‘+1 355 Chapter 5 - The open economy Macroeconomics. Problems and Questions Question 1 a. Country Iota’s Statistical Office announces that, between year π‘π‘ and year π‘π‘ + 1, gross investment has increased by €100bn, while public saving has fallen by €50bn. a.1 Knowing that Iota is a closed economy, by how much has private saving changed between the same two years? a.2 How would your answer change, were Iota an economy open to trade in goods, services and financial assets with the rest of the world? a.1 In a closed economy, in goods market equilibrium national saving − the sum of private savings and public savings − is equal to investment. If the latter has increased by €100bn, and public savings has decreased by €50bn, private savings will necessarily have increased by €150bn. a.2 In an open economy, in goods market equilibrium national saving is equal to the sum of investment and the trade balance (NX). Since we do not know what has happened to net exports NX, in this case it is not possible to say anything definite about the change in private savings. 358 The open economy b. Explain, briefly but rigorously, what is meant by ‘J-curve’ [in your answer, make explicit reference to the so-called ‘Marshall-Lerner condition’]. “J-curve”: graphical representation of the response of the trade balance to a real depreciation (or, more generally, to a change in the real exchange rate). When this response follows the J-curve, a depreciation initially worsens the trade balance. This happens because, immediately after the depreciation, the quantities exported and imported change little, while each unit of goods imported from abroad is now more expensive in terms of domestic goods (in other words, in the short run the Marshall-Lerner condition may not hold). The trade balance starts improving only after some time, when the quantities of goods exported and imported begin to adjust to the change in relative prices, and the Marshall-Lerner condition will eventually hold (as it empirically does, at least from the medium to long run, for most countries and time periods). 359 Macroeconomics. Problems and Questions Question 2 Consider an economy open to foreign trade, where only the goods market exists and prices are constant. Assume that foreign trade is initially balanced. a. Suppose the government decides to cut net taxes, πποΏ½. In the two-panel diagram that allows one to analyze simultaneously the determination of the goods market equilibrium and the trade balance associated with that equilibrium position, represent the effects of this fiscal policy decision on equilibrium income and net exports. In the graph, denote by πποΏ½ the initial level of equilibrium income, and by πποΏ½′ its new level. Why do equilibrium income and net exports change? Explain. π·π·π·π·′ 45° Z π·π·π·π· ππππ′ ππππ πποΏ½ πποΏ½′ NX πποΏ½′′ Trade deficit 360 ππ ππ The open economy By increasing disposable income, a tax cut leads to higher consumption at any given level of income. The increase in C shifts upwards, and by the same amount, both the ZZ curve and the DD curve. It follows that the intersection between the two curves will take place for the same level of income as before the change in taxes, and that, in the lower panel, the NX curve does not move. The equilibrium level of income (that is, that value of ππ for which the ZZ curve crosses the 45-degree line) increases. Since net exports were initially zero, by raising the demand for imports this increase in income leads to a trade deficit. b. In the same graph used above, find the equilibrium level of income that would have been reached had the same tax cut been implemented in a closed economy. Denote this new, closed-economy equilibrium income by πποΏ½′′. Will πποΏ½′′ be less than, equal to, or greater than πποΏ½′? Why? Were the economy closed, the new equilibrium income would be the one corresponding to the intersection between the DD curve and the 45-degree line, ra........................................................................................................................... ther than between this latter line and the ZZ curve. In the graph, the new ........................................................................................................................... equilibrium income would therefore be πποΏ½′′, greater than πποΏ½′. It follows that the effect of a tax cut on income will be greater in a closed economy than in an ........................................................................................................................... open economy. The reason is that, while in a closed economy the increase in ........................................................................................................................... demand due to the decrease in taxes falls entirely on domestic goods, in an open economy part of the additional demand falls on foreign goods, thus rais........................................................................................................................... ing foreign income, rather than domestic income. For this reason, the multi........................................................................................................................... plier is smaller in an open economy than in a closed economy. 361 Macroeconomics. Problems and Questions Question 3 a. Consider a country that is open to trade in goods and services with the rest of the world, where prices are fixed and in which only the goods market exists. Initially, the country is in goods market equilibrium with a trade surplus. Draw in the two-panel diagram below the initial equilibrium position, and show how it changes following a reduction in the domestic price level, ππ [Hint: assume that, after this reduction, ππ remains forever constant at its new, lower level, so that the economy is still described by a model with fixed prices]. In particular, explain if and why income and net exports will change in the new equilibrium. 45° Z 0 NX 1 β¦ β¦β¦ πποΏ½0 πποΏ½1 A B A’ B’ ππππππ β¦ ′ ππππππ DD ππ ππ NX 362 ZZ’ ZZ NX’ The open economy A decrease in ππ leads to a fall in the real exchange rate ππ = πΈπΈπΈπΈ⁄ππ∗ (a real depreciation), a change that increases the price-competitiveness of domestically produced goods. The ZZ curve shifts up, and the NX curve does the same (since, for any given ππ, net exports are now higher). This increase in net exports raises aggregate demand and equilibrium income. As the graph clearly shows, although in the new equilibrium ππ has increased, the trade surplus is larger (the impact of the real depreciation, which tends to increase net exports, prevails on that coming from the increase in ππ, which tends to lower them). b. How would your answer to the previous point of this question change, if the quantities of goods and services exported and imported by the country did not depend on the real exchange rate, but only on foreign and domestic income levels, respectively? Explain. ........................................................................................................................... In this case, net exports are given by the following expression: ........................................................................................................................... 1 ππππ = ππ(ππ ∗ ) − πΌπΌπΌπΌ(ππ), ........................................................................................................................... ππ increasing in the real exchange rate ππ. In this economy, the real depreciation brought about by the fall in ππ lowers net exports, since it does not change ππ or πΌπΌπΌπΌ, but only causes an increase in the price of each unit of goods imported from abroad in terms of domestic goods. It follows that the Marshall-Lerner condition does not hold, and a fall in ππ shifts the ZZ curve downwards, rather than upwards. In the new equilibrium, income is lower, and the trade balance worsens. 363 Macroeconomics. Problems and Questions Question 4 Consider a country that is open to trade in goods and services with the rest of the world, where the exchange rate and prices are fixed and in which only the goods market exists. In the initial equilibrium, the country has a trade surplus. a. Show in the graph the effects of an exogenous increase in investment on equilibrium income and on the trade balance. Explain. 45° Z 0 NX πποΏ½0 π·π·π·π·′ ππππ′ π·π·π·π· 1 πποΏ½1 ππππ0 ππππ1 ππππ ππππππ ππ ππ 364 The open economy Starting from an initial equilibrium at point 0, with income πποΏ½0 and a trade surplus ππππ0 , an exogenous increase in investment leads to an increase in aggregate demand and equilibrium output. Both the ZZ and the DD curves shift upwards by an amount equal to the exogenous increase in investment (π₯π₯πΌπΌ Μ > 0). Equilibrium income becomes πποΏ½1 in the graph. In turn, this increase in income leads to higher imports, thus worsening the trade balance, that now becomes ππππ1 (< ππππ0 ). [Notice that, had investment increased by more than assumed in the graph, net exports could have become negative.] b. Does the increase in investment change the level of output for which trade is balanced? Why, or why not? ........................................................................................................................... ππππππ does not change, because the ZZ and the DD curves shift upwards in a ........................................................................................................................... parallel fashion by the same amount (π₯π₯πΌπΌ Μ ). It follows that they will keep intersecting for the same ππππππ (the level of income for which foreign trade is bal........................................................................................................................... anced). One can reach the same conclusion by noticing that the NX does not ........................................................................................................................... shift, since − for any given ππ − the change in πΌπΌ Μ does not affect net exports. ........................................................................................................................... 365 Macroeconomics. Problems and Questions Question 5 Eta is an economy open to international trade in goods and services, and in which financial markets do not exist. Assume that prices are constant both in Eta and in the rest of the world. Eta is initially in goods market equilibrium, with πΊπΊΜ = 60, πποΏ½ = 10, ππ = 70, πΌπΌ = 20, where πΊπΊ is government spending on goods and services, ππ net taxes, ππ private saving, and πΌπΌ investment. a. Using the information provided above, represent the initial equilibrium of Eta in the two-panel diagram below, denoting it by ‘1’ [Hint: you do not need to compute the equilibrium level of income; just show clearly where the initial equilibrium position is located in each of the two panels below, and explain why]. Z 2 1 NX 0 ππ1 π·π·π·π·′ 45° π·π·π·π· 1′ ππ1′ 1 ππππ1 ππ2 2 366 ππ ππ ππππ2 ππππ1 ππππ2 ππππ1′ The open economy In an open economy, the goods market equilibrium condition can be written as follows: ππ + (ππ − πΊπΊ) − πΌπΌ = ππππ. Since, in Eta, ππ + (ππ − πΊπΊ) − πΌπΌ = 70 + (10 − 60) − 20 = 0, in the initial equilibrium net exports are therefore equal to zero. It follows that, in the first panel, the ZZ and the DD curves will intersect at a point (1 in the figure) lying on the 45° line, and that equilibrium income πποΏ½1 , will also be the level of income for which trade is balanced. b. Suppose now that Eta’s policy-makers wish to raise equilibrium income, leaving net exports unchanged. Having discussed why resorting to fiscal policy only, or to an exchange rate policy only, does not allow the government to achieve both its objectives, discuss the combination of the two policies which would allow the country to reach its two goals. In the two panels of the graph, denote by ‘2’ the new equilibrium that will be reached following the policy-mix you are suggesting, and explain. An expansionary fiscal policy raises income, but worsens the country’s net exports; a real depreciation increases income, but also improves net exports. To raise ππ and keep net exports equal to zero, policy-makers will have to resort to an appropriate mix of fiscal expansion and real depreciation. In particular, an expansionary fiscal policy will shift up, and by the same amount, the ππππ and the π·π·π·π· curves. They will therefore keep crossing for the same level of production, and the new equilibrium income will become ππ1′ in the figure. However, for this level of output ππππ < 0. To prevent net exports from becoming negative, policy-makers will have to engineer a real depreciation of appropriate size – one that will shift the ZZ curve up, so that it will cross the new DD curve exactly on the 45° line, and that, in the lower panel, will lead to an upward shift in NX, so that it will cross the horizontal axis for the new equilibrium output, ππ2 . As planned, at point 2 ππππ = 0 and equilibrium income is higher. 367 Macroeconomics. Problems and Questions Question 6 Eta is an economy open to international trade in goods and services, and in which financial markets do not exist. Assume that prices are constant both in Eta and in the rest of the world. a. Eta is initially in goods market equilibrium, with πΊπΊΜ = 50, πποΏ½ = 50, ππ = 70, πΌπΌ = 20, where πΊπΊ is government spending on goods and services, ππ net taxes, ππ private saving, and πΌπΌ gross investment. Using this information, represent the initial equilibrium of Eta in the two-panel diagram below [Hint: you do not need to compute the equilibrium level of income; just show clearly where the initial equilibrium position is located in each of the two panels below, and explain why]. Z 0 NX πποΏ½0 ππππππ π΄π΄ π΅π΅ 368 45° 1 πποΏ½1 DD ′ ππππππ ZZ’ ZZ ππ π΄π΄′ π΅π΅′ NX’ NX ππ The open economy In an open economy, the goods market equilibrium condition can also be written as ππ + (ππ − πΊπΊ) − πΌπΌ = ππππ. Since, in this case, ππ + (ππ − πΊπΊ) − πΌπΌ = 70 + (50 − 50) − 20 = 50, in the initial equilibrium Eta runs a trade surplus, ππππ > 0. In the first graph, the ZZ and the DD curves will therefore intersect each other below the 45-degree line, and equilibrium income, πποΏ½0 , will be lower than the income level for which foreign trade is balanced, ππππππ . The magnitude of the initial trade surplus is equal, in the lower panel, to the length of the segment AB. b. Suppose now that real exchange rate ε falls – a real depreciation. Assuming that the Marshall-Lerner condition holds true, represent the effects of this change in the graph above. In particular, explain if it will affect the level of income at which Eta’s foreign trade is balanced, ππππππ , and – if your answer is positive – in what direction ππππππ will change. Finally, explain if and why the trade balance will vary. Since the Marshall-Lerner condition holds true, a fall in ε leads, other things the same, to an improvement in Eta’s net exports. In the upper panel of the figure, the ZZ curve shifts upwards, while the DD does not move. It follows that the two curves will intersect for a higher level of income, and that ππππππ (the income level for which foreign trade is balanced) will rise. One could reach the same conclusion by noticing that, in the lower panel, the NX curve shifts upwards. In the new equilibrium, income is higher, and the trade balance has improved (although by less than it would have done had domestic income, and therefore imports, remained constant). 