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?b. In part a, suppose the auctioneer chooses a non-negative pair (61, by}, by > by that induces participation of all bidders regardless of their valuations and that forms a strictly increasing, symmetric Bayesian Nash equilibrium. What would be the expected revenue maximizing choice of by and by for the auctioneer? What is the maximum expected revenue? c. Suppose that the auctioneer conducts the first price auction as in part a. But now suppose that a bidder (of any type) participates in the auction if and only if he can guarantee himself a nonnegative surplus regardless of the other bidder's bid. Find all values of {b1, by} that make by a dominant strategy for type 1 (any bidder with valuation 1) and by(> bi) a dominant strategy for type 2 (any bidder with valuation 2) and that induce all bidders of all types to participate. Draw the set of such {61, by}. d. In part c, suppose that the auctioneer chooses non-negative {61, by} that induces participation of all bidders of all types and that satisfies all the other conditions listed. Derive the expected revenue maximizing choice of (61, by} by the auctioneer and the expected maximum revenue. e. One can compare the maxima obtained in parts b and d without computing them. Explain why. 4. Consider /-player normal form games in which each player i has a (non-empty, finite) pure strategy set S, and a utility (payoff ) function ;. a. Define a strictly dominated (pure) strategy. b. In the following two person game, the row player has strategies {7, M, B} and the column player has strategies { L, R). Solve the game by applying the iterated elimination of strictly dominated strategies (IDSDS). Justify each step carefully. L R T (2,0) (0, 1) M (0,0) (3,1) B (1,1) (1,0) c. Show that a pure strategy played with positive probability in a mixed strategy Nash equilibrium survives any number of rounds of iterated deletion of strictly dominated (pure) strategies. d. Show that if IDSDS produces a unique strategy profile then it is a Nash equilibrium. e. Consider a Cournot duopoly model with linear demand P = a - Q and P = 0 if Q > a, where @ = q1 + 92 and a > 0. Both firms have zero fixed cost and constant marginal cost c and 0 & c 0, and $" 0, d' > 0, c'(0) by regardless of whether the bidder is H or L