AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Advanced Macroeconomic Theory
June 2013
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This examination has two Sections, I and II. You must answer both sections; follow the
directions in each section carefully. There is some choice in section II, but no choice in section I.
Each section receives equal weight in the overall grading. Therefore, plan to spend an equal
amount of time (about 2 hours and 30 minutes) on each section, I and II. Please make sure that
all math and graphs are intuitively explained, all diagrams are clearly labeled, and all answers are
responsive to the specific questions asked (no credit for extraneous material). The time limits
suggest the expected length and depth of your answers. No books, notes, calculators, or electronic devices of any kind are permitted. All answers are expected to demonstrate mastery of the
material at the PhD level. Good luck!
SECTION I
Directions: This section consists of two (2) problems; there is no choice and you must answer
all questions for each problem. Each of the ten questions (five per problem) carries the
same weight. Allow 1 hour and 15 minutes per problem, or approximately 15 minutes per
question.
Problem 1. Consider the following version of the 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 the typical budget constraint (and the no
Ponzi game condition):
1
1
. .
∙
→ max
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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 .
,
1. Consider the profit maximization problem of firm . Write down the first order conditions
, where
and show that all firms operate at the same level of capital-labor ratio
≡ / and ≡ / .
2. Write down (if you remember it) or derive the consumption Euler equation. Using the
equilibrium conditions in factor markets derive the law of motion for physical capital stock
per person.
3. Show that on the balanced growth path consumption, income, 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 (and also such that the utility functional converges). What is
the ratio / on the equilibrium path? Hint: Recall that on a balanced growth path, by
definition, all variables grow at constant rates.
4. Draw a phase diagram showing the co-evolution of and on the balanced growth path.
How is the equilibrium dynamics in this model different from the standard Ramsey model
in which the production technology is independent of the aggregate capital stock?
5. 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. 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 to
rear each 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
,
,0
1.
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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, furthermore, that the
rate of technological progress depends only on the population size (via the scale effect).
≡
/
, where
is the level of technology in period and
Specifically,
0. To complete the model, recall that per capita income in period is given by
,
≡
/ and is the fixed amount of land.
where
1. Set up the utility maximization problem of the parents. Write down the first order
conditions with respect to
and
(for the interior solution). Derive the optimal level
of investment in children’s education,
, on part of the parents. How does it depend on
the expected rate of technological progress
? Hint: You should arrive at
max 0,
1
.
2. Derive the optimal fertility rate . How does it depend on
? Derive the dynamics of
population, using the fact that
. Assuming that
, sketch the diagram
showing the evolution of population and find its steady-state level. Hint: The schedule will
consist of two segments and you can assume without providing a proof that the second
segment is a strictly increasing, concave function satisfying the Inada conditions.
3. Illustrate the co-evolution of technology and education (conditional on the level of
population) on the ,
plane. Specifically, depict two loci: the optimal educational
investment as a function of the rate of technological progress (from part 1) and the current
rate of technological progress. Show how the diagram changes as population size increases
and converges to its steady-state level. Hint: The second locus is obviously just a flat line,
given the population size.
4. Find the steady-state values of education, the rate of technological progress, and the
fertility rate. Explain intuitively the mechanism driving the onset of the demographic
transition in this model.
5. Find the growth rate of income per capita on the balanced growth path. How does it
depend upon the cost of raising children ?
(Section II starts on the next page)
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SECTION II
Directions: Choose any two (2) of the following four questions; answer all parts (a, b, c, ...) for
each question you choose. Time allocation: 1 hour and 15 minutes per question.
1. The following is a fairly general “new consensus” macro (NCM) model for a closed
economy:
(1)
Yt g  a0  a1Yt g1  a2 E (Yt g1 )  a3 [ Rt  Et ( pt 1 )]  s1
(2)
pt  b1Yt g  b2 pt 1  b3 Et ( pt 1 )  s2 , with b2 + b3 = 1.
(3)
Rt  (1  c3 )[ RR *  Et ( pt 1 )  c1Yt g1  c2 ( pt 1  pT )]  c3 Rt 1  s3
where Yg is the output gap, E is the expectations operator, R is the nominal interest rate, p is
the inflation rate (log change in the price level), and the si (i = 1, 2, 3) are shocks to aggregate
demand, aggregate supply, and monetary policy, respectively.
a. Interpret each equation—explain the underlying assumptions and intuition for each term
(including shocks)—and be sure to discuss the rationale for any lags or leads included.
b. Analyze intuitively, but in some depth and detail, how the economy is supposed to react
to a negative aggregate demand shock (s1 < 0) according to this model. Be very specific
about the mechanisms by which the shock is propagated and how “equilibrium” (Yg = 0)
is restored in the “medium run.” Be sure to state any assumptions you are making about
expectations and discuss any lags involved in the process.
c. What assumption(s) and conclusions of this model would be changed if agents exhibit
“near rational” behavior about inflation expectations at low inflation rates, as argued by
Akerlof, Dickens, and Perry (2000)? Do not give the details of the Akerlof et al. model,
but rather assess qualitatively the implications of “near rational” behavior as they define
it (be sure to give the definition!) for this NCM model and its results. Especially, think
about what assumption(s) need to be changed, and how Akerlof et al.’s results for the
NAIRU unemployment rate translate into the output gap used here.
d. It is now five years since the outbreak of the Great Recession of 2008-9, in which a
financial crisis led to a huge negative shock to aggregate demand, yet the U.S. and many
other economies still have negative output gaps (Yg < 0) and have not come back to
macro equilibrium. Do you think that this experience is consistent with a NCM model of
this type? Why or why not? If not, what features of the real-world macro economy and
the recent crisis and response are ignored or assumed away in this model, which are
important for explaining the slow recovery from the 2008-9 recession? Be as specific and
detailed as possible.
