高级微观经济理论_英文参考课件-info3.pdf

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1、III.Hidden Action and Moral HazardInformation Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-IntroductionDef.:Principalagent relationshipDef.:asymmetric information in principalagent problem:hidden in-formation and hidden action.Hidden action:Moral hazard:Solut

2、ion:Incentive mechanism.PA models:The principal-agent models deal with the tradeoffbetweenthe efficient risk allocation and incentives.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Two Base Models Principal is risk neutral;agent is risk averse and

3、has unlimited liabil-ity.If the agent is also risk neutral,then the incentive problem can besolved by using“Franchise Contract”.Both Principal and agent are risk neutral,however agent is subjectto limited liability.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxi

4、t PDF Editor-Model I:Principal risk neutral;agent risk averse,unlimited liability1.A simple model(2 2 model)2.Generalization of the model(n m model).Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.The 2 2 Model1.1 The model:A risk-neutral principal

5、 hires a risk-averse agent.After a contract issigned,the agent chooses an action from a,ah,aha,which ishis private information.The principal earns a random output or revenue Y yg,yb,ygyb.The agents choice of effort affects the probability distribution of Y.p(ai)=pi:=PrY=yg|ai,i h,1php0.The cost of e

6、ffort to the agent is C(ai),C0(ai)0.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.2.Market game Principal offers contract C=wg,wb,ai Agent either accepts or declines(and earns his“reservation utility”u=0).If contract is accepted,agent spends effo

7、rt a ah,a.Nature either draws state g or b,with probability piresp.1 pi,output is observed,payments are made,game over.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.3.Payofffunctions Expected profit of principal i h,i:=pi(yg wg)+(1 pi)(yb wb)Expe

8、cted utility of agent i h,U(ai):=piu(wg)+(1 pi)u(wb)c(ai)Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Marginal rates of substitutiondwbdwg?U(ai)=const.=pi(1 pi)u0(wg)u0(wb)dwbdwg?i=const.=pi1 piAlso,MRS(ah)MRS(a)Information Economics(3):Principala

9、gent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.4.Optimal allocation under complete informationProposition 1.The optimal contract is as follows:a=ah,wg=wb,Uh=u,=h.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Proof 1).The optimal c

10、ontract solvesmaxwipi(yg wg)+(1 pi)(yb wb)s.t.piu(wg)+(1 pi)u(wb)C(ai)0(IR).L=pi(ygwg)+(1pi)(ybwb)+(piu(wg)+(1pi)u(wb)C(ai).2).Which action is optimal,ahor a?3).What does the agent get in equilibrium?Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Eq

11、uilibrium requirementsThe optimal contract satisfies the following conditions:(1).MRSP=MRSA(2).U(ai)=u.(3).aimaximizes the principals expected profit.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Illustration:Equilibrium under complete information-

12、6.Agentwgwb45hBAUh=uUwbwgfixed-wagecontractsInformation Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 1.Consider a situation in which a principal hires an agentto perform some task.Suppose u=9.The agent derives utility fromhow much he is paid,w

13、and how hard he works,a.The agents overallutility from w and a is given by U(w,a)=wa.a=ah,a,whereah=5 and a=0.His output Y=yg,yb,where yg=$270 andyb=$70.1.To get the agent to choose a,how much should the principal pay?2.To get the agent to choose ah,how much should the principal pay?3.What is the op

14、timal contract?Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.5.Optimal contract with hidden action 1.5.1 Illustration of the optimal contract with hidden action;1.5.2 Solving for the optimal contract;1.5.3 What is the optimal action for the princ

15、ipal.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.5.1 Illustration:optimal contract with hidden actionQ1:Under hidden action,what kind of contracts can implement ah?Q2:Among all the contracts that can implement ah,which one is theoptimal?Q3:What

16、 is the main property of the optimal contract?Q4:What is the cost of delegation?Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-6.Agentwgwb45h0hlABDCUh=uUlWhat is the cost of delegation?Information Economics(3):Principalagent problem(c)by Foxit Softw