369 Macroeconomics. Problems and Questions Question 7 Consider a country that is open to trade in goods and services with the rest of the world, where prices are fixed and in which only the goods market exists. Initially, the country is in goods market equilibrium, and trade is balanced. a. Draw in the two-panel diagram below the initial equilibrium position and show how it changes following a reduction in foreign output, ππ ∗ . In particular, explain how and why net exports will have changed in the new equilibrium. Z 1 NX ′ ππππππ πποΏ½1 0 45° DD πποΏ½0 (= ππππππ ) ZZ ZZ’ ππ ππ 370 NX NX’ The open economy ................................................................................................................. A decrease in foreign output leads to a fall in the demand for domestically ......... produced goods. The ZZ curve shifts down, and the same does the NX curve, since net exports are now lower for any given ππ. This fall in net exports low........................................................................................................................... ers aggregate demand and equilibrium output. The lower panel show that, alt........................................................................................................................... hough in the new equilibrium ππ is lower than before, net exports will nevertheless be smaller than in the initial equilibrium – they were initially zero, and ........................................................................................................................... are now negative. ........................................................................................................................... b. Suppose the government wants to return to a situation of balanced trade, keeping however domestic output at the new level reached after the decrease in ππ ∗ . In order to achieve these objectives, economist Paul advocates the use of fiscal policy, while economist Edie thinks that one should necessarily resort to some combination of fiscal and exchange rate policies. Do you agree with Paul (in this case, explain if the appropriate fiscal policy is an expansionary or a contractionary one), or with Edie (if this is the case, describe the specific combination of fiscal policy and exchange rate policy you deem appropriate)? [Hint: just provide the economic intuition behind your answer; you are not requested to carry out any graphical analysis when answering this point.] A contractionary fiscal policy would succeed in improving the trade balance, returning net export to zero. However, it would also lead to a further decrease in domestic output. On the other hand, a real depreciation would improve net exports, but also raise Y. To achieve the two objectives of constant output and balanced trade, one should implement an appropriate combination of contractionary fiscal policy and real exchange rate depreciation. Edie is right. 371 Macroeconomics. Problems and Questions Question 8 a. Consider a country that is open to trade in goods and services with the rest of the world, where prices are fixed and in which only the goods market exists. Initially, the country is in goods market equilibrium, and trade is balanced. Draw in the two graphs below the initial equilibrium position, and show how this equilibrium changes following an increase in πΈπΈ, the nominal exchange rate [Hint: assume that E will remain forever at the new, higher level, and that domestic and foreign prices do not change]. Finally, explain if and why, in the new equilibrium, income and net exports will have changed. Z 1 NX ′ ππππππ 0 πποΏ½1 πποΏ½0 (= ππππππ ) 45° DD ZZ ZZ’ ππ ππ 372 NX NX’ The open economy .......................................................................................................................... An increase in the nominal exchange rate causes an appreciation of the real ........................................................................................................................... exchange rate ππ ≡ πΈπΈπΈπΈ/ππ∗ , and therefore reduces the price-competitiveness of domestically produced goods. The ZZ curve shifts downwards, and the same ........................................................................................................................... happens to the NX, since net exports are now lower for any given ππ. This fall in net exports lowers aggregate demand and equilibrium output. As it is clear from the lower panel of the graph above, although ππ is now smaller, in the new equilibrium net exports will be lower – the country will now be running a trade deficit (the adverse impact of the real appreciation, which tends to decrease net exports, prevails on the one coming from the decrease in domestic income, which tends to lower imports, and therefore to improve the trade balance). b. Consider now a different economy, where also financial markets exist and under a flexible exchange rate regime. Show in the graph below, and discuss, how an increase in domestic money supply ππ will affect the domestic interest rate ππ, the country’s level of income and the exchange rate of its currency, assuming that investment is exogenous, πΌπΌ = πΌπΌ ,Μ and therefore not a function of ππ. ππ π€π€Μ 0 π€π€Μ 1 0 πποΏ½0 1 πποΏ½1 πΏπΏπΏπΏ0 πΌπΌπΌπΌ0 πΏπΏπΏπΏ1 ππ ππ 1 0 πΈπΈ1 πΈπΈ0 πΈπΈ Even though investment is exogenous, aggregate demand still depends nega........................................................................................................................... tively on the interest rate, as a decrease in ππ causes a nominal exchange rate depreciation and an increase in net exports. It follows that the IS curve will ........................................................................................................................... have the usual, negative slope in the (ππ, ππ)-plane. In the figure above, the ex........................................................................................................................... pansionary monetary policy brings the equilibrium from point 0 to point 1, where the interest rate is lower (due to the increase in ππ), the exchange rate ........................................................................................................................... has depreciated (since the decrease in ππ amounts to a fall in the return on do........................................................................................................................... mestic assets, that leads to an increase in the demand for foreign assets and therefore in the demand for the currency in which they are denominated), and ........................................................................................................................... domestic income is higher, thanks to the exchange rate depreciation. ........................................................................................................................... 373 Macroeconomics. Problems and Questions Question 9 a. The euro-dollar exchange rate (number of dollars needed to purchase one euro) is expected to be 1.30 next year (i.e., πΈπΈ ππ = 1.30), and the one-year interest rates are 3% in Europe and 1% in the Unites States. Assuming that the (uncovered) interest parity condition holds, compute the current eurodollar exchange rate, πΈπΈ. Do financial markets participants expect the euro to appreciate or to depreciate in one year’s time? From the non-approximated version of the (uncovered) interest parity condition, (1 + πππ‘π‘ ) = (1 + πππ‘π‘∗ ) οΏ½ πΈπΈπ‘π‘ ππ οΏ½ , πΈπΈπ‘π‘+1 it follow that the current exchange rate can be written as: πΈπΈπ‘π‘ = (1 + πππ‘π‘ ) ππ πΈπΈ . (1 + πππ‘π‘∗ ) π‘π‘+1 Given the numerical values that πΈπΈ ππ , ππ and ππ ∗ take on in this case, it follows that: Being πΈπΈπ‘π‘ = 1.03 · 1.30 = 1.3265. 1.01 ππ πΈπΈπ‘π‘+1 − πΈπΈπ‘π‘ 1.30 − 1.3265 = β −0.02, πΈπΈπ‘π‘ 1.33 the euro is expected to depreciate by about 2% between today and next year. 374 The open economy b. Write down the approximated version of the interest parity condition, and check how good an approximation it is by repeating the computations you carried out when answering the previous point of this question. Finally, suggest two possible extensions that would likely increase the degree of realism of the interest parity condition in its simplest form (be it approximated or not) considered in this question. In its approximated version, the interest parity condition is: ........................................................................................................................... ππ πΈπΈπ‘π‘+1 − πΈπΈπ‘π‘ ........................................................................................................................... ∗ πππ‘π‘ = πππ‘π‘ − . πΈπΈ ........................................................................................................................... From the previous equation, it follows that the current exchange rate can be ........................................................................................................................... written as: ππ ........................................................................................................................... πΈπΈπ‘π‘+1 πΈπΈπ‘π‘ = . ........................................................................................................................... 1 + πππ‘π‘∗ − πππ‘π‘ Plugging the assigned values for the domestic interest rate, the foreign inter........................................................................................................................... est rate and the future expected exchange rate in the right-hand side, yields: ........................................................................................................................... 1.30 ........................................................................................................................... πΈπΈπ‘π‘ = = 1.3257. 1 + 0.01 − 0.03 ........................................................................................................................... Furthermore, and as before, ππ ........................................................................................................................... − πΈπΈπ‘π‘ 1.30 − 1.3257 πΈπΈπ‘π‘+1 = β −0.02, ........................................................................................................................... πΈπΈπ‘π‘ 1.33 One can therefore conclude that, when the values that interest rates and the ........................................................................................................................... expected appreciation term take on are, as in the case we are analyzing, small ........................................................................................................................... enough (close to zero), the version of the interest parity used to answer this second part of the question is a very good approximation to the non........................................................................................................................... approximated one considered when answering the previous point. ........................................................................................................................... Finally, both versions of the interest rate parity are based on the assumption that, in making their portfolio choices, financial investors only look at the expected returns of the available assets, and not also to the variability of those returns. In addition, both ignore the existence of transaction costs. Since, in the real world, financial market participants are typically risk averse, and transaction costs are nonzero, this explains why more sophisticated versions of this condition do exist (for instance, versions allowing for the existence of a nonzero risk premium) 375 Macroeconomics. Problems and Questions Question 10 The one-year interest rate in Japan is 1%, and the nominal exchange rate between the yen and the euro (number of yen needed to purchase one euro) is 140. Finally, assume that financial markets participants expect the euro to appreciate by 5% against the yen in the following year. a. For a European investor, what is the expected return from holding oneyear Japanese assets? ........................................................................................................................... πΈπΈπ‘π‘ = 140 (π¦π¦π¦π¦π¦π¦ ππππππ ππππππππ) = current exchange rate. ........................................................................................................................... Market participants expect the euro to appreciate by 5% against the yen, or: ........................................................................................................................... ππ ππ πΈπΈπ‘π‘+1 − πΈπΈπ‘π‘ πΈπΈπ‘π‘+1 − 140 ........................................................................................................................... = = 0.05. πΈπΈπ‘π‘ 140 ........................................................................................................................... ππ Solving for πΈπΈπ‘π‘+1 , ........................................................................................................................... ππ πΈπΈπ‘π‘+1 = 140(1 + 0.05) = 147. ........................................................................................................................... To compute the expected return from holding Japanese bonds, notice that, at the current exchange rate, with one euro a European investor can purchase 140 yen. Using these 140 yen to buy Japanese bonds, paying an interest of 1%, next year the European investor will receive 140(1 + 0.01) = 141.40 ππ = 141.40/147 = 0.96 yen, which are expected to be worth 141.40/πΈπΈπ‘π‘+1 euro one year from now. The expected return from investing in Japanese one-year bonds is therefore negative, equal to −4% [(0.96 − 1)/1 = −0.04]. 376 The open economy b. Suppose that financial investors care only about the expected rate of return, and therefore want to hold only the assets with the highest expected yield. Assuming that, in Europe, the one-year interest rate is 2%, would a European investor purchase European assets or Japanese assets? In this example, does the (uncovered) interest parity condition hold? A European investor who invests today one euro in European one-year bonds will receive, next year, 1(1 + .02) = 1.02 euros, for a rate of return of 2%. Using the same euro to purchase one-year Japanese bonds, we already know that the investor would receive, next year, 0.96 euros, so that the rate of return from investing in foreign bonds is negative (−4%). The investor is clearly better off by purchasing euro-denominated bonds. In this example, the interest parity condition does not hold, and there are therefore arbitrage opportunities between the two types of investments. 