Claremont Graduate University
Department of Economics
Microeconomics Qualifying Examination
May 28, 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. [25 pt.] Consider an economy with two consumers: A and B, and two commodities: X and
Y . Consumer A’s utility function is
0.5 0.5
UA (XA , YA ) = XA
YA
and consumer B’s utility function is
0.1 0.1
UB (XB , YB ) = XB
YB .
The consumers initial endowments are
ωA = (2, 3)
ωB = (3, 2).
(a) [5 pt.] Compute the marginal rate of substitution (MRS) for each consumer. Do the
two utility functions represent the same preferences? Explain why or why not.
(b) [4 pt.] Compute the set of Pareto optimal allocations (also called the “contract curve”).
(c) [4 pt.] Assuming that the price of X is equal to 1 (pX = $1), compute the competitive
equilibrium of this economy. (i.e., prices and quantities)
(d) [4 pt.] Show that the competitive equilibrium is on the contract curve. What does it
mean in terms of welfare?
(e) [4 pt.] Now assume that consumer A’s preferences have changed. Consumer A’s utility
function is now
UA (XA , YA ) = min{2XA + 4, 4YA + 2}
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Claremont Graduate University
Department of Economics
whereas consumer B’s utility function is the same as before
0.1 0.1
UB (XB , YB ) = XB
YB .
The endowments are the same as before. Find the contract curve and graph it.
(f) [4 pt.] Using the utility functions in part (e), compute the competitive equilibrium of
this economy (assume pX = $1).
2. [25 pt.] Define a set of outcomes
C = (1, 2, 3, 4)
and two lotteries
L1 = (0.25, 0.25, 0.25, 0.25)
L2 = (x, 0, y, 0.25)
(a) [9 pt.] Over what values of x and y, if any, is L1 first order stochastically dominated by
L2 ?
(b) [9 pt.] Over what values of x and y, if any, is L1 second order stochastically dominated
by L2 ?
(c) [7 pt.] If L1 is second order stochastically dominated by L2 , does it mean that for any
utility function, L2 is preferred to L1 ? Explain.
3. (25 pts) Consider the three-player normal-form game below. Player 1’s pure-strategies are
{U, D}, Player 2’s pure-strategies are {L, R}, and Player 3’s pure-strategies are {W, E}.
W
U
D
L
3, 5, −1
−1, 7, 3
E
R
0, 0, 4
1, 2, 4
U
D
L
9, 1, −2
6, 4, 2
R
2, 1, 1
3, 2, 3
Figure 1: Three-player Game
(a) Solve for all the pure-strategy Nash equilibria.
(b) Now suppose that the players move sequentially in order: Player 1, then Player 2, then
Player 3. There is perfect observation of the past history of play. Draw the game tree.
How many subgames are there if you count the full game as a subgame? How many
pure strategies does Player 3 have?
(c) Find all subgame-perfect equilibria to part (b).
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Claremont Graduate University
Department of Economics
(d) Now suppose that the players play the stage game an infinite number of times and
discount next round’s payoffs exponentially with discount factor δ. For what values of δ
can (D, L, E) be played every round as part of a subgame perfect equilibrium.
4. (25 pts) Consider the cheap-talk model from class. Assume that the loss function of both the
sender and the receiver is linear: l(x) = x. The bias b =
1
120 .
(a) In a partition equilibrium, the length of each partition is increasing by how much? (You
may give the answer from memory or re-derive the result. No explanation is necessary.)
(b) Construct the Pareto optimal perfect Bayesian equilibrium. You only need to solve for
the partitions.
(c) Calculate the receiver’s welfare in this equilibrium.
(d) Calculate the receiver’s welfare in a babbling equilibrium.
(e) What’s the lowest bias for which the Pareto optimal PBE has the same number of
partitions that you found in part (a).
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