Question

Suppose Mary can have good health with probability 0.8 and bad health with probability 0.2. If the person has a good health her wealth will be $256, if she has bad health her wealth will be $36. Suppose that the utility of wealth come from the following utility function: U(W)=W^0.5 Answer each part:

A. Find the reduction in wealth if Mary bad health.

B. Find the expected wealth of Mary if she has no insurance.

C. Find her utility if she has bad health and she has no insurance.

D. Find her utility if she has good health and she has no insurance.

E. Find the expected utility of Mary if she has no insurance.

F. Find the certain equivalent of the lottery.

G. If she has full insurance, find the payment the insurance company made to her if she has bad health.

H. Find the maximum premium she is willing to pay for full insurance.

I. Find the fair premium if she is full insured.

J. Find her expected utility if she paid the fair premium and has full insurance.

Answer 1

A. Find the reduction in wealth if Mary bad health.

Wealth in case of good health=Wg=$256

Wealth in case of bad health=Wb=$36

Reduction in wealth in case of bad health=256-36=$220

B. Find the expected wealth of Mary if she has no insurance.

Probability of good health=p=0.80

Probability of bad health=1-p=0.20

Expected wealth=p*Wg+(1-p)*Wb=0.80*256+0.20*36=$212

C. Find her utility if she has bad health and she has no insurance.

U(W=36)=36^0.5=6 utils

D. Find her utility if she has good health and she has no insurance.

U(W=256)=256^0.5=16 utils

E. Find the expected utility of Mary if she has no insurance.

Expected utility=p*U(W=256)+(1-p)*U(W=36)

Expected utility=0.80*16+0.20*6=14 utils

F. Find the certain equivalent of the lottery.

Certainty equivalent of wealth is the certain amount that gives the same utility as lottery gives.

So,

U(W)=14

W^0.50=14

W=14²=196

CE=$196

G. If she has full insurance, find the payment the insurance company made to her if she has bad health.

In case of full insurance, insurance company would given him the amount calculated in part a. i.e

Amount paid by insurer in bad health=Reduction in wealth=$220

H. Find the maximum premium she is willing to pay for full insurance.

Let she is willing to pay a maximum amount of X towards full insurance.So

Expected utility in case of insurance=Expected utility in case of no insurance

U(256-X)=14

(256-X)^0.5=14

256-X=14²

256-X=196

X=$60

I. Find the fair premium if she is full insured.

Fair premium=probability of loss*Loss amount=0.2*220=$44

J. Find her expected utility if she paid the fair premium and has full insurance.

Wealth in case of insurance at fair premium =256-44=$212

Utility in case of fair premium=U(212)=212^0.50=14.56 utils