377 Macroeconomics. Problems and Questions Question 11 Consider an economy freely trading goods, services and financial assets with the rest of the world, in a flexible exchange rate regime. Domestic and foreign prices are constant, and equal to one (ππ = ππ ∗ = 1). a. Using the pair of graphs ‘open economy IS-LM – interest parity condition’, show the initial equilibrium of the country, denoting it by ‘1’ (hint: assume that the economy is described by a standard open economy IS-LM model, with a central bank the central bank that chooses the interest rate). Suppose now that the government raises its purchases of goods and services and that, to keep equilibrium income constant, the central bank varies the interest rate by implementing an open market operation. To achieve its aim, should the central bank buy or sell bonds? In the figure, denote by 2 the new equilibrium that will prevail after the increase in πΊπΊ and the central bank intervention. In the move from 1 to 2, how will the interest rate, the exchange rate, investment and net exports change? Explain. ππ π€π€Μ 2 π€π€Μ 1 1 2 πποΏ½1 1′ πΌπΌπΌπΌ2 πΌπΌπΌπΌ1 πΏπΏπΏπΏ2 πΏπΏπΏπΏ1 ππ 378 ππ 1 2 πΈπΈ1 πΈπΈ2 πΈπΈ The open economy If the central bank did not intervene, the increase in πΊπΊ would raise equilibrium income. To prevent ππ from changing, the central bank will have to increase the policy rate, something that requires a sale of bonds in the open market. In the new equilibrium (point 2 in the figure) the interest rate is higher, and this leads to an exchange rate appreciation. Furthermore, investment is lower (because of the increase in ππ), consumption is unchanged (income and net taxes have remained constant), and net exports are smaller (due to the appreciation of the exchange rate). b. Suppose now that the country is in a fixed, rather than in a flexible, exchange rate regime. Can a central bank wishing to keep the exchange rate fixed also intervene to keep income at the initial level (the one prevailing at equilibrium 1, before the increase in πΊπΊ), as it did in the flexible exchange rate case considered before? Why, or why not? In the figure, denote by 1′ the equilibrium that will be reached in this case (increase in πΊπΊ, and a central bank that behaves in a way consistent with the constancy of the exchange rate). Finally, explain how the interest rate, consumption, investment and next exports will have changed in the move from 1 to 1′. From the answer to the previous point we know that, to prevent income from changing, the central bank should raise the interest rate, thus causing an appreciation of the exchange rate. It is therefore straightforward to conclude that a central bank wishing to keep the exchange rate fixed at the chosen level will have to accept the rise in income caused by the fiscal expansion implemented by the government. The interest rate will have to be kept unchanged, and in the figure the new equilibrium becomes in this case point 1′, where income has gone up, consumption and investment are higher (ππ is higher, and net taxes and the interest rate unchanged), and net exports lower (due to the rise in ππ, and taking into account the fact that the exchange rate has remained constant). 379 Macroeconomics. Problems and Questions Question 12 Consider Alpha, an open economy with a flexible exchange rate and in which investment is a function of ππ and of the borrowing rate ππ + π₯π₯. Aside from this, the country is described by a standard ‘open economy IS-LM – interest parity’ model with constant domestic and foreign prices, so that ππ = ππ. a. Assuming that the central bank chooses the interest rate, represent the initial equilibrium of the economy in an ‘open economy IS-LM – interest parity’ diagram, denoting it by ‘1’, and by ππ1 , ππ1 and πΈπΈ1 the associated values of the interest rate, output and exchange rate. Suppose now that the risk premium on which the borrowing rate depends, π₯π₯, increases. In the graph, denote by ‘2’ the new equilibrium that will be reached. In the move from the initial equilibrium ‘1’ to the new one ‘2’, how will production, consumption, investment, the exchange rate, net exports and the country’s money supply and money demand change? Explain. ππ π€π€Μ 1 π€π€Μ 3 πΌπΌπΌπΌ1 πΌπΌπΌπΌ2 ππ 2 πποΏ½2 3 1 πΏπΏπΏπΏ1 πΏπΏπΏπΏ3 ππ 3 1 πΈπΈ3 πΈπΈ1 πΈπΈ For a given interest rate ππ chosen by the central bank, an increase in the risk premium raises the borrowing rate and lowers investment. The IS curve shifts to the left, and the new equilibrium becomes point 2 in the figure, where production is lower. Consumption has gone down (due to the fall in income), and the same is true about investment (income is lower, and the borrowing rate higher). 380 The open economy Since the central bank keeps the interest rate at the initial level, the exchange rate remains constant and net exports improve (due to the decrease in income). Finally, since in the new equilibrium money demand has gone down (the interest rate is unchanged, but income is lower), money supply must be lower, too. In fact, if the central bank did not change ππ, the interest rate would fall (recall that the decrease in ππ lowers money demand). To keep ππ at the chosen level, the central bank will have to decrease money supply by the same amont by which money demand has fallen. b. How would your answer to the previous point change, should Alpha’s central bank decide to intervene in order to prevent the rise in π₯π₯ from affecting production, and therefore in order to keep ππ at the same level prevailing in the initial equilibrium (‘1’)? In the graph, denote by ‘3’ the equilibrium that would be reached in this case and explain if and why, in the move from ‘1’ to ‘3’, consumption, investment, the exchange rate, net exports and the country’s money supply and money demand will change. To keep production at the initial level, the central bank must lower the policy rate till π€π€Μ 3 , thus shifting downwards the πΏπΏπΏπΏ curve to πΏπΏπΏπΏ3 . In the new equilibrium that will be reached in this case, compared to 1 both money supply (the central bank has lowered the policy rate) and money demand (ππ is lower, ππ unchanged) will be higher, the exchange rate has depreciated (due to the decrease in the domestic interest rate), net exports have improved (thanks to the decrease in πΈπΈ) and consumption will be unchanged. Finally, and although the rise in π₯π₯ and the fall in ππ tend to push it in opposite directions, investment will necessarily be lower. In fact, in the two equilibria 1 and 3 the supply of goods is the same; this means that aggregate demand will have to be the same, too, and that, therefore, investment will have gone down by the same amount by which net exports have gone up. 381 Macroeconomics. Problems and Questions Question 13 Consider Zeta, an open economy with a fixed exchange rate and in which investment is a function of ππ and of the borrowing rate ππ + π₯π₯. Aside from this, the country is described by a standard ‘open economy IS-LM – interest parity’ model with constant domestic and foreign prices, so that ππ = ππ, in wh ich th e foreign interest rate is, as usual, denoted by ππ ∗ , and where the value at which the central bank keep the exchange rate constant is πΈπΈοΏ½ = πΈπΈ1 , with πΈπΈοΏ½ ππ = πΈπΈοΏ½ = πΈπΈ1 . a. Assuming that the central bank chooses the interest rate, represent the initial equilibrium of the economy in an ‘open economy IS-LM – interest parity’ diagram, denoting it by ‘1’, and by ππ1 , ππ1 and πΈπΈ1 the associated values of the interest rate, output and exchange rate. Suppose now that the risk premium on which the borrowing rate depends, π₯π₯, decreases. In the graph, denote by ‘2’ the new equilibrium that will be reached. In the move from the initial equilibrium ‘1’ to the new one ‘2’, how will production, consumption, investment, net exports and the country’s money supply and money demand change? Explain. ππ ππ ∗ πΌπΌπΌπΌ1 ππ πΌπΌπΌπΌ2 1,3 2 πΏπΏπΏπΏ1 πποΏ½1 πποΏ½2 ππ 1,2 3 πΈπΈ1 πΈπΈ3 πΈπΈ Given the interest rate chosen by the central bank, ππ = ππ ∗ , a decrease in the risk premium lowers the borrowing rate and raises investment. The IS curve shifts to the right and the equilibrium becomes point 2 in the figure, where production is higher. Consumption has gone up (because of the higher equilibrium income), and the same is true about investment (income is higher, and the borrowing rate lower). 382 The open economy Since the central bank keeps the interest rate constant at the initial level, the exchange rate, too, remains constant, while net exports are lower (due to the increase in domestic income). Finally, since in the new equilibrium money demand has gone up (the interest rate is unchanged, but income is higher), money supply, too, must be higher. In fact, if the central bank did not change ππ, the domestic interest rate would rise above ππ ∗ (remember that the increase in ππ raises money demand), and the exchange rate above πΈπΈ1 . To prevent this from happening, the central bank will have to raise money supply by the same amount by which money demand has gone up. b. Suppose now that, to bring back income to the level prevailing before the fall in π₯π₯ (that is, to ππ1 ), Zeta’s central bank decides to change from πΈπΈ1 to πΈπΈ3 the value at which the exchange rate is kept fixed, and that individuals revise accordingly the value of the exchange rate they expect to prevail in the future, so that the new future expected exchange rate now becomes οΏ½ = πΈπΈ3 . Having explained if, to achieve the central bank’s aim, πΈπΈ3 πΈπΈοΏ½ ππ′ = πΈπΈ′ will have to be greater or smaller than πΈπΈ1 , in the pair of graphs used before denote by ‘3’ the new equilibrium the economy will reach. Comparing the initial equilibrium (‘1’) to the final one (‘3’), how will consumption, investment, net exports and the country’s money supply and money demand will change? Why? Explain. To bring back income to the initial level, the central bank will have to resort to an exchange rate revaluation of appropriate size. In particular, the new value at which the exchange rate is kept fixed, πΈπΈ3 , will have to be consistent with an IS curve that returns to πΌπΌπΌπΌ1, and this due to the adverse impact of the revaluation on the country’s net exports. The new equilibrium is at point 3 in the two panels [notice that, in the right panel, the line representing the interest parity condition will now go through – as always it must do – the point in which the interest rate is ππ = ππ ∗ and the exchange rate πΈπΈ = πΈπΈοΏ½ ππ′ = πΈπΈ3 ]. As the economy goes from the initial equilibrium ‘1’ to the final one ‘3’, consumption is unchanged, investment is higher (income is unchanged, but the borrowing rate is lower), and net exports have gone down (domestic and foreign income levels are unchanged, but the real exchange rate is higher). Finally, since income and the interest rate are unchanged, also money demand will be unchanged; needless to say, the same must be true about money supply. 383 Macroeconomics. Problems and Questions Question 14 Consider an economy freely trading goods, services and financial assets with the rest of the world. Furthermore, assume that domestic and foreign prices are constant, and equal to one (ππ = ππ∗ = 1). a. Using the pair of graphs ‘open economy IS-LM – interest parity condition’, show the initial equilibrium of the country, assuming a flexible exchange rate regime and, as usual, that the domestic central bank chooses the interest rate, keeping it constant at the level it deems appropriate given the state of the economy. In the graph, denote by ‘1’ the initial equilibrium, by π€π€Μ 1 , ππ1 ed πΈπΈ1 the associated values of the interest rate, income and the exchange rate, and assume that, in this initial equilibrium, πΈπΈ ππ = πΈπΈ1 and π€π€Μ 1 = ππ ∗ (where, as usual, ππ ∗ is the foreign interest rate). Consider now an increase in πΈπΈ ππ , which now ′ becomes πΈπΈ ππ > πΈπΈ ππ = πΈπΈ1 . In other words, investors now expect the domestic currency to be ‘stronger’ than they initially thought. Show in the graph, and explain, how the economy’s income, interest rate, exchange rate, and money supply will be affected by this change. ππ π€π€Μ 1 = ππ ∗ 2 πποΏ½2 ππ 1 οΏ½οΏ½11 ππππ πΌπΌπΌπΌ2 πΌπΌπΌπΌ1 ππ 384 1 2 πΈπΈ ππ = πΈπΈ1 πΈπΈ ππ ′ πΈπΈ The open economy For given π€π€Μ 1 and ππ ∗ , the increase in πΈπΈ ππ lowers the expected return on foreign assets, thus raising (appreciating) the exchange rate πΈπΈ consistent with the interest parity condition. The exchange rate appreciation lowers, for any given ππ, net exports and shifts the IS curve to the left. The new equilibrium becomes point 2, where income is lower, ππ unchanged, πΈπΈ greater (equal to πΈπΈ ππ′ in the figure) and the supply of money lower. In fact, the fall in ππ lowers money demand. To keep the domestic interest rate at the chosen level, π€π€Μ 1 , the central bank will have to lower the supply of money by the same amount by which money demand has gone down. b. Suppose that the government intends to return domestic output to the level prevailing before the increase in πΈπΈ ππ by changing net taxes, πποΏ½. To achieve its aim, should the government raise or lower net taxes? Compare the composition of aggregate demand in the new equilibrium that will be attained after the fiscal policy intervention to that prevailing in the initial equilibrium (that is, the equilibrium in which the economy was before the increase in πΈπΈ ππ and the change in πποΏ½). Explain. To return output to the initial level, the government will have to shift the πΌπΌπΌπΌ curve to the right, till πΌπΌπΌπΌ1. This requires a cut in net taxes. Once the economy is back to the equilibrium corresponding to point 1, consumption will be higher than before the increase in πΈπΈ ππ and the cut in πποΏ½ (income has returned to the initial level, but net taxes are lower), investment will be unchanged (both ππ and ππ are unchanged), πΊπΊ unchanged (it is exogenous) and net exports lower (due to the exchange rate appreciation). 385 Macroeconomics. Problems and Questions Question 15 a. The world economy consists of just two countries, Alpha and Beta. International capital mobility is perfect, and the exchange rate between the currencies of the two countries is kept fixed at the level πΈπΈοΏ½ . Alpha has just entered a recession. To revive its economy, would you recommend Alpha the use of monetary policy or of fiscal policy? Explain your answer and illustrate graphically, using the pair of graphs 'open economy IS-LM'-'interest parity condition'. ππ π€π€Μ 0 = ππ∗ π€π€Μ 0 ′ 0 0 ′ 1 πΌπΌπΌπΌ0 πΏπΏπΏπΏ0 πΏπΏπΏπΏ1 πΌπΌπΌπΌ1 ππ ππ ππ ∗ 0 ′ 0 1 πΈπΈ0′ πΈπΈοΏ½ πΈπΈ Under a fixed exchange rate regime, monetary policy cannot be used to affect ........................................................................................................................... the equilibrium level of income. In fact, if individual expect the exchange rate to remain in the future at the current level πΈπΈοΏ½ , the value at which the central ........................................................................................................................... bank is pegging it, the interest parity condition implies that the domestic in........................................................................................................................... terest rate has to be equal to the foreign one. In the initial equilibrium, point 0 in the graph, this equality holds, with a domestic interest rate π€π€Μ 0 (= ππ ∗ ) and ........................................................................................................................... an exchange rate equal to πΈπΈοΏ½ . If, starting from this initial equilibrium, to boost ........................................................................................................................... economic activity the central bank decided to lower the interest rate to π€π€Μ 0 ′ (for instance, by purchasing bonds in the open market), the exchange rate ........................................................................................................................... would immediately tend to depreciate, falling to the level πΈπΈ0′ (< πΈπΈοΏ½ ). ........................................................................................................................... 