AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Econ 01B
January 2007
Page 1 of 2
COMPREHENSIVE EXAMINATION IN PRELIMINARY THEORY
Instructions: Please answer all questions and show all of your work.
1. (a) 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.
i. Derive the Marshallian demand functions.
ii. Derive the indirect utility function.
(b) Derive the cost function c(w, q) and conditional factor demand functions z(w, q) for
each of the following production functions:
i. f(z) = z1 + z2
ii. f(z) = min {z1, z2}
2. Consider an industry of N firms producing a homogeneous output, where one firm is
the “Stackelberg” leader and the remaining N - 1 firms are follower firms. The followers
are so numerous that they behave as competitors, taking prices as given (and producing at
a point where price equals marginal cost). Let industry demand be P = 100 - 0.5 Q,
where Q denotes total industry output. Note that Q = qL + (N - 1)qF, where qL and qF are
the supplies of the leader and each follower respectively. Let the marginal cost of each
follower firm be MC = 0.5qF and let the total cost function of the leader firm be TC =
0.25qL2.
(i) Calculate the leader firm’s optimal output taking into account the behavior of the
follower firms. How does the leader’s optimal level of output change as the number of
followers increases?
(ii) Find the equilibrium industry price and industry output.
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3. Consider the following simple Solow model:
k& = sf(k) - (n + g + δ) k
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).
4. 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 graphically.