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1、Lecture on Contract Theorv2. Complete Contracts I: Static Bilateral ContractingHuihua NIESchool of EconomicsRenimn Uinveirsity of Chinav w ChmaES. o【g cnniehinhua2000yahoo. com cn2.3 Adverse Selection I: Screening23.1 Typical case: lemon marketAlceriof (1970). Mirrlees (1971) formally analyzed the p
2、roblem of hidden lnfonmuon.TmiLDg: Ex ante adverse selectiont=0t=lt=2t=311i,喚N decidesP designA acceptscontractstatecontractor rejectsimplementsFig. 1-3 Whats the type of this game theory, static game with lnipeifect information or dynamic game with lmpeifect Liifomiation? Beycsian equilibrium.2.3.2
3、 Basic model: nonlinear pricingConsider a transacuon between a buyer (customer) and a seller (Tclcom), where the seLler doesnt know perfectly how much the buyer is willing to pay for his goodThe buyers preferences is i(q、7&) = 0u(q)-T , uhere m(0) = 0 , if (q) 0 , and u (l) USolve two equations, we
4、get= c and Q(C)= 7 + (/ 7 = -/ +.The total profit of the seller is 0(% - cqL) + (1- 一 cqH )Note 1 By having the buyer select a quantity7 such that marginal utility equals marginal cost and setting the payment, meanwhile by binding IR constraint via T the seller appropriates the full surplus and leav
5、es no rent to the buyerNote 2 The best contract is implemented by type-specific nvo-part tariffs.(2) Second best: optimal nonlinear pricingWith asymmetnc information, type 0H will pretend to be 0t and get information rent, so theseller will fail to peifectlv puce discnniinate An illustration is Ohii
6、(cit) - 7； (/ and 7； (/ can not hold simultaneously. What he can do。Without loss of generality% the optimal price can be descnbed as a schedule (q,T(q) The problem isMaxJ3T(qL) - eg + (1 - fl)T (qH ) - cqHT(q)(IC) q. g- T(qt) for i = L、HqAlso, we can treat U as exogenous and normalize it to be 0. No
7、tice that the problem involves optimization over a schedule T( mechanism design revelation pnnciple direct revelation mechanisin/truth-tellLng lmplementa tionWith revelation pnnciple, we can restrict each schedule T(qj to the pair of optinul choicesmade by the two types of buy eirs 7佈人佈and T (qH ),
8、qH . We define T(q J 三刁，thenthe problem can be rexx ntten asMaxp(rL -cqL)+(-URL) Oluc1l)-TlIJ_ /(IRH) 0Hu(qH)-TH U(ICL) OLu(qL)-Tl OLu(ciH) - Th(ICH) O/fu(q)-T OHii(qL)一 7； /Why we need not revelation prmciple in moral liazard problem?Step 2: Eliminate the “high M type 9s participation constraint.0H
9、 0L and ICH, /. OHu(qH)-TH &”(旺)-TL 九“(仇)-TL U.Step 3: Check Spence-Mirrlees single-crossing condition, and eliminate the low types incentive compatibilities constraint.The SM condition isNote This condition require that the imrginal utility of consumption vary with type, such that a separating equi
10、libnuni exits. So, we just need S-M condition H 0. not necessanlv 0 (for example 0 stands for costs)S-M condition implies the monotonicity condition. Adding ICH to ICL, we can get (3h - 0L)u(qH) - w(z) 0. which means that qH qL .(/ 3H 3L ). That is monotonicity and implementation condition. S-M cond
11、ition in this setting means diat the low type has no incenuve to mutate lugh type, te., ICL is not binding Specifically, because IRL is binding (and IRH is not), Tl = -U +, and -U + OLu(qH) -U + Olu(qH) We will check itlatter.Step 4: Check that the two remaining constraints of the relaxed problem wi
12、ll bind at the optimum.Max卩G -cqL) +(1-P)(TH-cqH)S.t.(IRL) 0Lu(qL)-TLU(ICH) OHu(qff)-T OHu(q, ) 一 T,If ICH is not bindings the seLler will raise TH until it does bind and miprove his welfare. So does IRL. Also, we can check die A and /z by Kuhn-Tucker theoremStep 5: Insert IRL and ICH constraints in
13、to sellers program, and perform the unconstrained optimization problemMaxcqH-U-(0H -qgNote 1 The low type has no consumer surplus.Note 2 The Last part of the second brackets Lugh types information rent, which come from rhefact that he can “mmiLC” the beliavior of the other of type and increases with qL FOC:T严P OLTl = 2心)-E、Th = OHu(qH) U-(0H- Q)心).It unplies that qHg； . ICH binds, we have &
