- Question 5 10 Marks A Consumer Gets Sick With Probability P And Stays Healthy With Probability 1 P Formally 0 1 (315.3 KiB) Viewed 20 times
Question 5 (10 marks) A consumer gets sick with probability p and stays healthy with probability (1 – p). Formally, 0 = {H, S} and P(H) = 1 - p and P(S) = p. The consumer is offered an insurance contract that requires her to pay a premium r in each state of nature (i.e. both if she gets sick and if she stays healthy), and in return pays the cost c of the medical bills if the consumer gets sick. The consumer has the option of purchasing only a fraction a E [0, 1] of the insurance contract, in which case she has to pay a premium of ar and, if she gets sick, gets paid ac. The consumer's wealth is w, and her von Neumann- Morgenstern utility function is given by: u(w') = (3w' + 2) Ź = for all w' > 0. Assume that w>c>r> 0. (a) (1.5 marks) Show that the consumer is strictly risk-averse. (b) (0.5 marks) Define when an insurance contract is fair. (c) (1 marks) When is the insurance fair in this example? (Your answer will be an equa- tion that depends on p, r and c.) (d) (5 mark) Write down the consumer's expected utility maximization problem. Derive the first-order condition of this problem and show that if the optimal choice of the consumer is to buy full insurance, then the insurance must be fair. (e) (2 marks) In economic terms, explain why the consumer fully insures herself if the insurance is fair but not otherwise. .