Exercise session #10
Lecturer: Dmytro Vikhrov
Date: April 24, 2012
Problem 1 (signaling game 1)
Find all separating and pooling equilibria in the following signaling game.
Problem 2 (signaling game 2)
Mr. Brown is involved in an accident with Mrs. Smith and knows whether he has
been negligent or not (for instance, he knows if he has used his cell phone or not while
driving). Mrs. Smith does not know for sure, but has a prior belief that the probability
that he has been negligent is p ∈ (0, 1). She has the opportunity to go to a court where it
can be determined whether Mr. Brown was negligent or not. Mr. Brown can try to settle
the case without going to a court by making a take-it-or-leave-it pre-trial compensation
offer. Suppose the offer can be either 5, 000 (high offer) or 3, 000 (low offer). Mrs. Smith
can accept or reject this offer. If she accepts, then she takes the money and does not take
the case to a court. If she rejects, then she takes the case to a court. If Mr. Brown is
found negligent in the trial, he has to compensate Mrs. Smith with an amount of 5, 000.
If he is found innocent (i.e. not negligent), then Mrs. Smith receives nothing. In either
case, Mr. Brown ends up paying 6, 000 for the trial expenses (so, in case of trial, he pays
11, 000 in total if he is found negligent, 6, 000 in total if he is not found negligent).
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a. Draw the extensive form of the game.
b. Show that no separating equilibrium exists.
c. Show that there are two pooling equilibria. Fully characterize them.
Problem 3 (education signaling 1)
An agent is born with a skill type η, η = {L, H} and she decides whether to obtain
education and, if yes, how many units. The cost function is given by c(η, e).
a. Impose reasonable restrictions on the cost function. Setup the agent’s utility function.
b. Setup the firm’s maximization problem.
c. Find the separating equilibrium, in which the employer can perfectly distinguish types.
Make sure to describe the out-of-equilibrium beliefs and draw a graph where necessary.
d. Find the pooling equilibrium, in which the employer cannot distinguish types. Make
sure to describe the out-of-equilibrium beliefs and and draw a graph where necessary.
Problem 4 (education signaling 2)
There are two identical firms who would like to hire a worker. There is a single worker
in the market whose productivity may be high (H) or low (L). Suppose that the worker’s
output is 2 if η = H, and is 1 if η = L. The worker may signal his actual productivity
by investing in education e. The cost of education level e depends on the type η of the
worker and is given by c(η, e) = e
η
. The utility of the worker is linear in wage net of the
education costs.
a. What would be the optimal education level chosen by the worker if the firms knew the
actual value of η?
Now suppose that the worker’s productivity is unobservable by firms, but his educational
level is. The prior belief of the firms that the worker is H-type is λ ∈ (0, 1). Furthermore,
suppose that firms believe that e ≥ e is a signal of high productivity, while e < e is a
signal of low productivity.
a. What are the optimal wages under these beliefs?
b. Given your answer above, calculate the educational level that each type of the worker
will choose (this will depend on the value of e).
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c. Find the necessary condition on e so that education is an effective signal of productivity
(i.e. for what values of e does a separating Perfect Bayesian equilibrium exist?).
Problem 5 (education signaling 3)
Find a separating Perfect Bayesian equilibrium in which the worker gets 0 years of
education if η ∈ [0, η∗
) for some η ∈ (0, η∗
) and is offered a low wage wL, but 4 years of
education if η ∈ [η∗
, 1] and is offered a high wage wH. Find η∗
, wL, wH and describe firms’
beliefs. If hired, worker’s output is 10η and costs of obtaining education are c(η, e) = e
η
.
Problem 6 (market for lemons)
There is a seller trying to sell his car, and there is only one potential buyer. The buyer
is uncertain about the quality of the car, which the seller knows. The buyer’s belief about
the seller’s value of keeping the car is described by a uniform distribution on [0, 1000].
The actual value of the car to the buyer is 50% more than its value to the seller.
a. Suppose the buyer offers to buy the car for pB = 600. What is the probability that
the seller will accept this offer? What is the buyers expected profit from bidding 600?
b. Find the bid for the buyer that maximizes her expected profit.
c. Now consider the game in which the seller can offer to sell his car at any price and
then the buyer decides to accept or decline this offer. Find such a price R the buyer
will surely agree to buy the car, but not for any higher price.
Now suppose the buyer’s belief about the seller’s value of the car is described by a
uniform distribution on [250, 1000]. Still consider the game in which the seller can offer
to sell his car at any price and then the buyer decides to accept or decline this offer.
Find an equilibrium in which the buyer would surely accept a price of 500, would
have some probability more than 0 but less than 1 of accepting a price of 1100, but would
reject any other price over 500.
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