Macroeconomics
Preliminary Examination
Fall 2014
Instructions: The preliminary exam has eight (8) questions. Answer two (2) questions in
Part I and one (1) question in Part II. Each question in Part I is worth twenty-five (25)
points. All parts are equally weighted. Part II is worth fifty (50) points.
PLEASE MAKE YOUR ANSWERS NEAT AND CONCISE.
Make whatever assumptions you need to answer the questions.
BE SURE TO STATE THEM CLEARLY.
Part I
2 Question I.1
(Lamar)
Consider a version of the Solow Model where households require a subsistence level of
consumption. The production function is Cobb-Douglas:
𝑌 = 𝐴𝐾 ! 𝐿!!!
as is the equation of motion for the aggregate capital stock,
𝐾 = 𝑠𝑌 − 𝛿𝐾
Labor (L) grows at a constant rate n. Labor-augmenting technical progress (A) is constant
at 1. Unlike the standard Solow Model, savings is not a constant, but depends on percapita income 𝑦 = 𝑌/𝐿.
If y is lower than a threshold level 𝑦, then the representative household does not save. Of
the income above 𝑦, the household saves a constant fraction s. Per-capita saving is
therefore given by sy = 0 if 𝑦 < 𝑦 and 𝑠𝑦 = 𝑠(𝑦 − 𝑦) if 𝑦 > 𝑦.
(a) Graph the saving rate s as a function of y.
(b) Draw the Solow diagram for this model, indicating the sign of the rate of growth in
each section of the diagram.
(c) How many steady states does the model have? (Assume that y is
not too large. Derive arrows of motion
(d) For various values of initial capital, characterize which steady
states the economy may converge to.
(e) This formulation of the Solow Model is sometimes called a “poverty
trap" model. Why? Provide the intuition for this result.
3 Question I.2
(Lamar)
Consider the classical model with perfect information, where a representative producer
1
(consumer) of good i supplies labor (in logs) according to li =
( pi − p) , where
1− γ
γ ≻ 1, pi and p are the log of Pi and P respectively.
(a) What does the labor supplied by each individual implies for the labor supply?
(b) Assume that demand for good i is represented by the following log-linear demand
function qi = y − η ( pi − p) . Find the equilibrium output for each producer.
(c) Now assume that information about prices is not perfect. Explain how labor supplied
by each individual changes and what it implies for the labor supply?
(d) Assume Aggregate Demand is represented in logs by y = m – p. What is the effect of
money in this model. Explain.
(e) Assume that the monetary rule is given by 𝑚! = 𝑚!!! + 𝑐 + 𝑢! ,
where c is the trend and ut is a white noise. Show that unexpected changes in m can
generate a tradeoff inflation-output.
4 Question I.3
(Rutledge)
A key recurring theme in the history of the macroeconomics literature is understanding
the impact of changes in money supply on output and prices. For each of the leading
thinkers listed below answer all 7 of the following questions:
(a) How would they analyze the effect of doubling the stock of money (measured in the
way each thought appropriate) on output/employment and on prices/wages in the
short-term and long-term?
(b) What would be their view on monetary neutrality?
(c) What would be their position on the case for activist policy vs. a policy rule.
(d) What theoretical or environmental factors led them to question the validity of
accepted wisdom and develop new lines of thought?
(e) What is their most important and lasting legacy or valuable contribution to
macroeconomic thought?
(f) What is the most important weakness or limitation in their thinking that allowed (or
would allow) it to be pushed aside by later work?
(g) What would be each of their reaction to the recent global financial crisis and adoption
of Quantitative Easing (QE) by the major central banks? Make your answers as
technical as you wish in answering the questions.
1. Ricardo/Marshall and the Classical dichotomy
2. Irving Fisher
3. J.M. Keynes
4. Hayek
5. Hicks/Tobin/Samuelson and the neoclassical synthesis
6. A. W. Phillips
7. Klein/Eckstein/Fair and the large-scale macro-econometric models of the 1970’s.
8. Friedman/Phelps and the Accelerationist hypothesis
9. Lucas/Sargent and rational expectations
10. Kydland/Prescott/Plosser and real business cycles
11. Mankiw/Summers/Krugman and new Keynesian DSGE models
12. Arthur/Schweitzer/Leijonhufvud and Post-Walrasian complexity and behavioral
work
5 Question I.4
(Rutledge)
Consider a Solow economy with a Cobb Douglas production function with laboraugmenting technological progress, Y = F(K, AL) = Ka (AL)(1-a) where 0 < a < 1, Y is
output, K is capital stock, L is labor, and A represents the effectiveness of labor. Output
is made up of consumer goods and capital goods, Y = C + I. The fraction of output that is
saved, s, is constant. All savings are used to produce capital goods. Assume that labor
grows at constant annual rate n, technology (A) grows at constant annual rate g, and the
capital stock depreciates at the constant annual rate d. There is no government and no
foreign sector. Output is assumed to be at full employment.
