Question: A risk-averse expected-utility maximizer has initial wealth w and utility function u. She faces a risk of a financial loss of L dollars, which occurs with probability

1. Answer all parts (a)-(e) of this question.

A risk-averse expected-utility maximizer has initial wealth w and utility function u. She faces a risk of a financial loss of L dollars, which occurs with probability . An insurance company offers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in the event of a loss). Denote by x the number of dollars of coverage.

(a) [5 marks] Give the formula for her expected utility V(x) as a function of x.

(b) [10 marks] Suppose that u(z) = e z ,  = 1=4, L = 100 and p = 1=3. Write V(x) using these values. There should be three variables, x; and w. Find the optimal value of x, as a function of  and w, by solving the first-order condition (set the derivative of the expected utility with respect to x equal to zero). (The second-order condition for this problem holds but you do not need to check it.) Does the optimal amount of coverage increase or decrease in ?

(c) [10 marks] Repeat exercise (b), but with p = 1=6.

(d) [7 marks] You should find that for either (b) or (c), the optimal coverage is increasing in , and that in the other case it is decreasing in . Reconcile these two results.

2. Answer all parts (a) - (e) of this question.

Demand for potatoes is given by p(q) = 11  q and supply by p(q) = 1+ q.

(a) [6 marks] What are equilibrium prices and quantities in this case? Calculate consumer surplus and producer surplus.

(b) [6 marks] Assume that there is a unit tax of t = 2. What price do producers get? What is the price that consumers pay? How high is tax revenue?

(c) [6 marks] Calculate consumer surplus, producer surplus and deadweight loss for the case in (b).

Nowsetthetaxtozeroagaint = 0. Assumethatbecauseofwarsupplyshiftstop(q) = 1+4q. The government puts in place a price cap such that price is regulated to be the same as in (a).

(d) [7 marks] What are the effects on producer surplus and consumer surplus of the price cap?

(e) [7 marks] In situations as in (d) price caps are often combined with rationing–a certain allotment per person. What might be an economic argument for rationing? Against?