long-term contracting with markovian consumers...long-term contracting with markovian consumers by m...

Long-Term Contracting with Markovian Consumers

By MARCO BATTAGLINI

To study how a firm can capitalize on a long-term customer relationship wecharacterize the optimal contract between a monopolist and a consumer whosepreferences follow a Markov process The optimal contract is nonstationary and hasinfinite memory but is described by a simple state variable Under general condi-tions supply converges to the efficient level for any degree of persistence of thetypes and along any history though convergence is history-dependent In contrastas with constant types the optimal contract can be renegotiation-proof even withhighly persistent types These properties provide insights into the optimal ownershipstructure of the production technology (JEL D23 D42 D82)

Advances in information processing and newmanagement strategies have made long-termnonanonymous relations between buyers andsellers feasible in an increasing number of mar-kets Many retailers can now store large data-bases on consumersrsquo choices and utilize themfor pricing decisions at a very low cost In partbecause of these new technologies recent man-agerial schools have stressed the importance ofcapitalizing on long-term relations with custom-ers (see eg Louis V Gerstner Jr 2002 andJack Welch 2001) When a long-term relation-ship is nonanonymous and types are persistentthe seller can mitigate the problem of asymmet-ric information by using consumersrsquo choices toforecast future behavior As a result howeverbuyers are more reluctant to reveal private in-formation that affects their consumption deci-sions their strategic reaction may limit or even

eliminate the benefits for the seller The existingliterature has studied this problem focusing onthose cases in which the consumerrsquos type isconstant over time1 Here it is well known thatthe seller finds it optimal to offer the optimalstatic contract period after period In a sense theseller commits not to use the information gath-ered from the consumerrsquos choices

A model of long-term contracting that as-sumes constant types however clearly missesan important dimension of the problem Con-sider the case of a monopolist selling to anentrepreneur whose type depends on the num-ber of customers waiting for service As is wellknown under standard assumptions on the ar-rival rate of customers the type of this entre-preneur follows a Markov process (see egSamuel Karlin and Howard M Taylor 1975)Or to give another example consider the caseof a company selling cellular telephones Thesecontracts often last for years and it would not bereasonable to assume that the telephone com-pany or the customer does not take into ac-count the likely but uncertain evolution ofpreferences (see eg Eugenio J Miravete2003 for evidence) In all these situations theassumption that the consumerrsquos type is constantis clearly not realistic Even if types are verypersistent it is reasonable to assume that theymay vary over time and follow a stochasticprocess

In this paper we characterize the optimalcontract offered in an infinitely repeated setting

Department of Economics Princeton University001 Fisher Hall Princeton NJ 08544 (e-mailmbattaglprincetonedu) I gratefully acknowledge finan-cial support from the National Science Foundation (GrantNo SES-0418150) and the hospitality of the EconomicsDepartment at the Massachusetts Institute of Technologyfor the academic year 2002ndash2003 I am grateful for helpfulcomments to Pierpaolo Battigalli Douglas Bernheim Ste-phen Coate Eddy Dekel Avinash Dixit Glenn Ellison JeffEly Bengt Holmstrom John Kennan Alessandro LizzeriRohini Pande Nicola Persico William Rogerson ArielRubinstein Jean Tirole Asher Wolinsky seminar partici-pants at Bocconi University Cornell University HarvardUniversity MIT Northwestern University Princeton Uni-versity the Stanford Institute of Theoretical EconomicsUCLA University College London and University ofWisconsin-Madison as well as to three referees MarekPicia provided valuable research assistance 1 The related literature is discussed in Section I

637

by a monopolist to a consumer whose prefer-ences evolve following a Markov process Inthis case even if types are highly persistentthe contract is very different from the contractwith constant types because the seller finds itoptimal to use information acquired along theinteraction in a truly dynamic way For thisreason the characterization of the optimal con-tract when there is heterogeneity within andacross periods allows a new understanding ofimportant aspects of a dynamic principal-agentrelationship that previous models could not cap-turemdashparticularly with regard to the memoryand complexity of the contract its efficiencyand its robustness to renegotiation Perhaps sur-prisingly it also provides insights into the op-timal ownership structure of the productiontechnology

As noted when types are constant the con-tract has no memory and the inefficiency of theoptimal static contract is repeated period afterperiod With persistent but stochastic typeseven in a simple stationary environment withone period memory (ie a Markov process) thecontract is nonstationary and has infinite mem-ory despite this however it can be representedin a very economical way by a simple statevariable Even if types are arbitrarily highlycorrelated and the discount factor is arbitrarilysmall the sellerrsquos optimal offer converges overtime to the efficient supply schedule along allpossible histories The speed of convergencehowever is state contingent and occurs in aparticular way which extends a well-knownproperty of the static model On the one hand infact we have a generalized no distortion at thetop (GNDT) principle after any history if theagent reveals himself to have the highest possi-ble marginal valuation for the good supply isset efficiently from that date onward in anyinfinite history that may follow On the otherhand and more importantly we have a novelvanishing distortion at the bottom (VDB) prin-ciple even in the history in which the agentalways reveals to have the lowest marginal val-uation for the good the contract converges tothe efficient menu offer One immediate impli-cation of this result is that in the ldquosteady staterdquoor even after a few periods the monopolistrsquossupply schedule may be empirically indistin-guishable from the outcome of an efficientcompetitive market moreover since higher ef-ficiency is associated with a higher consumer

rent it explains why ldquooldrdquo customers should betreated more favorably than ldquonewrdquo customers2

In a stochastic environment the incentivesfor renegotiation are also very different Asshown in the received literature (see discussionbelow) when types are constant over time themonopolist benefits from the ability to committo not renegotiating the contract because theoptimal contract is never time-consistent Withvariable types in contrast this is not the caseindeed even when types are highly correlated asimple and easily satisfied condition guaranteesrenegotiation-proofness Interestingly whentypes are constant the optimal renegotiation-proof contract always requires the agent to usesophisticated mixed strategies with correlatedbut stochastic types the optimal renegotiation-proof contract has an equilibrium in pure strat-egies and simply requires the agent to report histype

There is an intuitive argument which explainsthe dynamics of the distortions in the optimalcontract and the efficiency result mentioned ear-lier Assume that the agentrsquos type can take twovalues high and low marginal valuation for thegood (respectively H and L ) Consider Fig-ure 1 which shows the impact on profits at timezero of a marginal increase q in the quantityq(ht) offered to the consumer after a history htOn the one hand this change increases thesurplus that can potentially be appropriated bythe seller if history ht is realized (which isrepresented by the ldquothick arrowrdquo on the left-hand panel in Figure 1)3 However as in a staticmodel this increase in supply increases the rentthat the principal must leave to the agent tosatisfy incentive compatibility In every periodoptimal supply is determined by this marginalcostndashmarginal benefit trade-off and the dy-namic properties of the contract are driven by itsevolution To determine optimal supply there-fore it is important to understand the impactover expected rents of this change at time t

To this goal consider the right panel of Fig-ure 1 and assume that the rent of the high type

2 See Igal Hendel and Alessandro Lizzeri (2003) andGeorges Dionne and Neil A Doherty (1994) for evidence ofthis phenomenon

3 Because consumption is always distorted below its firstbest level starting from the sellerrsquos optimum a marginalincrease in supply after history ht corresponds to an increasein efficiency of the contract at that node

638 THE AMERICAN ECONOMIC REVIEW JUNE 2005

increases by Rt at time t At time t 1 theexpected utility of the agent in the history im-mediately preceding ht increases as well be-cause although the agent is a low type at t 1he can become a high type in the followingperiod and then benefit from the increase inrent Part of this extra expected rent can beextracted by the seller at t 1 but not all sinceincentive compatibility must be satisfied at thattime as well At time t 1 the high type cannotreceive less than what he would receive if hechose the option designed for the low typeEven if the seller extracts all the expected in-crease in consumption of the low type with anincrease in price pt1 Pr(HL )Rt at t 1the change in rent of the high type at t 1Rt1 would be equal to Pr(HH)Rt pt1 that is (Pr(HH) Pr(HL ))Rt