(a) Express the production function in intensive terms, per unit of effective labor, AL.
(b) Show that the production function has constant returns.
(c) Show that the production function has positive and diminishing returns to increases in
K.
(d) Show that the production function satisfies the Inada conditions.
(e) Show the equation describing the net growth in the capital stock.
(f) Derive the Solow fundamental equation for the model.
(g) For a given set of parameters, show a graphical depiction of the steady state, or
balanced growth path.
(h) Starting from the initial steady state in question 7, show the effect of a permanent
increase in the rate of population growth. Is the change in growth temporary or
permanent?
(i) What share of output is earned by owners of capital? What share by labor? What
assumptions underlie your answers?
(j) What savings rate would lead to a steady state capital stock that would maximize
income per effective unit of labor.
(k) How is growth accounting used to decompose growth into contributions from
population growth, capital accumulation, and technological progress? In empirical
studies, what fraction of growth is explained by capital accumulation in the simple
Solow model?
(l) What predictions does the Solow model make about the possibility that international
differences in income per capita will converge over time? What assumptions are
necessary to imply convergence? To what degree will international differences in
return on capital lead to capital flows that would accelerate convergence? Are you
aware of any empirical evidence on the convergence question? What are the main
results?
(m) What other factors has the growth theory literature identified that may be important
for determining a country’s long-term growth rate?
(n) How would you revise the production function to extend the basic Solow model to
include the contribution of human capital as well as physical capital to output? How
would you think about estimating the share of income earned by physical capital,
6 human capital, and labor? Are you aware of the results of any empirical work doing
this?
7 Part II
8 Question II.1
(Lamar)
Assume an economy with government that produces with the following technology,
Yt = AK tα Gt1−α . A is a constant, and G represents government spending which is
assumed to be desirable. To finance G the government collect income tax from
individuals, which are taxed at a rate τ. Tax revenue is proportional to income.
Capital accumulates according to K! (t ) = sY (t ) − δK (t ) . Assume exogenous
L!
population growth, which is given by = n .
L
G
Yd
(a) Using the following definitions, g ≡
and y d ≡
, where Yd is disposable
L
L
income do the necessary transformations to write down the capital accumulation
equation in per-capita terms.
(b) Assume that government spending is equal to tax collection in any period and
write down the government budget constraint in per-capita terms.
(c) Using what you did in part (b) find and expression for economic growth and
briefly explain your result.
(d) Show that y, c, g, and Y grow at a constant rate.
(e) What is effect of taxes on economic growth? Provide a brief explanation.
(f) Find the tax rate that maximizes economic growth.
(g) Provide a brief explanation why the introduction of the government in this model
is desirable.
9 Question II.2
(Lamar)
Assume a closed economy where firms produce output using only labor, i.e., Y= F(L)
under the assumptions of diminishing and positive marginal product of labor. Households
live forever and derive utility from consumption and holding real money balances, and
disutility from working. The representative household’s utility function is given by
𝑈=
!
!
!!! 𝛽
𝑈 𝐶! + Γ
!!
!!
0<β<1, with 𝑈 ! (.) >0, 𝑈 !! . < 0,
− 𝑉(𝐿! ) ,
Γ ! . > 0, Γ !! . < 0, 𝑉 ! > 0, 𝑉 !! > 0. U and Γ have relative-risk-aversion forms,
𝑈 𝐶! =
!!!!!
!!!
,
θ > 0, and
Γ
!!
!!
=
!
( ! )!!!
!!
!!!
, v > 0.
There are two assets: Money, which pays no interest and Bonds, which pay an interest
rate of 𝑖! . Households have a labor income WtLt per period, its consumption expenditures
are PtCt., W and P represent nominal wages and the price level respectively. At is the
household’s wealth at the start of period t.
(a) Write down an equation describing the evolution of wealth.
(b) Set up the maximization problem of the household.
(c) Using First Order Condition find, graph, and interpret the IS and LM curves.
(d) Show and explain the effect of an increase in nominal Money supply.
10 Question II.3
(Rutledge)
Lucas (1972) presented a simple macro model that has had a major impact on the
subsequent course of macroeconomic thinking based on the assumption that people make
output decisions using imperfect information. In his model, competitive, price-taking,
producer (i) has perfect information about the current price (Pi) of his own good but
imperfect, perhaps delayed, inflation on the general price level (P). When he receives
information about a change in Pi he cannot know if it represents an increase in the
relative price of his product, in which case it is optimal to increase output, or a general
increase in the price level, in which case he should leave output unchanged. A simplified
version of that model is presented below.
