Question

2. Imagine a firm with 2 types of workers: half are “sick” and half are “healthy.” Both types have a 20% chance of getting sick in a year, but their medical costs when sick are different.

If a “healthy” person gets sick, she will have medical costs of $1,000. If a “sick” person gets sick, she will have medical costs of $10,000. Both types of people have starting wealth of $40,000 and utility of wealth = sqrt(W)

a) What is the actuarially fair price of insurance if everyone buys? (Think of what the insurance company will expect to spend on average.)

b) How much are the healthy people willing to pay for insurance? How much are the sick people willing to pay? Who will buy insurance if the premium is set at the actuarially fair price?

c) Based on who actually buys insurance in part b), what will the actuarially fair premium be?

d) Will anyone buy insurance at that price? Explain why or why not.

Answer 1

a) 20% chance of falling sick for both kinds of workers.  

Hence, actuarially fair price of insurance will be

0.20*1000 + 0.20*10000 = 200 + 2000 = 2200

b. Both individuals will choose a price that maximizes their utility level

For healthy, let w be the price of insurance

Then he will maximize

0.80 ( 40000- w)^1/2+ 0.20 ( 40000+ 1000 - w) ^1/2

Differentiating with respect to w

1/2*0.80 ( ( 40000- w) ^ -1/2 )* -1 + 1/2* 0.20 (( 41000 - w) ^ -1/2) * -1 = 0

Since the actuarial fair price is less than what the two workers are willing to pay, both will buy the insurance