Question

1. Suppose Bob has income of $18,000. There is a 20% chance that Bob will get sick and have to spend $10,000 of his income on a treatment. Suppose Bob’s income-utility relationship is given by:   

Where I is Bob’s income.

U(I) = square root I

Complete the table below to find Bob’s total certain utility for various levels of wealth, along his marginal utility associated with increases in his wealth.

Wealth	Certain Utility	Marginal Utility = change in utility per extra $2,000
0	0.0
2000	44.7	22.36
4000	63.2	9.26
6000	77.5
8000	89.4
10000	100
12000	109.5
14000	118.3
16000	126.5
18000	134.2
20000	141.42
22000	148.3

a. What is Bob’s expected income, given that he faces a chance of illness? What is the actuarially fair premium for Bob’s circumstance?

b. How happy is Bob if he doesn’t buy any insurance? (Hint: expected utility!) Compare this to how happy Bob is if he can buy insurance at the actuarially fair premium.

Answer 1

a. Bob's income=$18,000

probablity of illness=0.2

probablity of being healthy=0.8

expenses during illness=$10,000

Bob's expected income=0.2*($18,000-$10,000)+0.8*($18,000)

= 0.2*$8,000+0.8*$18,000

= $1,600+$14,400

=$16,000

Acturially fair premium=0.2*$10,000=$2,000

b. Bob's expected utility with insurance=0.2 sqrt18000-2000 +0.8sqrt18000

=0.2* sqrt16000 + 0.8sqrt18000

=0.2*126.5+0.8*134.2=132.66

Bob's expected utility with no insurance=0.2*sqrt8000 + 0.8sqrt18000

=0.2*89.4+0.8*134.2

=125.24

As Bob's expected utility is more when he takes insurance of fair premium so, he should take insurance policy.

thanx