4

which is positive if types are positively corre-lated If the seller tries to extract this extra rentat t 1 then repeating the same argument shestill must provide an increase in rent to the hightype at time t 2 equal to Rt2 (Pr(HH) Pr(HL ))Rt1 which can bewritten as (Pr(HH) Pr(HL ))2Rt Pro-ceeding backward we arrive at an increase inthe rent left to the consumer at time 1 propor-tional to (Pr(HH) Pr(HL ))t1 (see thedashed arrows in the right panel of Figure 1)

While the marginal impact of the change insupply on expected surplus evaluated at timezero is proportional to the probability of thehistory ht (ie LPr(LL )t1) the impact on

the agentrsquos expected rent is proportional to theldquocumulative effectrdquo of the difference in expec-tations of the types [Pr(HH) Pr(HL)]t15

Accordingly the marginal costndashmarginal bene-fit ratio at time t is proportional to

(1) PrHH PrHL

PrLL t 1

The dynamics of the optimal contract dependson the evolution of this cost-benefit ratio Whentypes are constant the term in parentheses isexactly equal to one so (1) is independent of tand the distortion is constant this explains whywith constant types it is optimal to offer thestatic contract repeatedly When types are pos-itively but imperfectly correlated even if typesare highly persistent optimal supply convergesto an efficient level along all histories as t3 because (1) converges to zero

In general any change in the contract at atime t has cascade effects on the expected rentsin the previous periods These effects dependnot only on the transition probabilities but alsoon the structure of the constraints that are bind-ing at the optimum As time passes these cas-cade effects become increasingly complicatedbecause the number of histories grows exponen-tially A methodological contribution of thispaper is in a novel characterization of the bind-ing constraints by an inductive argument which

4 The probability that a type i in period t becomes a typej in t 1 is denoted Pr(ji)

5 The expected change in the agentrsquos rent at time one isH[Pr(HH) Pr(HL )]t1Rt where H is the proba-bility that the agent is a high type in the first period Theconstant H however is irrelevant for our argument

FIGURE 1 MARGINAL COST AND BENEFIT OF A CHANGE IN THE QUANTITY OFFERED AFTER HISTORY ht

Notes The arrows represent the history tree an arrow pointing up (respectively down) represents a high (respectively low)type realization The horizontal axis is the time line

639VOL 95 NO 3 BATTAGLINI LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS

allows a substantial simplification of theproblem

The particular features of the optimal contractdescribed above also have implications for theoptimal ownership structure of the monopolistrsquosbusiness It is indeed interesting to ask why themonopolist keeps control of the productiontechnology after all only the consumer benefitsdirectly from it and has information for its ef-ficient use We show that the optimal contractcan be interpreted as offering the high-type con-sumer a call option to buy out the technologyused by the monopolist The sale of the tech-nology however is state contingent and themonopolist tends to retain control more oftenthan what would be socially optimal by keep-ing the ownership rights the monopolist cancontrol future rents of the high types and thisimproves surplus extraction because types havedifferent expectations for the future This in-sight seems relevant to understand the owner-ship structure of a new technology The initialowner of a new technology generally has mo-nopoly power on its use thanks to a patent andmust decide if it is more convenient to use thetechnology directly selling its products or tosell the patent

The paper is organized as follows Section Isurveys the related literature In Section II wedescribe the model In Section III we character-ize the optimal contract and discuss its effi-ciency properties Section IV discusses thetheory of property rights that follows from thecharacterization Section V discusses the prop-erties of the monetary payments in the optimalcontract Section VI studies renegotiation-proofness Section VII presents concludingcomments

I Related Literature

As mentioned above in dynamic models ofprice discrimination it is generally assumed thatthe agentrsquos type is constant over time6 In thiscase we have a ldquofalse dynamicsrdquo in which themonopolist finds it optimal to commit to a con-tract in which past information is ignored and

the optimal static menu is repeated in everyperiod (see eg Laffont and Tirole 1993)With constant types the dynamics becomesinteresting only when other constraints arebinding in particular when a renegotiation-proofness constraint must be satisfied Seminalpapers in this literature are Dewatripont (1989)Oliver Hart and Tirole (1988) and Laffont andTirole (1990)7 In contrast to our findings withvariable types a common result in this literaturewith constant types is that the ex ante optimalcontract is never renegotiation-proof

Kevin Roberts (1982) and Robert MTownsend (1982) are the first to present re-peated principal-agent models with stochastictypes In these frameworks however types areserially independent realizations and thereforeincentives for present and future actions caneasily be separated Indeed in this case exceptfor the first period there is no asymmetric in-formation between the principal and the agentbecause both share the same expectation for thefuture8 David Baron and David Besanko(1984) and Laffont and Tirole (1996) extendthis research presenting two period procure-ment models in which the type in the secondperiod is stochastic and correlated with the typein the first period Because these models haveonly two periods however they cannot capturesuch important aspects of the dynamics of theoptimal contract as its memory and complexityafter long histories or its convergence to effi-ciency Aldo Rustichini and Asher Wolinsky(1995) characterize optimal pricing in a modelwith infinite horizon and Markovian types asours However in their model consumers arenot strategic and ignore that future prices de-pend on their current actions demand more-over can assume two values zero or one None

6 For excellent overviews of the literature on dynamiccontracting see Patrick Bolton and Mathias Dewatripont(2005) and Jean-Jacques Laffont and Jean Tirole (1993)

7 These papers study the optimal renegotiation-proofcontract with constant types under different assumptionsHart and Tirole (1988) and Dewatripont (1989) presentmodels with many periods the first paper assumes thatsupply can have two values zero or one the second focuseson pure strategies and assumes some simplifications in thenature of the contractual agreement Laffont and Tirole(1990) solve a model in which supply can assume more thantwo values assuming two periods

8 Because Townsend (1982) is specifically interested inmodelling risk sharing he assumes that the principal is lessrisk averse than the agent In this case even with iid typesthe contract depends on the cumulated wealth of the agent


of these papers with variable types considersrenegotiation-proofness9

II The Model

We consider a model with two parties abuyer and a seller The buyer repeatedly buys anondurable good from the seller He enjoys aper-period utility tq p for q units of the goodbought at a price p In every period the sellerproduces the good with a cost function c(q) 1frasl2 q2 The marginal benefit t evolves over timeaccording to a Markov process To focus on thedynamics of the contract we consider the sim-plest case in which each period the agent canassume one of two types L H with H L 0 The probability that state l isreached if the agent is in state k is denotedPr(lk) (0 1) the distribution of types con-ditional on being a high (low) type is denotedH (Pr(HH) Pr(LH)) (L (Pr(HL )Pr(LL ))) We assume that types are positivelycorrelated ie Pr(HH) Pr(HL ) How-ever we do not make assumptions on the degreeof correlation indeed an environment withconstant types can be seen as a limiting case ofour model in which the probability that a typedoes not change converges to one In each pe-riod the consumer observes the realization of hisown type the seller in contrast cannot see itAt date 0 the seller has a prior (H L ) onthe agentrsquos type10 For future reference notethat the efficient level of output is equal toqe(t) t in all periods and after any history ofrealizations of the types

We assume that the relationship between thebuyer and the seller is infinitely repeated and thediscount factor is (0 1) In period 1 the selleroffers a supply contract to the buyer The buyercan reject the offer or accept it in the latter casethe buyer can walk away from the relationshipat any time t 1 if the expected continuationutility offered by the contract falls below thereservation value u 0 In line with the stan-dard model of price discrimination the monop-olist commits to the contract that is offered inSection VI we relax this assumption allowingthe parties to renegotiate the contract

