Tackle the following questions.

b. Formulate the monopolist's protmaidmizing problem: write down the objective function and all the constraints including incentive constraints (105) and individual rationality constraints (IRS) for both types. Call this the original problem (OP). c. Consider the seller's problem but without the type 2 buyer's [R constraint (IR2) and without the type 1 buyer's IC constraint (1C1). Call this the relaxed problem (RP). Find the optimal set of contracts (qr, ti) that maximize the seller's prots in this relaxed problem. d. Show that the solution in part c also satises the additional constraints, 1R2, 101, in the original problem. Argue that based on this result the optimal contracts found in HP in fact are also optimal for the Original problem OP. e. Explain how the distribution of the two types (1,2) affects the optimal contracts. IGlzlaraieterise each pure strategy Nash equilibrium of this game. Compare the resulting total output to what it is in a Pareto optimal allocation. How do the workers' utility levels in Nash equilibrium compare to what they are in the competitive equilibrium in part d? 3. An auctioneer auctions off one indiyisible object to two potential bidders in a sealed bid auction. The bidders' valuations are distributed independently as follows: Bidder L has valuation 1 with probability 1}? and valuation 2 with probability 1,12. Bidder H has valuation i with probability lid and valuation '2 with probability 31/4. Each bidder luiows his valuation only but the prior distribution of valuations is common knowledge. a. Suppose that the auctioneer conducts a rst price sealed bid auction. If there is a tie in the bids, each bidder wins with probability If? and the winner pays the (highest) bid. A bidder (after knowing his valuation] participates in the auction if and only if his ex- pected surplus is non-negative. Find the set of nonnegative {151, a} that form a strictly increasing, symmetric Bayesian Nash equilibrium in which both bidders participate re- gardless of their valuations. Draw the set of feasible {51,52}. At a strictly increasing, symmetric Bayesian Nash equilibrium, a type 1 bidder (any bidder with valuation 1} bids a and a type 2 bidder bids ()2 7: b1 regardless of whether the bidder is H or L. b. In part a, suppose the auctioneer chooses a non-negative pair {51,52}, h; 1': in 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 [11 and 32 for the auctioneer?' What is the maximum expected revenue? c. Suppose that the auctioneer conducts the rst 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 {t1I E33} that make 51 a dominant strategy for type 1 (any bidder with valuation 1] and babe in} a dominant strategy for type 2 (any bidder with valuation 2) and that induce ell bidders of all types to participate. Draw the set of such {151, be}. d. In part c, suppose that the auctioneer chooses non-negative {this} that induces participation of all bidders of all types and that satises all the other conditions listed. Derive the expected revenue maximising choice of {this} by the auctioneer and the expected maximum revenue. e. One can compare the mac-rims. obtained in parts b and d without computing them. Explain why. 4L Consider Nplayer normal form games in which each player i has a (non-empty, nite] pure strategy set .5\".- and a utility {payoff} function to. a. Dene a strictly dominated (pure) strategy. h. In the following two person game, the row player has strategies {'11 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 l l l } 1| ) 1' T (20 M {as B {11 -|J In.\" I