#provide references where possible.

g. Is there a procedure that would take reported rankings by students and hospitals and determine a stable matching such that every hospital would have an incentive to report its true ranking listed above when the students report their true preferences? Explain. h. What economic reason might explain why? in the absence of a formal organised matching procedure? hospitals typically offer residencies to students rat her than students proposing to hospitals? 3. Five medical students 2' = 11, . . . y 5 are candidates to be placed in residencies with four hospitals.I j = 1} . . . 3.4. No student wants more than one residency position and no hospital wants to hire more than one student. The tables below show the students' rankings of hospital residencies and the hospitals rankings of the students. Each student 1-5 [respectivelyj hospital j] ranks the hospitals [resp. students] as listed in the row beginning with i [resp. j]? with more preferred listed rst? reading left to right. For example? student 3 ranks hospital 4 rst} then hospital 31 then hospital 1? then 2. Student 5 ranks 1 first1 then hospital 21 then 41 and prefers no residency to a residency at hospital 3. Each hospital would rather ll its residency with any student instead of leaving the position empty. 11234 . 123145 StUdEHti 2 4 2 31 Hospitals: 2 312 4 5 34312 354123 41432 414523 5124 a. Dene a matching of students to hospital residencies. Note that no hospital can be matched with more than one student? so there must be at least one student without a residency. b. Dene a stable matching in this environment. c. Find the outcome of the studentproposing deferred acceptance algorithm {EDA} applied to the preference rankings above. Show all your work. d. Is the outcome matching in part c stable? Show that your answer is correct. e. Is there a group of students who could each get a more preferred outcome than the one in part c if? for the SBA? they all reported different from their true preferences listed above?I Assume that the agents outside the group report their preferences listed above. Justify your answer. You do not need to prove it is correct. f. Is there a hospital that could get a more preferred outcome than the one in part c by reporting different preferences to be used in the SBA when all other agents report their _tr_ue_uqeferences listed above? Show that vour answer is correct. 4. A monopolistic seller sells products to a continuum of buyers of size 1. There are two types of buyers, 1' = 1, 2. Every type 2 buyer receives utility equal to 15512 (9,2,2 is if it purchases 1}; units of the good for a total payment of 2 dollars. Every type 1 buyer receives utility equal to 129'1 (91F t1 if it purchases ql units of the good for a total payment of ti dollars. It costs the seller a constant average cost of $2 to produce one unit of the good. Let m- be the fraction of buyers of type i, so 711 + n3 2 1. Each buyer receives a reservation utility equal to zero if the buyer does not purchase anything from the seller. {a} First suppose the seller can observe each buyer's type and can offer each typei buyer a contract (-11,, ti]. Each type 3' buyer can either choose the contract [qh ti} or walk away. 1What is the sellers protmaximizing [i.e. optimal} contract (qr, ti} for each type i? For the rest of the problem, suppose buyer types are not observable to the seller. The seller can offer a menu of two contracts {[14], 1], (:32, t} to all buyers. Each buyer may choose one of the contracts or walk away. If a buyer selects contract {air} then the buyer is entitled to receive q.- units of the good by paying the total price i;- to the seller regardless of the buyer's true type. For parts [b] through {e}, assume n1 2 n2 2 1,32. {b} Formulate the seller's protmaximizing problem: write down the seller's total prots as the objective function and all the constraints including incentive constraints (10s) and individual rationality constraints {1R5} for both types. Call this the original problem {OP}. {c} Next consider the seller's prot optimization problem but without type 2 buyer's IR constraint (IR?) and without type 1 buyer's IC constraint {1131:}. Call this the relaxed problem (HP) because the constraints of the original problem have been relaxed. With only constraints [[02] and (1R1), nd the optimal menu of contracts {[q1,t1}, [(32, tag\" that maximize the seller's prots in this relaxed