AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Advanced Macroeconomic Theory
June 2014
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General Instructions: You must answer both sections, I and II; follow the directions in each section carefully (there is some choice in section II, but not in section I). Each section receives equal
weight in the overall grading, so plan to spend equal time (about 2 hours and 30 minutes) on
each. 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 and points (200 total points for both sections together) 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: This section consists of two (2) problems; there is no choice and you must answer
all questions for each problem. Allow 1 hour and 15 minutes per problem.
Problem 1. (50 points) Consider the following endogenous growth model with learning-bydoing and knowledge spillovers. The production technology of each firm is given by
∙
,
where 0
1 and is the aggregate stock of capital. 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 constant and equal to . Markets are perfectly competitive.
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
. Physical capital does not depreciate, that is,
The initial stock of capital per person is equal to .
,
0.
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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 ≡ / .
2. (5 points) Write down the Hamiltonian for consumption optimization problem and derive
the consumption Euler equation. Using the equilibrium conditions in factor markets derive
the law of motion for physical capital stock per person,
.
3. (5 points) Prove that on the balanced growth path (on which, by definition, and grow at
constant rates) consumption, output, and capital, all in per person terms, grow at the same
constant rate . Find this rate and assume that the parameters of the model are such that
0.
4. (10 points) Using the Euler equation, write down
as a function of , , , , , and (yet
to be found) 0 . Combine it with the law of motion for capital stock per person and
solve for
as a function of , , , , , , and 0 .
5. (10 points) Write down the transversality condition for the consumer optimization
problem. Combine it with the results from previous parts to find the optimal value of 0 .
Derive the equilibrium trajectories of
and
.
6. (5 points) Draw the ,
phase diagram for this model. Mark the initial point, the
balanced growth path, and indicate equilibrium dynamics with arrows. Discuss (without
doing any calculations) how the equilibrium dynamics of this model are different from the
standard Ramsey model with a neoclassical production function, as well as the AK model.
7. (10 points) Now 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 optimal
growth rate of consumption. How does it compare to that in the decentralized equilibrium?
Is the decentralized equilibrium Pareto optimal? If not, why?
Problem 2. (50 points) Consider the following version of the Galor-Weil (2000) model.
Households (parents) split the available unit of time between supplying labor and rearing
children. A time unit of labor supplied earns the income . Assume that there is a fixed cost of
units of time per child, plus a variable cost of one unit of time per each unit of education given to
each child. Parents consume the income earned on the labor market. To summarize, the budget
constraint is given by
, where
is consumption,
is the number of
children, and
is the educational level of each child. The preferences of parents are given by
1
where
,
ln
ln
ln
,
is the human capital formation function assumed to take the form
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,
.
That is, the level of human capital depends positively on the amount of education and negatively
on the rate of technological progress
(due to obsolescence of skills). Note that, unlike in the
original model, there is no subsistence consumption constraint. Assume that the rate of
technological progress depends only on the population size (via the scale effect). Furthermore,
the scale effect only kicks in once the population level crosses a critical threshold
0.
Specifically,
0, ;
≡
/
, ,
where
is the level of technology in period and
capita income in period is given by
amount of land.
0. To complete the model, recall that per
, where
≡
/ and
is the fixed
1. (15 points) Set up the utility maximization problem of the parents. Write down the first
order conditions with respect to
and
. Derive the optimal level of investment in
children’s education,
. How does it depend on the expected rate of technological
progress
?
2. (15 points) Derive the optimal fertility rate . How does it depend on
? Derive the
dynamics of population using the functional form of
given above. Assuming that
, sketch the diagram showing the evolution of population and find its steady-state
level, . Hint: The diagram will consist of two segments and you can assume that the
second segment is a strictly increasing, concave function satisfying the Inada conditions
(specifically,
√ ).
3. (10 points) Illustrate the co-evolution of technology and education (conditional on the
level of population) on the
,
plane. Depict two loci: the optimal educational
investment as a function of the rate of technological progress (from part 1) and the rate of
technological progress itself (the latter is obviously just a flat line, given the population
size). Show these loci for three generic cases: 1 , 2) , and 3)
.
