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ECON 3410/4410: Seminar exercises, spring 2006 Revised and complete version: 3 March, 2006 The zipped data sets referred to in the exercises can be downloaded from. http://folk.uio.no/rnymoen/rnyteach.html, follow the Teaching-spring-2006 link. Exercise 1 1. Consider the dynamic model made up of equation (1.3) and (1.4) in IDM. Assume that in the initial situation (in period t = 0) εd,0 = εs,0 = 0 and P0 = P̄ and X0 = X̄ where P̄ and X̄ are stationary values. (a) Assume c < −a, and draw a figure similar to figure 1.3 in IDM. Assume that εd,1 < 0 and that all other values of εd,t and εs,t are zero, ie a temporary demand shock. Use the graph to illustrate the behaviour of Pt and Xt in period t = 1, 2 and 3. What happens when t becomes really large? (b) Re-do the analysis, but with one change in the set of assumptions: that c > −a. 2. Using the same model as in question 1, illustrate the behaviour of Pt and Xt in period t = 1, 2 and 3 in the case of a permanent demand shock, i.e., εd,t < 0 for t = 1, 2, ....... Examine both constellation of slopes of the two curves. 3. Consider the model defined by equation (1.5) and (1.6) in IDM. (a) What is the expression for the slope of the long-run demand curve? (b) What happens to the slope of the long-run demand curve if b = 1? Can you think of an interpretation? (c) Assume that in the initial situation (in period t = 0) εd,0 = εs,0 = 0 and P0 = P̄ and X0 = X̄ where P̄ and X̄ are stationary values. Assume that b = 1 and c1 = 0. Assume that εs,1 > 0 and that all other values of εd,t and εs,t are zero, ie a temporary supply shock in period 1. i. What are the long-run eﬀects of this postive supply shock? ii. Show, graphically, that in period 1: X1 > X0 and P1 < P0 ; and in period 2: X2 < X1 and P2 > P1 . iii. If, instead of a < 0, we set a = 0, what are the eﬀects of the supply shock on X1 and P1 ? What about period 2 and 3? 4. Classify the following as either stock or flow variables 1 (a) The labour force (b) The rate of unemployment (c) The trade surplus (d) The money stock (e) Private consumption expenditure (f) Government dept (g) Government deficit (h) The price level (i) Inflation (j) The supply of foreign currency to the central bank. 5. Stock variables ususally change gradually: at each point in time their rate of change is finite. But some stock variables can jump from one value to another at any given point in time. In such instances the derivative of the variable with respect to time is infinite–at least in principle: with actual data (quarterly, monthly, daily, hourly) the rate of change is finite, but for the observations when jumps occur, that rate will be huge. Such stock variables are dubbed jump-variables. Modern economic theory implies that the nominal exchange rate is a jump variable. From NORKVAR.zip, which can be downloaded from the course internet page, obtain the datafile NORKVAR.xls.The dataset contains a quarterly time series CPIVAL, which is an eﬀective exchange rate index for Norway, and two bilateral exchange rates in the dataset: SPEURO and SPUSD, between kroner and Euro and kroner and USD respectively. (a) What is the diﬀerence between a bilateral exchange rate and an eﬀective exchange rate. (b) The so called base year of the series is 2001. What does that mean? (c) Are there any examples of jump-behaviour in these series? (d) Consider the correlation between CPIVAL and SPEURO on the one hand, and CPIVAL and SPEURO on the other.1 Which one is largest–and why? (e) Derive a series for the bilateral exchange rate between USD and EURO. What are the most conspicuous developments over the available period? 6. Calculate the time series of the quarterly rate of change of CPI,the oﬃcial Norwegian consumer price index (published by Statistic Norway). What is the common name of this series? Calculate also the 4-quarter rate of change in CPI. Plot the two series in a graph. Why is the annual rate of change “smoother” than the quarterly rate of change? 1 Before the introduction of the EURO (and the abolition of national currencies in most EU countries), the SPEURO series represent the exchange rate between kroner and the European Currency Unit, ECU. 2 7. Calculate also the approximate growth rate, based on the log transform of CPI (see e.g., the appendix to IDM). How good is the approximation? Exercise 2 1. In the dataset CY.xls on the web pages, there are observations of two variables C and Y . Try to decide whether the underlying relationship between these two variables are linear or log-linear. What is the approximate value of the derivative/elasticity of C with respect to Y ? 