2 is possible that some firms do not produce in a PE allocation. What can be said about which firms they are? c. Compare the levels of food consumption by different consumers in a given PE allocation. Compare the levels of food consumption that any given consumer gets in different PE allocations for the same economy. Explain how the different PE allocations differ from each other. d. Define a competitive (Walrasian) equilibrium (CE) for this economy. Characterize a CE in which money is numeraire (assuming that such a CE exists). Be as specific as possible. For each firm : that produces in a CE, find an inequality that relates K; to the output level and the marginal and total cost of firm i in equilibrium. e. Is a CE allocation necessarily Pareto efficient in this economy? f. Show that a CE allocation does not necessarily exist in this economy. Explain why CE might not exist. Use the notation above to specify an allocation that might plausibly arise if no CE exists. What can be said about how this allocation compares to PE allocations? 3. Consider the pricing problem faced by a monopolistic seller. There is a continuum of potential buyers of size 1. Each buyer demands at most one unit of the good. Buyers are of three types, i = 0, 1, 2. Each type i buyer values one unit of the good at V, = 0;VI, where r 2 0 is the quality of the good and @, is a parameter that describes the buyer's taste for quality. Assume 0 0 is constant, to produce one unit of the good of quality z. The monopolist's payoff is its expected profit. The buyers get payoffs equal to the value to them of what they buy minus what they pay. a. Suppose the monopolist can directly observe buyer type and can offer contracts contin- gent on type. Characterize the profit maximizing set of contracts for the monopolist. For the rest of the problem, suppose that types are not observable to the monopolist. The monopolist offers a menu of contracts of the form (po, a;) where a type i contract is meant for type i buyers. b. Formulate the monopolist's pricing problem with incentive and participation constraints, assuming each buyer has a reservation payoff equal to zero. c. Consider the relaxed monopoly pricing problem (RP) in which only the following down- ward adjacent incentive constraints (DAIC) and a participation constraint (PO) for type i = 0 are imposed. even - p2 2 02V/21 - Pi. (DAIC2) 01val - P1 2 01V/20 - po, (DAICI) GovID - Po 2 0 (PO) Show that all these constraints bind in a solution to this relaxed problem. d. Show that if the solution obtained in the relaxed problem (RP) satisfies monotonicity, i.e. r; 2 71 2 26, then all of the incentive constraints and participation constraints in the original problem are satisfied and the solution to the relaxed problem is also a solution to the original problem. e. Solve the relaxed problem (RP). Compare the optimal quality levels (25, 21, a;) to the quality levels the monopolist would choose in part a. Discuss any differences. f. Based on the solution to (RP) in part e, provide a sufficient condition on buyers' preferences such that the solution to (R.P) in part e is indeed a solution to the original problem in part b. Interpret this condition. Is the monopoly better off when this condition holds than when a solution to (RP) is not a solution to the original problem in part b?3 4. Two producers can grow food for a consumer who cares only about food and money. Producer j (j = 1,2) can plant q, > 0 units of seed at a cost of (1/3) + q, (in units of money) and then produce q, units of food from the seed at no additional cost. If a producer plants no seed, it has no cost. The agents' interaction can be described as a game in which the producers independently plant q, 2 0 units and pay any cost of planting. Next, they independently choose food prices p, 2 0 (in terms of money) and then the consumer chooses to buy of = [0, q,] units of food from each producer j. The payoff of producer y is its revenue pic, minus its cost. (It can borrow to pay for seed at no interest and repay its loan from its revenue.) The consumer's payoff is 3c-(1/2)c' - Pic - pace, where c = c + 02. All this information is common knowledge. a. How many pure strategies does the consumer have? Give an example of one. b. Show that in the consumer's best response to (91, 92, pi, p2), if pi 0 and q2 > 0. c. Show that there is no pure subgame perfect equilibrium (SPE) of this subgame in which p1 0. Hint: Use part b and consider deviation by producer 1. d. Use parts b and c to show that this subgame has no pure SPE of this subgame in which 0 0) but con- sume no food (y = 0)? Explain.e. Suppose that here, too, resources were allocated via the Walrasian mechanism with each consumer owning half of the firm. Prove that the equilibrium wage would be lower than in the equilibrium in part b. Explain why it is. f. Under these circumstances, could there be an asymmetric Walrasian equilibrium in which the consumers get different consumption bundles? Explain. g. Explain what is meant by the statement, "the decreasing portion of the production function is economically irrelevant." h. Suppose that instead of L = In + Ly it were the case that L = In to 2, for 0