Consider an agent with wealth w facing the possibility of an

Consider an agent with wealth w facing the possibility of an accident. The probability of the accident is p, and if it happens he looses an amount of money equal to L: The agent can buy insurance that pays q if the accident happens, and costs qz, i.e., z is the premium per dollar of coverage.

(a) Derive the first order condition for the agent\'s optimal level of coverage.

(b) Describe the expected payoff of the insurance company, and find the z that makes his profits 0.

(c) Substitute the you found in part b in part a. What is the optimal level of coverage for the agent?

Solution

Given that the total wealth is w

Probability of accident is p with looses amount is L

So, the remaining amount after the accident is w-L

(a)

the first order condition for the agents optimal level of coverage is

p*L*u(w+p(1-q)-L)+(1-p)*u(w-qL)

The expected wealth is

p*(w-L)+(1-p)*w

premiun per dollar of coverage is q when the accident happens

(b) The expected payoff of the insurance company is pq. It would have a positive expected prot at any price that exceeds pq for the policy. At this price, the company has an expected prot of 0.