Mel has a utility function in the form of U(w)=sqrt(w) where w denotes his wealt

Question

Mel has a utility function in the form of U(w)=sqrt(w) where w denotes his wealth. His initial wealth is $9, but he owns a lottery ticket that is worth $25 with probability .25 and $0 with probability .75 (that is, the lottery ticket gives him 25 percent chance of winning $16). What is his expected utility if he hold onto this lottery ticket? Suppose that Ari wants to buy the lottery ticket from Mel; what is the lowest price that Mel would be willing to accept for the lottery ticket? Explain. What are Mel’s level of absolute and relative risk aversion at his current level of wealth?

Explanation / Answer

Expected Utility=0.25sqrt(25)+0.75(0)=1.25

Now this utility when no risk is hedged under then he needs to pay some premium which provides him with at least as good as utility with no insurance

U(with insuranceE)>=EUwith no insurance)=1.25

sqrt(9-optimum premium)=1.25

9-p=1.5625

p=7.4325

Hence willingness to accept the lottery =7.43

Now let's find absolute risk aversion=-U''(w)/U'(w)=1/4*w^(-3/2)=0.25*/27

Relative risk aversion=absolute risk aversion/w=0.25*w^(-5/2)=0.25*(9)^(-5/2)

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