17、are Company,2004 Foxit PDF Editor-1.5.2 Solving for the optimal contractThe optimal contract that implements ahsolves:maxwg,wbph(yg wg)+(1 ph)(yb wb)s.t.phu(wg)+(1 ph)u(wb)C(ah)0(IR)phu(wg)+(1 ph)u(wb)C(ah)pu(wg)+(1 p)u(wb)C(a)(IC)Information Economics(3):Principalagent problem(c)by Foxit Software C

18、ompany,2004 Foxit PDF Editor-1.5.3 Choice of the optimal actionQ:Is it always optimal for the principal to implement ah?Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 2.Suppose u(w):=w,c(a):=a2,ah=2,a=1,ph=0.5,p=0.25,u=2.Then,1.what i

19、s the optimal contract when the principal implements a?2.what is the optimal contract when the principal implements ah?3.Under what condition will the principal choose to implement ah?Solution:wg=wb=9whg=144,whb=0yg yb+252.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2

20、004 Foxit PDF Editor-Why implementing ahis not always optimal?So far we have analyzed under what condition it is optimal to implementah.Now lets consider more questions:What is the reason that the principal sometimes chooses to imple-ment a?Why is it more expensive to implement ah?Information Econom

21、ics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-2.Generalization(n m)We allow for more than two states and more than two actions.The main objective of generalization is 2.1 to give a general account of the optimal implementation of agiven action,and 2.2 to find suffic

22、ient conditions for the monotonicity of the associ-ated wage schedule.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Assumptions n 2,Y y1,.,yn,yn yn1 y1.m 2,am am1 a1.ps(ai):=PrY=ys|ai;i=1,.,m,s=1,.,n ps(ai)0 and is twice continuously differentiable

23、 in ai.Monotone likelihood-ratio condition(MLRC):Likelihood-ration:LRs=ps(ai)ps(aj),ai aj.MLRC:ai aj,s rps(ai)ps(aj)pr(ai)pr(aj).Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Example:MLRCExample 3.Suppose that the agent is a salesman,who be represe

24、nt-ing the principal to a particular client.There are 3 possible outcomesto this interaction:The client can place no order with the principal,oran order that is worth$100 or worth$400.The agents level of effortaffects the probabilities of each of these 3 outcomes:ps(a)ps(ah)y1=00.60.1y2=1000.30.3y3=

25、4000.10.6What is the likelihood ratios?Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-2.1 Optimal ContractWage contract:C=w1,.,wn,aiThe agents expected utility:U(ai):=nXs=1ps(ai)u(ws)c(ai).The principals expected profit:nXs=1ps(ai)?ys ws?.Informatio

26、n Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Solving for the optimal contractThe optimal contract solves:maxws,ainXs=1ps(ai)?ys ws?s.t.U(ai)u(participation constraint),U(ai)U(aj)j=1,.,n(incentive constraints).This can be solved in two steps:1)the solution o

27、f the expected-wage-cost-minimizing implementa-tion of a given effort.2)the choice of the effort level that maximizes the principalsexpected-profit function.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1).Optimal implementation of a given actionma

28、xwsnXs=1ps(ai)wss.t.U(ai)u(participation constraint),U(ai)U(aj)j=1,.,m(incentive constraints).Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Technical TrickDefine xs:=u(ws);h:=u1.maxxsnXs=1ps(ai)h(xs)s.t.nXs=1ps(ai)xs c(ai)u(IR)nXs=1ps(ai)xs c(ai)nX

29、s=1ps(aj)xs c(aj)j=1,.,m(IC).Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Equilibrium requirementsThe optimal contract satisfies the following conditions:The associated first-order conditions areh0(xs)=+mXj=1j 1 ps(aj)ps(ai)!=1u0(ws),s=1,.,n.(1)U(

30、ai)=u.U(ai)=U(aj).Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Why are the constraints binding?The incentive constraints are binding,i.e.,j 0.Proof:The participation constraint is binding,i.e.0.Proof:Information Economics(3):Principalagent problem