4. (5 points) Find the steady-state values of the rate of technological progress, education, and
the fertility rate. Explain intuitively the mechanism driving the onset of the demographic
transition in this model.
5. (5 points) Find the growth rate of income per capita on the balanced growth path.
(Section II starts on the next page)
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SECTION II
Directions: Choose any two (2) of the following three questions; answer all parts (a, b, c, ...) for
each question you choose. Time allocation: 1 hour and 15 minutes per question.
1. (50 points) In the aftermath of the financial crisis and Great Recession of 2008-9, saving
rates (broadly defined) have increased in the U.S. and many other countries as households
seek to reduce their debts, corporate cash flows (gross retained profits) are at record levels,
and governments are pursuing austerity policies. This question asks you to analyze the
impact of higher saving rates in alternative “closures” of a one-sector model of growth and
distribution, assuming an inverse wage-profit relation given by
1 a
w  1 r
a0 a0
where w is the real wage, r is the rate of profit, a0 is the labor coefficient (hours per unit of
output), and a1 is the capital coefficient (capital per unit of output). Combining math, graphs,
and intuition, determine and explain the impact of the following changes on the endogenous
variables w, r, profit share , and/or accumulation rate g (as appropriate to each model—do
NOT analyze consumption per worker c) in each of the following cases:
a. (10 points) A rise in the propensity to “accumulate” α (0 < α < 1) in a Classical-Marxian
model with an exogenously given wage share ( 1   ), where capital accumulation is
determined by the function g = α(r  rmin) [you may assume a solution for which r > rmin].
b. (10 points) A rise in the propensity to “accumulate” α (0 < α < 1) in a Classical-Marxian
model with a constant unemployment rate (full employment/natural rate of growth, g = n)
where the growth of labor supply is endogenous (n = n0 + n1w, n0, n1 > 0) and you may
use the same accumulation function as in part a.
c. (10 points) An increase the saving rate out of profits sr (0 < sr < 1) in a Neo-Keynesian
model, in which the classical accumulation function is replaced by the saving function 
 S/K = srr and an independent investment function g  I/K = f0 + f1r, f0, f1 > 0. Be sure
to derive and explain any necessary stability condition.
d. (10 points) An increase the saving rate out of profits sr (0 < sr < 1) in a Kalecki-Steindl
model, which allows for variable capacity utilization u = Y/Y* (where Y is actual output
and Y* is potential output), with 0 < u < 1, and the investment function is g = g0 + g1r +
g2u (g0, g1, g2 > 0), while the saving function is the same as in part c. For this model, you
may assume that prices are set by P = (1+)Wa0, where W is the nominal wage rate, and
the inverse w-r relation (with w = W/P) is modified to incorporate variable utilization. Be
sure to derive results for u as well as the other endogenous variables, and again derive
and explain any necessary stability condition.
e. (10 points) Finally, discuss (compare and contrast) your results from parts a.-d., focusing
on the different implications for growth and also for whether there is a positive or inverse
(trade-off) relationship between growth and distributive equity (as reflected in either the
wage share or real wage) and especially what differences in assumptions lead to the different results. What assumptions would have to hold for an increased saving rate to lead
to more rapid growth? Under what assumptions would increased saving rates reduce
growth or leave it unchanged in the long run?
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2. (50 points) The following is a fairly standard expectations-augmented Phillips Curve (PC), of
the general type often used as the short-run “aggregate supply” relationship in a new consensus macro (NCM) model (similar to what Stockhammer calls the “Neoclassical New
Keynesian Synthesis or NNKS):
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. (6 points) On the simplifying assumption that expected inflation equals the one-period lag
of inflation (pte = pt1), solve for the “non-accelerating inflation rate of unemployment”
(NAIRU) that is supposed to characterize a medium (or “long”) run equilibrium.
b. (8 points) What are the policy implications of the NAIRU solution from part a. 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 )?
c. (36 points) The standard NAIRU model has been criticized, for various reasons, by each
of the following economists. What features would each of these author(s) add to the PC
equation, and how do these changes affect the model’s solution for a NAIRU and/or other
implications of the model? Be sure to discuss the policy implications of each alternative
model—how are these different from the policy implications of the standard PC/NAIRU
model in parts a.-b.?
