Question

b) (2+5 + 1 + 2 +2) Consider an insurance market with insurance firms being competitive and risk neutral (Zero expected profits in equilibrium) and a risk averse customer with a von-Neumann Morgenstern utility function u(x) = x1/3. The customer is born with a wealth of $150. There is a probability of an accident p=0.3 in which case the customer will suffer a damage of $50. An insurance contract is (Q1, Q2), where az = premium paid in the no accident state, and a = net payout received in case of accident (0) (iii) (iv) What is the customer's utility without insurance? If all Zero profit insurance contracts are offered, write down the customer's utility maximization problem. What contract would she choose? What will be the premium? What is the payout?| Is the customer happier with insurance? Show why or why not. If there was another customer with p =0.5, (a high-risk customer) what contract will she choose? (assume everything else for this new customer is the same as the old one, and the customers are identifiable). Show your work. Suppose both contracts (for part (ii) and (iv)) are being offered by insurance companies, and all customers are free to choose whatever contract they prefer. Which one will the high type customer choose? Show your work. (v)

Answer 1

0 Y3 u(x) = x EU = -3 (100)'s +7 (150) 5:117 (150) XG = 150-21 x B' = 100-4, + Benitit = 100+ 42 Eu : 07 (150 -4, JUB + 13 (100 +4, 3/3. Expected gain in good state = Expected loss in bod state (iPk, P.dz Eu= P(100+ Cr-plks. ) 134 (0.6) (150 - 4,²3. P both side then we get on By derivating Az = (1-P) 50 P = 3 50x13= 15, ·7x50 = 35 2 Eu with Insurance 51299 (1383's (150-15)' 13 (100+30)3 +7 so happy with So Eu increas/paise post Insurance contract the

(1259'3 iv) P=5 X = 5850 x=25 925.5x50 02 = 50 L = L2 = 50 13 Eu 100 prefer 4, = 15 4&2=35 high type Soi