31、(c)by Foxit Software Company,2004 Foxit PDF Editor-2)Choice of the optimal effort levelThe second step is a simple maximization problem.Given the optimal contract ws,the optimal aimaximizesmaxainXs=1ps(ai)?ys ws?.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit

32、PDF Editor-2.2 Monotonicity of the optimal wage contractProposition 2.MontonicityThe optimal wage schedule implement-ing the action ai,is monotone nondecreasing:wn wn1 w1.Proof:Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 4.Suppose

33、u(w):=ln(w),n=3(three states,y3 y2y1)and m=2(two effort levels,a2 a1),and assume the conditionalprobability distributions summarized in the following tablep1(ai)p2(ai)p3(ai)i=1/31/31/3i=h1/95/93/9Show that the optimal wage schedule implementing ahis not monotone.Information Economics(3):Principalage

34、nt problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Model II(Limited Liability)Now suppose the agent is risk-neutral but faces a liability constraint.This also gives rise to an incentive problem.If the agent is risk neutral and is subject to unlimited liability,then theincentive problem has

35、 a simple solution:franchise contract.Franchise contract is efficient because it makes private and social ben-efits of agents effort coincide.However,in many cases franchising is not feasible.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-1.The mode

36、l Consider a principal hires an agent.After the contract is signed,theagent chooses action a A:=0,a,a 0.The agent is risk-neutral,u(w,a):=w c(a),where c(a)is twicecontinuously differentiable,c(0)=0,c0(a)0,c00(a)0.The agent has a limited liability with an initial wealth L.The output y Y:=yg,yb,yg yb.

37、The conditional probability is p(a):=PrY=yg|a,a A,p(a)(0,1),p0(a)0,p00(a)0,p(0)0.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-2.Optimal contract to implement a given actionRisk neutral:U(a)=p(a)wg+(1 p(a)wb C(a)The optimal contract wg,wb solvesmax

38、wg,wb(p(a)wg+(1 p(a)wb)(2)s.t.p(a)wg+(1 p(a)wb C(a)0,IR(3)a argmaxaAU(a),IC(4)wb+L 0,L1(5)wg+L 0.L2(6)Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 5.Suppose a salesman represents the principal to go to aparticular client.His utility

39、 function is U(w,a)=w a,a=ah,a,ah=25 and a=0.u=81,and L=100.There are 3 possibleoutcomes:y1=0,y2=100,y3=400.a affects the probabilities ofeach of these outcomes:ps(a)ps(ah)y1=00.60.1y2=1000.30.3y3=4000.10.61.Suppose the agent is subject to unlimited liability,what is the optimalcontract?2.Suppose L=

40、100,is the liability constraint binding?Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-The optimal wage contractProposition 3.The following contract optimally implements the giveneffort level a 0wg=wb+c0(a)p0(a)wb(7)wb=maxL,c(a)p(a)p0(a)c0(a).(8)The

41、 associated minimum expected wage cost Cw(a,L)isCw(a,L)=maxL+p(a)p0(a)c0(a),c(a).(9)Proof:Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Inefficiency with limited liabilityIs the optimal contract implementing a given action efficient?Efficient imple

42、mentation means:Cw(a,L)=c(a).EUSB:=CW(a,L)C(a):The agents expected rent from limited lia-bility.Proposition 4.The optimal contract implementing a given action a is(i)efficient if and only if liability constraint(5)is not binding(if L issufficiently large);(ii)inefficient if the liability limit L is

43、sufficiently small.Proof:Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-Conclusion The optimal contract pays a fixed wage plus a bonus.The value of the fixed wage depends upon whether the liability orthe participation constraint is binding.If the liability constraint does not bind,one has efficient first-bestoutcome.If the liability constraint binds,there will be an incentive problem.Information Economics(3):Principalagent problem(c)by Foxit Software Company,2004 Foxit PDF Editor-