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Problem Set 2 29 November 2007 1 of 4
Problem Set 2
2.1. (7%) Consider a screening model where a monopolist banker is providing a loan for a
privately informed entrepreneur. To be specific, assume that the opportunity cost of
funds for the monopolist is R per unit of capital and hence the cost of lending k units is
Rk. The entrepreneur uses the capital in a production process with a production function
y = θf(k). Assume that θ ∈ {θ H , θ L } where θ H > θ L . The banker sets a schedule of
repayments, t i and amounts to be lent k i for i ∈ {H, L}. The profit of the bank is then
t i − Rk i from contract i and the payoff to the entrepreneur is θ i f(k i ) − t i ≥ 0. Assume
that the production function is increasing, concave and satisfies the Inada conditions, i.e,
lim k=0 f ′ (k) = ∞ and lim k=∞ f ′ (k) = 0.
(a) Assume first that the type θ i is observable to the banker and solve for the optimal
contract to offer. Which type will be offered a higher loan?
(b) Assume next that θ i is private information of the entrepreneur. Let µ denote the
probability that θ = θ L . Formulate the banker’s problem and solve for the optimal
capital-repayment pairs for all values of µ. Clearly write all steps of your reasoning
and proof all claims you need for solving the problem.
(c) Compare solutions from (a) and (b), which type of entrepreneur is better off and
for which values of µ?
(d) Assume now that the competition between banks drives their expected profits down
to zero. Find the optimal contracts under full and asymmetric information.
2.2. (8%) Consider screening with continuum of types. A monopolist producer of some good
is faced with a continuum of consumers indexed by θ ∈ [θ, θ] (where θ − θ = 1 and θ > 0)
and distributed according to uniform distribution. If consumer of type θ buys good of
quality q for price p, her utility is
U(q, θ, p) = θq − p.
Every consumer buys at most one unit of the good. The reservation utility level for all
types of consumers is zero. The monopolist has a quadratic cost function with respect
to quality of the good given by C(q) = 0.5cq 2 .
(a) Let q F B (θ), θ ∈ [θ, θ] be the consumption profile that corresponds to an interior
first best efficient allocation. Find (q F B (·), p F B (·)) and show that both q F B (·) and
p F B (·) increase in θ
Assume asymmetric information. Let {q(θ), p(θ)} be a differentiable revelation mechanism
that induces consumer θ to reveal his type truthfully. More precise, if he announces
θ, he receives a good of quality q(θ) for the price p(θ).
(b) Write the optimization program for the monopolist who maximizes expected profits
and who is required to supply his product to all consumers and solve it by finding
the optimal contract ( q SB (·), p SB (·) ) . In particular:
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Problem Set 2 29 November 2007 2 of 4
(i) Write down IC constraint and show, by deriving FOC and SOCs, that it is
equivalent to LIC and monotonicity of q(·).
(ii) Write down P constraint and show that it has to be binding for the lowest type
and it is irrelevant for any higher type.
(iii) Derive the formula for p(·) from LIC and put it into the objective function of
the monopolist.
(iv) Find optimal q(·) by ignoring monotonicity constraint.
(c) For the optimal contract ( q SB (·), p SB (·) )
(i) Check if the optimal profile q SB (·) is increasing in θ.
(ii) Check if information rent is increasing with θ
(iii) Check if ‘no distortion at the top’ rule is preserved at the optimum.
2.3. (5%) Consider the moral hazard model with two outcomes and two effort levels discussed
during the lecture and do one modification: the Agent is risk neutral. All other
assumptions are preserved.
(a) Characterize all optimal contracts for the P if he wants the A to choose high effort
level.
Assume additionally that high effort level produces more social surplus than the low
effort level.
(b) Show that it is the best for the P if the A chooses high effort level.
(c) Show that there exists the optimal contract which involves the P getting a fixed
rent from the A, who becomes then a residual claimant, bearing all the risk and
taking all the profits and losses.
2.4. (7%) Consider a principal-agent problem with three exogenous states of nature, θ 1 , θ 2 ,
and θ 3 ; two effort levels, e L and e H ; and two output levels, distributed as follows as a
function of the state of nature and the effort level:
State of nature θ 1 θ 2 θ 3
Probability 0.25 0.5 0.25
Output under e H 18 18 1
Output under e L 18 1 1
The state of nature is not verifiable (it cannot be contracted on). The principal is risk
neutral. The agent’s utility when choosing effort e and receiving monetary compensation
w is √ w − g(e), where g(e L ) = 0 and g(e H ) = 0.1. The agent’s reservation utility is 0.1.
