I need help for the Econ Assignments.

ECON 417: Economics of Uncertainty

The Pennsylvania State University, Fall 2014

Problem Set 3

Monday, November 3, 2014, in class

Problem Set #3

For problems 1-4, circle your final answer. Provide explanations for each solution. (Each

question is worth 5 points). u(x) = log x is the natural logarithm.

1. Demand for Insurance

Consider the utility function u(x) = log x.

(a) Set up the individual’s expected utility maximization problem. Derive the first-order

condition.

(b) Find the optimal insurance coverage, C∗, when insurance is actuarially fair (i.e. q = p).

(c) Find the optimal insurance coverage when q > p.

(d) Comparative Statics. Use the first order condition from part (a) to find change in

C∗ = C(W, L, q, p) with respect to

(a) Probability

(b) Loss

(c) Wealth (Hint: consider IARA, CARA, DARA)

2. Supply of Insurance

Suppose there are two risk averse individuals, Cate and Dirk. They both face an identical

independent risky prospect: each individual has a 50% chance of earning $100 and a 50%

chance of earning $10. Let u(x) = log x be the utility function.

(a) Find Dirk’s expected utility from this prospect.

(b) Suppose Cate and Dirk decide to pool their incomes. They pay their realized income

into the pool and they each get half of the total income of the pool. Find Dirk’s

expected utility under the pooling scheme. (Hint: Since the two prospects are identical

and independent, there are four possible outcomes).

(c) Show that Dirk’s expected utility under the pooling scheme is greater than his expected

utility without the pooling scheme.

(d) Compare the variance of the risky prospect with the pooling scheme and without the

pooling scheme.

3. Adverse Selection

Consider the Rothschild and Stiglitz (1976) insurance model under asymmetric information.

Suppose that insurance companies offer price-quantity contracts. There are two types of

agents with type i = H or L. The initial wealth for all agents is W . An agent of type i has

probability pi of losing an amount L when the bad event happens. All agents have the same

utility function. Let W = 24, L = 16, u(x) = 2 √ x, pL =

1 2 , and pH =

3 4 .

(a) Compute the marginal rates of substitution for the two types.

(b) Compute the wealth in the good and bad states, i.e. Wg and Wb, for each type in the

separating equilibrium.

4. Moral Hazard

An individual has initial wealth of $80,000 and faces a potential loss of $36,000. The prob-

ability of loss depends on the amount of effort the individual puts into trying to avoid it.

If the individual puts a high level of effort, then the probability of loss is 5%, while if she

exerts low effort, the probability is 15%. The individual’s utility is u(x) = √ x if low effort

and u(x) = √ x− 1 if high effort.

(a) If the individual remains uninsured, what level of effort will be chosen, i.e. low or high?

(b) If the individual is offered full insurance with a premium of $2,250, will the individual

accept? (Hint: compare no insurance with full insurance without the level of effort)

(c) If the individual accepts the insurance offered in part (b), what level of effort will be

chosen, i.e. low or high?

(d) What will be the insurance company’s expected profit from the full insurance contract

with premium $2,250? (Hint: you must consider the probability of the effort that the

individual selects from part (c))

5. Reading: Einav and Finkelstein (2011). Selection in Insurance Markets: Theory and

Empirics in Pictures.

(a) How does the downward-sloping MC curve represent the well-known adverse selection

property of insurance markets?

(b) What is the fundamental inefficiency created by adverse selection?

(c) What are three common public policy interventions in insurance markets?

(d) Define advantageous selection and how is this different from adverse selection?

(e) What are some of the limits to using positive correlation tests for adverse selection?