2. In this course we learn how to analyse the dynamics of linear economic models. Hence it is important to be able recognize a linear model when you see it! The key property is that the models are linear in parameters, and linear models may therefore be non-linear in the variables. For example, consider the three alternative models of the relationship between the two variables Y and X: (1) (2) (3) Y ln Y ln Y = α + βX = α + β ln X = α + βX (a) A model which is linear (also) in variables has the property that the first derivative is a constant (independent of X). Which of the three equations has this property? (b) A model which is linear (only) in parameters has the property that the first derivative is itself a function of X. Describe how the first derivatives of (1)-(3) depend on X. (c) Which equation implies a constant elasticity of Y with respect to X? (d) There are other important linear model specifications that you need to be aware of as well. Let for example Y denote the rate of inflation in %, and let X denote the rate of unemployment (also in %). What is the qualitative diﬀerence between the ‘linear Phillips curve’ in (1) and the two ‘concave Phillips curves’ given by the two alternative specifications: (4) (5) Yt = α + β ln Xt , and 1 Y = α+β ? X (e) Choose values of the coeﬃcients of equation (4) and (5) in way that makes both models consistent with the following: At a 4% initial unemployment, a 1 percentage point increase in unemployment reduces inflation by 1 percentage point. (f) Equation (5) is called the reciprocal model. Using for example the β value form question (e), sketch the Phillips curve (i.e., for diﬀerent values of U ). Discuss in class: What could become main policy issues if the reciprocal model was the correct model of inflation in an economy? 3 3. Using the variables CPIVAL, CPI and CPIKONK in the datafile NORKVAR.xls, calculate a real exchange rate index for Norway. Draw a graph of the series, call it REX for example. 4. Answer question 2-5 in the exercises to chapter 2 in IDM. 5. Consider the dynamic model made up of equation (1.3) and (1.4) in IDM. Assume that the models parameters are known with certainty. Give the necessary and suﬃcient conditions for (a) a unique solution, and (b) an asymptotically stable solution. 6. Answer question 6 of the exercises to chapter 2 in IDM. 7. Consider the model given in question 3 of Exercise 1. (a) For the case of c1 = 0, find the final equation for Xt . (b) Is the system stable? (c) How is stability aﬀected by setting i. b = 1? ii. a = 0? iii. b = 1 and a = 0? 8. Answer question 8 in the exercises to chapter 2 in IDM Exercise 3 1. Inflation is measured in diﬀerent ways, using for example diﬀerent price indices. Which operational definition of the consumer price index is used by Norges Bank (The Central Bank of Norway)? What about Bank of England? Use the internet for information! 2. A ready-made dataset is available in the file wage price prod.zip on the course page. (a) Show inflation and unemployment in a scatter plot, i.e., an empirical Phillips curve. (b) Draw a line which, intuitively, represents the average relationship between the rates of inflation and unemployment. (Hint: in GiveWin: choose Graphics properties and click 1 (sequential) regression line). (c) Are there signs of a non-linear relationship in your data set? Explain your findings. (Hint: make a scatter plot with inflation and the log of the rate unemployment). 4 (d) Are there periods (“sub samples”) where the Phillips curve “fits better” than in other periods? If so, do you have you any explanation for this phenomenon? 3. Assume that the rate of inflation is given by the augmented price Phillips curve (1) ∆pt = β 0 + β 1 Ut + α∆pt−1 + εt , t = 1, 2, ..... where the subscript t denotes time period (e.g., quarter or year) and ∆ denotes the diﬀerence operator, i.e., ∆pt ≡ pt − pt−1 where pt denotes the (natural) logarithm of the domestic price level. Ut denotes the unemployment rate (i.e., this variable is a rate, it is not log-transformed). εt is the disturbance. (a) Give (a least) one economic theoretical argument for inclusion of the lagged inflation rate on the right hand side of the equation. (b) Show that (1) is an example of an ADL model of the relationship between inflation and the of unemployment. (c) Assume that the impact multiplier of ∆pt with respect to a change in the rate of unemployment is −0.01. What does this imply for the value of β1? (d) Assume that the long run multiplier is −0.20. Using the answer to c., what is the implied value of α? (e) In your own words: explain the concept of long-run multiplier in this application. (f) A majority of modern economists now routinely sets α = 1. What is their rationale? (g) Assume that α = 1, and that the NAIRU rate of unemployment is 0.05. What is the corresponding value of β 0 ? 