It is easy to show that in the environment thatwe will study a form of the revelation principleis valid and allows us to consider without loss ofgenerality only contracts that in each period tdepend on the revealed type at time t and on thehistory of previous type revelations In this casethe contract p q can be written as p q (pt(ht) qt(ht))t1

where ht and are respec-tively the public history and the type revealedat time t and qt and pt are the quantities andprices conditional on the declaration and thehistory11 In general ht can be defined recur-sively as ht t1 ht1 h1 A wheret1 is the type revealed in period t 1 The setof possible histories at time t is denoted Ht theset of histories at time j following a history ht(t j) is denoted Hj(ht) A strategy for a sellerconsists of offering a direct mechanism p q asdescribed above The strategy of a consumer isat least potentially contingent on a richer his-tory ht

C t1 t ht1C h1

C 1 becausethe agent always knows his own type For agiven contract a strategy for the consumerthen is simply a function that maps a history ht

C

into a revealed type htC b(ht

C)In the study of static models it is often as-

sumed that all types are served ie each type isoffered a positive quantity which is guaranteedby the assumption that is not too large Thesame condition that guarantees this property inthe static model also guarantees it in our dy-namic model therefore to simplify notationwe assume that this condition is verified in ourmodel12 This assumption can easily be relaxed

9 Dynamic environments with adverse selection and sto-chastic types have recently been used to study models ofleasing insurance and other applications See PascalCourty and Hao Li (2000) and Hendel and Lizzeri (19992003) John Kennan (2001) has studied a model with vari-able types but in which only short-term contracts with oneperiod length can be offered Battaglini and Stephen Coate(2003) apply the techniques of the present paper to charac-terize the Pareto optimal frontier of taxation with correlatedtypes

10 The fact that the agentrsquos type follows a Markov pro-cess can be modelled in many natural ways The agent maybe a firm whose type depends on its list of customerswaiting for services which according to the ldquoinventorymodelrdquo follows a Markov process (see Karlin and Taylor1975 sect22d) Or the agentrsquos type may depend on hisinvestment opportunities if these follow a branching pro-cess then they are described by a Markov process (seeKarlin and Taylor 1975 sect22f)

11 Note therefore that p(h) is not the per-unit pricepaid after history h but the total monetary transfer atthat history

12 The condition that guarantees that all types are servedis (LH)L As we will see the distortion introduced


but this would complicate notation with no gainin insight

In the first part of the analysis we focus on thecase with unilateral commitment in which themonopolist can commit but the consumer canleave the relationship anytime This assumptionseems the most appropriate in many markets13

On the other hand there are many situations inwhich renegotiation is an important componentof the problem in Section VI we show thatunder general conditions the optimal contract isrenegotiation-proof and therefore it can be ap-plied to these environments too

III The Optimal Contract

The monopolistrsquos optimal choice of contractmaximizes profits under the constraint that afterany history the consumer receives (at least) hisreservation utility and also after any historythere is no incentive to report a false type

PI maxpqHpHh1 q2Hh12

E h1 Ht H13]

LpLh1 q2Lh12

E h1 Lt L13]

st IChtH ICht

L IRhtH

IRhtL ht

where E[(h1 i)t i] i H L is theexpected value function of the monopolist afterhistory h1 i The incentive constraintsICht

(i) for i H L are described by

IChti qiht i piht

EUht i t i 13

qjht i pjht

EUht j t i 13

i j i j H L where U(ht i) is the valuefunction of a type after a history ht iThese constraints guarantee that type i does notwant to imitate type j after any history ht Andthe individual rationality constraint IRht

(i) sim-ply requires that the agent wants to participatein the relationship each period U(ht i) 0for any i and ht

The classic approach to characterize the so-lution to this problem in a static environment isin two steps First a simplified program inwhich the participation constraints of the hightype and the incentive compatibility constraintsof the low type are ignored is considered (theldquorelaxed problemrdquo) Then it is shown that thereis no loss of generality in restricting attention tothis case In a static model the remaining con-straints of the relaxed problem are necessarilybinding at the optimal solution this simplifiesthe analysis because it allows us to substitutethem directly in the objective function

It is easy however to see that in a dynamicmodel this cannot be true Given an optimalcontract we can always add a ldquoborrowingrdquoagreement in which the monopolist receives apayment at time t and pays it back in the fol-lowing periods If the net present value of thistransaction is zero then neither the monopo-listrsquos profit changes nor any constraint wouldbe violated so the contract would remain opti-mal but the individual rationality constraintsneed not remain binding after some historiesMore importantly the incentive compatibilityconstraints may also not be binding In order toprovide incentives to the high type to reveal hisprivate information the monopolist may find ituseful to use future payoffs instead of presentpayoffs to screen the agentrsquos types If this werethe case there would be a history after whichthe contract leaves to the high type more surplusthan what a binding incentive compatibilityconstraint would imply

The following result generalizes the ldquobindingconstraintsrdquo result of the static model showingthat in a dynamic setting although constraints

by the monopolist is declining over time in all histories andin the first period it is equal to the distortion of the staticmodel Therefore if the monopolist serves all customers inthe static model then she serves all customers after allhistories in our dynamic model too

13 Discussing the life insurance market Hendel and Liz-zeri observe that the term value contracts in the insurancemarket which account for 37 percent of ordinary life insur-ance ldquo are unilateral the insurance companies must re-spect the terms of the contract for the duration but the buyercan look for better deals at any time [] These features fita model of unilateral commitmentrdquo (Hendel and Lizzeri2003 p 302) Moreover there is evidence that firms seemaware that the possibility to commit is important to winexclusive long-term contracts


need not bind in every optimal scheme there isno loss of generality in assuming that con-straints in the relaxed problem are satisfied asequalities Let us define PII as the program inwhich expected profits are maximized assum-ing that the incentive constraints of the hightype and the participation constraints of the lowtype hold as equalities after any history and noother constraint is assumed We say that a sup-ply schedule qt (ht) is a solution of a givenprogram if there exists a payment schedulept (ht) such that the menu qt (ht) pt (ht)is a solution of the program

LEMMA 1 The supply schedule qt (ht)solves PI if and only if it solves PII

The result that the constraints may be assumedto hold as equalities without loss may be intu-itively explained in a two-period version of themodel (the complete argument presented in theAppendix is by induction on t) Assume that attime t 2 the incentive compatibility constraint ofthe high type is not binding after a history h2 LConsider this change in the contract reduce theextra rent at t 2 and reduce the price paid by thelow type at t 1 so that his participation con-straint is satisfied as an equality after the changeThe rent of the high type at time 1 depends on hisoutside option (the utility obtained by reportinghimself untruthfully to be a low type) so it isaffected by both these changes Even if the netchange in payments has a neutral effect on the lowtypersquos expected utility however it will reduce theoutside option of the high type because the hightype is more optimistic about the future realizationof his type the reduction in future rents if hereports his type untruthfully will be larger than the

increase in payments at time t If the value of thehigh typersquos outside option goes down then hisequilibrium rent goes down as well Expectedprofits therefore would be larger after the changein the contract and all constraints would be re-spected but this is not possible if the contract isoptimal so we have a contradiction After a his-tory h2 H we proceed in a similar way in thiscase profits remain constant after the change inprices so the constraint needs not necessarily bebinding at the optimum but it can be reduced to anequality without loss The argument for the par-ticipation constraints is analogous