386 The open economy ........................................................................................................................... To avoid this depreciation, the central bank must immediately intervene in the ........................................................................................................................... foreign exchange market, buying the domestic currency and selling the foreign ........................................................................................................................... one. This intervention, that reduces ππ, must continue until ππ returns to its original level, and the LM curve to its initial position. But if, to make sure that the exchange rate remains constant, the money supply and the domestic interest rate must be returned to their initial levels, aggregate demand and output will be unaffected. Under fixed exchange rates, to increase equilibrium output one can only resort to an expansionary fiscal policy that, by shifting the IS to the right (πΌπΌπΌπΌ1 ), leads to a new equilibrium such as point 1 in the figure. b. Discuss the effects of the policy intervention you have just suggested on the trade balance of Alpha and on that of Beta, always assuming a fixed exchange rate. Because of the expansionary fiscal policy adopted in Alpha and described in the preceding paragraph, Alpha’s income increases, while the exchange rate remains constant. The trade balance of Alpha worsens, because of the increase in imports caused by the rise in equilibrium income. In a two-country world such as the one we are considering, Beta's trade balance can only improve by the same amount by which Alpha's worsens. 387 Macroeconomics. Problems and Questions Question 16 Consider an economy that trades goods, services and financial assets with the rest of the world, under a fixed exchange rate regime. Domestic and foreign prices are constant and, for simplicity, both equal to 1 (ππ = ππ ∗ = 1). In the pair of graphs ‘open economy IS-LM’-‘interest parity condition’, show the initial equilibrium, denoting by ππ0 , ππ0 and πΈπΈ0 the values of the interest rate, output and exchange rate in that equilibrium. Market participants expect that the exchange rate will be kept fixed at the current level πΈπΈ0 also in the future, so that πΈπΈ ππ = πΈπΈ0 . a. Suppose that there is a fall in the foreign interest rate, ππ ∗ , while foreign output ππ ∗ and all the other exogenous variables remain constant. Assuming that the country’s central bank keeps fixing the exchange rate at πΈπΈ0 , show in the graph, and explain carefully, the effects of this exogenous decrease in ππ ∗ on equilibrium output, interest rate, money supply and net exports of the country, denoting by ‘1’ the new equilibrium that will be reached. ππ ππ ∗ ππ ∗ ’ πΌπΌπΌπΌ0 πΌπΌπΌπΌ2 0 1 2 πποΏ½0 πποΏ½1 πποΏ½2 πΏπΏπΏπΏ0 ππ 0 πΏπΏπΏπΏ1 1, 2 ππ πΈπΈ ππ = πΈπΈ0 388 πΈπΈ The open economy In the right panel, the fall in the foreign interest rate leads to a downward shift in the line representing the interest parity condition; the new line will now go through – as it has to do by construction – the point (πΈπΈ = πΈπΈ ππ , ππ = ππ ∗ ′), where ππ ∗ ′ is the new level of the foreign interest rate. Absent any intervention by the domestic central bank, the exchange rate would appreciate (it would rise above πΈπΈ0 ). To keep it at πΈπΈ0 , the central bank will have to intervene. This intervention will have to lower the domestic interest rate by the same amount by which the foreign one has gone down, thus shifting the LM curve downwards till πΏπΏπΏπΏ1 . In the new equilibrium (point 1), the decrease in the domestic interest rate has raised aggregate demand and equilibrium output. As for net exports, they will now be lower, since domestic income is higher and the exchange rate unchanged. b. How would your answer have changed if, when ππ ∗ falls, foreign output ππ ∗ had also gone up, rather than remaining constant? In the same graph used before, denote by ‘2’ the equilibrium that the economy would have reached in this case (lower ππ ∗ and higher ππ ∗ ) and compare it to the equilibrium (‘1’) where the economy settles when to vary is just the foreign interest rate. Finally, explain in which of the two cases the intervention of the central bank (consisting in a change in the money supply aimed at keeping the exchange rate constant) will have to be quantitatively larger. In this case, the interest parity condition line will still shift downwards, while ........................................................................................................................... the increase in ππ ∗, that – other things the same – leads to higher net exports, ........................................................................................................................... shifts the IS curve to the right in the first panel (πΌπΌπΌπΌ2 ). The increase in domestic output is now larger, since in this case the demand for domestically pro........................................................................................................................... duced goods goes up not only because of the fall in the interest rate, but also ........................................................................................................................... because of the rise in the demand by foreigners. The central bank’s intervention, aimed at keeping πΈπΈ constant, must therefore be larger than before. More ........................................................................................................................... specifically, the domestic money supply will have to be raised by more than ........................................................................................................................... when the only change taking place was the one in ππ ∗ − now it will have to go up not only to make sure that the domestic interest rate falls by the same ........................................................................................................................... amount as the foreign one, but also to prevent the increase in income, and ........................................................................................................................... therefore money demand, caused by the rise in ππ ∗ from pushing the domestic interest rate above the new level ππ ∗′ consistent with the fixed exhange rate. 389 Macroeconomics. Problems and Questions Question 17 Consider an open economy under a fixed exchange rate regime. Domestic and foreign prices are constant and, for simplicity, both equal to 1 (ππ = ππ∗ = 1). In the pair of graphs ‘open economy IS-LM’-‘interest parity condition’, show the initial equilibrium, denoting it by ‘0’, and by ππ0 , ππ0 and πΈπΈ0 the associated values of the interest rate, output and the exchange rate. Market participants expect that the exchange rate will be kept fixed at the current level πΈπΈ0 also in the future, so that πΈπΈ ππ = πΈπΈ0 . a. Suppose the central bank announces a devaluation of the currency − that is, it announces that, effective immediately, the value at which the exchange rate is kept fixed is lowered to πΈπΈ1 < πΈπΈ0 . In addition, assume that individuals, who did not expect this announcement, revise accordingly the value of the exchange rate they expect to prevail in the future, so that now πΈπΈ ππ falls to πΈπΈ ππ ′ = πΈπΈ1 . Analyze in the graph, and explain, the effects of the devaluation on domestic output, interest rate and money supply, denoting by ‘1’ the new equilibrium that the economy will reach. ππ ππ0 = ππ ∗ πΌπΌπΌπΌ0 πΌπΌπΌπΌ1 0 1 πΏπΏπΏπΏ0 πποΏ½0 πποΏ½1 ππ 390 ππ 1 πΈπΈ ππ′ = πΈπΈ1 0 πΈπΈ ππ = πΈπΈ0 πΈπΈ The open economy In the right-hand panel, the curve representing the interest parity condition ′ shifts upwards, now going through the new point (πΈπΈ = πΈπΈ ππ = πΈπΈ1 ; ππ = ππ ∗). In the first panel, there is a rightward shift of the IS curve, due to the fact that, for given domestic and foreign prices, the devaluation depreciates the real exchange rate, thus raising net exports and therefore the demand for domestically produced goods. In both panels, the new equilibrium is at point 1, where output is higher and the domestic interest rate still equal to the foreign one (as it must necessarily be the case if, as in the present setting, individuals expect the exchange rate to take on in the future the same value it is taking on today). In the move from the old to the new equilibrium, the central bank must have increased the money supply. In fact, the increase in income caused by the devaluation raises money demand. If ππ were kept constant, the domestic interest rate would raise above the foreign one, and the exchange rate above the value πΈπΈ1 at which the central bank now wants it to remain constant. Money supply will have to be raised by the amount needed to make sure that, even though income is higher, the domestic interest rate remains at its initial level, ππ0 = ππ ∗ . b. Suppose that, before the devaluation, both the domestic country and the rest of the world were in a medium-run equilibrium, with income at its natural level and zero inflation. In addition, assume that the inflation rate is determined according to the following Phillips curve: ππ = (πΌπΌ/πΏπΏ)(ππ − ππππ ) Once price adjustment − as implied by the previous equation − is taken into account, do you think that a devaluation can permanently affect the level of income of a country? Why, or why not? Discuss [Hint: no formal analysis is required; just describe the likely dynamic adjustment of the economy following a devaluation]. As we have just concluded, the devaluation will rise output above its natural ........................................................................................................................... level. Domestic prices will start rising and, given that foreign prices are constant, the real exchange rate will start appreciating, thus hurting the price........................................................................................................................... competitiveness of domestically produced goods and worsening the trade bal........................................................................................................................... ance. It follows that, in the figure above, sooner or later the curve labeled πΌπΌπΌπΌ1 will start shifting to the left. As long as output is above its natural level, this ........................................................................................................................... process will continue. Eventually, the economy will converge to the same me........................................................................................................................... dium-run equilibrium prevailing before the devaluation (‘0’), with a real exchange rate − on impact lower because of the fall in πΈπΈ − back to its initial level, due to the rise in domestic prices. 391 Macroeconomics. Problems and Questions Question 18 Consider an open economy under a fixed exchange rate regime. Domestic and foreign prices are constant and, for simplicity, both equal to 1 (ππ = ππ∗ = 1). In the pair of graphs ‘open economy IS-LM’-‘interest parity condition’, show the initial equilibrium, denoting it by ‘0’, and by ππ0 , ππ0 and πΈπΈ0 the associated values of the interest rate, output and the exchange rate. Initially, market participants expect that the exchange rate will be kept fixed at the current level πΈπΈ0 also in the future, so that πΈπΈ ππ = πΈπΈ0 . a. Suppose that, in the initial equilibrium, income is below its natural level. Given that, as discussed in the answer to the previous question, a decision to devalue leads a higher equilibrium output in a country under a fixed exchange rate regime, individuals start expecting an impeding devaluation. It follows that the future expected exchange rate decreases from πΈπΈ ππ = πΈπΈ0 to πΈπΈ ππ′ = πΈπΈ1 , with πΈπΈ1 < πΈπΈ0 . Analyze in the graph the effects of this downward revision in the expectations on the exchange rate that will prevail in the future on the levels of domestic income and interest rate, assuming that the central bank intends to keep the exchange rate fixed at πΈπΈ0 . ππ ππ1 πΌπΌπΌπΌ0 πΏπΏπΏπΏ1 1 πποΏ½1 πποΏ½0 ππ 392 1 0′ πΏπΏπΏπΏ0 0 ππ0 = ππ ∗ ππ πΈπΈ ππ′ = πΈπΈ1 0 πΈπΈ ππ = πΈπΈ0 πΈπΈ The open economy In the right-hand panel, the curve representing the interest parity condition ′ shifts upwards, now going through the new point (πΈπΈ = πΈπΈ ππ = πΈπΈ1 ; ππ = ππ ∗ ). In the first panel, the IS curve does not shift, as net exports do not depend on the value that the exchange rate is expected to take on in the future, but rather on the value the exchange rate is taking on today (a value that the central bank is keeping constant at πΈπΈ0 ). To make sure that πΈπΈ remains at πΈπΈ0 even after expectations of a devaluation have emerged, the central bank will have to raise the domestic interest rate to ππ1 . In fact, if ππ remained at the initial level, ππ0 , in the right-hand side panel we would go from point 0 to 0’, with an exchange rate that depreciates immediately. As implied by the interest parity condition, to prevent this drop in the exchange rate, the central bank will have to set the domestic interest rate to a level that exceeds the foreign one by an amount equal to the expected devaluation of the domestic currency, ππ1 − ππ0 = − (πΈπΈ1 − πΈπΈ0 ) . πΈπΈ0 b. Which type of economic policy (monetary or fiscal; exansionary or contractionary) could prevent the expectations of an impending devaluation studied before from changing equilibrium output? Explain. As we already know, under a fixed exchange rate regime fiscal policy is the only macroeconomic policy that can affect equilibrium income. In this case, since equilibrium output tends to fall, the fiscal policy intervention that is called for is an expansionary one. As for the role of monetary policy, and as discussed before, it will have to make sure that the domestic interest rate takes on a value consistent with the fixed exchange rate, given ππ ∗ and the new value of πΈπΈ ππ . 