2. This question concerns theories of the expectations-augmented (short-run) Phillips Curve and
the “non-accelerating inflation rate of unemployment” (NAIRU).
a. Briefly (but precisely) explain the difference between the NAIRU “model” and NAIRU
“stories” according to Stockhammer.
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b. Solve for the short-run Phillips Curve (SRPC) and the NAIRU in a basic “model” of
distributional conflict, and represent them on a diagram or diagrams, assuming that the
target and actual wage and profit shares are given as follows:
Workers’ target wage share: (1   )W  0  1u
Workers’ actual wage share: 1    0  1u  2 p u
Firms’ target profit share:  R  h0
c.
d.
e.
f.
Firms’ actual profit share:   h0  h2 p u
where  is the profit share, u is the unemployment rate, p is the inflation rate (percentage
change in the price level), and pu indicates “unexpected” inflation. Be sure to give the
rationale for each equation.
What is it that reconciles the inconsistent distributional claims (target shares) of workers
and firms in the short run? In the medium [“long”] run? Explain intuitively.
In what Stockhammer calls the “Neoclassical/New Keynesian Synthesis” (NNKS) story,
we add the aggregate demand or IS curve relationship yIS = y(y0, p), where y/y0 > 0 and
y/p < 0, and an inflation-targeting monetary policy rule specified as
iCB = i0 + p + i2(p – pT ).
In this version of the model, is the NAIRU equilibrium stable in the medium run if the
central bank does not follow an inflation-targeting monetary policy rule? Is it stable if
the central bank does follow such a rule? Analyze graphically and discuss intuitively.
In Stockhammer’s “Post-Keynesian” (PK) story, the aggregate demand or IS curve is
specified as yIS = y(y0, p, ), where both y/p and y/ have ambiguous signs. First,
explain what determines whether the sign of each of these partial derivatives is positive
or negative. Then, show how this version of the model compares with the NNKS story in
part d. in terms of the slopes of the curves, stability or instability of the NAIRU, etc.
Suppose the firms’ target for the profit share is also an increasing function of the interest
rate,  R  h0  h1i , and the actual profit share is   h0  h1i  h2 p u . How are the results of
the PK story changed in this case?
3. If a central bank adopts an inflation targeting (IT) monetary policy, does this mean that it
should only respond to deviations of inflation from its target rate, or should it also pay
attention to the output gap? Why? Answer using the following model from Svensson:
πt+1 = πt + α1 yt + α2 xt + εt+1
yt+1 = β1 yt – β2(it – πt) + β3 xt + 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 (i.e., the output gap expressed as a log difference), x is an exogenous supply-side variable, i is the nominal policy 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 singleperiod loss function is as follows:
L(πt) = ½ (πt – π*)²
where π* is the central bank’s target for inflation.
Specifically, answer the following questions:
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a. Derive an expression for inflation in terms of the other variables and shocks in the model.
b. Define and interpret the “control lag” and explain its significance.
c. Assume that the central bank targets inflation. Given this model, derive the central bank’s
optimal policy rule for the nominal interest rate it sets (“repo” or overnight rate). Then
explain the rationale for each term in this rule.
d. Assuming the central bank conducts monetary policy as described above, how will actual
inflation deviate from expected inflation? How should the central bank’s success be
assessed in this model? Related your answer to Svensson’s proposed “intermediate
target” for monetary policy.
e. Suppose a central bank follows an interest rate-setting rule (reaction function) in which it
only responds to the gap between actual and target inflation (and possibly information on
aggregate supply factors, x), and ignores information on the output gap, such as
it = t + b1(t – *) + b3xt
Would that be an optimal way of targeting inflation, in Svensson’s view? Why or why
not? What would be the cost of such a policy?
f. Based on the analysis in parts a-d., now summarize and explain your response to the
question posed at the beginning: If a central bank adopts an IT monetary policy, does this
mean that it should only respond to deviations of inflation from its target rate, or should it
also pay attention to the output gap?
4. Consider a neo-Kaleckian model of aggregate demand; you may assume a closed economy
with no government for simplicity. Suppose that there is positive saving out of wages, but at
a lower rate than out of profits (0 < sw < sr < 1), so that the saving rate is  = S/K =
[sr + (1)sw]u, and investment depends on the profit rate so that g = I/K = 0 + 1r, where
0, 1 > 0, r = u is the profit rate,  is the profit share, and u is the output-capital ratio used
as a proxy for the rate of capacity utilization. Also for simplicity there is no depreciation of
capital. Be sure to find explicit, reduced-form solutions and partial derivatives in all parts!
a. Find the (explicit, reduced form) solutions for the equilibrium utilization rate u* and
growth rate g*. Also find and interpret the (goods market) stability condition.
b. Determine whether aggregate demand is wage-led or profit-led in this economy. Find the
condition for u*/ to be positive or negative and interpret intuitively.
c. Determine whether economic growth is wage-led or profit-led in this economy. Find the
condition for g*/ to be positive or negative and interpret intuitively. How can you
reconcile your result with what you found for demand in part b.?
d. How important is the assumption of positive saving out of wages for your results in parts
b. and c.? What happens in the special case where sw = 0? Analyze and explain.
e. Continuing to assume zero saving out of wages (sw = 0), suppose that the investment
function also contains a “strong accelerator condition” or a separate effect of capacity
utilization in addition to the profit rate, so that g = 0 + 1r + 2u, with 0, 1, 2 > 0. Redo your analysis from parts a., b., and c. and explain your results.
f. Now, tie all your results together and compare them: What do these various models tell
us about the parameter conditions for demand and growth to be wage-led or profit-led in
a closed economy setting? Summarize intuitively, and also comment briefly on what the
empirical research shows in this regard.