It is important to point out that Lemma 1 doesnot claim that any solution p q of a relaxedproblem in which the incentive constraint of thelow type and the participation constraint of thehigh type are ignored is a solution of PI In PIIwe assume that the constraints are satisfied asequalities so it is not just a relaxed version ofPI Indeed such a claim would not be truesome solutions of the relaxed problem wouldimply future rents for the high type that wouldviolate the incentive compatibility constraint ofthe low type after some histories However ifp q solves the relaxed problem then thereexists a p such that p q solves PI and if pq solves PI then there exists a p such that pq solves PII and because of this solves therelaxed problem as well

We can now focus the simpler problem withequality constraints PII from the first-orderconditions we obtain

PROPOSITION 1 At any time t the optimalcontract is characterized by the supplyfunction

(2) qt ht H if H

L H

LPrHH PrHL

PrLL t 1

if L and ht htL

L if L and ht HthtL

where htL L L L the history along

which the agent always reports himself to be alow type in the first t 1 periods

From (2) we can see that the optimal contractis nonstationary and has unbounded memoryfor any T 0 we can always find two historiesthat are identical for the last T periods but that

induce different menus in the optimal mecha-nism14 This fact however does not imply that

14 Consider two histories that differ only in the firstrealization of types the first being high the second beinglow and which have low realizations in any period follow-ing date two If these histories are longer than a positiveparameter T say they have T 1 length then they coincide


the contract has a complicated structure FromProposition 1 we can see that the only thing thatmatters for the contract is whether we are on thelower branch or not Since this depends only onthe current type and if in the previous periodsthe agent reported himself to be a high type thestate can be described by a simple 0-1 variablewhich can be defined recursively

(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else

for t 1 and X0 1 This variable starts withvalue one and remains one if the agent persistsin reporting a low type once the agent hasreported himself to be a high type the stateswitches to zero and remains constant foreverLet us define [Pr(HH) Pr(HL )Pr(LL )] we have

PROPOSITION 2 The optimal solution is afunction of time and the 0-1 state variable de-scribed by (3) qt (t Xt1) t (HL )X(t Xt1)t1

The length of the memory of the optimalcontract is a central issue in the literature ondynamic moral hazard (see William P Roger-son 1985) but it has not been studied in ad-verse selection models because when theagentrsquos type is perfectly constant we know thatthe contract is also constant over time and in-dependent of past histories so it has no mem-ory In the moral hazard literature the memoryof the contract is a direct consequence of theagentrsquos risk aversion With risk aversion it isoptimal not only to smooth consumption overstates of the world as in the static moral hazardframework but also to smooth consumptionover periods To this end the contract mustkeep track of the past realizations of the agentrsquosincome In the model presented above how-ever the agent is risk neutral the persistence of

the distortion therefore does not depend onconsumption smoothing but it is a necessaryfeature of dynamic price discrimination In adynamic environment the principal has morefreedom to redistribute distortions over timeand states in order to screen the agentrsquos typesPropositions 1 and 2 characterize the optimalway to redistribute the distortion proving thatthe principal finds it optimal to introduce dis-tortions even in the far future potentially for anunbounded number of periods This is perhapssurprising since the agentrsquos taste follows aMarkov process and therefore the relevant eco-nomic environment has a memory of only oneperiod15

We now turn to the particular pattern inwhich distortions are introduced In Sections IIIA and III B we discuss the dynamics of distor-tions and the asymptotic properties of the con-tract as 3 1 In Section III C we discuss thekey assumptions of the model

A Efficiency The GNDT and VDB Principles

In order to interpret (2) it is useful to com-pare it with the benchmark with constant typesIn this case there are only two possible histo-ries either the agent is always a high type inwhich case the contract is efficient or the agentis always a low type and the contract is dis-torted below the efficient level in all periods bya constant (HL )

When types follow a Markov process thecontract instantly becomes efficient as soon asthe agent reports himself to be a high type butnow efficiency ldquoinvadesrdquo the histories in whichthe agent subsequently reports himself to be alow type This is the generalized no distortionat the top (GNDT) principle Its intuition is thefollowing Distortions are introduced only toextract more surplus from higher types there-fore there is no reason to distort the quantityoffered to the highest type After any history htthe rent that must be paid to a high type toreveal himself is independent of the quantitiesthat follow this history since the incentive com-patibility constraint for the high type is binding

for at least the last T periods At time 1 the monopolistoffers an efficient contract in the first history ie regardlessof the realizations in the following periods the quantityoffered is efficient in any period following the first Not sofor the second history Therefore even if there is no intrin-sic economic reason in the environment to offer differentmenus at date T 1 the contracts are different

15 As we discuss in greater detail in Section IV the factthat the seller wants to distort supply for a unlimited andstate-contingent number of periods has important implica-tions for the allocation of the property rights of the produc-tion technology


he receives the same utility as if he falselyreported himself to be a low type thereforeonly the quantities that follow such a historyaffect his rents This implies that the monopolistis residual claimant on the surplus generated onhistories after a high type report and thereforethe quantities that follow such histories are cho-sen efficiently In our dynamic framework thissimple principle has strong implications be-cause it forces the contract to be efficient notonly in the first period in which the agent truth-fully reveals himself to be a high type but alsoin all the following periods

A distortion persists on the lowest branchof the history tree (ie when the agent al-ways declares to be a low type) By a sim-ple manipulation of the formula in Proposition2 the distortion can be written as L qt(Lht) (HL ) X(t Xt 1)1 [Pr(LH)Pr(LL )]t 1 since the efficient level ofoutput with a low type is L Given that typesare positively correlated we have [Pr(LH)Pr(LL )] (0 1) and it follows thatlimt3qt(ht) qe( ) which proves

PROPOSITION 3 For any discount factor (0 1) the optimal contract converges overtime to an efficient contract along any possiblehistory

This is the vanishing distortion at the bottom(VDB) principle The monopolist introduces adistortion along the ldquolowestrdquo history becausethis minimizes the cost of screening the agentrsquostypes however even this distortion convergesto zero as t 3

The optimal distortion simply equalizes themarginal cost of a decrease in supply (in termsof reduced surplus generated in the relationship)and its marginal benefit (in terms of reducedrent to be paid to the high type) With constanttypes after any history ht in which the agentdeclares to be a low type the marginal benefitof increasing surplus with a higher q(ht) is in-dependent of the length of the history it isproportional only to L because once the typeis low in the first period then it is low foreverSimilarly the marginal cost of an increase inq(ht) is proportional to H the probability thatthe high type receives the increase in the rentSince therefore the marginal costmarginalbenefit ratio is time-independent it is not sur-prising that the optimal distortion is also con-

stant over time Indeed the case with constanttypes is not asymptotically efficient because1 [Pr(LH)Pr(LL )]t1 is exactly oneand therefore independent from t

Clearly as the persistence of types convergesto one we have that [Pr(LH)Pr(LL )]3 0Not surprisingly this implies that ceteribus pa-ribus the contract converges in every period tothe optimal static contract as Pr(HH) andPr(LL ) converge to one There are howevertwo important observations First convergenceto efficiency appears to be relatively fast even iftypes are highly correlated16 Second as wediscuss below the results with fixed and sto-chastic types are very different when 3 1regardless of the level of persistence of thetypes

B Distribution of Surplus with LargeDiscount Factors

All the results presented above are valid forany (0 1) if we assume that 3 1 evenstronger results emerge In this case we caneasily bound the inefficiency and determine thedistribution of surplus between the seller andthe buyer17

With constant types the average utility of theconsumer is bounded away from zero and inde-pendent of the average payoff of the monop-olist is equal to the profit that would be achievedin a static model and independent of as wellHowever even an arbitrarily small reduction inthe persistence of the types has a very highimpact on surplus and payoffs when the dis-count factor is high

