Question

There are two individuals, Scooter and the Big Man. There are two different streets in which each

individual can choose to live: Tenth Avenue and Thunder Road. It costs nothing to live or move

into each location. However, the probability of getting sick (and losing $100 to medical bills) on

Tenth Avenue is 0.25, while the probability of getting sick on Thunder Road is 0.5. Each individual

is endowed with $100 and faces a utility function of

U

=

√

w

where w is wealth. Scooter values living

on Thunder Road at $80 and living on Tenth Avenue at $40. The Big Man values living on Tenth

Avenue at $80 and living on Thunder Road at $40. Each person can choose where he wants to live

only once.

(a) Assuming each individual only cares about maximizing expected utility, where will each indi-

vidual want to live?

(b) Scooter and the Big Man can buy insurance for their medical bills from a firm. The firm can

break even, but cannot expect to make a positive profit because of perfectly competitive markets

and entry and exit. Due to a failure to collect accurate information, the firm hedges its bets

and assumes each individual will choose to live on the risky Thunder Road and values living on

Thunder Road at $80. What is the premium (p) the firm will charge each individual?

(c) Assume the firm charges each individual the premium as found in part b. Still assuming Scooter

and the Big Man only care about maximizing their expected utilities, what will each individual

choose to do? That is, where will they live? Will they choose to buy insurance?

(d) What is the name of the market failure that takes place in part c? How does it apply here?

(e) The firm hires a consultant who is good at acquiring accurate market information. Assume the

firm knows that Scooter values living on Thunder Road at $80 and the Big Man values living

on Tenth Avenue at $80. What premium should the firm charge each individual?

(f) Assume the firm charges the premiums found in part e. Still assuming Scooter and the Big

Man only care about maximizing their expected utilities, what will each individual choose to

do? That is, where will they live? Will they choose to buy insurance?

(g) Did the results from parts e and f solve the market failure identified in part e? If so, how? If

not, why not?

Answer 1

Utility function of Scooter and the Big Man: U = √w

Scooter’s utility from living on Tenth Avenue = √(initial endowments+ value of living at Tenth Avenue - expected loss of living at Tenth Avenue) = √(100 + 40 – (0.25× 100)) = √115 = 10.72

Scooter’s utility from living on Thunder Road = √(initial endowments+ value of living at Thunder Road - expected loss of living at Thunder Road) = √(100 + 80 – (0.5× 100)) = √130 = 11.4

Big Man’s utility from living on Tenth Avenue = √(initial endowments+ value of living at Tenth Avenue - expected loss of living at Tenth Avenue) = √(100 + 80 – (0.25× 100)) = √155 = 12.45

Big Man’s utility from living on Thunder Road = √(initial endowments+ value of living at Thunder Road - expected loss of living at Thunder Road) = √(100 + 40 – (0.5× 100)) = √90 = 9.49

1. Scooter will live on Thunder Road as he gets higher utility from living on it. Big Man’s utility from living on Tenth Avenue is higher than his utility from living at Thunder Road. Therefore, he will live on Tenth Avenue.
2. Assuming that each person will live at Thunder Road and value living on it at $80, the insurance company will expect everyone to live on Thunder Road and will charge a premium of $50 (assuming that the firm has no other cost than the insurance claim).

Scooter’s utility from living on Tenth Avenue with insurance = √(initial endowments+ value of living at Tenth Avenue – insurance premium) = √(100 + 40 – 50) = √90 = 9.49

Scooter’s utility from living on Thunder Road = √(initial endowments+ value of living at Thunder Road - insurance premium) = √(100 + 80 – 50) = √130 = 11.4

Big Man’s utility from living on Tenth Avenue = √(initial endowments+ value of living at Tenth Avenue - insurance premium) = √(100 + 80 –50) = √130 = 11.4

Big Man’s utility from living on Thunder Road = √(initial endowments+ value of living at Thunder Road - insurance premium) = √(100 + 40 – 50) = √90 = 9.49

A comparison between the utility of living on Tenth Avenue and Thunder Road shows that Scooter will still prefer to live on Thunder Road and Big Man on Tenth Avenue. The comparison of Scooter and Big Man’s utilities with and without insurance shows that Scooter is indifferent between buying insurance. However, given that the utility function is concave (i.e. he is risk-averse), he may buy the insurance. Big Man, on the other hand, will not buy insurance as his utility from buying insurance will be lower.

4. The name of market failure in part c is adverse selection. Since insurance company does not know about the individuals, they keep their premium based on the worst outcome. However, it drives out the people choosing safer options from the market. In the present case, it means that Big Man will not buy the insurance.