AMERICAN UNIVERSITY
Department of Economics
Comprehensive Exam
Preliminary Theory
June 2013
Total Exam Pages: 3
COMPREHENSIVE EXAM IN PRELIMINARY THEORY
Directions: Complete all questions in both Part I and Part II. Read the questions
carefully. Please show all of your work, and make sure to explain and discuss if the
question indicates. Your grade depends on the quality of your explanations. Answers
should be expressed in your own words.
Part I: Basic Concepts
Define and discuss the significance of each of these terms. Keep your discussions
concise. Limit your discussion of each term to no more than two paragraphs (or a
maximum of 125 words).
Moral Hazard
Involuntary Unemployment
Pareto Efficiency
Golden rule of capital accumulation
Part II: Microeconomics
Answer both questions in this part.
1. Consider the following two-good Cobb-Douglas Utility function:
U(x1 , x 2 )  x1 x12  , where 0    1
and the budget constraint p1x1 + p2x2 = I, where I denotes income and p prices.
a. Derive the Marshallian demand functions.
b. Derive the indirect utility function.
c. Compute the Hicksian demands (by minimizing expenditures subject to utility).
d. Derive the Expenditure function.
e. Define Roy’s Identity and Shepard’s lemma, and explain their significance.
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2. Suppose the market demand for tablets is given by P = 500 - Q, and that there are
only two firms in the world industry, Apple and Samsung, both of which have constant
average and marginal costs: AC = MC = 200 dollars. To simplify, assume only one
(homogenous) type of tablet.
a. Suppose the two firms competed as Cournot duopolists. Derive the two firms’ reaction
functions and find the equilibrium output per firm, price, and profits per firm.
b. If Apple were a Stackelberg leader, what would be the equilibrium output per firm,
price, and profits per firm?
c. If Apple and Samsung were to merge as one company, what would be the equilibrium
output produced, price, and profits?
d. If the two firms competed as price takers, what would be the equilibrium output per
firm, price, and profits per firm?
e. Suppose Apple and Samsung competed as Bertrand duopolists. Characterize the two
firms’ reaction functions and find the equilibrium output per firm, price, and profits.
f. Summarize the above results by ranking the different models according to the total
output produced.
Part III: Macroeconomics
Answer both questions in this part.
3. Consider the following closed-economy Keynesian model:
Y=C+I+G
C = C(Y),
I = I(r),
(M/P) = L(r, Y),
0 < CY < 1
Ir < 0
Lr < 0, LY > 0
where Y denotes output, C consumption, I investment, L liquidity preference, and r the
interest rate. Government spending G, money supply M, and the price level P are
exogenously given.
a. Derive the IS curve and LM curve.
b. Linearize the IS and LM curves, and rewrite the system of equations in 2 x 2 matrix
form; namely, AX = B, where “X” is a vector of endogenous variables (dY and dr), “A” a
matrix of coefficients, and “B” a vector of exogenous changes (dM, dG, and dP).
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c. Using Cramer’s rule, derive the following comparative static effects (holding other
factors constant): Y/G, Y/M, r/G, and r/M. Discuss how the efficacy of
monetary and fiscal policies depends on the interest elasticities of investment and money
demand. Illustrate graphically.
4. Suppose that a consumer maximizes:

U   e t ln c t dt
0
subject to :
a t  ra t  w t  c t
where c denotes consumption, a financial assets, w wage income, r the real interest rate,
and  the time preference rate. Do not assume r =  (yet).
a. Use the method of Hamiltonian to derive the optimal path of consumption (i.e. the
Euler equation).
b. Solve the flow budget constraint forward by the method of integrating factors, and
apply the transversality condition to derive the intertemporal budget constraint.
c. Solve the Euler equation to derive the level of consumption at time t as function of
consumption at time 0. (Again, do not assume r =  yet). Combine your result with the
result in part b to derive the permanent income hypothesis (PIH) consumption function.
Concisely explain the PIH. What does the marginal propensity to consume depend on?
d. Now assume that, in the aggregate economy, r = . Why would consumption follow a
random walk? Discuss using the discrete-time counterpart of the Euler equation. What
would cause changes in consumption over time?
e. Discuss the testable implications of the PIH. How should consumption react to lagged,
current income? If u represents innovation in permanent income, which is greater: the
variance of consumption changes or the variance of u, and why?
f. Are the empirical findings consistent with the theoretical predictions? Explain what
might cause the PIH model to fail empirically.
g. From a policy perspective, why should it matter whether consumption follows the PIH
or the Keynesian model?