16 Assume for example that types are ex ante equallylikely and the types are very much correlated (for examplethe type is persistent 80 percent of the time) Then theexpected inefficiency of the contract after ten periods willbe 003779 the expected inefficiency after 50 periodswill be 50517 1012

17 In this comparative statics exercise we change thediscount factor keeping the transition probabilities con-stant Another interesting exercise would be to modify thelevel of persistence of the types or the frequency of theirchanges Increasing the frequency of changes would rein-force the effects of an increase in But when we simulta-neously increase types persistence and the discount factorthe result depends on which of the two (persistence and )converges faster to one The case considered in the paper inwhich only 3 1 corresponds to the case in which thediscount factor converges more quickly than persistence


PROPOSITION 4 When types are imperfectlycorrelated even if correlation is not positivethen as 3 1 the average profit of the monop-olist converges to the first-best level of surplusand the average utility of the consumer con-verges to zero regardless of the renegotiation-proofness constraint

When the discount factor is high it does notmatter what happens in the first T periods for Tfinite However because we are working with aMarkov process in the long run the distributionof types converges to a stationary distributionwhich is independent from the initial valueThis implies that at time one the ability of theconsumer to predict his own type realizations inthe far future is almost as good as the sellerrsquosability For this reason when is high themonopolist can separate the agents paying onlya minuscule rent to the higher type18

C Discussion

Before presenting further results we now dis-cuss the assumptions of the model emphasizingthe issues that are still open for future researchIn particular we focus on the stochastic processthe utility function cash constraints and thetime horizon

As noted any change of the contract at timet has a ldquocascaderdquo effect on expected utilities inthe previous histories These effects dependboth on the structure of the constraints that arebinding at the optimum and on the transitionprobabilities which determine the conditionalexpectation of the consumer at each historynode This is the reason the properties of thestochastic process are important in the charac-terization A key assumption of the model is

that types are positively correlated19 When thisis the case a ldquohighrdquo type has not only a highermarginal valuation for the good today but alsoa higher expected valuation for a contract in thefuture Without this assumption a type wouldbe ldquohighrdquo or ldquolowrdquo depending on which of thesetwo components of utility prevails Along withthe rest of the literature on dynamic contract-ing20 we also assume that at any point in timethe type t can assume one of two values Whenthere are n possible values the conditional dis-tribution of future realization of t is a n 1dimensional vector so the characteristics ofeach agent are n 1 dimensional In this casebesides the problem of dynamic screeningwe would have an additional problem of multi-dimensional screening As is well known in thiscase types are not ldquonaturallyrdquo ordered and theset of constraints that are binding can be morecomplicated The environment studied abovehas the advantage of separating the study ofthe dynamics of the contract from the studyof the multidimensionality of the types whichis a conceptually distinct problem and there-fore provides a better understanding of thedynamics21

A related issue regards the transition proba-bilities in the stochastic process Clearly manydifferent assumptions can be made regardingthese probabilities In this paper we have con-sidered the case in which the transition proba-bilities do not change over time This howeveris not essential for the characterization indeedeven if the degree of positive correlationchanges over time (but remains positive) wewould be able to perform the same simplifica-tion of the incentive constraints as inLemma 122 Another assumption of the model isthat the transition probabilities between types

18 It is worthwhile to point out the differences betweenthis result and the results in Proposition 1 and 2 because thelogic of their proofs is different The proof of Proposition 4does not require the assumption that types are positivelycorrelated For this reason Proposition 4 is stronger than theresult that would have followed from taking the limit in theformula of Proposition 1 as 3 1 However while Prop-osition 1 characterizes the optimal contract for any Prop-osition 4 is only a limit result Even in the limit case inwhich 3 1 Proposition 4 shows that the contract con-verges to an efficient contract in probability but it is silenton the behavior of the contract in any single history

19 This assumption is used in the characterization of theoptimal contract but not in Proposition 4

20 See eg Hart and Tirole (1988) Laffont and Tirole(1990) and Rustichini and Wolinsky (1995)

21 At the cost of higher complexities the model can beextended to multidimensional types Indeed even in a mul-tidimensional environment types can be ldquoendogenouslyrdquoordered to simplify the set of incentive constraints (see forexample Jean-Charles Rochet 1987)

22 In this case the optimal contract would not depend onthe likelihood ratio raised to the t as in Proposition 2 buton the multiplication of the changing likelihood ratios alongthe lowest history For this extension we would also need tocontinue assuming that a high type remains more likely tobe high in the future


are all positive although they may be small thisprecludes a case in which there is a type i whichwill never become some other type j23 Aninteresting extension of the model could be toconsider a process with more than two types anda form of long-term heterogeneity in whichdifferent transition probabilities correspond toeach initial type A systematic analysis of theproperties of the stochastic process and the ex-tension to the case of dynamic screening withmultidimensional types is left for futureresearch

Regarding the utility and the cost functionthe results can easily be extended to the case inwhich the cost function is a generic convexfunction and utility is a generic function u(t q)provided that the usual single crossing condi-tion is assumed24 A relevant assumption how-ever is that the utility is quasilinear (asgenerally assumed in the literature on nonlinearpricing) When the utility function is not quasi-linear we have an additional issue of consump-tion smoothing over time In this case too theanalysis of the quasilinear case allows us toseparate the dynamic screening problem fromthe conceptually different problem of consump-tion shooting25

As standard in the literature we do not im-pose cash constraints on the consumerrsquoschoice26 At the cost of additional complica-tions it would be a simple exercise to incorpo-rate these constraints in our model Underplausible assumptions however these con-straints would be irrelevant for the analysis Inthe model monetary transfers can always bebounded above in all periods by a finite upper

bound which is generally very small For exam-ple the upper bound on monetary transfersdepends (among other variables) on the persis-tence of agentsrsquo types as persistence convergesto one the per-period payments converge to thesame payments as in the static model If weassume that cash constraints are satisfied in thestandard static version of the model then whentypes are sufficiently persistent (as perhaps rea-sonable to assume when consumption is fre-quent) cash constraints would not be binding inour dynamic model either

Finally we turn to the time horizon Besidesa direct theoretical interest the analysis of astationary model with infinite periods is usefulfor two reasons First with this assumption wecan study long-term behavior and convergenceof the contract which would be impossible ina two-period model It is also instrumentalhowever in the study of price dynamics Forexample we will show that the transfer price ofthe monopolistrsquos technology is declining overtime Since the model is stationary the truevalue of the technology is constant and identicalin any period Therefore this decline in pricearises purely for strategic reasons in a nonsta-tionary model with finite periods we would notbe able to separate the strategic effect from thenatural decline in value due to the shorter hori-zon It is however easy to show that our char-acterization would be valid even in a modelwith T periods

IV Property Rights

Before presenting results on the monetarytransfers it is useful to discuss property rightssince their allocation typically (although notnecessarily) influences the flow of monetarytransfers Up to this point we have assumed thatthe monopolist has the right to decide the quan-tity supplied in every period Instead of sellingoutput on a period-by-period basis howeverthe monopolist may decide to sell the propertyrights of her exclusive technology to the con-sumer Only the consumer benefits directlyfrom the technology and has information for itsefficient use It is therefore natural to expect thatthe property rights are ultimately acquired bythe agent who has a superior valuation of itsfuture use The decision to transfer propertyrights however depends on the history of theagentrsquos types

23 This environment however can be approximatedsince transition probabilities can be arbitrarily small

24 This requires the cross derivative uq to be positiveWhen the utility and cost functions are generic functionshowever the first-order conditions do not necessarily yieldclosed form solutions All the results however continue tohold in this more general environment (see Battaglini andCoate 2003 for details)

25 The results however are robust to changes in thedegree of risk aversion In Battaglini and Coate (2003) weshow that when risk aversion is below a critical value thecharacterization with risk aversion is the same as the char-acterization without risk aversion and a small change in riskaversion would imply only a small change in the contract

