Claremont Graduate University
Department of Economics
Microeconomics Qualifying Examination
Jan 22, 2014
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. Define a set of outcomes
C = (10, 20, 30, 40)
and two lotteries
L1 = (0.5, 0, 0, 0.5)
L2 = (0.25, 0.25, 0.25, 0.25)
Assume a rational consumer and that the independence axiom is satisfied.
(a) If the consumer’s utility function is
u(c) =
√
c
which lottery is preferred?
(b) If instead the consumer’s utility function is
u(c) = c2
which lottery is preferred? Is your answer in part a different than in part b? Explain
why.
(c) Now define a third lottery
L3 = (x, 0.2, y, 0.3)
Over what values of x and y, if any, L2 is first order stochastically dominated by L3 ?
Over what values of x and y, if any, L2 is second order stochastically dominated by L3 ?
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Claremont Graduate University
Department of Economics
If L2 is second order stochastically dominated by L3 , does it mean that for any utility
function, L3 is preferred to L2 ? Explain.
2. Consider a pure exchange economy, representable by an Edgeworth Box, with endowments
1/2 1/2
ω1 = (4, 2), ω2 = (2, 3), and utility functions u1 (x11 , x21 ) = x11 x21 , and u2 (x12 , x22 ) =
4/5 1/5
x12 x22 .
(a) Find the competitive equilibrium of this economy.
(b) Show that the equilibrium allocation is Pareto optimal.
(c) Show that the endowment is not Pareto optimal.
(d) Now suppose consumer 2’s endowment is ω2 = (0, 6). Solve for the new competitive
equilibrium. How have prices changed? Explain the intuition.
3. Consider the extensive-form game below.
1
L
R
M
2
a
2
1, 0
b
c
d
1
w
3, 3
1
x
w
−4, −4
y
x
1, 1
−5, −5
y
z
0, 1
2, 6
z
8, 5
1, 0
Figure 1: A game.
(a) Find all the Nash equilibria.
(b) Find all the subgame perfect equilibria.
(c) Now suppose that the player 1 has perfect information. More specifically the game tree
is now displayed in Figure 2. What is the subgame perfect now?
1
L
R
M
2
a
2
1, 0
b
1
r
3, 3
c
1
t
−4, −4
u
−5, −5
d
1
v
w
1, 1
2, 6
1
x
0, 1
Figure 2: A game with perfect information.
2
y
1, 0
z
8, 5
Claremont Graduate University
Department of Economics
(d) Are there any SPE where player 2 is worse off when player 1 cannot observer her actions?
Why or why not? Explain your answer.
4. There are four bidders each with independent valuations for a good distributed uniformly on
[0, 200]. All bidders are risk neutral and have quasilinear utility given by ui = xi θi − Ti where
xi is the probability of getting a good and Ti is the payment, for individual i.
(a) Suppose there is only one good and the auction format is a second-price auction. What
is the expected revenue of the auction?
(b) Now suppose there are two goods and each bidder has only unit-demand. Is the secondprice auction – in which the winning bids are the two highest and the payments both
equal the second-highest bid – incentive compatible? Prove your answer.
(c) Now suppose there are five bidders and three goods and each bidder has only unitdemand. Design an incentive compatible auction format in which it is optimal for all
bidders to bid their true valuation.
(d) Again suppose there are five bidders and three goods and each bidder has only unitdemand. The auction format assigns the goods to the the two highest bidders and the
winners pay their bids. What is the expected revenue generated in this auction?
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