Question

Eve insures her car with flood damage. The probability of a flood is 5%. The car is valued 𝐿 = $20,000 and her income is 𝑚 = $100,000. Let 𝑐𝑦 denote her consumption in case of a flood, where she must buy a new car for the same price, and 𝑐𝑛 her consumption if there is no flood. Her utility function for both states 𝑐 = 𝑐𝑛,𝑐𝑦 is 𝑈(𝑐) = √𝑐. [25%]

a. From the simple utility function, construct the von Neumann-Morgenstern utility function (𝑐𝑛,𝑐𝑦) for the two states with their given probabilities.

b. How much insurance Eve would buy if the insurance price is unfair? Briefly explain without solving the consumption problem.

c. If the insurance price is fair, what is Eve’s rational consumption under uncertainty? Make appropriate assumptions and explain your answer.

d. Assume that the probability of a flood is now 2%. Explain the change in Eve’s budget constraint and her consumption in both states if the insurance industry is perfectly competitive.

Answer 1

(a)

Von Neumann Morgenstern utility function $U(cn,cy)$ is simply the expected value of the two utility functions.

Hence,

$U(cn,cy)=P(NoFire)$

$U(cn,cy)=P(No Fire)\times U(cn)+P(Fire)\times U(cy)$

$\implies U(cn,cy)=95$ %$\times \sqrt cn+ 5$ %$\times \sqrt cy.$

(b)

Eve is a risk averse person, as can be seen from her utility function. Eve will purchase insurance cover full value of the car valued ,L i.e. she would buy insurance worth $20,000.

If the insurance price is fair, it should be same as the expected loss that Eve may suffer due to flood damage.

Hence, Insurance price that Eve would pay, I=5% of L.

$\implies I=\frac {5}{100}\times 20,000=1000$

I=$1000.

(c)

Let S be the amount spent on purchasing insurance.

Eve will purchase insurance only if Expected utility with insurance is greater than Expected utility without insurance.

$\sqrt (cn-S)>95$%$\times\sqrt cn +5$%$\sqrt cy$

Assuming no saving, $cn=m=100,000$ and $cy=m-l=80,000$

$\sqrt (100,000-S)>95$%$\times \sqrt 100,000+5$%$\ times \sqrt 80,000$

$\implies \sqrt 100,000-S>314.5586$

$\implies 100,000-S>98947.11283$

$\implies S<1052.8872$

If the price is fair, Eve consumption $=m-I=100,000-1000=99,000$

=$99,000.