Question

Suppose Nadine owns an apartment in San Francisco worth $108 and does not have earthquake insurance. Nadine’s wealth is $144, including the value of the apartment. Nadine’s utility of wealth function is U (W ) = 10W 0.5 . If an earthquake hits, Nadine’s apartment will be destroyed. The probability that an earthquake will hit is 0.5.

(a) Is Nadine risk averse?

(b) What is Nadine’s expected wealth?

(c) What is the largest amount that Nadine is willing to pay for comprehensive earthquake insurance?

(d) Suppose her neighbor Patricia also owns an apartment in the same building that is also worth $108. Patricia has the same utility of wealth function as Nadine but has total 1 wealth of $169, including the value of her apartment. Is Patricia wiling to pay the same premium that you solved for in part (c)? Without any calculations, how do you know?

(e) Draw Nadine’s utility of wealth function. Indicate her expected utility if she does not purchase earthquake insurance. Also indicate her wealth and certain utility if she pur- chases earthquake insurance at the premium you solved for in part (c).

Answer 1

a)

Given

U=10W^0.5

Marginal Utility=5*W(-0.5)

We can see that marginal utility is decreasing. We can say that Nadine is Risk Averse

b)

Wealth in case of earthquake=W1=144-108=$36

Probability of earthquake=p=0.50

Wealth in case of no earthquake=W2=$144

Probability of no earthquake=1-p=1-0.50=0.50

Expected wealth=p*W1+(1-p)*W2=0.50*36+0.5*144=$90

c)

Utility in case of earth quake=U(36)=10*36^0.5=60 utils

Utility in case of no earth quake=U(144)=10*144^0.5=120 utils

Expected Utility=p*U(36)+(1-p)*U(144)=0.5*60+0.5*120=90 utils

Let Nadine is willing to pay a maximum sum of X towards insurance. then

Utility with insurance=Expected utility without insurance

U(144-X)=90

10*(144-X)^0.5=90

(144-X)^0.5=9

144-X=81

X=$63

Nadine will a maximum of $63 towards comprehensive insurance.

d)

In this case, expected utility will be higher. So, certainty equivalent will be higher. So, lower will be the Patricia's maximum willingness to pay for comprehensive insurance.

Patricia will not pay the same premium as calculated in part (c)

e)