AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Advanced Macroeconomic Theory
January 2015
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General Instructions: You must answer both sections, I and II; follow the directions in each section carefully. Each section receives equal weight in the overall grading, so plan to spend equal
time (about 2 hours and 30 minutes) on each section. Make sure all math and graphs are
intuitively explained, all diagrams are clearly labeled, and all answers are focused on the specific
questions asked (there is no credit for extraneous material). Time limits suggest the expected
length and depth of answers. No calculators or electronic devices of any kind are permitted.
Answers are expected to demonstrate mastery of the material at the PhD level.
SECTION I
Directions: Answer one (1) question in each part (A-B-C); be sure to observe the time limits
strictly! Points will be in proportion to the minutes allotted for each question.
Part A – Answer one (1) of the following (40 minutes):
1. Using a classical-Marxian model of growth and distribution based on the inverse wageprofit relation
1 a
w  1 r,
a0 a0
where a0 is the labor-output ratio and a1 is the capital-output ratio, and the “accumulation” (saving) function g = K/K= r (where 0 <  < 1, assuming rmin = 0 for simplicity),
analyze and compare the effects of the three major types of technological change
(Harrod-neutral, Hicks-neutral, and Marx-biased) on the long-run equilibrium real wage
w, profit rate r, and growth rate g (or whichever of these are endogenous) [you do not
need to solve for consumption c] in each of the following “closures”:
a. Fixed wage share, wa0  1   .
b. Constant unemployment rate or “natural rate of growth” (g = n), where the growth
of labor supply is exogenous (n = n0).
Be sure to provide a brief intuitive explanation for the differences in the results.
2. This question asks you to compare the effects of an increase in the capitalists’ “animal
spirits” in the (i) neo-Keynesian and (ii) Kalecki-Steindl closures of a model of growth
and distribution. The two models are specified as follows:
(i) Neo-Keynesian: the saving and investment functions are
 = sr r
and
g = f 0 + f 1 re
( f0, f1 > 0)
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where   S/K is the ratio of saving to capital, 0 < sr < 1 is the saving rate out of profits
(there are no savings out of wages), g  I/K is the ratio of investment to capital, and re is
the expected profit rate; you may assume that re = r (the actual profit rate) in equilibrium.
The profit rate is related to the real wage through the same inverse relation shown for
question 1, above, and capacity is fully utilized.
(ii) Kalecki-Steindl: the saving function is the same as for the neo-Keynesian model, but
the investment function is
   Y u
,
g = g0 + g1r + g2u (assuming g0, g1, g2 > 0), with profit rate r  
 
 1    K a1
where  > 0 is the exogenously given mark-up rate from the mark-up pricing equation, 
is the profit share, P = (1 + )Wa0, W is the nominal wage rate, and the inverse w-r relation (with w = W/P) is modified to incorporate variable capacity utilization, u = Y/Y*
(where Y is actual output and Y* is potential output):
1  a  r 
w    1   .
a0  a0  u 
For each model (i) and (ii), analyze the effects of a rise in “animal spirits” modeled as an
increase in the shift factor in the investment function ( f0 or g0). Solve mathematically and
show graphically the effects on the profit rate r, real wage w, growth rate g, and utilization rate u (or whichever of these are endogenous in each model). Based on your results,
answer the following specific questions:
a. Why is the effect on growth g referred to as the case of a “widow’s cruse”?
b. What is the (other) key adjusting variable that enables the effect that you find on
the profit rate r in each model? Why is this different in the two models?
