1. P.T. Barnum is concerned about the health of his star trapeze artist. If the artist is capable of performing a triple somersault, PT's revenues will be m2. If however his star crashes and is unable to perform, revenues will only be m1. Lloyds of London is willing to offer PT insurance on the following terms: the premium for every dollar of insurance is γ. The insurance policy pays only if there is an accident. PT's expected utility function is

EU(c1,c2)=p1(c1)^(2/3) +p2(c2)^(2/3) (1)

where c1 is his consumption in the event of an accident; c2 his consumption if his trapeze star performs the triple somersault, p1 is the probability of an accident, and p2 is the probability of no accident.

1. (a) If x is the number of dollars of accident insurance purchased, derive equations which indicate the amount of consumption if there is an ac- cident and if there is not an accident. Use these equations to determine the budget constraint between c1 and c2. What is the price of an extra dollar of c1 in terms of c2? Why?

2. (b) Determine PT's demand functions for c1 and c2. PT's marginal utility of wealth function is: MU(c) =2/(3c^(1/3)).

3. (c) If the probability of an accident is 20%, γ = .20, m1 = 8, and m2 = 18, What are c1 and c2? Is PT fully insured? On a diagram draw PT's budget constraint, his utility maximizing indifference curve, and his optimal bundle.

4. (d) How much better off is PT for the ability to buy insurance? Show this on an expected utility diagram.

5. (e) What happens if γ = .8? What is x? Would you expect the insurance company to agree. (Explain).  And what happens if γ = .1? What is x? Is PT over or under insured?

2. Monopoly. There are two types of gold mines in the world. The short-run cost function for the only low cost gold mine is c^l(q) = cq^2. The marginal cost of a low cost producer is MC^l = 2cq. There is a perfectly elastic supply of high cost producers and the cost function for a high cost producer is c^(q)=sq. The demand curve for gold is P =A-Q.

1. (a)  For what parameter values could the low cost gold mine exercise market power? Why for high values of s does the low cost producer have market power?

2. (b)  For A = 30, c = 1 and s = 4 what is the equilibrium price? Output and profits of the low cost gold mine? And for A = 30, c = 1 and s = 25 what is the equilibrium price? Output and profits of the low cost gold mine?