393 Macroeconomics. Problems and Questions Question 19 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. An expansionary monetary policy rises equilibrium income under flexible exchange rates, but does not affect income under fixed exchange rates. True. Under flexible exchange rates, the decrease in the interest rate caused by an expansionary monetary policy stimulates not only investment, but – by inducing a depreciation of the exchange rate − also net exports, thus leading to an increase in aggregate demand and equilibrium income. On the contrary, under fixed exchange rates authorities must oppose to the potential depreciation and intervene in the exchange rate market. This intervention, which must necessarily consist in the purchase of the domestic currency and in the selling of the foreign currency, must go on until money supply and the domestic interest rate are back to the initial levels. It follows that equilibrium income will not vary either. With fixed exchange rates, monetary policy cannot be used to stimulate the economy. 394 The open economy b. If financial investors only care about the expected rate of return, and therefore want to hold only the assets with the highest expected rate of return, then from the interest parity condition it follows that, under fixed exchange rates, the domestic interest rate can only be equal to the foreign interest rate, ππ = ππ ∗ . False. Under fixed exchange rates, ππ = ππ ∗ if πΈπΈοΏ½ = πΈπΈ ππ , that is, if individuals expect that the exchange rate will be kept fixed at the current level πΈπΈοΏ½ also in the future. If, however, there are expectations of a future revaluation or devaluation, πΈπΈοΏ½ ≠ πΈπΈ ππ , and the domestic interest rate ππ will necessarily differ from ππ ∗ . If, for instance, there are expectations of an imminent devaluation of the exchange rate, πΈπΈοΏ½ > πΈπΈ ππ , and the domestic interest rate will have to be greater than the foreign interest rate in order to compensate investors for the expected depreciation of the domestic currency. 395 Macroeconomics. Problems and Questions Question 20 a. Psi is an open economy with a flexible exchange rate. Illustrate graphically, and explain, the effects of a decrease in the reserve ratio, θ, on output, the exchange rate, and the trade balance of Psi. When answering, assume that – starting from an initial interest rate that you will denote by π€π€Μ 0 in the figure − the central bank will not intervene to keep ππ equal to π€π€Μ 0, but that it will allow the interest rate to take on the new equilibrium value – to be denoted by π€π€Μ 1 in the graph − that will prevail after the fall in θ. Once ππ has become equal to π€π€Μ 1 , the central bank will however keep it at this new level from then on. ππ π€π€Μ 0 π€π€Μ 1 ππ πΌπΌπΌπΌ0 πΌπΌπΌπΌ2 2 0 πποΏ½0 1 πποΏ½1 πΏπΏπΏπΏ0 πΏπΏπΏπΏ1 ππ 1 0 πΈπΈ1 πΈπΈ0 πΈπΈ A reduction in the reserves ratio (i.e., a decrease in the parameter θ) raises the the money supply (banks will lend out a larger fraction f any given ........................................................................................................................... amount of deposits received by their customers; this will lead to more depos........................................................................................................................... its, more loans, more deposits ... and therefore to a larger ππ). If the central bank does not intervene, this increase in ππ lowers – let's say, from π€π€Μ 0 to π€π€Μ 1 < ........................................................................................................................... π€π€Μ 0 − the interest rate for which the money market is in equilibrium Since this ........................................................................................................................... new, lower ππ by assumption becomes the level at which the central bank will keep the interest rate, the LM curve shifts downwards to πΏπΏπΏπΏ1, and the new ........................................................................................................................... equilibrium becomes point 1. ........................................................................................................................... 396 The open economy ........................................................................................................................ In the new equilibrium, the interest rate is lower, the exchange rate has depreciated, and income is higher. The direction in which net exports will change is however uncertain – the exchange rate depreciation tends to rise them, but the rise in income tends to worsen them. b. Suppose now that the government of Psi implements a fiscal policy aimed at bringing equilibrium output back to the level it was taking on before the decrease in θ. What happens to the trade balance in this case? Explain. In order to bring back income to its initial level, the government must implement a restrictive fiscal policy, thus shifting the IS curve to the left, up to πΌπΌπΌπΌ2 . In the final equilibrium, point 2 in the first graph and point 1 in the second one, income will be at the same level as than at point 0, but the interest rate will be lower, and the exchange rate weaker. Due to the exchange rate depreciation, in the new equilibrium (point 1) net exports will be higher than in the initial equilibrium (point 0). 397 Macroeconomics. Problems and Questions Question 21 Consider an economy freely trading goods, services and financial assets with the rest of the world, in a flexible exchange rate regime. Domestic and foreign prices are constant, and equal to one (ππ = ππ ∗ = 1). a. Using the pair of graphs ‘open economy IS-LM – interest parity condition’, show the initial equilibrium of the country, denoting it by ‘1’, assuming that the central bank chooses the interest rate, that investment is entirely exogenous (πΌπΌ = πΌπΌ )Μ , and finally that net exports depend positively on foreign income and negatively on the domestic one, but that they do not depend on the real exchange rate, ππ. Suppose now that the central bank sells bonds in the open market, in order to raise the domestic interest rate. Show in the graph how the economy’s income, interest rate and exchange rate will be affected by this change, denoting by ‘2’ the new equilibrium that the economy will reach. Explain. ππ π€π€Μ 2 π€π€Μ 1 πΌπΌπΌπΌ1 2 1 πποΏ½1 πΏπΏπΏπΏ1 πΏπΏπΏπΏ2 ππ ππ 1 2 πΈπΈ1 πΈπΈ2 πΈπΈ If investment is exogenous and net exports do not depend on the real exchange rate, then no component of aggregate demand for domestically produced goods depends on the domestic interest rate, and the IS is a vertical line in the (ππ, ππ) space. The central bank’s decision to sell bonds in the open market lowers money supply and raises the domestic interest rate. The LM curve shifts up and, for given levels of the foreign interest rate and of the expected future exchange rate, the interest parity condition implies a nominal (and real) appreciation of the currency. In the two panels, the equilibrium point moves from 1 to 2. Income does not change, because − as already mentioned – no component of aggregate demand depends on ππ. 398 The open economy b. Suppose now that, rather than being independent of the real exchange rate, net exports depend positively on ε – in other words, a real depreciation (appreciation) worsens (improves) the country’s trade balance. Assuming that the remaining assumptions (exogeneity of investment, a flexible exchange rate, a central bank that chooses the interest rate, etc.) still hold true, discuss the effects of the same monetary policy studied above on the equilibrium levels of income, interest rate and exchange rate. Show in the figure the new equilibrium that will be reached in this case, and explain. ππ π€π€Μ 2 π€π€Μ 1 1 2 πΌπΌπΌπΌ1 πΏπΏπΏπΏ2 πΏπΏπΏπΏ1 πποΏ½1 πποΏ½2 ππ ππ 1 2 πΈπΈ1 πΈπΈ2 πΈπΈ The IS curve is now positively sloped – an increase in ππ appreciates the exchange rate and raises net exports and the demand for domestically produced goods; for the economy to return in goods market equilibrium, ππ must rise. As before, the open market sale of bonds by the central bank lowers the money supply and raises the domestic interest rate. For given ππ ∗ and πΈπΈ ππ , the interest parity condition implies that the exchange rate will appreciate. Since in this economy net exports depend positively on the exchange rate, this appreciation will increase the demand for domestically produced goods, and therefore equilibrium output. 399 Macroeconomics. Problems and Questions Question 22 Gamma is an economy that trades goods, services and financial assets with the rest of the world, under a flexible exchange rate regime. In Gamma, changes in the autonomous components of the demand for domestic goods are the main cause of the observed fluctuations in real GDP. Starting from an initial equilibrium like point ‘0’ in the graph, sometimes the level of autonomous demand is high, leading to an πΌπΌπΌπΌ curve such as πΌπΌπΌπΌ1 in the figure; sometimes it is however low, so that the πΌπΌπΌπΌ curve shifts to the left in the graph (πΌπΌπΌπΌ2 ). Economists Jack and Jill are asked which monetary policy rule − the one assumed in the 'standard' version of the IS-LM model, in which the central bank chooses a value for the interest rate, or the alternative one, in which the central bank chooses the nominal money supply − would, in their opinion, help minimize the variability of output around its initial level, ππ0 , caused by such demand fluctuations. Jack thinks that the best option is for the central bank to choose the interest rate, and keep it at the chosen value; on the contrary, Jill suggests that the central bank should choose a level for the money supply, and then let the interest rate to take on any value that, given this choice of ππ, is consistent with the macroeconomic equilibrium. a. In the pair of graphs ‘open economy IS-LM - interest parity condition’ below, show the equilibrium levels of output that, under flexible exchange rates, will prevail when the central bank chooses the interest rate and, following the fluctuations in demand just discussed, the IS curve shifts to the right (πΌπΌπΌπΌ1 ) or to the left (πΌπΌπΌπΌ2 ). In the figure, make sure to denote the corresponding equilibrium levels of output by ππ1 and ππ2 , respectively. ππ ππ1 π€π€Μ 0 ππ2 πΌπΌπΌπΌ0 πΌπΌπΌπΌ2 πΌπΌπΌπΌ1 ππ2 0 ππ2′ ππ0 πΏπΏπΏπΏ0ππ ππ1′ πΏπΏπΏπΏ0ππ ππ1 400 ππ ππ 0’ πΈπΈ2 πΈπΈ0 πΈπΈ1 πΈπΈ The open economy In this case, the LM curve (πΏπΏπΏπΏ0ππ in the graph) is, as usual, horizontal at the level of the interest rate chosen by the central bank (denoted by π€π€Μ 0). The equilibrium levels of output ππ1 and ππ2 prevailing as autonomous demand fluctuates are shown in the figure. In the case of a positive (adverse) demand shock – that is, IS curve that shifts to the right (left) – the central bank makes sure that the interest rate remains at π€π€Μ 0 by raising (lowering) the nominal money supply. And, since the domestic interest rate does not change, the exchange rate will remain constant, too. b. In the same graph, denote by ππ1′ and ππ2′ the equilibrium levels of production that, following the same changes in the autonomous components of aggregate demand and the same shifts in the πΌπΌπΌπΌ curve discussed before, would prevail should the central bank choose the nominal money supply. On the basis of your answers to this and to the previous point, do you agree with Jack or with Jill? Provide the intuition for the lower output variability in the central banks behaves in the way suggested by the economist whose view you share. The equilibrium levels of output ππ1′ and ππ2′ prevailing when the central bank chooses the money supply are shown in the figure. In the case of a positive (adverse) demand shock, the rise (fall) in the demand for goods and in equilibrium income will increase (decrease) the demand for money. Being money supply constant at the level chosen by the central bank, the change in money demand will lead to a higher (lower) interest rate. In turn, this change in the interest rate will mitigate the impact on equilibrium output of the changes in autonomous demand both directly, through its effect on investment, and indirectly, by affecting the exchange rate. For instance, following a positive demand shock, both ππ and πΈπΈ will go up; this leads to an increase in output smaller than the one that would prevail in the case studied in the previous point of this question, where the central bank was keeping constant the interest rate, • rather than the money supply. Jill is right. 401 Macroeconomics. Problems and Questions Question 23 a. We are at time π‘π‘. Use the non-approximated form of the interest parity condition to write the current period exchange rate, πΈπΈπ‘π‘ , as a function of the current and future expected interest rates for each year over the next ππ ππ . years, as well as of the expected exchange rate for time π‘π‘ + ππ + 1, πΈπΈπ‘π‘+ππ+1 The (uncovered) interest parity condition is, in its non-approximated form, Solving for πΈπΈπ‘π‘ , one gets: (1 + πππ‘π‘ ) = (1 + πππ‘π‘∗ ) οΏ½ πΈπΈπ‘π‘ = πΈπΈπ‘π‘ ππ οΏ½ . πΈπΈπ‘π‘+1 (1 + πππ‘π‘ ) ππ πΈπΈ . (1 + πππ‘π‘∗ ) π‘π‘+1 (∗) Updating by one period and taking expectations of both sides, yields: ππ πΈπΈπ‘π‘+1 ππ ) (1 + πππ‘π‘+1 = πΈπΈ ππ . ∗ππ (1 + πππ‘π‘+1 ) π‘π‘+2 (∗∗) Plugging in the right-hand side of (∗), one can write the current exchange rate as: πΈπΈπ‘π‘ = ππ ) (1 + πππ‘π‘ )(1 + πππ‘π‘+1 πΈπΈ ππ . ∗ππ (1 + πππ‘π‘∗ )(1 + πππ‘π‘+1 ) π‘π‘+2 (∗∗∗) ππ Following the same steps to eliminate πΈπΈπ‘π‘+2 [that is, updating equation (∗∗) ππ in the right-hand by one period, and plugging the resulting expression for πΈπΈπ‘π‘+2 ππ ππ side of equation (∗∗∗)], and then πΈπΈπ‘π‘+3 , πΈπΈπ‘π‘+4 , etc., yields: πΈπΈπ‘π‘ = ππ ) ππ ) (1 + πππ‘π‘ )(1 + πππ‘π‘+1 β― (1 + πππ‘π‘+ππ πΈπΈ ππ . ∗ππ ∗ππ (1 + πππ‘π‘∗ )(1 + πππ‘π‘+1 ) β― (1 + πππ‘π‘+ππ ) π‘π‘+ππ+1 (∗∗∗∗) The current exchange rate therefore depends positively on the current and future expected short-term interest rates over the next ππ years, negatively on the foreign short-term interest rates expected over the same period, and positively on the exchange rate expected to prevail at time π‘π‘ + ππ + 1 – or, if you want, on the value of the exchange rate expected to prevail in the long-run. 402 The open economy b. Explain how each of the following announcements – which, made at time π‘π‘, are unexpected, and believed, by financial markets participants – affects the time π‘π‘ exchange rate, πΈπΈπ‘π‘ [Hint: assume that, in each period, the functioning of the economic system is described by a static IS-LM model, with consumption and investment depending on contemporaneous variables only]: b.1 b.2 a permanently more expansionary monetary policy abroad, implemented from time π‘π‘ + 2 onwards; a permanently more expansionary fiscal policy, and a permanently more restrictive monetary policy, at home, from time π‘π‘ + 1 onwards; b.3 the emergence of expectations of a progressive, lasting worsening of the country's current account balance. b.1 Abroad, the interest rate will be lower from time π‘π‘ + 2 onwards. From equation (∗∗∗∗) it follows that πΈπΈπ‘π‘ will rise – the domestic currency will appreciate. ππ ππ b.2 The policy-mix will lead πΏπΏπΏπΏ0 to an increase in the domestic interest rate from πΏπΏπΏπΏ2 time π‘π‘ + 1 onwards, the current exchange 0 rate. 0 thus appreciating 1 sustainable indef1 widening current account deficits are not b.3 Persistent and 2 initely, as they imply 2 that the country keeps borrowing from abroad, thus increasing its foreign debt. It is therefore reasonable to assume that indiπΌπΌπΌπΌ0 viduals will start expecting that, sooner or later, the exchange rate will πΌπΌπΌπΌ2 fall − that is, that eventually a depreciation will be needed to restore the πΌπΌπΌπΌ price -competitiveness of1 domestic goods and bring the dynamics of the πΈπΈ1 ofπΈπΈthe πΈπΈ ππ This decrease in the level current account exchange πποΏ½0 control. πποΏ½1 under 0 ππ rate that financial investors expect to prevail in the long-run, πΈπΈπ‘π‘+ππ+1 , leads to an immediate depreciation of the currency, as clear from equation (∗∗∗∗) above. 