Im a fairly riskaverse person my utility function over wealt

I\'m a fairly risk-averse person; my utility function over wealth is U = W^5, where W is my wealth in dollars. Suppose I currently have $100,000 in wealth, but I have the opportunity to have an experimental surgery. If it is successful-and it is 10 percent of the time-I will become tall, fit, and handsome, have an awesome career in film, and my wealth will skyrocket to $10 million. If it fails, however. I remain myself, but with chronic pain that is the equivalent of being $50,000 less wealthy. Would I buy this surgery if it cost $50,000? Show me why or why not. What\'s the fair price for this surgery?

Solution

Utility function: U=W^0.5

Probability of success = 10% or 0.1

Wealth after success = $10,000,000

Probability of failure = 90% or 0.9

Wealth after failure = $50,000

a)

Calculate expected utility in case of experimental surgery being undertaken

That is,

E(U) = ps.Us + pf.Uf

Where, ps and pf are probabilities in case of success and failure

Us and Uf are utilities in both cases

Thus,

E(U)=0.1(10,000,000)^0.5 + 0.9(50,000)^0.5

E(U)=316.22+201.24

E(U)=517.46

Or, W=$267764.85

Since $50,000 is less than $267764.85, the person should buy the surgery.

b)

Fair price of surgery is anything less than or equal to $267764.85