Suppose you have $50 and you would like to purchase some lottery tickets. Assume

Question

Suppose you have $50 and you would like to purchase some lottery tickets. Assume that you can win with 40% probability, and if you win, you can earn 100% of your investment (i.e., double your investment). On the other hand, if you lose, you lose 100% of your investment. Further assume that your utility function is given by Now, suppose your utility function is u(w) = e^w - 1 (a) Are you risk-averse, risk-neutral, or risk-loving? How do you know? (b) What is the expected value of the lottery if you invest all $50? (c) If you want to maximize your expected utility, how much should you invest?

Explanation / Answer

2 . u(w) = ew -1

(a) If du/dw is positive in nature, the person is risk- loving. If it is negative in nature, the person is risk- averse. or if it is zero than the person is risk - neutral.

so du(w)/dw = ew which is always positive so the man is risk - loving in nature.

(b) Expected Value of lottery if i invest all $ 50

E(Lottery) = P(WIn) * win amount + P(loose) * Win amount = 0.4 * 100 + 0.6 * 0 = $40

(c) :Let say the optimum investment quantity is X dollars.Here the terms ( 50-X+ 2X) means that if we put X dollers in lottery if we win then my total wealth will be (50-X) + 2X. Similarly if i loose the lottery, i will be having wealth equals to (50-X)

E [U] = 0.4 * U [(50-X + 2X)/50] + 0.6 * U[( 50-X)/50]

E(U) = 0.4 U((50+X)/50) + 0.6 ((50-X) /50)

E(U) = 0.4 [ e1+0.02X -1] + 0.6 [ e1-0.02X -1]

E(U) = 0.4 e.e0.02X + 0.6 ee-0.02x - 1

for maximum utility

d E(U)/dU = 0

so, d E(U)/dU = e( 0.4 * 0.02 * e0.02x - 0.6 * 0.02 e-0.02x) which should be equal to 0

so 0.4 e0.02x - 0.6 e-0.02x = 0

e0.04x = 3/2

0.04X = ln(3/2) = 0.4055

X = $10.14

so he must invest 10.14 dollers for maximum utility.

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