Each household uses its own labor Li to produce its own unique product Yi according to
the production function Yi = Li. Each household maximizes the objective function
Ui = Ci – (1/a)Li = (Pi/P)Yi – (1/a)Yi, where Ci = (Pi/P)Yi is consumption paid for with
household revenues obtained from selling its output.
(a) Set up the producer’s constrained optimization problem to choose the level of output
Yi that maximizes welfare under the initial assumption that the producer has perfect
information about both Pi and P.
(b) Under the assumption in question 1, derive the first order condition for the optimal
level of output Yi as a function of Pi and P.
(c) Express your answer to question 2 in terms of yi, pi and p, where lower-case letters
refer to the natural logarithms of the corresponding upper-case letter.
(d) Let demand for good i be given by yi = y + zi - b(pi-p) where y is economy-wide
output and zi is a demand shock specific to good i. Let y = m – p represent aggregate
demand where m is money supply so that the demand for good i can be expressed as
yi = m – p + zi – b(pi-p) = m – p + zi – bri, where m is log money supply and ri = pi – p
is the relative price of good i. Express optimal output in your answer to question 3 as
a function of ri.
(e) Lucas assumes the producer knows the price of his own good pi, but cannot observe
m, zi, or p. He assumes that m and z are independent and distributed, respectively, as
N(E[m], Vm) and N(0,Vz).
(f) Lucas also assumes certainty equivalence, that the producer determines E[ri | pi], the
expected value of the relative price ri given knowledge of the price pi, then produces
the amount that would be optimal if ri were known with certainty. Express optimal
output in your answer to question 4 as a function of E[ri | pi].
(g) With these assumptions, a result from statistics is that E[ri | pi] = E[ri] + (Vr/(Vr + Vp))
(pi – E[p]) = (Vr/(Vr + Vp)) (pi – E[p]). Use this to re-express optimal output in your
answer to question 6 as a function of pi and E[p].
(h) Derive the Lucas Supply Curve by averaging your result in question 7 across all
producers to express y as a function of unexpected inflation (p – E[p]).
(i) What is the relation between the Lucas Supply Curve and the expectations-augmented
Phillips Curve?
11 (j) Now examine the 2 equation system made up of the Lucas Supply Curve from
question 8 and the aggregate demand equation y = m – p. Derive solutions for the
equilibrium price level p and the equilibrium level of total output y in terms of m and
E[p].
(k) Take the expected value of your equation for equilibrium price in question 10 and
solve for E[p] as a function of E[m].
(l) Using your answer from question 11 and your answer from question 10, express
solutions for the equilibrium price level p and the equilibrium level of total output y
in terms of expected money E[m] and unexpected money (m – E[m]).
(m) Based upon your answer to question 12, what is the impact of a 1% increase in E[m]
on the price level? On output? What is the impact of a 1% increase in unexpected
money growth (m – E[m]) on the price level? On output?
(n) Can you cite any empirical studies that confirm or disconfirm these findings?
(o) Finally, what it the total impact of an increase in money, m, on nominal output (p +
y)?
12 Question II.4
(Rutledge)
Consider a problem where households maximize the expected value of the following
utility function:
!
𝑈 = !!!
𝑢(𝑐! , 1 − 𝑙! )
(1 + 𝜌)!
where 𝑐! is consumption in period t, (1 − 𝑙! ) is time devoted to leisure, and 𝜌 is the
discount rate. Further assume that u! = ln c! + b ln (1 − l! ) and b > 0. In the two-period
problem, the household’s two-period budget constraint is:
!
!
!
𝑐! + !!!
= 𝑤! ℓ𝓁! + !!! 𝑤! ℓ𝓁! , where 𝑤! is the wage rate in period t and r is the one-period interest rate.
(a) Set up the household’s constrained maximization problem.
(b) Show the first order conditions for 𝑐! , 𝑐! , ℓ𝓁! , 𝑎𝑛𝑑 ℓ𝓁! .
(c) Show that an increase in 𝑤! and 𝑤! that leaves 𝑤! /𝑤! unchanged does not change
optimal leisure in either period.
(d) What is the impact of an increase in 𝑤! on 𝑐! , 𝑐! , ℓ𝓁! , 𝑎𝑛𝑑 ℓ𝓁! ?
(e) How does the intertemporal relative demand for leisure depend on the interest rate?
(f) What is the intertemporal marginal rate of substitution between 𝑐! 𝑎𝑛𝑑 𝑐! ?
(g) Now assume initial wealth in period 1 of Z > 0. Does the result in question 3 continue
to hold?
13