4. Using wage price prod.zip: is there a wage Phillips curve in this data set? Note that there are three unemployment series, you have to choose one of them for your analysis. Are there sub-periods where the relationship is more pronounced? Does it matter whether the rate of unemployment is in log or not? Exercise 4 1. Use the data in Norw wage shares.zip, and formulate a view on the following issues (a) The degree of correlation between the exposed sector wage rate and the components of the “main-course” (labour productivity and the product price) in the long-run (use a scatter plot). (b) What about the short-run correlation? 5 (c) Comment on the degree of constancy over time in the two wage shares (you may want to use the smoothed version of the series) (d) Are there any evidence that the rate of unemployment is correlated with the e-sector wage-share, and that it can explain some of the shifts in the wage-share? Note that the file Norw wage shares.txt explains the variable definitions. 2. What would you say is the Norwegian model’s counterpart to the modern concept of “core inflation”? 3. In the ECM version of the Norwegian model of inflation, with a direct link from (lagged) profitability to wage increases, there is no implied natural rate of unemployment. This is diﬀerent from the Phillips curve version of the model, and also from any other Phillips curve variant. Does this mean that if the ECM main-course model is correct, no long-run rate of unemployment exists? Discuss. 4. Answer question 1-6 in the exercises to chapter 2 in IDM. 5. Exercise 5 to chapter 18 in IAM. Exercise 5 1. Exercise 2 in chapter 15 in IAM. 2. Exercise 3 and 4 in chapter 16 in IAM. Exercise 6 Assume that the rate of inflation and the output-gap of an economy can be represented by the following two equations: (1) (2) ∆pt = as0 + asy yt + asz zs,t yt = ad0 + adp ∆pt−1 + adz zdt where the subscript t denotes time period (e.g., quarter or year) and ∆ denotes the diﬀerence operator, i.e., ∆pt ≡ pt − pt−1 where pt denotes the (natural) logarithm of the domestic price level. yt denotes the output-gap in period t (deviation from full employment output). zs,t and zd,t are catch-all indicators of important exogenous supply-side and demand-side shocks. We could have included disturbances εs,t and εd,t , but omit them for simplicity. 1. Explain, intuitively, how you would sign the two slope coeﬃcients asy and adp . 2. In (1), substitute yt by the right hand side of (2) to derive the so called final form equation for the rate of inflation, and show that it takes the form of an ADL model with two exogenous variables, zs,t and zd,t . 6 3. Once you have found the final form equation for ∆pt , and have used that equation to calculate the inflation multipliers (for example ∂∆pt+j /∂zs ), it is possible to also find the multipliers for yt by taking the derivative of (2) with respect to zs,t or zd,t . Use this method to answer the following: (a) Assume a permanent increase in zs,t . Calculate the impact multiplier, the first four cumulated dynamic multipliers and the long-run multiplier, for both the rate of inflation and for the output gap. Use the following coeﬃcient values for the calculations: asy = 0.1, asz = 0.5 and adp = −0.01. (b) Are the multipliers of y with respect to zd very diﬀerent from the multipliers in a.? 4. Try to illustrate the dynamics in a diagram with AD/AS curves (i.e., after a shift in the AS curve). 5. Using the data set in Dp.zip, investigate whether the inflation dynamics that you found in your answer to question 4 is realistic for that data set. (note: read the text file carefully, it contains explanations and essential hints!). 6. Discuss in seminar: How can the model (1)-(2) be modified so that it can (logically) accommodate more realistic inflation response to a change in zs ? 7. Exercise 3 in chapter 19 in IAM. Exercise 7 1. Exercise 3 in chapter 20 in IAM 2. Exercise 1 in chapter 20 in IAM. Exercise 8 1. Exercise 2 in chapter 21 in IAM 2. Answer the first three questions in exercise 1 in chapter 23 in IAM. 3. Using the variables CPIVAL, CPI and CPIKONK in the datafile NORKVAR.xls,calculate a real exchange rate index for Norway. Draw a graph of the series, call it REX. (a) According to an economic hypothesis called purchasing power parity (PPP), real exchange rates typically have constant means. Is there any evidence in support of PPP to be hauled from the real exchange rate REX that you have calculated? 