Al, a Risk- Averse Expected- Utility Maximizer with Utility Over

question 80

Essay

Al, a risk- averse expected- utility maximizer with utility over wealth given by U(W), has W₀ dollars that he can save or invest in a risky asset. With probability p the rate of return on the asset is r_g>0, and with probability 1 - p the rate of return on the asset is r_b <0 . (Note that if you invest x dollars in an asset with rate or return r, you end up with (1 + r)x dollars.)
i)What is Al's expected utility he invests x in the risky asset?
ii)Now suppose that Al's utility for wealth takes the form U(W)=- e^{- aW}. Show that the optimal choice of x does not depend on W_0.