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Contract Theory (Summer 2007)
F-1
Prof. Dr. Klaus M. Schmidt
Final Exam
You have to solve both questions. For each problem you can get a maximum of 20 points.
Your answers may be in English or in German. You have 120 minutes. Good luck!
1. Adverse Selection
Consider the following regulation problem. The catering firm Beetle Inc. provides catering
services x in the Nymphenburger Schloss, which is owned by the Bavarian government. Beetle
Inc. produces quantity x with cost function c(x, θ) = θx, where the cost parameter θ ∈
{θ, θ̄}, 0 < θ < θ̄ < 1. Consumers of food at the Nymphenburger Schloss have linear demand
x = 1 − p, where p is the price charged by Beetle Inc. It follows that consumer surplus
CS = 21 (1 − p)2 . At time 0, the Bavarian government makes a “Take-it-or-leave-it-offer”
to Beetle Inc. and they sign a contract consisting of a transfer t to the government and a
price p, which Beetle Inc. has to charge the consumers. The profits of Beetle Inc. are thus
given by Π(θ) = (p − θ)(1 − p) − t and the government’s revenues are given by R = t. The
government cannot force the firm to produce but has to offer a regulation scheme that gives to
each type of Beetle Inc. at least its reservation utility of 0. The government maximizes total
welfare but it overweighs government revenues by the factor 1 + λ, λ > 0 because to raise one
Euro, distortionary taxation inflicts a disutility of 1 + λ Euros on the consumers; hence, the
government’s objective function is given by
W = CS + Π + (1 + λ)R
a) Suppose both, the government and the firm, know the type of the firm. What is the
optimal contract (pF B (θ), tF B (θ)) for the government to offer either to the low or the
high cost firm. How does pF B (θ) vary with λ? Calculate the optimal pF B (i) if λ goes to
zero and (ii) if λ goes to infinity. Give a short intuitive explanation for both, the change
with λ and the limiting contracts.
From now on, suppose that the cost parameter θ is private information of the firm. The
government does not know the realization of θ but believes that P rob(θ = θ) = q and Prob(θ =
θ̄
θ̄) = 1 − q. Let 0 < q < 1−
< 1, which ensures an interior solution, i.e. 0 < p < 1, in the
1−θ
following subsections.
b) State the second best problem of the government. Show which constraints are binding
in the optimal solution, which constraints can be ignored, and which should be ignored
first but later checked for any optimal contract.
c) Compute the price schedule pSB (θ̂) that the government uses in the second-best optimal
direct mechanism. You do not have to plug in for tSB (θ̂) and you do not have to check
the remaining constraints.
Contract Theory (Summer 2007)
F-2
Prof. Dr. Klaus M. Schmidt
d) Calculate for both types the difference ∆p between pF B and pSB . How does ∆p vary
with λ? Calculate ∆p (i) if λ goes to zero and (ii) if λ goes to infinity. Give a short
intuitive explanation for both, ∆p’s change with λ and its behavior in the limit.
2. Moral Hazard
A risk neutral principal wants to hire a risk averse agent to increase the profits of her firm.
If the agent increases the profits of the firm, the net profit is x̄ = 21
, otherwise net profit is
2
x = 3. At time 0 the principal makes a “Take-it-or-leave-it-offer” to the agent. If the agent
accepts, he is employed by the principal. Otherwise, both parties get a payoff of zero. At time
1 an employed agent can either provide low, medium, or high effort. If the agents exerts low
effort a1 , the probability of a profit increase is p1 = 0. Medium effort a2 results in a success
probability of p2 = 21 . By exerting high effort a3 , the agent can increase the probability of a
profit increase to p3 = 1. The agent’s costs of exerting effort are

 ln 1, if a = a1 ;
ln 2, if a = a2 ;
c(a) =

ln 6, if a = a3 .
The principal’s profit is ΠP = x − w and the agents utility UA = ln w − c(a).
a) Characterize the first best effort level; i.e., the effort level the parties would agree upon
if effort was contractible.
Suppose in the following that effort is not observable by the principal and that only the profit
realization is contractible.
b) What is the cheapest incentive scheme that implements the low effort a1 ?
c) What is the cheapest incentive scheme that implements the high effort a3 ?
d) State formally the principal’s problem if he wants to implement medium effort a2 . Argue
which of the three constraints must be binding in the optimum and which must be slack.
e) Derive the cheapest wage scheme that implements medium effort a2 .
f) Compute the profit maximizing effort level.
g) Compare the profit maximizing effort level to the first best effort level and explain
intuitively why they differ.
h) Now, suppose c(a3 ) = ln α,
principal to implement a2 ?
α > 2. For which values of α is it impossible for the