Part B – Answer one (1) of the following (50 minutes):
3. Suppose a country’s economy is described by the following functions (with all variables
in growth rate form):
imports:
m = m(e + p*  p) + my
exports:
x = x(e + p*  p) + xy*
y   (x x  a a)
prices:
p=+w–q
output:
labor productivity (Verdoorn’s Law):
q = q0 + y
balance-of-payments (BP) equilibrium assuming no capital flows: p  x  p*  e  m
a. Briefly define all variables and explain each equation.
b. Solve for the export-led, cumulative causation (ELCC) growth rate, yE, using the
appropriate equations (which equations and what assumptions are required to
obtain this solution? which one(s) is (are) ignored and why?). Illustrate your
solution on a graph, and be sure to state and interpret the condition for stability of
the equilibrium at y = yE.
c. Using any additional equations that you need, solve for the balance-of-paymentsconstrained growth (BPCG) rate yB. First, give a general solution that incorporates
Verdoorn’s Law (but still omitting capital flows). Then, show how this solution is
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simplified if we impose the assumption of long-run relative purchasing power
parity (PPP), e + p*  p = 0.
d. According to each model (ELCC and BPCG), will a country’s long-run equilibrium growth rate be raised by (i) an upward shift in the Verdoorn equation (rise in
q0)? (ii) a shift in the composition of exports to higher-quality products that have a
higher income elasticity x? Discuss the similarities and differences in what these
two models imply about the long-run impact of each type of shift (for BPCG, use
the version assuming that PPP holds).
4. Consider the following neo-Kaleckian macro model, which uses a linear Marglin-Bhaduri
investment function and (initially) assumes that there are no savings out of wages:
Investment function: g = h0 + h1u + h2, with h0, h1, h2 > 0
 = sr r
Saving function:
where 0 < sr < 1; r = u is the profit rate;  is the profit share (taken as determined by an
exogenous mark-up z); u is the utilization rate (proxied by the output-capital ratio, Y/K); g
= I/K is the investment-capital ratio or “growth rate,” and  = S/K is the saving-capital
ratio. For simplicity, there is no foreign trade, no government, and no depreciation of
capital.
a. Solve for the equilibrium utilization rate u and growth rate g. (Show your work,
and be sure to get reduced form solutions with only exogenous variables and
parameters on the right-hand side). Is demand (u) profit-led or wage-led, or can it
be either? What about growth (g)? Take the appropriate derivatives (again, be
sure to get reduced form solutions!) and analyze/explain.
b. Now, suppose that there are also positive savings out of wages, so the saving
function becomes
 = [sr + sw(1−)]u
where 0 < sw < sr < 0. How does this change your findings for demand? Re-solve
the model for u (you do not need to solve for growth g in this part) using this saving function and compare your results with what you found for u in part a. (Again,
get explicit, reduced form solutions for u and the relevant derivative.) What does
this tell us about the role of workers’ saving? Discuss briefly.
Part C – Answer one (1) of the following (60 minutes):
5. Svensson describes an economy in which the central bank pursues an inflation target as
follows:
πt+1 = πt + α1 yt + α2 xt + εt+1
yt+1 = β1 yt – β2 (it – πt) + ηt+1
xt+1 = γ xt + t+1
where πt = pt – pt–1 is the inflation rate in year t, p is the log of the price level, y is the log
of output relative to potential, x is an exogenous variable that affects inflation, i is the
nominal interest rate, and εt, t, and t are i.i.d. shocks in year t that are not known in year
(t – 1). Assume that α1 and β2 are positive, and all the other coefficients are non-negative.
Assume also that β1 < 1 and γ < 1. The central bank’s single-period loss function is
L(πt) = ½ (πt – π*)²
where π* is the central bank’s target for inflation.
a. Derive an expression for inflation in terms of the other variables and shocks in the
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b.
c.
d.
e.
model. Define and interpret the control lag.
Given this model, derive the central bank’s rule for setting the interest (“repo”)
rate it. Explain why the output gap appears in the policy rule even though it is not
in the central bank’s objective function.
What is the “intermediate target” for monetary policy implied by this model?
Explain what an intermediate target is in general, carefully define Svensson’s
preferred intermediate target, and discuss why he advocates its use.
Assuming the central bank follows Svensson’s recommended policy approach,
how will actual inflation deviate from expected inflation? How should the central
bank’s success be assessed if it follows his recommendations?