26 Cash constraints limit the maximum amount of per-period monetary transfers between the principal and theagent Clearly such constraints can be incorporated in botha dynamic model and a static model


PROPOSITION 5 Without loss of generalitythe optimal contract offers a call option to buyout the firm to the agent as soon as he revealshimself to be a high type However the monop-olist never finds it optimal to sell the firm to anagent who has always revealed himself to be alow type

The first part of this result should not besurprising After the agent reveals himself to bea high type there is no residual asymmetricinformation At this stage and before the real-ization of the type in the following period weshould expect no reason for the monopolist tokeep the ownership of the technology27 Theinteresting observation however is in the sec-ond part of the proposition after a history inwhich the agent has never revealed he is a hightype the monopolist finds it strictly suboptimalto sell the firm and prefers to introduce a dis-tortion in the value of the firm not only in theperiod in which the type is revealed but also inthe subsequent periods28 Indeed as we dis-cussed in Section III A the distortion is intro-duced to extract surplus from the high typeThis suggests that it is natural to observe adistortion in the period in which the agent re-veals his type But this does not explain why themonopolist still wants to introduce a distortionin the following periods given that the agenthas revealed his low type there is no asymmet-ric information anymore in this case either Thischaracteristic of the optimal contract dependson the dynamic nature of the incentive con-straint and it is instructive to see why it is true

Consider a simple two-period example As-sume that after the declaration in period 1 themonopolist sells the firm to the agent irrespec-tive of the type In the second period the agentwould receive all the surplus ie 1frasl2 H

2 if he isa high type and 1frasl2 L

2 if he is a low type Thisimplies that the high type receives a rent ie anextra payoff with respect to the lower type

equal to Rown 1frasl2 (H L ) This rent ishigher than the minimal rent that would guar-antee truthful revelation the incentive compat-ibility constraint requires only a rent equal toRIC L Rown Imagine now that themonopolist after the agent reveals himself to bea low type keeps the ownership in order toreduce the rent of the high type at t 2 insteadof selling the firm Assume in particular thatinstead of selling the good at cost in the secondperiod she sells to the high type H units atprice 1frasl2 H

2 ie she reduces the extra rentof the high type by in case in period 1 theagent declares to be a low type For small thecontract remains incentive compatible in thesecond period In order to satisfy the constraintsat t 1 suppose that the monopolist reducesthe price paid by the low type by Pr(HL )dollars The low typersquos incentives in period 1are unchanged if he reports himself to be a lowtype he receives Pr(HL ) dollars more int 1 and he expects to receive Pr(HL ) lessat t 2 moreover the contract does not changeif the agent chooses to report himself to be ahigh type Consider now the impact of thischange on the incentive compatibility constraintof the high type at t 1 If the high typedeviates and reports himself to be a low type hereceives Pr(HL ) more the same as the lowtype since this is paid ldquoin cashrdquo at t 1 with areduced price However the expected loss forthe high type is Pr(HH) because he is moreoptimistic than the low type about the futureSince [Pr(HL ) Pr(HH)] is negativethis implies that the outside option of the hightype ie the utility of reporting untruthfullyhas a lower value and the monopolist can inducetruthful revelation by leaving a lower rent to thehigh type at t 1 The monopolist thereforeprefers to keep strict ownership of the firmownership enables control of the rent of theagent in the second period and this control isimportant to extract surplus in the sale of thetechnology to the high type in the first periodThe characterization of the optimal contract in(2) goes beyond this observation In our infiniteand stationary environment in fact the monop-olist finds it optimal to reduce the efficiency ofthe firm for potentially infinite periods until shehears a ldquohigh-typerdquo report Moreover as we willprove in Section V the dynamics of the transferprice of the technology will be dictated by thedynamics of the optimal inefficiency in supply

27 Note that this result is different from the classicalresults by Jeremy Bulow (1982) concerning the trade-offbetween the sale and the rental of a durable good In thisliterature in fact if a durable good is sold then the quantityremains constant in the following periods in our frame-work instead the firm is selling the technology to producethe good and the future quantities depend on the realizedtype

28 I am grateful to Bengt Holmstrom and Asher Wolin-sky who have independently suggested this point


V The Dynamics of Monetary Payments

As mentioned above two payment scheduleswith the same present value can give the sameincentives to an agent therefore the pricescharged in the optimal contract are not uniquelyidentified Indeed although it is true that we canassume without loss of generality that the opti-mal contract keeps the lowest type at his reser-vation utility in any period we can constructequilibrium contracts that do not have this fea-ture an example is the contract in which themonopolist sells the technology to the agent Ingeneral when we have many periods we canfind optimal contracts in which the monopolistreceives a large payment at some date t and shecommits to pay it back in installments Theinstallments can in principle follow any timepattern In this section we focus on two types ofoptimal monetary transfers that seem more in-teresting from a theoretical and empirical pointof view

For any contract in which the monopolistborrows money and repays it in an arbitrarytime pattern we can distinguish two parts asupply contract in which the relevant ICht

(H)and IRht

(L ) constraints are satisfied as equali-ties and a residual lending contract in whichthe monopolist borrows some amount of moneyand pays it back over time to the agent Areason why the supply contract is more inter-esting than other contracts is that if we assumethat the monopolist is even slightly more patientthan the agent she would never find it optimalto ask the agent to anticipate payments for fu-ture supply (as occurs when the technology issold to the agent) and therefore all the con-straints would be binding after all histories29

The lending contract can take (almost) any formbecause the monopolist can commit to repay itaccording to any time pattern The supply con-tract however is uniquely determined by the

incentive structure of the model We can there-fore ask what is the dynamics of prices andmore importantly the dynamics of the consum-errsquos utility in the optimal supply contract

There is one particular case in which themonopolist receives anticipated payments fromthe consumer which has special significancefrom an empirical and theoretical reason thecontract discussed in Section IV in which themonopolist as soon as compatible with profitmaximization sells the firm to the consumerwho reports himself to be a high type We callthis arrangement the sale-of-the-firm contractIn this case too the monopolist can add on topof a sale-of-the-firm contract a lending contractas defined above in which she borrows moremoney than the value of the firm and repays theextra amount over time Since we are not inter-ested in this case we assume without loss ingenerality that the IRht

(L ) constraint is satisfiedas equality in all periods Again if this condi-tion is satisfied the sale-of-the-firm contract isuniquely determined The interesting questionin this case is the dynamics of the strike price ofthe call option on the technology30

Regarding the supply contract we have

PROPOSITION 6 In the optimal supply con-tract the average per-period utility of the agentstarting from any date t is nondecreasing in t inall possible histories and strictly increasing insome history therefore the expectation at timezero of the average rent of the agent from datet is strictly increasing in t

Recent empirical work has highlighted that insome important markets long-term contracts arefront-loaded prices are initially high and de-cline over time31 A consequence of this effecttherefore is that the expected utility of a con-sumer from continuing to remain a monopo-listrsquos customer increases over time Dionne andDoherty (1994) explain this phenomenon as aconsequence of the possibility to renegotiatecontracts over time Building on Milton Harrisand Holmstrom (1982) Hendel and Lizzeri

29 If the monopolist is more patient than the agent thenthe incentive compatibility for the low type would be bind-ing in all periods and the monopolist would not find itoptimal to lend money to the agent because the agent wouldnot be able to commit to repay the debt Note that themonopolistrsquos objective function is continuous in her dis-count factor and the constraints do not depend on it there-fore an infinitesimal reduction in the monopolistrsquos discountfactor would have only an infinitesimal impact on the opti-mal quantities qt(ht) but would eliminate equilibria inwhich the monopolist borrows money