(a) Derive the optimal contract assuming that the agent’s action is (observable and)
verifiable.
(b) Derive the optimal contract assuming that the agent’s action is not verifiable.
Advanced Microeconomics I, Part 2

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Problem Set 2 29 November 2007 3 of 4
(c) Assume the principal can buy, for a price of 0.1, an information system that allows
the parties to verify whether or not state of nature θ 3 happened. Will the principal
buy this information system?
2.5. (3%) During the lecture we showed that in moral hazard model with two effort levels
and continuum of outcomes if both parties are risk neutral, selling the project to the A
(making him residual claimant) is optimal for the P. Show that such a contract is also
optimal when the A is risk neutral and the P is risk-averse.
2.6. (5%) Consider the following variant of Clarke-Groves mechanism discussed in the class:
• Every agent reports a valuation ˆθ i .
• The bridge is built if and only ∑ I
i=1 ˆθ i ≥ 0.
• Each agent receives a payment equal to the sum of other bids if the bridge is built,
i.e, T i = − ∑ j≠i ˆθ j . Hence, if ∑ j≠i ˆθ j > 0 agent i receives payment; otherwise she
has to pay.
• If bridge is not built then all T i = 0.
(a) Show that the truth telling is a dominant strategy for every agent, hence the mechanism
leads to efficient outcome.
(b) Is this mechanism balanced?
Since the above mechanism is not balanced, to improve it we may to introduce “tax”
on agents to assure balanced budget. The basic idea is to require each agent i to make
an additional payment that depends only on what the other agents do without affecting
any of i’s incentives. Let ˆθ −i be a I − 1-dimensional vector of reports omitting the
report of agent i and let h i (ˆθ −i ) be an extra payment made by the agent i always, i.e.,
independently if the bridge is built or not. Hence, we have, compared to the previous
problem: T i = − ∑ ˆθ j≠i j + h i (ˆθ −i ) if bridge is built; T i = h i (ˆθ −i ) if bridge is not built.
(c) Show that the truth telling is again a dominant strategy for every agent.
(d) Assume that h i (ˆθ −i ) = ∑ j≠i ˆθ j if ∑ j≠i ˆθ j ≥ 0 and h i (ˆθ −i ) = 0 if ∑ j≠i ˆθ j < 0. Show
that under this scheme every agent may be taxed but never subsidized. Is this new
scheme balanced?
2.7. (3%) Show that in the model of ‘cheap talk’ game analyzed during the lecture, the
equilibrium with two different messages is better for both sender and receiver than the
‘babbling’ equilibrium.
2.8. (6%) Consider a variant of the cheap talk game analyzed in the class in which payoff
functions are: for Receiver −|y − t|; for Sender −|y − (t + b)|. Find the 2-report and
3-report Perfect Bayesian Equilibrium for this game. How they differ from analogous
equilibria from the original model?
Advanced Microeconomics I, Part 2

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Problem Set 2 29 November 2007 4 of 4
2.9. (4%) Consider a variant of cheap talk model discussed in the class when decision maker
consults two experts who observe t but have different positive biases: b 1 , b 2 > 0, b 1 ≠ b 2 .
Both experts report to the decision maker simultaneously and then the decision maker
chooses the action that is the more ’conservative’ of the two reports. Precisely, if m 1 < m 2
he chooses action m 1 and vice versa if m 2 < m 1 .
(a) Model the expert’s game as a static game with complete information, what are
strategy spaces and payoff functions?
(b) Consider the following pair of strategies m 1 (t) = m 2 (t) = t + k where k ≥ 0. Find
all values of k for which the above pair is a Nash equilibrium of the expert game.
(c) What is the Pareto ranking of the Nash equilibria from (b); which equilibrium is
the best from expert’s point of view?
2.10. (10%) Consider the communication game analyzed in the class, with full commitment
contract proposed by the decision maker. Hence, he proposes the compensation scheme
x(ˆt) ≥ 0 to the expert, based on the expert report ˆt, moreover, he commits ex-ante to the
choice y(ˆt) based on the expert report. If the expert does not accept the report, there is
no communication.
(a) Model this mechanism as an example of screening with continuum of types. Who
is the Principal and who is the Agent? Is Single Crossing Property fulfilled in this
model?
(b) Solve for optimal contract assuming for simplicity that the reservation utility of the
A is zero. You may use all results and methods derived in the class. If you have
problems, you may solve for special values of b, say b = 1/4, b = 0.5.
(c) Show that although truthful reporting by the A is preserved (due to the Revelation
Principle and incentive compatibility), the Principal never implements y(t) = t.
Advanced Microeconomics I, Part 2