Question

1) Contract Law - Insurance

Mario has purchased a new kart, and would like to get some insurance on it. The kart is

worth $1,000, and there is a 10% chance he will be in an accident and do $500 worth of

damage to his kart. Mario's utility function is U(V) = ln(V), where V is the value of his

kart.

a) Calculate Mario's expected utility with no insurance.

b) Now suppose Mario can buy auto insurance from Yoshi for a premium of $I that will completely compensate him for his $500 loss if it occurs. Find an expression Mario's

expected utility with insurance.

c) What is the most Mario is willing to pay for insurance?

d) Repeat steps a) - c) with a utility function of U(V) = V. Comment on why any differences are present in willingness to pay for insurance.

Answer 1

a)

Utility(at w=$1000)=U(1000)=Ln(1000)=6.907755

Utility(at w=$500)=U(500)=Ln(500)=6.214608

Probability of accident=p=10%

Probability of no accident=1-p=1-10%=90%

Expected utility=p*Ln(1000)+(1-p)*Ln(500)=0.1*6.907755+0.9*6.214608=6.838441

b)

Let Mario buys the insurance worth I that will pay her $500 i.e. equivalent to loss

In case of accident, Mario's worth with insurance I=1000+500-500-I=1000-I

In case of accident, Mario's Utility=U(1000-I)

In case of accident, Mario's worth with no insurance I=1000-I=1000-I

In case of no accident, Mario's Utility=U(1000-I)

Mario's expected utility in case of insurance=0.1*U(1000-I)+0.9*U(1000-I)=U(1000-I)=Ln(1000-I)

c)

Mario would be willing at most I, such that expected utility in case of insurance is same as expected utility in case of no insurance. i.e.

U(1000-I)=6.838441

1000-I=e^6.838441=933.03

I=1000-933.03=$66.97

d)

Utility(at w=$1000)=U(1000)=1000

Utility(at w=$500)=U(500)=500

Probability of accident=p=10%

Probability of no accident=1-p=1-10%=90%

Expected utility=p*U1000)+(1-p)*Ln(500)=0.1*1000+0.9*500=550

Let Mario buys the insurance worth I that will pay her $500 i.e. equivalent to loss

In case of accident, Mario's worth with insurance I=1000+500-500-I=1000-I

In case of accident, Mario's Utility=U(1000-I)

In case of accident, Mario's worth with no insurance I=1000-I=1000-I

In case of no accident, Mario's Utility=U(1000-I)

Mario's expected utility in case of insurance=0.1*U(1000-I)+0.9*U(1000-I)=U(1000-I)=1000-I

Mario would be willing at most I, such that expected utility in case of insurance is same as expected utility in case of no insurance. i.e.

U(1000-I)=550

1000-I=550

I=1000-550=$450

We find that willingness to pay is much higher in new case.

It is because of difference in utility function i.e. attitude towards wealth. In first case marginal utility is decreasing and now it is constant. So, in case of linear utility, Mario is willing to pay much higher for insurance.