Problem 1 (25p). (Uncertainty and insurance) You are an owner of a luxurious sailing boat, worth $6, that you use for recreation on Mendota lake. Unfortunately, there is a good (50%) chance of a tornado in Madison (probability is equal to ?) that completely destroys it. Thus, your boat is in fact a lottery with payment (6, 0). a) What is the expected value of the "boat" lottery? (give one number) b) Suppose your Bernoulli utility function is given by u(c) = c". Give von Neuman-Morgenstern utility function over lotteries U(C1; C2). (formula) Are you risk averse, neutral or risk loving? (two words). Find the certainty equivalent (CE) of the "boat lottery" (one number). Which is bigger, CE or the expected value of a lottery from a)? Why? (one sentence) c) Your Bernoulli utility function changes to u(c) = Inc. Give von Neuman-Morgenstern utility function. (give a formula). Are you risk averse now? 1) You can insure your boat by buying insurance policy in which you specify coverage . The insurance contract costs y . T where the premium rate is equal to y =;. Find analytically and depict in the graph your budget constraint. Mark the point that corresponds to no insurance. e) Find optimal level of coverage T. Are you going to fully insure your boat? (one number and yes-no answer). Depict optimal consumption plan on the graph. f) Propose a premium rate y for which you will only partially insure your boat. (one number) Problem 2 (30p). (Edgeworth box, and equilibrium) Consider an economy with apples and oranges. Andy is initially endowed with wa= (40, 0) and Bob's endowment is w = (0, 40). The utility function of both Andy and Bob is the same and given by U (71, T2) = 2 In1+2Inc2 a) Plot the Edgeworth box and mark the allocation representing the initial endowment. b) Provide general definition of Pareto efficiency (one sentence starting with: Allocation is Pareto efficient if ... ). c) Prove, that an allocation is Pareto efficient if and only in such allocation satisfies MRSA= MRS. Start with necessity by showing that if the MRS condition does not hold then allocation is not Pareto efficient. Then proceed to sufficiency by showing that if the condition MRS is satisfied then indeed allocation is efficient (use a graph and write two sentences for each of the two conditions). 1) Find analytically a collection of all Pareto efficient allocations (contract curve) and depict it in the graph. e) Find the competitive equilibrium (give six numbers). g) Give some other prices that are consistent with competitive equilibrium (give two numbers). f) Using MRS condition verify that equilibrium allocation is Pareto efficient and hence an invisible hand of a free (and competitive) market guides selfish Andy and Bob to a socially optimal outcome