403 Chapter 6 - Government debt and economic growth Macroeconomics. Problems and Questions * Question 1 [A graphical analysis of the evolution of the debt-to-GDP ratio] Write down the government budget constraint as an equation that, for given values of the real interest rate (ππ), of the growth rate of the economy (ππ) and of the primary deficit-to-GDP ratio (ππ), relates the debt-to-GDP ratio (π΅π΅⁄ππ) at time π‘π‘ to the value of the same ratio at time π‘π‘ − 1. Use that equation to graphically analyze the evolution of the ratio π΅π΅⁄ππ in the following four cases, providing the intuition underlying your conclusions: a. ππ < ππ and ππ > 0. π΅π΅π‘π‘ πππ‘π‘ 45° πποΏ½ debt/GDP line ππ2 ππ1 ππ ππ0 ππ1 ππ2 πποΏ½ π΅π΅π‘π‘−1 πππ‘π‘−1 The debt-to-GDP ratio evolves over time as implied by the following equation: ........................................................................................................................... π΅π΅π‘π‘ π΅π΅π‘π‘−1 π΅π΅π‘π‘−1 πΊπΊπ‘π‘ − πππ‘π‘ − = (ππ − ππ) + , (∗) ........................................................................................................................... πππ‘π‘ πππ‘π‘−1 πππ‘π‘−1 πππ‘π‘ ........................................................................................................................... or, defining ππ ≡ π΅π΅⁄ππ, solving for πππ‘π‘ and assuming that the primary deficit-toGDP ratio remains constant over time, so that (πΊπΊπ‘π‘ − πππ‘π‘ )⁄πππ‘π‘ ≡ ππ ∀π‘π‘, ........................................................................................................................... πππ‘π‘ = (1 + ππ − ππ)πππ‘π‘−1 + ππ. (∗∗) ........................................................................................................................... In the (πππ‘π‘−1 , πππ‘π‘ ) plane, (∗∗) is the equation of a straight line with vertical in........................................................................................................................... tercept equal to ππ and slope (1 + ππ − ππ). Being ππ > 0, this line – the ........................................................................................................................... "debt/GDP line" in the figure – crosses the vertical axis for the positive value ........................................................................................................................... that, by assumption, the primary deficit-to-GDP ratio takes on in the economy under consideration. 406 Government debt and economic growth ........................................................................................................................... Furthermore, its slope is positive, but smaller than one (for realistic values of ππ and ππ, the term 1 + ππ − ππ is, as we shall always assume, greater than zero ........................................................................................................................... even when ππ < ππ), and will therefore cross the straight, 45° line going through ........................................................................................................................... the origin drawn in the figure for a positive value, let's call it πποΏ½, of the debt-toGDP ratio. ........................................................................................................................... To understand how the figure we have just drawn helps determine the dynam........................................................................................................................... ics of the debt-to-GDP ratio starting from any value of the same ratio inherit........................................................................................................................... ed from the past, let us suppose that we are at time π‘π‘ = 1 and that, at time zero, the debt-to GDP ratio was ππ0 . One uses the debt/GDP line to read, on the vertical axis, the implied value for the debt ratio that will prevail today, ππ1 . At π‘π‘ = 2, the value of ππ1 so determined becomes the debt ratio inherited from the previous period, and can be used to determine ππ2 in the very same way we used ππ0 to determine ππ1 . In particular, using the 45° line we can transfer ππ1 from the vertical to the horizontal axis, and then use the debt/GDP line to read, on the vertical axis, the implied value for ππ2 . Proceeding in a similar fashion time after time, one can determine the whole sequence of the values that ππ will take on from today onwards. As clear from the figure, in the case under consideration the debt-to-GDP ratio increases over time, but at a decreasing rate, and ends up converging to the constant value πποΏ½. When ππ takes on this latter value, corresponding to the intersection between the debt/GDP line and the 45° line, today's ππ is the same as yesterday's, and the debt ratio will remain constant at this common vaue over time. Because of this, πποΏ½ is referred to as the steady state value of the debt-toGDP ratio. Analytically, its expression can be derived by setting πππ‘π‘ = πππ‘π‘−1 = πποΏ½ in equation (∗) or equation (∗∗), and solving, to get: πποΏ½ = ππ . ππ − ππ To understand why, in this economy, ππ converges to a constant value, remember that the budget constraint of the government implies that, were the primary budget balanced, the stock of debt would grow at the rate ππ. Since income ππ grows at the rate ππ, and given that in this economy ππ < ππ by assumption, if the primary budget were balanced the debt ratio π΅π΅/ππ would converge to zero. However, by assumption ππ is not equal to zero, but positive; it follows that ππ will not converge to zero, but to that positive value such that the increase in the debt ratio due to the fact that ππ, the second term on the right-hand side of equation (∗), is positive is exactly offset by the decrease in the debt ratio due to the first term (remember that ππ < ππ). 407 Macroeconomics. Problems and Questions b. ππ > ππ and ππ > 0. π΅π΅π‘π‘ πππ‘π‘ debt/GDP line 45° ππ2 ππ1 πποΏ½ ππ ππ0 ππ1 ππ2 π΅π΅π‘π‘−1 πππ‘π‘−1 In this case, ππ tends to +∞ – the dynamics of the debt ratio is unsustainable – for any positive value of the debt ratio inherited from the past (that is, for any ππ0 > 0). In fact, being ππ > ππ, ππ would grow over time even if the primary budget were balanced; since ππ > 0 (the government is running primary deficit), it will grow at an even faster rate. c. ππ > ππ and ππ < 0. ........................................................................................................................... In this economy, the fact that ππ exceeds ππ tends to raise the debt ratio over ........................................................................................................................... time, while the primary surplus (ππ < 0) tends to lower it. As clear from equation (∗), which of the two, opposing forces will prevail depends on the value of the ratio π΅π΅π‘π‘−1 ⁄πππ‘π‘−1 inherited from the past. As shown in the next figure, if this ratio is high, greater than the steady state value of the debt ratio, ππ will keep rising over time, while it will keep shrinking over time if its initial level, ππ0 , is ‘small’, less than the steady state value πποΏ½. 408 Government debt and economic growth debt/GDP line π΅π΅π‘π‘ πππ‘π‘ ππ0′ πποΏ½ ππ 45° π΅π΅π‘π‘−1 πππ‘π‘−1 ππ0 d. ππ < ππ and ππ < 0. In this final case, ππ converges to a negative steady state value [to check your understanding of the analysis carried out in this question, use equation (∗) to explain why] – in the steady state, the government will be a creditor. This, of course, provided that it will keep running primary surpluses, and will not transform – maybe well before ππ has turned negative – those surpluses into deficits, eventually putting an end to the tendency of the debt ratio to fall over time. π΅π΅π‘π‘ πππ‘π‘ πποΏ½ 45° debt/GDP line ππ0 ππ 409 π΅π΅π‘π‘−1 πππ‘π‘−1 Macroeconomics. Problems and Questions Question 2 a. Consider a country that, in period π‘π‘ = 1, inherits from the previous period, π‘π‘ = 0, a debt-to-GDP ratio of 100%: ππ0 ≡ π΅π΅0 ⁄ππ0 = 1. Moreover, the real interest rate and the rate of economic growth are constant and equal to 3% and to 5%, respectively (ππ = 0.03, ππ = 0.05). Finally, the ratio of the primary deficit to GDP is 4%, assumed to be constant over time. Write down the equation that gives the dynamics of the debt ratio ππ (≡ π΅π΅⁄ππ) and use it to calculate the value of the debt-to-GDP ratio at times π‘π‘ = 1 and π‘π‘ = 2, and the steady state level of ππ, πποΏ½. Will the debt ratio diverge over time, or converge to its steady state value? Why, or why not? The equation that allows us to analyze the change of the debt ratio over time is: π΅π΅π‘π‘−1 πΊπΊπ‘π‘ − πππ‘π‘ π΅π΅π‘π‘ π΅π΅π‘π‘−1 − = (ππ − ππ) + , πππ‘π‘ πππ‘π‘−1 πππ‘π‘−1 πππ‘π‘ which, defining ππ ≡ π΅π΅⁄ππ, can be equivalently written as: πππ‘π‘ = (1 + ππ − ππ)πππ‘π‘−1 + ππ, where (πΊπΊπ‘π‘ − πππ‘π‘ )⁄πππ‘π‘ (≡ ππ) is the ratio of the primary deficit to GDP. Using in the latter equation the quantitative information we have been provided with, it is straightforward to compute ππ1 = 1.02, ππ2 = 1.0396 e πποΏ½ = 2. The debt ratio clearly converges over time to its steady state value. Indeed, since in this economy ππ < ππ, the debt ratio would decrease over time towards zero, were ππ = 0. Given that, however, ππ is positive (the government is running a primary deficit), ππ increases, but at a decreasing rate, tending over time to its steady state level (πποΏ½ = 2). 410 Government debt and economic growth b. Suppose now that the government intends to stabilize ππ at the value observed at π‘π‘ = 0. In other words, the government wants ππ to continue to take on the value 1 both in period π‘π‘ = 1 and in all subsequent periods. To achieve this goal, the Government is considering the possibility of generating a permanent change in the ratio of the primary deficit to GDP. Compute the value that this ratio should take on to stabilize ππ at the value 1 forever, and explain whether it implies that the Government should implement a restrictive or an expansionary fiscal policy. Since πππ‘π‘ = (1 + ππ − ππ)πππ‘π‘−1 + ππ, in order to stabilize at 1 the debt-to-GDP ratio at time π‘π‘ = 1 – that is, in order to make ππ1 = ππ0 – the government must set ππ at the unique value that solves: 0 = (ππ − ππ)ππ0 + ππ. Since ππ = 0.03, ππ = 0.05, and ππ0 = 1, in this economy the above condition becomes 0 = − 0.02 · 1 + ππ. Solving, we get ππ = 0.02 – the government will have to lower the primary deficit-to-GDP ratio from 4% to 2%. This of course requires a restrictive fiscal policy. 411 Macroeconomics. Problems and Questions Question 3 At time π‘π‘, a country inherits from the previous period a stock of government 0 debt greater than zero, corresponding to the debt-to-GDP ratio πππ‘π‘−1 in the graph below. Assuming that the real interest rate is smaller than the rate of growth of the economy, and that the government runs a primary surplus, a. show in the graph the steady state debt-to-GDP ratio in this economy and explain if, absent any intervention, the debt-to-GDP ratio of the country will converge or not to this stationary level; π΅π΅π‘π‘ πππ‘π‘ πποΏ½ debt/GDP line ππ′ ππ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 The steady state debt-to-GDP ratio, πποΏ½ < 0 in the graph, is that value of π΅π΅⁄ππ for which the debt/GDP line (the continuous, bold line in the figure) crosses the 45° line going through the origin. The debt-to-GDP ratio will converge to that steady state following the arrowed path in the graph. 412 Government debt and economic growth b. show in the graph the change in the ratio between the primary balance 0 and GDP needed to stabilize the debt-to-GDP ratio at the value πππ‘π‘−1 from time π‘π‘ onwards. Explain. 0 To stabilize the debt-to-GDP ratio at the level πππ‘π‘−1 , at time π‘π‘ the government will have to run a primary deficit. As a ratio to GDP, the primary balance will have to go from the value ππ < 0 (primary surplus) in the graph to a level ππ′ > 0 (primary deficit) such that the new debt/GDP line (the bold, dashed line in the figure) crosses the 45° line exactly for a debt-to-GDP ratio equal 0 to πππ‘π‘−1 . 413 Macroeconomics. Problems and Questions Question 4 a. Consider the following graph: π΅π΅π‘π‘ πππ‘π‘ debt/GDP line 45° πππ‘π‘+1 πππ‘π‘ ππ 0 πππ‘π‘−1 πππ‘π‘ πππ‘π‘+1 π΅π΅π‘π‘−1 πππ‘π‘−1 In the figure, the bold straight line, that gives the time π‘π‘ debt/GDP ratio as a function of the same ratio in the previous period, is parallel to the 45° line going through the origin. In this economy, what is the relative size of the growth rate of real GDP and of the real interest rate? Is the government running a primary deficit or a primary surplus? Explain. The equation of the debt/GDP line is: π΅π΅π‘π‘ π΅π΅π‘π‘−1 πΊπΊπ‘π‘ − πππ‘π‘ = (1 + ππ − ππ) + . πππ‘π‘ πππ‘π‘−1 πππ‘π‘ Its vertical intercept is therefore (πΊπΊπ‘π‘ − πππ‘π‘ )/πππ‘π‘ (≡ ππ) and its slope 1 + ππ − ππ. From the graph it is easy to conclude that the vertical intercept is positive; it follows that ππ > 0 (the government is running a primary deficit). Moreover, since the slope of the line is 45°, we can conclude that 1 + ππ − ππ = 1, so that ππ = ππ. 414 Government debt and economic growth 0 , show in the b. Assuming that the debt-to-GDP ratio at time π‘π‘ − 1 was πππ‘π‘−1 graph the values that this ratio will take on at times π‘π‘ and π‘π‘ + 1. Will the debt-to-GDP ratio converge to a steady state value πποΏ½? If not, why? If yes, show in the figure this steady state value and explain whether it is stable (that is, if ππ will converge to it independently of the value of the debt-to0 ) or unstable (in which case, ππ will GDP ratio inherited from the past, πππ‘π‘−1 0 take on the steady state value if and only if πππ‘π‘−1 = πποΏ½). ........................................................................................................................... Since in this economy ππ = ππ, the dynamics of debt/GDP ratio is given by the ........................................................................................................................... equation: ........................................................................................................................... π΅π΅π‘π‘ ⁄πππ‘π‘ − π΅π΅π‘π‘−1⁄πππ‘π‘−1 = ππ. ........................................................................................................................... Being ππ > 0, absent any policy intervention the debt/GDP ratio will keep ris........................................................................................................................... ing over time, and therefore will not converge to a stationary value (in the economy under consideration, a steady state value of the debt/GDP ratio does ........................................................................................................................... not exist). .............................................................................................................. 