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 markups 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 c., you can use any combination of equations, graphs, and intuition to
explain each model i-ii-iii (intuition is crucial!); you do not need to give the full math for
these models, and any equations you do provide can be in the authors’ original notation
(don’t worry about making the notation consistent). What you should focus on is how
each of these alternative models modifies the short-run PC analysis and/or the mediumrun results for the NAIRU, and the policy implications of each model.
3. (50 points) Consider a “conflicting claims” model of inflation combined with a neo-Kaleckian model of aggregate demand. The wage share is  = a0W/P, where W is the nominal wage,
a0 is the labor coefficient (hours per unit of output), and P is the price level. The profit share
is  = 1 – . The price is set via mark-up pricing: P = (1 + z)a0W (labor is the only variable
cost, there are no raw materials, and the mark-up rate is z > 0). The capacity utilization rate is
proxied by the output-capital ratio, u = Y/K. Answer the following questions:
a. (12 points) First, assume that wage and price increases are governed by the reaction functions Wˆ   ( w  )  u and Pˆ   (   f )   u , where w and f are the workers’
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and firms’ respective targets for the wage share, with , ,  , > 0. Briefly explain the
behavior represented by these equations, including the rationale for the u and u terms.
Then, assuming also that labor productivity grows at the exogenously given rate  â0   ,
find the equation for the “distributive curve” (DC, or ˆ  0 ) and determine its slope in
u   space. Does DC necessarily slope upward or downward, or can it slope either way,
and if so what is the condition for a positive or negative slope? Solve and draw all possible cases on an appropriate graph (or graphs), given the functions as specified in part a.;
briefly explain the intuition for each possible case.
b. (14 points) On the aggregate demand side, the economy is closed (no trade), and there is
no depreciation of capital for simplicity. There are no savings out of wages, so the saving
function is  = S/K = srr, where r =  u is the profit rate and sr is the saving rate out of
profits (0 < sr < 1). The investment function is g = I/K =  +  +  u, with , ,  > 0.
Also, the government has a budget deficit, which is exogenously fixed as a ratio to the
capital stock:  = (G – T)/K.
i. Solve the aggregate demand side of the model for the goods market equilibrium condition (“IS curve”). Find an explicit (reduced form) solution for the equilibrium utilization rate u as a function of the profit share  and fiscal deficit .
ii. What do you have to assume for stability of the goods market and for an economically meaningful solution?
iii. Is aggregate demand (measured by u) wage-led or profit-led, or can it be either? If it
can be either, find and interpret the condition for wage-led vs. profit-led demand.
Explain your result in terms of the parameters of this specific model.
c. (12 points) Combine the two curves (IS and DC) on a series of diagrams in u   space.
Show all possible cases (in terms of the relative slopes of the curves) given the functions
specified above (show only possible cases). Analyze (graphically) the conditions on the
slopes of the two curves for the economy to be stable or unstable, assuming that output
(goods market) adjustment is instantaneous but wage/price (distributional) adjustment is
slow and gradual. What is the intuition for the unstable case(s), if any are possible?
d. (12 points) Analyze the effects of each of the following graphically using IS and DC
curves for all possible, stable cases—show the dynamics of adjustment for u and  on
your graphs (again assuming faster adjustment of the goods market), analyze the effects
on medium-run equilibrium inflation ( P̂ ), and explain intuitively.
i. An increase in the bargaining power of labor caused by new legislation (for example,
a rise in the minimum wage and easier rules for forming unions), so that the workers’
target for the wage share f rises.
ii. A fiscal stimulus (positive shock to  ).