Question

N. If Parsifal does not have an accident, then he will earn $20,000, if he does, he will be reduced to earning only $4,000. The probability of having an accident is 0.02. His expected utility function is u(x) = √ x where x represents his earnings. [Ch. 7 will cover the expected utility approach to choice under uncertainty, this is a first instance.]

1. Give the cdf of Parsifal’s random wealth.

2. Give the expected value and the variance of Parsifal’s random wealth.

3. Give Parsifal’s expected utility, that is, give E u(X).

4. What income, W, if delivered to Parsifal for sure, would give him the same expected utility.

5. In expected terms, how much could an insurance company make off of Parsifal by offering him income replacement insurance?

Answer 1

1.

CDF is given by

F_X(X)=0.02 for X4000

F_X(X)=1.00 for X20000

2.

Probability of an accident=p=0.02

Wealth in case of accident=W₁=$4000

Probability of no accident=1-p=0.98

Wealth in case of no accident=W₂=$20000

Expected wealth=p*W₁+(1-p)*W₂=0.02*4000+0.98*20000=$19680

3.

Probability of an accident=p=0.02

Utility in case of accident=U(4000)=4000^0.5=63.2456 utils

Probability of getting no accident=1-p=0.98

Utility in case of no accident=U(20000)=141.4214 utils

Expected utility=E[U(X)]=p*U(4000)+(1-p)*U(20000)=0.02*63.2456+0.98*141.4214=139.8579 utils

4)

At certain income of X, agent's utility will be same as expected utility i.e.

U(X)=139.8579

X^0.5=139.8579

X=19560.23

5)

Maximum revenue for insurance company=TR=Maximum willingness to pay for insurance

=W₂-X=20000-19560.23=$439.77

Loss amount=L=20000-4000=$16000

Expected Cost=EC=p*L=0.02*16000=$320

Profit of insurance company=TR-EC=439.77-320=$119.77