7 (b) In macroeconomics we often make the assumption that REX is constant in each period (a stronger version of the PPP hypothesis). Comment on the realism of such an assumption. (c) Take the (natural) log of REX, and obtain its (approximate) growth rate. Comment on how much the three components have contributed to the rate of change of REX. Exercise 9 (exam, May 2004) 1. Consider the ADL model (1) yt = β 0 + β 1 xt + β 2 xt−1 + αyt−1 + εt , where εt denotes the disturbance term in period t. The other Greek letters denote coeﬃcients. Assume that xt is an exogenous variable (i.e., not influenced by yt or yt−1 ). (a) Consider a permanent shock to the exogenous variable. Give the expression for the impact multiplier, the 2nd multiplier, and the long run multiplier. (b) What is the condition for stability of (1)? (c) Figure 1 contains graphs of two solutions of an ADL equation of the type given in (1). The solutions are based on identical coeﬃcient values, and the same numbers for the exogenous variable have been used in both cases. The thicker line shows a solution where the first solution period is 1990. The thinner line has 2000 as the first solution period. Consider the following statement: The ADL model is unstable, or even explosive, since both graphs show persistent growth in yt . Do you agree? Give a brief justification for your answer. 8 2.2 y t solution, start 1990 y t solution, start 2000 2.1 yt 2.0 1.9 1.8 1990 1995 2000 time 2005 2010 Figure 1: Two solutions of an ADL equation for yt . 2. Consider the case where yt in (1) denotes the hourly wage rate in the exposed sector of an open economy, and where xt is an exogenous main-course variable. Assume that the main-course model’s hypothesis about a constant long run wage share holds. (a) Re-write the ADL model (1) as an error-correction model for the hourly wage rate. (b) Give a brief economic interpretation of wage setting in the exposed sector. 3. Make use of the AD-AS model given below. Analyse the short and long run eﬀects of a permanent reduction in the exogenous tax level. Consider both a fixed and a floating exchange rate regime. Y = C(Y − T ) + I(ρ) + G + P CA(Y, Y ∗ , σ) 0 < CY < 1, Iρ < 0, P CAY < 0, P CAY ∗ > 0, P CAσ < 0 SP σ= ∗ P ρ = i − π̄, M = L(Y, i), LY > 0, Li < 0 P i = i∗ − se (S), s0e (S) S 0 (i.e., unsigned) π = π̄ + a(Y − Ȳ ) + z, a > 0 9 Volumes Y GDP (output) Ȳ full employment trend output T net taxes G government consumption and investments Prices, interest rates i nominal interest rate ρ real interest rate σ real exchange rate S nominal exchange rate, foreign currency per unit of domestic currency P price level π inflation π̄ “core inflation” z variable representing a supply side shock. Exercise 10 (exam May/Aug 2005) 1. Assume that the rate of unemployment (Ut ) and inflation (π t ) is endogenous in the dynamic system made up of equation (1) π t = β 0 − β 1 Ut + β 2 π t−1 , β 1 ≥ 0, 0 ≤ β 2 ≤ 1. and Ut = γ 0 + γ 1 (it−1 − πt−1 ) + γ 2 (π t−1 − π ∗t−1 ), γ 2 ≥ 0, γ 1 ≥ 0, (2) where it denotes the domestic interest rate and π ∗t is the foreign rate of inflation, both taken as exogenous. It is also assumed that the nominal rate of foreign exchange is exogenous and constant (for simplicity, it is not specified as a separate variable in equation (2)). (a) Suggest an economic interpretation of equation (2). (b) Explain what is meant by a solution to the system defined by (1) and (2). (c) Show that the condition for a stable solution of the system is: (3) β 2 + β 1 γ 1 − β 1 γ 2 < 1. (d) Is a vertical long-run Phillips curve consistent with a stable solution of the system? Explain. (e) Assume that condition (3) holds. Denote the steady-state (long-run) solution of the two endogenous variables by π̄ and Ū. Draw a figure that illustrates how π̄ and Ū are determined. (f) Give the expression for Ū. How does this expression compare to your answers to question 1 (c)? 10 (g) Can inflation be controlled by monetary policy in this model? 2. Try to answer the following questions concisely but without the use of mathematics. (a) Consider an open economy of the type studied in the course. What will be the eﬀect of an increase in government spending? How does the answer depend on the exchange rate regime in this economy? (b) Explain briefly what are: i) Rational expectations, ii) The Lucas critique, and iii) Ricardian equivalence 11