ECN103 – The Economics of Uncertainty

Subject: Economics    / General Economics
Question
ECN103: The Economics of Uncertainty
Problem set 6
Exercise 1. Consider the canonical insurance problem, where any individual faces two states
of the world. In state of the world 1, her income is w; in state 2, she suffers a loss and her
income is w ? `. There are two types of agents, H and L, with the agents of type H having a
higher risk of suffering the loss, pH > pL . Ex-ante preferences over consumption plans are
U? (x) = (1 ? p? ) ?
?
x1 + p? x2 , for agents of type ? = H, L. There is perfect competition between insurance companies.
a. Argue that if the type of each individual was observable, then two insurance contracts would
be offered, (?H , ?H ) and (?L , ?L ), with premium ?? = p? ` and coverage ?? = (1 ? p? )`.
b. Now, suppose that the type of each individual is unobservable, but the market reaches a
separating equilibrium. With respect to this equilibrium,
(i) Argue that the contract for individuals of type H is still the one described in part a.
(ii) Argue that the contract for individuals of type L satisfies
(1 ? pL )?L ? pL ?L = 0
and
p w ? pH ` = (1 ? pH ) p w ? ?L + p H p
w ? ` + ?L . Exercise 2. Consider the canonical insurance problem, where any individual faces two states
of the world. In state of the world 1, her income is w; in state 2, she suffers a loss and her
income is w ? `. There are three types of agents, H, M and L, distributed in proportions ?H ,
?M and ?L . Agents of type H have the highest probability of suffering the loss, pH > pM ,
while those of type L have the lowest one, pL < pM . Ex-ante preferences over consumption
plans are
U? (x) = (1 ? p? )u(x1 ) + p? u(x2 ),
for agents of type ?, where u is an increasing Bernoulli index displaying strict risk aversion.
Insurance contracts are as in class: (? , ?), with premium ? and net coverage ?. There is perfect
competition between insurance companies.
a. For the case of a pooling contract, write conditions under which individuals of all types
take this contract and the insurance company breaks even. 1 b. For the case of separating contracts, where there is a different insurance contract for each
type of costumer, write conditions under which individuals of type M take the contract
designed for them, and the insurance company breaks even in this contract.
c. Consider now the case of “partially separating” contracts, where there is one contract intended for individuals of type L, and a different contract for those whose type is either
M or H. Write conditions under which individuals of the latter types choose the contract
intended for them, and the insurance company breaks even in this contract. 2