Claremont Graduate University
Department of Economics
Microeconomics Qualifying Examination
May 29, 2013
Instructions: You have one hour to read and outline your thoughts and another four hours to
answer the questions. Please answer ALL four questions. Each question is worth equal weight.
You may NOT use notes or a calculator. Read the questions carefully before answering them.
1. (30 pts.) A consumer has a utility function of u(x) = x31 x22 , with wealth w and faces prices p.
(a) Solve for the Walrasian demand function. If p = (2, 3) and w = 240 what is the demand?
(b) Use the Slutsky equation,
∂hl (p,u)
∂pk
=
∂xl (p,w)
∂pk
+
∂xl (p,w)
∂w xk (p, w),
to find the slope of the
Hicksian demand function.
(c) Now suppose the government levies a tax on good 1 that raises the price to p1 = 2.2.
Approximate the compensating variation using the slope you found in the previous part.
Hint: the CV is approximately the trapezoidal area to the right of the Hicksian demand
curve.
(d) Draw a figure depicting the price change in with x1 on the horizontal axis and p1 on
the vertical axis. Be sure to show the Walrasian demand curve, the Hicksian demand
curves, and the deadweight loss as measured by CV. The curves need not be perfect,
but should have the broad characteristics correct (e.g. slope of curves have the correct
sign).
2. Consider a pure exchange economy representable by an Edgeworth Box. There is one physical
good and two states s ∈ {1, 2}. Both consumers have expected utility with the Bernoulli
utility function ui (xi ) = ln(xi ). The individuals’ endowments are ω1 = (3 − a, 3 + a) and
ω2 = (2 − a, 4 + a), where a ∈ [0, 2].
(a) Suppose that both consumers have the same subjective probabilities π11 = π12 =
π21 = π22 =
2
3.
1
2
and
Solve for the price and allocation in the Arrow-Debreu equilibrium when
a = 0.
(b) How do the prices and allocations change as a function of a? Explain the intuition.
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Department of Economics
(c) Find the set of Pareto optimal allocations. Show that your solution in part (a) is in this
set. In the light of risk preferences and insurance interpret this result in three sentences
or less.
(d) Suppose there are no forward markets, only spot markets. Will this lead to an efficient
allocation? Please explain in four sentences or less.
3. Consider the normal-form game below
X
1, 4
5, 5
2, −3
A
B
C
Y
3, 5
−2, 3
0, 0
Z
−2, −4
0, 0
−1, 4
Figure 1: Solid Game
(a) Find all the Nash equilibria.
(b) Harsanyi Purification: Now suppose they are playing a perturbation of Solid Game of
the form below where d ∼ U [0, x] and e ∼ U [0, x]. Players know their own payoffs but
not the payoffs of their opponents. Find the pure Bayesian Nash Equilibrium of this
game that uses a threshold strategy as a function of the type.
(c) What is the probability distribution that players play a given strategy as a function of
x?
(d) Solve for the probability distribution that players place on their strategies x approaches
0 and show that it converges to one of your solutions in part (a).
A
B
C
X
1, 4
5, 5 + e
2, −3
Y
3 + d, 5
−2, 3
0, 0
Z
−2, −4
0, 0
−1, 4
Figure 2: Perturbed Game
4. Career Concerns: Indiana is an assistant professor in an archeology department. His ability θ
is distributed on [0, ∞) with an average of θ̄. His output in a period is equal to yt = 2et + 4θ
where et is effort in period t. Indiana’s cost of effort in period t is c(et ) = e2t . Assume that
neither the universities that employ Indiana nor the professor himself know his own ability.
The professor does observe his own effort but the university does not. Assume there are two
universities that interact in perfect competition to employ the professor. The university’s
payoff is yt − wt where wt is the wage. The wage is given to the professor in each period
before the professor chooses effort. The professor works for three periods and then retires.
He does not discount the future.
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(a) What is the efficient level of effort in each period?
Department of Economics
(b) What will be the equilibrium effort e∗3 and wage w3 as a function of y1 , y2 , e∗1 , and e∗2 in
period 3?
(c) Solve for the equilibrium effort and wage in period 2 as a function of y1 and e∗1 and solve
for the equilibrium effort and wage in period 1. Is effort Pareto optimal?
(d) Now suppose there are long publication delays and this is common knowledge. Period 1
output is now not observed until the beginning of t = 3. Solve for the equilibrium effort
and wages in all periods.
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