Finally, assess the usefulness of this model of monetary policy formulation for
post-financial crisis conditions in which the central bank interest rate is at or near
the zero lower bound. What else should the central bank be looking at and doing
in this situation, beyond what is included in Svensson’s model? Discuss.
6. The following is a fairly standard expectations-augmented Phillips Curve (PC), of the
type often used as the short-run “aggregate supply” relationship in a new consensus
macro (NCM) model:
pt = pte +   Ut + εt
where p is the inflation rate (log difference of the price level), U is the unemployment
rate, t indexes time, a superscript e indicates an expected variable, and  is a random error
(aggregate supply shock).
a. Assuming that expected inflation equals one-period lagged inflation (pte = pt1),
solve for the “non-accelerating inflation rate of unemployment” (NAIRU) for a
medium (or “long”) run equilibrium. What are the policy implications of this
NAIRU solution in regard to (i) the efficacy of expansionary aggregate demand
policies, such as a fiscal or monetary stimulus, for lowering unemployment; and
(ii) the impact of stronger workers’ bargaining power in wage negotiations with
firms (assuming that this is reflected positively in the parameter )?
b. For any two (2) of the following three alternative models, how would each one
modify the short-run PC analysis and/or the solution for a NAIRU? Also, how are
the policy implications of each alternative model different from the standard
NAIRU model in part a.? [Answer only for two out of these three!]
i. Akerlof, Dickens, and Perry in their model of “near rational” behavior.
ii. Ball and Moffitt in their model of “wage aspirations”.
iii. Hein and Stockhammer in their post-Keynesian model in which firms’
target mark-ups are an increasing function of the interest rate (i) set by the
central bank (in their notation, hFT  h0  h1i ), assuming a central bank
reaction function that targets zero unexpected inflation.
Note: In part b., you can use any combination of equations, graphs, and intuition.
You do not need to give the full math for these models, but be sure to define any
key concepts, and equations can be written in the authors’ notation (don’t worry
about making notation consistent). Focus on how each of these alternative models
modifies the short-run PC analysis and/or the medium-run results for the NAIRU,
and the policy implications of each one.
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SECTION II
Directions: This section consists of two (2) problems; there is no choice and you must answer
all questions (parts) for each problem. Allow 1 hour and 15 minutes per problem.
Problem 1. (50 points) Consider the following growth model with learning-by-doing and
knowledge spillovers. The production technology of each firm is given by
∙
/
,
where 0
1, 0
1
, and / is the aggregate capital-labor ratio. Each firm is
small enough to neglect its own contribution to the aggregate capital stock and, therefore, treats
/ as given. The aggregate supply of labor (population size) is growing at a constant rate
0. Markets are perfectly competitive. There is no technological progress.
The household side of the economy is standard. Specifically, a representative infinitely-lived
consumer is maximizing her lifetime utility subject to a typical budget constraint (and the no
Ponzi game condition):
. .
ln
→ max
.
As usual, one unit of labor is supplied by each person inelastically at the market wage rate
,
and assets earn the market interest rate
. Capital depreciates at rate and its initial stock is
equal to in per capita terms.
1. (5 points) Consider the profit maximization problem of firm . Write down the first-order
,
conditions and prove that all firms operate at the same level of capital-labor ratio
where ≡ / and ≡ / . Derive the expressions for rand w as functions of k.
2. (10 points) Write down the Hamiltonian for consumer optimization problem and derive the
Euler equation. Using the equilibrium conditions in factor markets derive the law of
motion for capital stock per person .
3. (10 points) Draw a phase diagram showing the co-evolution of and and find their
steady-state levels. Use arrows to show the qualitative dynamics of and in different
quadrants of the diagram and mark the equilibrium path. Is the dynamics in this model
qualitatively different from the basic Ramsey case (with
0)? What are the growth rates
of output per person and total output in the steady state?
4. (5 points) Assume that the economy is in the steady state found in part 3. Examine the
immediate and long-run consequences of a permanent drop in the rate of population
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growth using the model’s phase diagram. What happens at the moment of shock and in
the long run?
