ECON4240 Game Theory and Economics of Information—Spring semester 2011
Problems for the fifth seminar: Main emphasis: Moral hazard
The problems will be discussed on Wednesday March 30 and Thursday March 31, 2011.
Problem 1 will primarily be discussed. To the extent time allows, problem 2 will also be
discussed.
Please direct any questions to Aanund Hylland (Room ES1147, Tel: 228 54271,
e-mail: aanund.hylland@econ.uio.no)
Problem 1
Consider a relationship between a principal and an agent in which the value of the
product, denoted x, can be either x1 = 50,000 or x2 = 25,000. The agent’s effort, denoted
e, takes on one of three possible values, e1 = 5, e2 = 20, or e3 = 40. The probability of the
good result, x1, is 0.25 for effort level e1, 0.5 for effort level e2, and 0.75 for effort level
e3.
Assume that the principal is risk neutral, so that the principal’s utility is given by
x - t, where t is the transfer from the principal to the agent. The agent is risk averse with
utility function
U (t , e=
)
t −e
The agent has a reservation utility of 120.
(a)
Find the optimal contract under symmetric information for each level of effort.
What is the principal’s expected profit in each case? Which effort level does the
principal prefer?
(b)
Find the optimal contract when there is a moral hazard problem. As usual, we
assume that the principal has all bargaining power and formulates the contract.
Which effort level will be induced? How does the moral hazard problem have its
influence?
(c)
Discuss briefly what might happen if both sides have some bargaining power.
Problem 2
Consider an insurance market where insurance is offered by a large number of riskneutral insurance companies and where consumers have private information about their
accident probabilities, which are either high or low. The accident probability is denoted
π, where π = πH if the accident probability is high, π = πL if the accident probability is
low, and 1 > πH > πL > 0. Consumers are risk-averse and maximize expected utility. In
case of no insurance, the expected utility is given by
EU =
(1 − π ) u ( w ) + π u ( w − T ) ,
where u′ > 0, u″ < 0, w is initial wealth, and T is the loss incurred in case of an accident.
An insurance contract specifies that a consumer pays α1 to her insurer in case of no
accident and receives a net payment of α2 from her insurer in case of an accident. (If we
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assume that the premium α1 is always paid, this means that in case of an accident, the
gross payment from the company to the consumer is α1 + α2.) Hence, the expected utility
of contract (α1, α2) equals:
EU = (1 − π ) u ( w − α1 ) + π u ( w − T + α 2 )
Hence there is a deductible equal to (w − α1) − (w − T + α2) = T − (α1 + α2). (If a loss
occurs, this is the part of the loss that has to be born by the customer.)
The insurance company can offer several contracts, among which the costumer can
choose.
Provide short answers to the following questions:
(a)
Assume that there exists an equilibrium in such an insurance market. Why do the
consumers with the low accident probability pay a deductible, while the
consumers with the high accident probability do not?
(b)
If you were faced with a hypothetical choice in this market, between being a
consumer with the high accident probability and consuming its contract without a
deductible and being a consumer with the low accident probability and consuming
its contract with a deductible, which kind of consumer would you choose to be
and why?
(c)
Why may an equilibrium fail to exist if there is a high proportion of consumers
with the low accident probability?
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