provide solution

e. Suppose that in the student proposing deferred acceptance algorithm, H1 misrepresents its preference as 5'1 >11, 32 reg, 33,. What is the resulting match and how does the misrep- resentation affect the welfare of H1? f. Suppose that in the student proposing deferred acceptance algorithm, S: misrepresents its preference as H1 bgj H2 a3, H3. What is the resulting match and how does the mis representation affect the welfare of 82? g. Suppose that starting item a match SIHI, Selig, SgHg, we ignore the preferences of stu- dents and let hospitals trade students according to the top trading cycle algorithm. What would be the outcome? l1. Suppose that starting from a match SlHl, SgHg, 3353, we ignore the preferences of hos- pitals and let students trade hospitals according to the top trading cycle algorithm. What would be the outcome? i. Continuing to treat the assignment of hospitals to students in SIH1,32H3,33H3 as the students' initial endowments, is the outcome in part 11 in the core for the hospitaltrading students? 4. An auctioneer auctions o' one indivisible object to two potential bidders in a. sealed bid auction. Initially, the biddersl valuations are distributed identically and independently as follows: Each has valuation 1 with probability :1 and valuation 2 with probability %. This prior distribution of valuations is common knowledge. Then each bidder learns his own valuation {not the other's valuation) and, with that knowledge, participates in the auction if and only if his expected surplus is non-negative. a. Suppose that the auctioneer conducts a rst price sealed bid auction but species a set {3)}, 6:} with b2 r. in 13 ll, from which each bid must come. A bidder with the higher bid wins and if there is a tie in the bids, each bidder wins with probabilityr %. Find the set of {bhbg} that form strictly increasing, syrmnetrie Bayesian Nash equilibria in which both bidders participate regardless of their valuations. b. In part a, what would be the expected revenue maximising choice of ill and h; for the auctioneer? Draw a graph representing the constraints the auctioneer faces. What is the auctioneers maximum expected revenue? a. Suppose that the auctioneer conducts a rst price sealed bid auction but species a set {b}, tn} with {12 a 31 13 D, from which each bid must come, A bidder with the higher bid wins and if there is a tie in the bids, each bidder wins with probabity %. Find the set of {thin} that form strictly increasing, synnnstrie Bayesian Nash equilibria in which both bidders participate regardless of their valuations. b. In part a, what would be the expected revenue maximising choice of in and b; for the auctioneer? Draw a graph representing the constraints the auctioneer faces. What is the auctionaer's maximum expected revenue? c. Now, suppose the auctioneer species a pair of allowed bidsI by or :53, with tag 2'} b1 33 i}, but conducts a second price sealed bid auction. In the case of a tie {say at 2:2], each bidder wins the object with probability % and pays the second highest bid if he wins [in this case, 52). Find all values of {ab :52} that induce participation of both bidders regardless of their valuations and make in a dominant strategy for type 1 (a. bidder with valuation 1} and is a dominant strategy for type 2 (a bidder with valuation 2). d. In part c, derive the expected revenue maximizing choice of {51, E92} by the auctioneer and the expected maximum revenue. c. How do the answers to parts c and d change if we assume (for this part only} that a bidder participates if and only if he can guarantee a nonnegative surplus regardless of the other bidder's bid? '1'. In part c, what are the values of in and fig available to the auctioneer in Bayesian Nash equilibria where both types participate and type 1 chooses b1 and type 2 chooses on? g. In part f, what is the choice of {11, 52} that maidniises the auctionaar's expected ravenne? What is the maximum expected revenue? \f4. Several rms in an industry compete to hire from a population of workeIs. One third of the workers {type 1 workers} can get e 33 [1 units of education at a cost of e and then produce a/E units of output if theyr are hired. The remaining workers {type 2} can get e units of education at a cost of Bef and then produce 53/5 units of output if hired. The workers know their types when they choose their education levels. The rms independently oer wages that can depend on the workers' education levels and the workers accept at most one oer. A worker who accepts an alter gets as a [net] payoff the wage minus the worker's education cost. Workers who reject all offers get (net) payo I]. For each worker who accepts a rm's oer, the rm gets a prot equal to the worker's output minus the worker's wage. The rms seek to maximise their expected total net prot. All of this information is common knowledge. a. 1Why could it be reasonable to model the two types of workers as having different education costs even if there is no discrimination in the provision of education and all workers pay the same tuition per unit of education? h. Find competitive equilibrium education levels and wages for both worker types if the rms know each worker's type and the workers know that they will be paid dierent equilibrium wages depending on their education levels. For the rest of this problem, assume that the rms cannot tell what type any particular worker is. Consider a game played by two of the rms and a randomly selectecl worker. After the worker learns its type and chooses its education level, rm 1 makes a wage offer, then rm 2 makes an offer without knowing rm 1's offer, then the worker accepts one offer or rejects both offers. Use graphs to illustrate your answers to the following problems. c. Show that the game has a separating perfect Bayesian equilibrium {FEE} in which the worker chooses 16 units of education if it is of type 2. Show that this PBE is constrained inefcient, i.e., that a regulator who could control the rms and interact with the worker the any they do, knowing what the rms know, could obtain a Pareto superior nal allocation. On what basis can it he argued that this PBE is more plausible than any PBE in which the type 2 worker gets more than 15 units of education