415 Macroeconomics. Problems and Questions Question 5 a. Consider a country that, in period π‘π‘ = 1, inherits from the previous period, π‘π‘ = 0, a debt-to-GDP ratio of 60%: ππ0 ≡ π΅π΅0 ⁄ππ0 = 0.6. In addition, the real interest rate and the rate of economic growth are constant and equal to 6% and to 0%, respectively (ππ = 0.06, ππ = 0). Finally, the ratio of the primary deficit to GDP is 3%. Write down the relation that describes the dynamics of the debt ratio (the government budget constraint), and use it to compute the debt-to-GDP ratio ππ for times π‘π‘ = 1 and π‘π‘ = 2. Will this ratio converge over time to a steady state value πποΏ½ > 0? Why, or why not? Represent this specific case in the graph that one uses to study the evolution of the debt ratio over time. π΅π΅π‘π‘ πππ‘π‘ 0.03 45° ππ0 ππ1 ππ2 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... The dynamics of the debt ratio in this economy is the one implied by the equa........................................................................................................................... tion πππ‘π‘ = (1 + ππ − ππ)πππ‘π‘−1 + ππ, where ππ ≡ π΅π΅⁄ππ, and ππ is the ratio of the primary deficit to GDP. Given the numerical values that ππ, ππ, ππ and ππ0 take on ........................................................................................................................... in this case, it is easy to compute ππ1 = 0.666, ππ2 = 0.736. The debt ratio ........................................................................................................................... does not converge to a steady state. Since ππ > ππ and ππ > 0, in the graph the . debt/GDP line does not intersects the 45-degree line in the first quadrant [its slope is 1.06 (> 1) ]. Being ππ0 positive, ππ tends to +∞. 416 Government debt and economic growth b. Suppose now that, rather than a deficit, the same country runs a primary surplus of 6% of GDP. How would your answer to the previous point of this question change? Compute the debt ratio at times π‘π‘ = 1 and π‘π‘ = 2, and its steady state value, πποΏ½. Will the debt ratio converge to πποΏ½? Explain, using the graph below to motivate your conclusions. π΅π΅π‘π‘ πππ‘π‘ 45° −0.06 ππ0 πποΏ½ = 1 π΅π΅π‘π‘−1 πππ‘π‘−1 ........................................................................................................................... If ππ = −0.06, following the same steps described above one can compute ππ1 = ........................................................................................................................... 0.576, ππ2 = 0.551. The steady state debt ratio is in this case: ........................................................................................................................... −ππ 0.06 ........................................................................................................................... πποΏ½ = = = 1. (ππ − ππ) 0.06 ........................................................................................................................... This steady state value will never be reached: ππ tends to decrease over time, ........................................................................................................................... since ππ0 < 1 and the slope of the debt/GDP line is still 1.06 (> 1). Notice that, in this economy, the fact that ππ > ππ tends, other things the same, to raise ........................................................................................................................... ππ over time; on the other hand, the primary surplus tends to lower it. Given ........................................................................................................................... the magnitude of the primary surplus, in this example the second of the two forces just mentioned prevails, and ππ ends up decreasing over time. ........................................................................................................................... 417 Macroeconomics. Problems and Questions Question 6 We are at time π‘π‘. The economy inherits from the previous period a debt-to0 GDP ratio πππ‘π‘−1 . Consider the following graph: 45° New debt/GDP line π΅π΅π‘π‘ πππ‘π‘ Debt/GDP line πποΏ½ 0 πππ‘π‘−1 π΅π΅π‘π‘−1 πππ‘π‘−1 a. In this economy, is the GDP growth rate higher or lower than the real interest rate? Is the government running a primary surplus or a primary deficit? Finally, is the debt ratio πποΏ½ a stable or an unstable steady state? Explain. The vertical intercept of the debt/GDP line is equal to the primary deficit-toGDP ratio. Since, in this economy, that intercept is greater than zero, the government is running a primary deficit. Furthermore, being the debt/GDP line flatter than the 45° one going through the origin, ππ < ππ (recall that the slope of the debt/GDP line is 1 + ππ − ππ, and that this slope is less than one in this case). It is straightforward to verify that the steady state πποΏ½ is stable – the debt-to-GDP ratio will always converge to πποΏ½, independently of the value of the same ratio inherited from the past. ........................................................................................................................... 418 Government debt and economic growth Finally, the steady state οΏ½ππ is unstable. As one can see from the graph, given any initial value of the debt ratio different from οΏ½ππ , the economy will feature a π΅π΅/ππ tending to +∞ if the initial value of the ratio is greater than the steady state one, or to −∞ in the opposite case. b. Suppose now that, at time π‘π‘, the Government intends to stabilize the debt0 to-GDP ratio at the level πππ‘π‘−1 by changing the economy’s growth rate. To achieve its aim, should it attempt to increase or to decrease that growth rate? Show how your answer is going to affect the graph used to answer the previous question, and discuss. 0 Absent policy interventions, and starting from πππ‘π‘−1 , the debt-to-GDP ratio 0 for this economy would converge to the steady state πποΏ½. To keep it at πππ‘π‘−1 by changing the economy’s growth rate, the government should lower ππ, so as to 0 lead to a new, steeper debt/GDP line crossing the 45° one for πππ‘π‘−1 , thus making this latter the new steady state value of the debt-to-GDP ratio. Notice that the debt/GDP line will rotate around an unchanged vertical intercept, since the primary deficit-to-GDP ratio has remained constant. 419 Macroeconomics. Problems and Questions * Question 7 a. Consider a country with a zero primary deficit-to-GDP ratio (ππ = 0), and in which the real interest rate and the rate of growth of the economy are constant and equal to ππ = ππΜ and ππ = ππΜ , respectively, with ππΜ < ππΜ and 1 + ππΜ − ππΜ > 0. Write down the equation that gives the dynamics of the debt-toGDP ratio for this economy (the government budget constraint) and derive the steady state value of that ratio. Assuming that at time π‘π‘ the econ0 > 0, explain omy inherits from the past a debt-to-GDP ratio equal to πππ‘π‘−1 if, and why, the economy will ever converge to that steady state. Use the graph below to motivate your answer. 45° π΅π΅π‘π‘ πππ‘π‘ 0 . 0 πππ‘π‘−1 debt/GDP line π΅π΅π‘π‘−1 πππ‘π‘−1 Being ππ = 0, the dynamics of the debt-to-GDP ratio is given by the equation πππ‘π‘ = (1 + ππΜ − ππΜ )πππ‘π‘−1 , the bold line in the figure above. As clear from the graph, in this economy there is just one steady state debt-to-GDP ratio, πποΏ½ = 0. Since, by assumption, ππΜ < ππΜ , the economy will converge to this value of ππ starting from any state debt-to-GDP ratio inherited from the past. 420 Government debt and economic growth b. Suppose now that the interest rate at which the government can borrow is no longer equal to ππΜ , but rather to ππΜ + π£π£πππ‘π‘−1 , where π£π£ is a positive parameter. In other words, the real interest rate is no longer constant, but increasing in the debt-to-GDP ratio (ππ) prevailing in the last period. The growth rate of the economy is however stiIl constant and, as before, ππΜ < ππΜ and 1 + ππΜ − ππΜ > 0. Repeat the analysis carried out in order to answer the previous point of this question. In particular, derive the expression of the steady state debt-to-GDP ratio, explain if and why ππ will converge to a steady state, and represent graphically. π΅π΅π‘π‘ πππ‘π‘ 0 debt/GDP line 0 πππ‘π‘−1 00 πποΏ½ πππ‘π‘−1 45° π΅π΅π‘π‘−1 πππ‘π‘−1 Since now ππ = ππΜ + π£π£πππ‘π‘−1 , the evolution of the debt-to-GDP ratio over time is given by the equation πππ‘π‘ = (1 + ππΜ + π£π£πππ‘π‘−1 − ππΜ )πππ‘π‘−1 2 , = (1 + ππΜ − ππΜ )πππ‘π‘−1 + π£π£πππ‘π‘−1 the bold curve in the figure. As usual, to find the steady state values of the debt-to-GDP ratio one has to set πππ‘π‘ = πππ‘π‘−1 in the equation above, and then solve. There are now two steady state values of ππ (two intersections between the parabola and the 45° line): zero, as before, and the new steady state πποΏ½ = (ππΜ − ππΜ )⁄π£π£ (a positive quantity, given the hypotheses on the values the parameters take on in this economy). If the initial debt-to-GDP ratio happens to be less than πποΏ½, as before there will be convergence to the steady state πποΏ½ = 0. But if the initial debt-to-GDP ratio is greater than πποΏ½, then ππ will tend to +∞. In fact, ππ = ππΜ + π£π£πππ‘π‘−1 and πππ‘π‘−1 > πποΏ½ = (ππΜ − ππΜ )⁄π£π£ imply ππ > ππ and therefore, with a zero primary deficit, an unsustainable dynamics of the debt-to-GDP ratio. 421 Macroeconomics. Problems and Questions Question 8 a. Consider the Solow growth model without technological progress. The 3 1 production function is ππ = πΎπΎ οΏ½4 ππ οΏ½4 and the rate of depreciation, πΏπΏ, is equal to 0.1. Calculate the propensity to save π π for which the steady state level of capital per worker is 100. Represent graphically this steady state. )∗′ (ππ⁄ππ (ππ⁄ππ)∗ πΈπΈ πΈπΈ ′ πΏπΏ · (πΎπΎπ‘π‘ ⁄ππ) ππ(πΎπΎπ‘π‘ ⁄ππ) ′ π π · ππ(πΎπΎπ‘π‘ ⁄ππ) π π · ππ(πΎπΎπ‘π‘ ⁄ππ) (πΎπΎ/ππ)∗(πΎπΎ ⁄ππ)∗′ πΎπΎ ⁄ππ When the steady state level of capital per worker is 100, steady state output per worker is: 3οΏ½ 4 ππ ∗ πΎπΎ ∗ οΏ½ οΏ½ = οΏ½οΏ½ οΏ½ οΏ½ ππ ππ = 100 3οΏ½ 4. In steady state, a situation in which capital and output per worker are constant, saving and investment per worker (recall that, in a closed economy goods market equilibrium, saving equals investment) are just enough to cover depreciation, and capital and output per worker are constant. 422 Government debt and economic growth ........................................................................................................................... It follows that, in steady state: ........................................................................................................................... ππ πΎπΎ ........................................................................................................................... π π = πΏπΏ , ππ ππ ........................................................................................................................... a condition that, in the economy we are studying, can be written as: ........................................................................................................................... 3 π π · 100 οΏ½4 = 0.1 · 100. Solving, π π β 0.32. b. Assuming that the economy is initially in the steady state just described, explain and represent graphically what happens to capital per worker, output per worker and the growth rate of the economy following a reduction in the marginal propensity to consume. A decrease in the marginal propensity to consume is equivalent to an increase in the savings rate. Assuming that this latter rises from π π to π π ′ > π π , in the graph the curve which represents saving/investment per worker as a function of capital per worker shifts upwards. At point E, investment per worker now exceeds depreciation per worker, and capital per worker starts to increase. This process continues until the economy reaches the new steady state E’, where both capital per worker and output per worker are once again constant (i.e., the growth rate of the economy is zero), but at a level higher than before. 423 Macroeconomics. Problems and Questions Question 9 a. Consider the Solow growth model without technological progress. The 1 1 production function is ππ = πΎπΎ οΏ½2 ππ οΏ½2 and the rate of depreciation is πΏπΏ = 0.05. Calculate the propensity to save π π for which the steady state level of capital per worker is 200. Represent graphically this steady state. (ππ⁄ππ )∗ πΈπΈ (πΎπΎ/ππ)∗ = 200 πΏπΏ · (πΎπΎπ‘π‘ ⁄ππ) ππ(πΎπΎπ‘π‘ ⁄ππ) π π · ππ(πΎπΎπ‘π‘ ⁄ππ) πΎπΎ ⁄ππ ........................................................................................................................... ........................................................................................................................... In steady state, output and capital per worker are constant, and investment per worker is equal to the depreciation per worker − that is, the following ........................................................................................................................... condition holds: ........................................................................................................................... πΎπΎ πΎπΎ π π · ππ οΏ½ οΏ½ = πΏπΏ , ........................................................................................................................... ππ ππ 1 ........................................................................................................................... where ππ(πΎπΎ ⁄ππ) = ππ⁄ππ = (πΎπΎ ⁄ππ) οΏ½2 . By substituting the values the parameters take on in this economy, and impos........................................................................................................................... 1 ing πΎπΎ ⁄ππ = 200, one gets π π · 200 οΏ½2 = 0.05 · 200. Solving for π π , the value of ........................................................................................................................... the saving rate we are looking for is π π ∗ β 0.71. . 424 Government debt and economic growth b. Assuming that the economy is initially in the steady state just described, explain, and show in the graph, what happens to capital per worker, output per worker and the growth rate of the economy following an increase in the rate of depreciation, πΏπΏ. πΏπΏ ′ · (πΎπΎπ‘π‘ ⁄ππ) (ππ⁄ππ)∗ (ππ⁄ππ)∗′ πΈπΈ πΈπΈ ′ (πΎπΎ ⁄ππ)∗′ (πΎπΎ/ππ)∗ πΏπΏ · (πΎπΎπ‘π‘ ⁄ππ) ππ(πΎπΎπ‘π‘ ⁄ππ) π π · ππ(πΎπΎπ‘π‘ ⁄ππ) πΎπΎ ⁄ππ Following the increase in δ, the line representing depreciation per worker becomes steeper. At point E, the initial steady state, investment per worker is now smaller than depreciation per worker, and capital per worker starts to fall. This process continues until the economy reaches the new steady state E’, where both capital per worker and output per worker are once again constant (i.e., the growth rate of the economy is zero), but at a level lower than before. 