John has a risk asset worth Y and derives utility from the consumption of his we

Question

John has a risk asset worth Y and derives utility from the consumption of his wealth. His Utility function is:

U(Y) = ln(Y)

There is the probability P that he will suffer a loss L and his wealth will be reduced to Y = W-L

A) Suppose W=100, L=36, P=0.5 , Find John's expected Utitily of income and Utility of expected income and explain that intuition behind each.

B) Now suppose that he has an offer from another person who is willing to pay him a definite amount to buy his risky property from him. How much would he be willing to accept?

C) Between buying fair insurance and selling out as in part (B) which option would he prefer? Give reason.

Explanation / Answer

Given that U(Y) = ln(Y)

Y = W-L

Y=100-36=64

U(Y)=ln(100)*0.5+ln(64)*0.5

=4.6*0.5+ 2.07

=4.42

It represents the utility an john gets when he plays this gamble.

Expected income E(Y)=0.5*100+0.5*64

=50+32

=82

utility of expected income =ln(82)=4.41

Both of the values are almost same.

b. He will accept anything above than 32.

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