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2. Let M and W be nonempty?r nite and disjoint sets of men and women. A function y, : MLJW } MUW is a matching if for each min Mrj nfm} E W or Mm} = m; for each n: in W: putts} E M or p,|[nr} = to; and for an}r pair (m, to} E M X W: Mn!) 2 m if and onlyr if Mm} = w. Assume that each man m has a complete, transitive and strict preference )Fmover W U {m}. Each woman is has a complete? transitive and strict preference over M U {to}. We assume that all men and women prefer being matched [to anyone of the opposite sex} to remaining single [matched to themselves}. A nonempty subset S of MUW blockspifthere isamatching pf : MUW > MUWsuch that for each i E 5'? p512} E S and EH) >1- Ms} A matching is in the core if it is not blocked by an}r set S. A matching n is stable if it cannot be blocked by an individual or by a pair. 1 3. A government needs to procure 10 units of a product. There are two rms it can procure the product from. Each rm independently draws constant marginal cost 1 with probability % and constant marginal cost 2 with probability %. There is no xed cost. These draw probabilities are common knowledge. The rms learn their marginal costs but the information remains private. The government announces its compensation plan that is contingent on reports of the rms of their marginal costs. The rms report their marginal costs truthfully or not. The payoff of a rm when it is assigned :1 units of procurement quota at procurement price p per unit is pg or}? where c is the rm's true marginal cost. a. If both rms report 2, the govermnent pays 2 per unit and splits the order evenly. If both report 1... government pays :r} 1 g :r E 2 per unit and splits the order evenly. If one rm reports 1 and the other reports 2? the government orders the whole procurement from the lower cost {reporting} rm and pays 3;, 1 i: y E '2 per unit. Find the set of I? 3;: that would induce the rms to report their true marginal costs in a Bayesian Nash equilibrium. b. Find the set of any among those given in part a that would minimize the expected procurement cost of the government. 1What is the minimum expected procurement cost?I c. In part a? nd the set of :11:1 3; that would make the rms1 reporting true marginal costs a dominant strategy equilibrium. d. In part c.J nd .5: and 3: that would minimise the expected govermnent procurement cost. What is the minimum procurement cost in this case":I e. Argue directly {without computation} that the minimum expected expenditure of the government in the case of dominant strategy equilibrium cannot be lower than that in the Bayesian Nash equilibrium case. 3. A government needs to procure 10 units of a product. There are two rms it can procure the product from. Each rm independently draws constant marginal cost 1 with probability % and constant marginal cost 2 with probability %. There is no xed cost. These draw probabilities are common knowledge. The rms learn their marginal costs but the information remains private. The government announces its compensation plan that is contingent on reports of the rms of their marginal costs. The rms report their marginal costs truthfully or not. The payoff of a rm when it is assigned :1 units of procurement quota at procurement price p per unit is pg or}? where c is the rm's true marginal cost. a. If both rms report 2, the govermnent pays 2 per unit and splits the order evenly. If both report 1... government pays :r} 1 g :r E 2 per unit and splits the order evenly. If one rm reports 1 and the other reports 2? the government orders the whole procurement from the lower cost {reporting} rm and pays 3;, 1 i: y E '2 per unit. Find the set of I? 3;: that would induce the rms to report their true marginal costs in a Bayesian Nash equilibrium. b. Find the set of any among those given in part a that would minimize the expected procurement cost of the government. 1What is the minimum expected procurement cost?I c. In part a? nd the set of :11:1 3; that would make the rms1 reporting true marginal costs a dominant strategy equilibrium. d. In part c.J nd .5: and 3: that would minimise the expected govermnent procurement cost. What is the minimum procurement cost in this case":I e. Argue directly {without computation} that the minimum expected expenditure of the government in the case of dominant strategy equilibrium cannot be lower than that in the Bayesian Nash equilibrium case. a. Show that the set of stable matchings is equal to the set of matchings in the core. b. Consider the preference prole 1111 H12 11.13 \"[114 m1 1,3 3,2 2,1 4,3 1712 1,4 2,3 3,2 4,4 where, [nihw = [1,3] means m1 ranks 1111 rst among 1713 3,1 1,4 2,3 4,2 1171.; 2,2 3,1 1,4 4,1 women and mi ranks m1 third among men, etc. Apply the men {M} proposing deferred acceptance algorithm to the above preference prole and obtain the resulting match. c. Showr directlyr {without resorting to a theorem} that the resulting match in part b is stable. :1. Start with the match you obtain in part b and then suppose that women apply the Top Trading Cycle algorithm to trade men, ignoring the preferences of men. What would be the outcome? Describe and explain the outcome. h. Draw a second graph of the feasible consumption set you drew in part d. Assume now that the economj,r has a competitive equilibrium and draw? in the graphJ a possible indifference curve through the consumer's equilibrium consumption vector (If? f} 35> I}. Draw the consumer's competitive equilibrium budget line in the space of (E: f} Explain why the equilibrium value of the consumer's food consumption in this equilibrium is greater than the value of the labor time the consumer supplies. i. Suppose that the consumer consumes a positive amount of food in a competitive equilibrium. Justify the claim that the consumer supplies at least 2 units of labor time in equilibrium. j. What properties of the fundamentals of the econom].r in part h account for the existence of competitive equilibrium and differ from the case in part e. where no competitive equilibrium exists?I Explain