Question

8. An individual with utility function over money u(w) = 8Vw has $C in cash and a lottery ticket that pays $W if it wins and nothing if it loses. The probability of winning is .. Suppose an insurance is available at price $p per unit, where each unit of insurance pays $1 if the ticket does not win and nothing if it wins. (a) Is the individual risk averse, risk neutral, or risk loving? (b) What is the fair price for this insurance and how much insurance will the individual buy at this price? (c) Suppose the price of insurance is p = 0.8. What is the optimal amount of insurance the individual will buy? You may assume that the solution is interior. (d) Is the insurance a normal good or an inferior good for this individual? What does this suggest about whether the individual becomes more or less risk averse as her income increases?

Answer 1

A) as utility function is concave in wealth , so agent is risk averse

B) fair price per unit of insurance = loss Probability

= 3/4 = $ .75

individual will buy full insurance, that is loss of C

.

C) p = .8

Then

Risk uerse Agent is Eu Maximizer, then of Insurance is bought, then payment in Both states = PB. =o8B where ß is optimal Amount of Insurance bought wealth in Good state = wto- •8B un Bad state = c-c-.8B + B = .QB 80 eu= 1 u( WTC-08P) & 3/4 UC • QB) = + x8h w te- o88 + 3 x 38 N •2 B EU = 2 Nutc-o8B + 6 T. 2B apu = x x -08 + 6 x 2 = ƏB Z NWt6.8B 2 J QB 663 = 84 04-88 . 9 (wtc- 08B = 16 (6QB) 9W 90-22B = 3 2 B. 9CW+C) = 10.4 B B" = q (W+C) B* = 0865 (w to) | 10.4 Ang d) Now Raw) Arrow Batt Measure of Risk Avession Raww)= - Uh(W) / ullw) 4'(W) = 8 = 4w 112 U"(w) = -2 (w) - 3/2 Ralw)= +2 (w/3/2 = 05/w. - 4 wt12 80. d Raw) 20, 8o Decreasino Absolute Risk Avension (MARA) dw as wealth in Individual gets less Risk Averse, puts more money Tun Risky Asset, & Buys Less Insurance

So insurance is an inferior good