425 Macroeconomics. Problems and Questions Question 10 The government of a closed economy, whose budget was previously balanced, starts running a budget deficit equal to the percentage ππ of the country’s GDP, with 0 < ππ < π π , where π π is the private saving rate. a. Assuming that there is no technological progress and that both the private saving rate and the population of the country are constant, show in the graph below the impact of the emergence of a budget deficit (that is, of the increase in ππ from zero to a positive value) on output per worker and capital per worker in the Solow growth model. Explain. )∗ (ππ⁄ππ (ππ⁄ππ)∗′ πΈπΈ ′ πΈπΈ (πΎπΎ ⁄ππ)∗′ (πΎπΎ/ππ)∗ πΏπΏ · (πΎπΎπ‘π‘ ⁄ππ) ππ(πΎπΎπ‘π‘ ⁄ππ) π π · ππ(πΎπΎπ‘π‘ ⁄ππ) (π π − ππ) · ππ(πΎπΎπ‘π‘ ⁄ππ) πΎπΎ ⁄ππ With a government that needs not to balance its budget, but that can now run surpluses or deficits, national saving is, in this economy, the following func................................................................................................................. tion of output: ................................................................................................................. πππ‘π‘ = π π πππ‘π‘ − πππππ‘π‘ , where π π πππ‘π‘ is private saving, and (−πππππ‘π‘ ) public (government) saving. .................. 426 Government debt and economic growth ................................................................................................................. Since, as always, πΌπΌπ‘π‘ = πΎπΎπ‘π‘+1 − πΎπΎπ‘π‘ + πΏπΏπΎπΎπ‘π‘ , by substituting in the goods market equilibrium condition (πππ‘π‘ = πΌπΌπ‘π‘ ) the expressions for ππ and πΌπΌ written above, dividing by the number of workers ππ, and finally rearranging terms, one gets: ......... ................................................................................................................. πΎπΎπ‘π‘ πΎπΎπ‘π‘ πΎπΎπ‘π‘+1 πΎπΎπ‘π‘ − = (π π − ππ)ππ οΏ½ οΏ½ − πΏπΏ , (∗) ππ ππ ππ ππ ................................................................................................................. where ππ(πΎπΎ ⁄ππ) = ππ ⁄ ππ. The first and the second term on the right-hand side of the equilibrium condition (∗) are drawn in the graph above. ........................... Initially, ππ = 0 (the government budget is balanced) and the economy is in ................................................................................................................. the steady state equilibrium E, with capital per worker (πΎπΎ ⁄ππ)∗ and output per worker (ππ⁄ππ)∗ . When ππ rises from zero to a positive value, the effect on the ................................................................................................................. economy is similar to that of a fall in the private saving rate, π π : capital and ................................................................................................................. output per worker start to decrease, and the economy converges over time to a new steady state (point E’ in the graph) where both capital and output per ................................................................................................................. worker are permanently lower. π‘π‘ π‘π‘ ................................................................................................................. b. Will the budget deficit permanently affect the growth rate of the economy? Explain. The fact that now the government runs a budget deficit leads to a permanent decrease in the saving rate of the economy. We know that, in the Solow model: .......................................................................................................................... 1) a change in the saving rate changes, temporarily and in the same direction, ........................................................................................................................... the growth rate of the economy; ........................................................................................................................... 2) however, the change in the saving rate does not affect the growth rate of output per worker in the long run; given the assumptions made about the ........................................................................................................................... economy, this growth rate will always be zero in the steady state, no matter ........................................................................................................................... what the saving rate is (and therefore, no matter whether the government is running a budget deficit, a surplus, or has a balanced budget). ........................................................................................................................... We therefore conclude that the deficit does not change the rate at which this ........................................................................................................................... economy will grow in the long run − this growth rate will remain equal to ze.......................................................................................................................... ro. However, as discussed before, the budget deficit causes the economy to converge to a new steady state where the levels of capital per worker and out.......................................................................................................................... put per worker will be permanently lower. 427 Macroeconomics. Problems and Questions Question 11 True or False? Explain whether the following statements are true or false. Motivate your answer in a brief but rigorous way, by making explicit reference to the relevant theory. Lack of proper explanations will result in zero points. a. “From the Solow model without technological progress it follows that, as the saving rate s increases from its minimum value (0) to its maximum value (1), steady-state capital per worker and output per worker first increase and then decrease”. The statement is false. As it can be verified for instance by using the graph employed in the Solow model to analyse the evolution of capital per capita ad income per capita over time, an increase in the saving rate always leads to an increase in the steady state levels of capital and output per capita. [To first increase and then decrease as the saving rate increases from 0 to 1 is the steady state level of consumption, and not capital or output per capita]. 428 Government debt and economic growth b. Using the Solow model without technological progress and assuming a constant population (πππ΄π΄ = ππππ = 0), explain if and why you agree, or do not agree, with the following statements: b.1 b.2 an increase in the saving rate π π will always raise steady state output per worker; an increase in the saving rate π π will always raise steady state consumption per worker. b.1 True. An increase in the saving rate leads to a higher investment per worker, and thus to an increase in capital per worker ΜΆ at least for a while, that is, during the transition to the new steady state. Since output per worker is an increasing function of capital per worker, an increase in π π will lead to a steady state in which both capital and output per worker are permanently higher. b.2 False. An increase in the saving rate will increase consumption per worker only if, in the initial steady state, capital per worker was less than the golden rule level (the ratio K/N that maximizes steady state consumption per worker). In the opposite case, an increase in the saving rate will reduce consumption per worker in the steady state. 429 Macroeconomics. Problems and Questions Question 12 a. Using the Solow model with technological progress, and assuming the economy was initially in a steady state equilibrium, study in the graph below the effects of a decrease in the saving rate on capital and output per effective worker, briefly explaining the reasons for the observed changes in these variables. (ππ/ππππ) [πΏπΏ + πππ΄π΄ + ππππ ] · ( πΎπΎ ) ππππ πΎπΎ π π · ππ( ) ππππ πΎπΎ π π ′ · ππ( ) ππππ ππ( ∗ (ππ/ππππ)∗′ πΈπΈ ′ πΈπΈ (πΎπΎ/ππππ)∗ ′ (πΎπΎ/ππππ)∗ πΎπΎ ) ππππ πΎπΎ⁄ππππ The saving rate decreases from π π to π π ′ < π π . Since in goods market equilibrium savings equal investment, a lower saving rate implies less investment per effective worker. Since at point E, the initial steady state, investment per effective ........................................................................................................................... worker was at the level needed to keep πΎπΎ ⁄ππππ constant, the reduction in investment caused by the decrease in the saving rate implies that now πΎπΎ⁄ππππ − ........................................................................................................................... and, with it, ππ⁄ππππ − will start to decrease. Over time, the economy will con........................................................................................................................... verge to a new steady state (point E’ in the graph) in which both capital and income per effective worker are permanently lower, and their rate of growth is ........................................................................................................................... once again zero (and the growth rate of these variables measured in ‘per ........................................................................................................................... worker’ − or ‘per capita’ − terms is equal to the rate of technological progress, as in the initial steady state). ........................................................................................................................... 430 Government debt and economic growth b. How would capital and output per effective worker change following an increase in the rate of technological progress? (ππ/ππππ) ∗ (ππ/ππππ)∗′ πΎπΎ [πΏπΏ + πππ΄π΄′ + ππππ ] · οΏ½ οΏ½ ππππ πΈπΈ πΈπΈ ′ (πΎπΎ/ππππ)∗ ′ (πΎπΎ/ππππ)∗ [πΏπΏ + πππ΄π΄ + ππππ ] · ( πΎπΎ ) ππππ πΎπΎ π π · ππ( ) ππππ ππ( πΎπΎ ) ππππ πΎπΎ⁄ππππ If, when the economy was in a steady state like point E in the graph above, the .................................................. rate of technological progress rises from πππ΄π΄ to πππ΄π΄′ > πππ΄π΄ , investment per effective worker falls below the level needed to keep capital per effective worker constant over time. Exactly as discussed before in connection with a fall in the saving rate, capital and output per effective worker will fall for some time, until the economy reaches a new steady state (point E’) in which both variables are once again constant, but at a level permanently lower than in the initial steady state. Notice that, in this economy, the balanced growth path (along which all variables in ‘per worker’, or ‘per capita’, terms grow at the rate of technological progress) is now steeper. Take, for instance, output per worker. Although ππ⁄ππ π΄π΄ is lower in the new steady state, the growth rates of ππ and of ππ⁄ππ will be higher – since, in balanced growth, ππππ = ππππ + πππ΄π΄ , ππππ⁄ππ = πππ΄π΄ and πππ΄π΄ has gone up. 431 Macroeconomics. Problems and Questions Question 13 Consider the Solow model, assuming positive rates of technological progress and population growth, so that πππ΄π΄ > 0 and ππππ > 0. a. Assuming the usual aggregate production function, and denoting by π π the saving rate and by δ the depreciation rate, represent in the graph below the steady state, or state of balanced growth, of the economy. Clearly indicate the levels of output per effective worker, investment per effective worker and consumption per effective worker in this steady state. πΎπΎ [πΏπΏ + πππ΄π΄ + ππππ ] · ( ) ππππ πΎπΎ ππ( ) ππππ (ππ/ππππ)∗ πΎπΎ π π · ππ( ) ππππ (πΆπΆ/ππππ)∗ (πΌπΌ/ππππ)∗ (πΎπΎ/ππππ)∗ πΎπΎ ⁄ππππ In the graph, the steady state value of the variables is indicated with an asterisk. 432 Government debt and economic growth b. Explain if and why you agree, or do not agree, with the following statements: b.1 b.2 an increase in the saving rate leads to a new balanced growth path characterized by a higher level of output per worker, but an unchanged growth rate of ππ⁄ππ; the Solow model implies that, if the growth rate of population ππππ increases, the economy will reach a new steady state where the growth rate of aggregate output ππ is unchanged, and the growth rate output per worker ππ⁄ππ is lower. b.1 True. In balanced growth, ππ⁄ππππ = (ππ⁄ππ)/π΄π΄ is constant. Therefore, in balanced growth ππ⁄ππ must be growing at the rate πππ΄π΄ , the growth rate of technological progress. This rate is not affected by changes in π π . An increase in π π leads to an increase in the steady state levels of capital and output per effective worker, though, as you can verify from the graph above. During the transition towards this higher level of output per unit of effective labor, output per worker grows temporarily at a rate greater than πππ΄π΄ . Once the new steady state has been reached, ππ⁄ππ will remain at a level higher than the one prevailing before the increase in the saving rate. b.2 False. As already noted, in balanced growth ππ⁄ππππ = (ππ⁄ππ)/π΄π΄ is constant. This implies that, in the steady state, ππ will grow at a rate equal to πππ΄π΄ + ππππ . This rate is increasing in ππππ . Moreover, in balanced growth ππ⁄ππ grows at the rate πππ΄π΄ , which does not depend on ππππ . 433 ADDITIONAL CONTENTS ONLINE You can find other problems and exercise at https://mybook.egeaonline.it To access the contents, you need to register on our website and insert the following code: 46EBOLOE73 The code has to be inserted only the first time you enter the platform Giuseppe Ferraguto MACROECONOMICS The manual includes about one hundred questions, most in multiple parts and drawn from several years of exams at Bocconi University, on the models (IS-LM, IS-LM-PC, etc.) and topics (the macroeconomic equilibrium of a closed economy, the labor market and unemployment, inflation, the open economy, government debt, economic growth) covered by most introductory courses on Macroeconomics. The main objective of the problems is to help readers grasp the economic reasoning and intuition underlying the main conclusions of the discipline – the aspect of Macroeconomics, and more in general of Economics, that students find the most difficult to master, but that will turn out to be the most useful in their future. MyBook MACROECONOMICS Problems and questions GIUSEPPE FERRAGUTO is Associate Professor of Economics at Bocconi University, and director of the course on Macroeconomics offered at the same institution. 6th EDITION MyBook http://mybook.egeaonline.it MyBook is the gateway to access accompanying resources (both text and multimedia), the BookRoom, the EasyBook app and your purchased books. tools 196-1c_2b.indd 1 Giuseppe Ferraguto 24 mm 17/01/20 12:14