5. (10 points) Consider again the economy that is in the steady state found in part 3. Now,
assume that new information arrives about forthcoming demographic change. As a result, a
permanent drop in is expected to happen at a certain date ∗ in the future. How does the
economy react to this news at the moment when new information arrives and in the long
run? Show your analysis using the model’s phase diagram.
6. (10 points) Consider the “centralized” version of this economy. What is the aggregate
production function from the social planner’s point of view? Formulate the lifetime utility
maximization problem subject to the economy’s resource constraint and derive the firstorder conditions. How does the steady-state level of capital per person compare to that in
the decentralized equilibrium? Is the decentralized equilibrium Pareto optimal? If not,
why? Note: for simplicity, don’t worry about no Ponzi game and transversality conditions,
just do enough work to derive the Euler equation.
Problem 2. (50 points) Consider a version of the Galor-Zeira (1993) model with an extreme case
of credit market imperfections. Specifically, individuals cannot borrow at any interest rate and,
hence, cannot cover any part of their human capital investment from borrowed funds. Other parts
of the model are exactly the same as in the original paper and are summarized in the following
paragraph.
Individuals live for 2 periods. They have two potential career options: 1) work as unskilled
earning a constant wage
in each period; 2) acquire human capital in the first period and work
as skilled earning a constant wage
only in the second period. Acquisition of human
capital costs and, since there is no borrowing, can only be covered from initial (inherited)
wealth. In the first period of their lives individuals work or acquire human capital and save,
while in the second period they work, consume, and leave bequests. The preferences of an
individual born at time are given by
ln
1
ln
,
where
is the second-period consumption and
is the bequest. Total population size is
constant and normalized to 1. The initial distribution of inheritance is summarized by the
with support 0, ∞ , so that
1.
cumulative distribution function
1. (5 points) Consider an individual who inherits in the first period of her life. What is her
total wealth at the end of period 2 if she works as unskilled? What is her total wealth at the
end of period 2 and if she works as skilled assuming that
? How is that total wealth
divided between consumption and bequest in either case?
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2. (5 points) Consider the occupational choice of an individual with inheritance
.
Under what parametric condition will she choose to acquire human capital in the first
period? Assume that this condition always holds.
3. (10 points) Derive the law of motion for bequests and construct the diagram showing the
evolution of bequests over time. In doing so, assume that the following parametric
restrictions hold, in addition to the condition derived in part 2:
1
1
1, 1
2
1
1
1
, 1
.
What are the long-run shares of skilled and unskilled workers in this economy?
and
4. (5 points) Calculate the steady-state bequest levels of unskilled and skilled workers,
. Write down the formula to calculate the steady-state level of average wealth at the end
of period 2?
5. (10 points) Now consider a different economy in which the financial system is more
developed. Specifically, the young can borrow in the first period of their lives to finance
human capital investment. However, the interest rate on borrowing, , is higher than the
interest rate on saving, . In other words, this economy operates according to the original
Galor-Zeira (1993) model with credit market imperfections. All the other parameters in
this economy, including initial distribution of wealth, are exactly the same as in parts 1-4.
Consider an individual with inheritance level
. Calculate the threshold such that,
if
, the individual decides to acquire human capital. Given this result, derive the law
of motion for bequests over time in this economy.
6. (10 points) Draw the diagram showing the evolution of bequests over time. Assume that, in
addition to the parametric conditions above, the following restrictions hold:
1
1
1, 1
1
,
just like in the original Galor and Zeira model. Calculate the threshold bequest level, ,
which separates dynasties that are skilled and unskilled in the long run and mark it on the
diagram. On the same diagram show, for comparison, the bequests schedule from part 3.
7. (5 points) What are the long-run shares of skilled and unskilled workers in the economy
described in part 5? How does the long-run level of average wealth at the end of period 2
in this economy compare to that found in part 4? Why?
End of Exam