AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Econ 01B
June 2006
Page 1 of 3
COMPREHENSIVE EXAMINATION IN PRELIMINARY THEORY
Instructions: Please answer all questions and show all of your work.
1. Consider the following n-good Cobb-Douglas Utility function:
n
u(x) = A ∏ x i αi
i =1
where α1 + α2 + … + αn = 1 and A > 0 is a constant.
a. Derive the Marshallian demand functions.
b. Derive the indirect utility function.
2. Consider the following simple Solow model:
k = sf(k) - (n + g + δ)
where k is capital per effective worker, s the exogenous savings rate, g exogenous growth
rate of technical efficiency, n the growth rate of labor, and δ the depreciation rate.
a. Assume that n = δ = 0, s = 0.5, and g = 0.10. Moreover, assume the production
function is f(k) = /k. Find the steady-state level of capital per effective worker k* and
output per effective worker f(k*).
b. Suppose that the savings rate is cut by half (i.e. s = 0.25), what exogenous change in g
would keep the steady-state level of capital per effective worker constant (that is, keep
the same k* that you found in part a)?
c. Using the Solow investment diagram, show the effects of a simultaneous decrease in s
and increase in g that keeps k* constant.
d. Provide economic intuition as to why the Solow model predicts long run economic
convergence? (Limit your answer to no more than 150 words).
e. What factors might explain divergence? (Limit your answer to no more than 150
words).
2
3. Consider the following market for used cars.
The figure above shows the reservation prices for both car buyers and car sellers.
Assume asymmetric information. Sellers know whether their cars are lemons or good
cars, but buyers cannot tell which is which.
a. Assume that 10% of all cars are lemons. Given this probability distribution of cars
and buyer preferences, what is the buyer’s expected value of buying a used car? Would
owners of good used cars sell their cars? Would owners of lemons sell their cars? In
each case, explain why. At what price will the cars be sold?
b. Suppose instead that 40% of all cars are lemons. Now what is the buyer’s expected
value of buying a used car? Would the owners of a good used car sell their cars? How
about the owners of lemons? Explain briefly why. At what price will the cars be sold?
Explain your reasoning carefully.
4. Suppose the representative consumer maximizes:
∞
U = ∫ e −ρt log c t dt
0
subject to the following capital accumulation equation :
k = rk + w − c
t
t
t
t
where c denotes consumption, k capital stock, w wages, r the real interest rate, and ρ the
time preference rate. Assume that r =f’(k), the marginal product of capital.
a. Use the Hamiltonian method to derive the optimal path of consumption.
b. Use the equation you derived in part a and the capital accumulation equation to draw a
phase diagram in (c, k) space, showing the arrows of motion and saddlepath to steadystate.
c. Using the phase diagram, analyze the effects of each of the following on the both the
steady-state and dynamic paths of c, k:
(i) A permanent, unanticipated change in w.
(ii) A temporary change in w.
(Remember to show the time paths of c and k for each of these experiments).
3
5. Consider the following model of two Cournot firms, each producing a differentiated
good. Assume zero costs of production. Let the demands facing each firm be:
p1 = α – β q1 – γ q2
p2 = α – β q2 – γ q1
where |β| > |γ|, and where p’s and q’s denote prices and quantities respectively.
a) Find the equilibrium quantities, prices, and profits of each firm.
b) Show that increased product differentiation increases firms’ profits, and explain the
underlying intuition.
6. 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 & LM curves.
c. Rewrite the system you derived in part (ii) 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).
d. Using Cramer’s rule, derive the following comparative static effects (holding other
factors constant): MY/MG, MY/MM, Mr/MG, and Mr/MM. Discuss how the efficacy of
monetary and fiscal policies depends on the interest elasticities of investment and money
demand. Illustrate graphically.
e. Suppose that we modify consumption and money demand to depend on wealth, and
assume that government bonds are net wealth. Would a bond-financed increase in
government spending increase equilibrium output (holding other exogenous variables
constant)? Discuss and show diagrammatically.