There are finitely many states s 2 S. The set of outcomes is [0; infinity), the amount of consumption. Consider an expected utility maximizer with utility function u (c) = p c. Suppose that for each state s 2 S, there is an asset As that pays 1 unit of consumption if the state is s and 0 otherwise (these are called Arrow-Debreu securities). Suppose also that we know the preference of the decision maker among these assets and the constant consumption levels; e.g., we know how he compares an asset As to consuming c at every state.
(a) Derive the decision maker's preference relation among all acts from the above information.
(b) Assume that the decision maker has a fixed amount of money M, which he cannot consume unless he invests in the Arrow-Debreu securities above, assuming that these securities are perfectly divisible, and the price of a unit of As is some ps > 0. Derive the demand of the decision maker for these securities as a function of the price vector p = (ps) s2S.
Alice has M dollars and has a constant absolute risk aversion alpha (i.e. u pxq"ex) for some 0. With some probability P p0, 1q she may get sick, in which case she would need to spend L dollars on her health.
There is a health-insurance policy that fully covers her health care expenses in case of sickness and costs P to her. (If she buys the policy, she needs to pay P regardless of whether she gets sick.)
(a) Find the set of prices P that she is willing to pay for the policy.
How does the maximum price P she is willing to pay varies with the parameters M, L, and
(b) Suppose now that there is a test t P t1, `1u that she can take before she makes her decision on buying the insurance policy. If she takes the test and the test t is positive, her posterior probability of getting sick jumps to and if the test is negative, then her posterior probability of getting sick becomes 0. What is the maximum price c she is willing to pay in order to take the test? (Take P P.)
1. Consider an industry with 3 firms, each having marginal costs equal to 0. The inverse demand curve facing this industry is P(Q) = 60 - Q, where Q = qi + 92 + 93 is total output. (a) If each firm behaves as a Cournot competitor, what is firm l's best response function optimal choice given other firms outputs? (b) Calculate the Cournot equilibrium. (c) Firms 2 and 3 decided to merge and form a single firm with marginal costs still equal to 0. Calculate new industry equilibrium. Is firm 1 is worse of or better of as a result? Was it a good idea for firms 2 and 3 to merge? Would it be a good idea for all three firms to organize the cartel? (d) Suppose firm 1 can commit to a certain level of output in advance. If the choice of firm 1 is q1, what would be the optimal choices of firms 2 and 3? (Hint: After observing q1 firms 2 and 3 would engage in (Cournot) duopolistic competition. What is the optimal level of q1? Calculate profits of firm 1, compare with (b). 2. Consider an economy with 3 firms and 2 consumers. Each consumer owns 10 units of Land. Firm 1 produces Food and Wood using technology (-L, F, W) = (-1, 1, 2). Firm 2 produces only Food with technology (-L, F) = (-2, 1) and firm 3 produces only Wood with technology (-L, W) = (-1, 1). Firms 2 and 3 are owned by consumer 1, firm 1 is owned by consumer 2. Consumers have identical utilities u(w, f) = wf. Calculate Walrasian equilibrium. 3. There is one consumer and one firm. The firm may have a high quality indivisible product (with probability q) or a low quality product (with probability 1 - q). The firm knows the value of the product, while the consumer cannot observe it prior to the sale. The consumer's utility from a product of given quality is v; - p, where un = 8, ; = 4, and p is the price paid. Costs of production are ch = 3, q = 1. (a) Under what conditions on q consumers will be willing to buy the product at a prespecified price p? What qualities of the product would be sold? (Hint: Analyze it case by case. E.g., if both qualities are sold, what is the expected utility of the consumer? Would she by? Would both types of the firm sell?) (b) Suppose a firm can spend some money A on advertisement of its product, and A is observable by the consumers. Present a separating equilibrium, where the high quality firm advertises, A; > 0, and sells the product at price p = 8, while the low quality firm does not advertise, A; = 0 and charges p = 4. (It would suffice if you check incentive compatibility and individual rationality constraints for only two choices of A, A; and A;. ) 3. Consider identical individuals i = 1, ..., n with constant absolute risk aversion a > 0 and dates t = 1 , ..., T. Each i has an asset that pays Yi = Xat ... + Xir at time T where the random variables Xit, i = 1, ..., n, t = 1,..., T, are all indepen dently and identically distributed with N (4, o?). Each Xat becomes publicly observable at time t. There is also risk-neutral company who wants to but these assets. At the beginning of each time t, before (Xit, . .. Xnt) becomes observable, the company offers a price Pit to each individual i who has not sold his asset to the company. If an individ ual i accepts the offer, the company owns the asset of i and pays Pit to the individual i at time T. (Note that individual i gets Y, if he holds on to his asset and Put if he sells his asset at some time t, getting the money at time T in both cases.) (a) For any t, any realization (Xa, ..., Xi(t-1) ) and any i, determine the lowest price Pa at which i is willing to sell his asset if he still owns it. Should the company offer Pit? (b) Consider a new identical individual (with costant risk-aversion parameter a). How much is he willing to pay for a 1th share of the company (getting 1th of the net proceedings of the company at T)? (c) What does the answer to part (b) change if the company is barred from trading at date t = 1? (d) Briefly discuss your results