30 I am grateful to William Rogerson who suggested thispoint

31 This effect has been shown with California auto in-surance data by Dionne and Doherty (1994) and morerecently by Hendel and Lizzeri (2003) in the life insurancemarket


(2000) present a model with no asymmetricinformation but in which both the principal andthe agent learn over time from a public signalthe type of the agent front-loading is thereforea consequence of reclassification risk Ourmodel suggests a new explanation for this phe-nomenon in which front-loading is precisely aconsequence of the commitment power ofthe seller32 Indeed in the optimal contracteven without imposing renegotiation con-straints she finds it optimal to promise an effi-cient contract to the agent if he reports himselfto be a high type or to provide a contract withdecreasing inefficiency Because of this shemust commit to pay a rent to the high type thatincreases over time because the higher the effi-ciency of the contract the more expensive it isto separate the agentsrsquo types

We now turn to sale-of-the-firm contracts Aswe discussed in Section IV the monopolistfinds it optimal to sell her technology only if theagent reveals himself to be a high type It istherefore natural to look at the evolution of thecall price of the option to buy the technologyalong the history in which it can be exercised(ie when the agent always reveals himself tobe a low type) This question is interestingbecause given the stationary structure of themodel the present value of the firm along thishistory is constant Remember that the modelhas infinite periods and because the preferencesof the consumer follow a Markov process thevalue of the firm depends only on the currentstate of the consumerrsquos type33 This fact maysuggest that the price of the firm is constant overtime We have however

PROPOSITION 7 In the optimal sale-of-the-firm contract the strike price of the call optionto buy out the technology is strictly decliningover time

What really matters in the determination ofthe transfer price of the firm is the outside

option of the high type (ie the value of report-ing untruthfully) This outside option changesover time because the contract becomes increas-ingly efficient along the ldquolowestrdquo history andthe improvements in the contract benefit thehigh type more than the low type The higherefficiency of the contract in fact increases theagentrsquos utility in the event in which he turns intoa high type and an agent who is a high typetoday has a higher probability of being a hightype tomorrow For this reason the price for theservice that the low type is willing to pay in-creases more slowly than the increase in utilityof a deviation for a high type For this reasonthe outside option of the high type increasesover time This implies that the only way for themonopolist to induce a truthful revelation is toreduce the strike price of the call option on theproperty rights of the firm

VI Renegotiation-Proofness

So far we have assumed that the monopolistcan commit to a contractual offer We discussedthis point above arguing that this is the mostappropriate assumption in many environmentsin particular when the monopolist is servingmany consumers and is interested in maintain-ing her reputation or when renegotiation costsare larger than the benefits There are situationshowever in which the seller cannot commit notto renegotiating the contract after some histo-ries The received literature has shown that iftypes are constant the optimal contract is neverrenegotiation-proof Perhaps surprisingly givena condition that is easily satisfied this is nolonger true when the agentrsquos type follows astochastic process34 We say that a contract isrenegotiation-proof if after no history ht there isa new contract starting in period t that the con-sumer would accept in exchange for the originalcontract and that is strictly superior for themonopolist This definition is standard in the

32 In our model we assume that the seller has monopolypower Clearly this assumption should be taken into con-sideration when using the model to interpret evidence inmarkets with more competitive environments

33 In an equilibrium of a direct mechanism the consumerwould reveal his type truthfully therefore the firm would besold only to high types This implies that whenever the firmis sold the expected present value of the firm is constantirrespective of the period in which it is sold

34 This result should not be confused with the findings inPatrick Rey and Bernard Salanie (1990 1996) Beside thefact that they consider a different model with moral hazardand constant types these authors show that a renegotiation-proof contract can be implemented by a chain of short-termcontracts (two-period contracts in which the principal cancommit) They do not characterize the renegotiation-proofcontract and they do not prove that the ex ante optimalcontract is renegotiation-proof (which indeed would not betrue in their models with constant types)


literature and natural when a contract isrenegotiation-proof either the monopolist or theconsumer would reject a revision of the initialagreement

PROPOSITION 8 The optimal contract isrenegotiation-proof if Pr(LL ) 1 HPr(HH) Moreover if this condition is notsatisfied there exists a t such that thecontract is not renegotiation-proof only in thefirst t periods

Figure 2 represents the condition of Proposi-tion 7 in an example in which there is a 30-percent initial probability that the agent is a hightype the contract is renegotiation-proof for anypoint below the straight ldquothick linerdquo As can beseen from the figure the set of parameters forwhich the contract is renegotiation-proof coversmost of the set of feasible parameters (Becausetypes are positively correlated Pr(LL ) andPr(HH) are both larger than 1frasl2 )35

The intuition of this result can be seen fromFigure 336 Assume for the sake of illustra-tion that at time t the monopolist is contem-

plating a change only in the quantity offeredafter the agent reports a type t j after ahistory Xt j1 keeping constant all otherquantities The complete argument clearlyneeds to consider a change on the entire se-quence of contingent menus this is a moresophisticated problem (solved in the Appen-dix) but this example provides a useful intu-ition First note that a contract is renegotiatedonly by a contract that is Pareto superiorotherwise either the seller or the buyer wouldnot accept the change Since welfare isstrictly concave in the quantity supplied andthe ex ante optimal quantity at time t (ieq(t j Xt j1)) is not larger than the effi-cient output qe(t j) it must be that welfareis strictly increasing in the interval [0q(t j Xt j1)] (see Figure 3) This im-plies that the new output q(t j Xt j1)prescribed by a renegotiated contract must bestrictly larger than q(t j Xt j1)

Consider now a history ht If the agent haspreviously reported himself to be a high typethen the contract is efficient and not renego-tiable assume therefore that the agent hasalways reported himself to be a low type Ifafter ht the monopolist could choose anyother continuation contract the quantity sup-plied after a history ht j following or equal toht in the new contract would be qR(t j

35 If we measure the persistence of types by maxPr(LL ) Pr(HH) an immediate implication ofProposition 8 is that given any initial prior there is anupper bound (H) on persistence such that for (H)the optimal contract is renegotiation-proof (area below thesemicircle in Figure 2)

36 The two concave functions in the figure represent theprofits and welfare generated in period t j after historyxt j1 q(t j Xt j1) is the optimal contract qe(t j) is

the efficient contract and qR(t j Xt j1) is the contractthat is ex post optimal after history ht

FIGURE 2 THE RENEGOTIATION-PROOFNESS CONDITION


Xt j1) t j [Pr(HL )Pr(LL )]X(tj Xj1)j1 This formula is identical to theformula of the ex ante optimal contract in Prop-osition 2 except that instead of the prior H weuse the appropriate posterior after ht Pr(HL )and the state Xt j is started afresh at time tComparing qR(t j Xt j1) with the contractin Proposition 2 it is easy to verify that the exante optimal q(t j Xt j1) is larger than theex post optimal level qR(t j Xt j1) if

(4)H

Lt j1

PrHL

PrLLj1

which is always satisfied if Pr(LL ) 1 HPr(HH) since t 1 (Obviously thecontract can be renegotiated only startingfrom the second period) Because the profitfunction is also strictly concave this impliesthat when (4) holds any quantity larger thanq(t j Xt j1) reduces expected profits atht But then any change that would be ac-cepted at ht by the customer would necessar-ily reduce profits implying that the quantityq(t j Xt j1) would not be renegotiatedat any time t

When types are constant the optimalrenegotiation-proof contract requires the con-sumer to play sophisticated mixed strategiesand this may appear unrealistic These strategiesare necessary to guarantee that after any pos-sible history the monopolistrsquos posterior be-liefs are such that there are no ex post Paretosuperior contracts The result presented abovehowever shows that when types are correlated

but follow a stochastic process even if thecorrelation level is very high (as in Fig-ure 2) consumers do not need to use mixedstrategies in equilibrium but simply truthfullyreport their type The conflict between optimal-ity and renegotiation-proofness and the sophis-tication of equilibrium strategies necessary toguarantee the latter property therefore are im-plications of the assumption that types are con-stant or very highly correlated37

VII Conclusion

This paper shows that a long-term contractualrelationship in which the type of the buyer isconstant over time is qualitatively differentfrom a contractual relationship in which thetype follows a Markov process even if the typesare highly persistent While in the first case thecontract is constant in the second the contract istruly dynamic and converges to the efficientcontract Even if the environment has only one-period memory and risk-neutral agents the op-timal contract is not stationary and hasunbounded memory The structure of the opti-mal contract however is remarkably simple Inanalogy with the static model we have a stron-ger version of the generalized no distortion atthe top principle which implies that the entirestate-contingent contract becomes forever effi-cient as soon as the agent reports himself to be

37 A complete analysis of the mixed strategy equilibriumin the optimal renegotiation-proof contract with variabletypes when (4) is not satisfied is presented in Battaglini(2005)

FIGURE 3 THE EX POST MAXIMIZATION PROBLEM AFTER A HISTORY ht


a high type In our dynamic setting howeverwe also have a novel vanishing distortion at thebottom principle which clearly could not beappreciated in a static model

With constant types there always is a conflictbetween optimality and renegotiation-proofnessand the latter property is guaranteed only ifconsumers use sophisticated mixed strategiesWith stochastic types in contrast even if thereis high persistence the optimal contract isrenegotiation-proof for natural specifications ofthe parameters Consumers moreover adoptsimple pure strategies

The dynamic theory of contracting presentedin the paper also provides insights into the own-ership structure of the monopolistrsquos exclusivetechnology and contributes to explaining someempirical findings The monopolist may find itoptimal to keep the ownership of the technologyeven when it would be inefficient in order tocontrol the agentsrsquo future rents and thereforemaximize rent extraction This inefficient reten-tion of property rights may potentially last foran arbitrarily large number of periods but theallocation of property rights will be efficientwith probability one in the long term

APPENDIX

PROOF OF LEMMA 1For a generic maximization program P we

define V(P) the value of the objective functionat the optimum Let us also define P I

R theprogram in which expected profit is maximizedonly under the incentive compatibility con-straints of the high type and the individual ra-tionality constraints of the low type ICht

(H)IRht

(L ) ht We proceed in two steps

CLAIM 1If p q solves P I

R then there exists a p suchthat p q satisfies IRht

(L ) and ICht(H) as

equality ht and achieves the same value as pq V(P I

R) V(PII)

PROOFGiven a solution p q we first show that the

price vector can be modified to guarantee thatall the incentive constraints are satisfied as equal-ities without reducing the value of the programthen we show that the resulting contract withincentive constraints satisfied as equalities can bemodified to make the individual rationality con-

straints satisfied as equalities as well while leav-ing the incentive constraints untouched andkeeping the value of the program constant

Step 1 We show the result by induction Let tbe a finite integer Assume that for any solutionp q of P I

R and any t t there is a pt suchthat pt q is also a solution of P I

R and allincentive compatibility constraints are satisfiedas equalities up to period t the value of theobjective function is unchanged and pt is iden-tical to p in any period j t pt( hj) p(hj) for j t This step is clearly satisfied for t 1 since in this period the incentive compatibil-ity constraint is necessarily binding at the opti-mum We now show that if pt exists t tthen there must be a price vector pt1 such thatall the incentive compatibility constraints aresatisfied as equalities up to period t 1 pt1 isidentical to p in any period j t 1 and thevalue of the objective function is unchangedGiven the induction step assume without lossof generality that the incentive constraints aresatisfied as equalities for any j t There aretwo cases to consider Assume first that at pe-riod t 1 after a history ht1 ht HL thehigh type receives an expected continuationutility equal to the utility he would receive if hedeclares to be a low type plus a constant 0Modify the contract so that the new prices afterhistories ht1 H and ht1 L are re-spectively pt1(H ht1) p(H ht1) andp(L ht1) p(L ht1)38 simultaneouslyreduce the price after history ht L so thatpt1(L ht) p(L ht) Pr(HL ) We callthis new price vector pt1 This change wouldnot reduce the monopolistrsquos expected profit itwould not violate IRht

(L ) in any period itwould not violate the incentive compatibilityconstraints for histories following ht and itwould satisfy ICht1

(H) as equality Considernow the ICht

(H) constraint at ht The utility of ahigh type that is truthful U(H ht) is unchangedif the high type reports himself to be a low typehowever he would receive

(A1) UH ht 1 PrHL

PrHH UH ht 1

38 Remember that p( ht) is the price charged afterhistory ht if the agent declared to be a type so it is theprice charged after a history ht


where the inequality follows from the fact thattypes are positively correlated It follows thatIChj

(H) are satisfied for any j t and pt1 qis a solution of P I

R By the induction step wecan find a new price vector pt1 which is suchthat the incentive compatibility constraints aresatisfied as equalities in all periods j t andthat is identical to pt1 for periods j t (whichis identical to p in periods l t 1) Since thischange does not affect prices at t 1 andfollowing periods in correspondence to pt1

the incentive compatibility is satisfied as equal-ity t t 1 prices are equal to the prices inp for periods j t 1 and the value functionis equal to the original

Assume now that at period t 1 after somehistory ht ht1 H the high type receivesa utility equal to the utility he would receive ifhe declares to be a low type plus a constant 0 Modify the contract so that the new pricesafter histories ht H and ht L are respec-tively pt1(H ht) p(H ht) and pt1(Lht) p(L ht) simultaneously reduce pricesafter history ht H so that pt1(H ht1) p(H ht1) Pr(HH) This new con-tract pt1 would leave all the constraints of therelaxed problem satisfied ht and the incentiveconstraint satisfied as equalities in the first t 1 periods And it would not reduce profits

Step 2 By the previous step we can assumewithout loss of generality that all the incentivecompatibility constraints are satisfied as equal-ities We now show that the individual rational-ity constraints can also be reduced to equalitiesIt can be verified that the individual rationalityconstraint must be binding at t 1 Again weprove the result for the remaining periods byinduction Assume that in all periods j tIRhj

(L ) holds as equality and that the expectedutility of a low type agent after history ht1 is13 0 Consider an increase by 13 of the pricescharged in the period t 1 pt1( ht i) p( ht i) 13 and a reduction of the priceat time t so that pt1( ht1) p( ht1) 13 Clearly this change would not violatethe constraints of P I

R would leave the incentivecompatibility constraints untouched and wouldsatisfy all the individual rationality constraint asequality up to period t 1 Profit would remainunchanged as well

We now prove

CLAIM 2Any solution of PII satisfies all the constraintsof PI and V(PI

R) V(PI)

PROOFSince V(PI

R) V(PII) we need only showthat in correspondence to the solution of PIIafter any history ht the low type does not wantto imitate the high type (ie the ICht

(L ) con-straint is satisfied) and the high type receives atleast his reservation value (ie the IRht

(H) con-straint is satisfied) This guarantees thatV(PI) V(P I

R) and hence the result

Step 1 the ICht(L ) constraints Note that by

ICht(H) and IRht

(L ) after any history ht

pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht

where p(i ht)i H L is the price charged afterthe agent declares to be a type i and U(H ht) U(H ht H) U(H ht L) the differencebetween the rent of a high type after a H and a Ldeclaration (the continuation value of a low type iszero in PII) As can be seen from (2) in corre-spondence to the solution of PII

39 after an agentdeclares to be a high type an efficient contract isoffered in the optimal solution of the relaxed prob-lem so using ICht

(H) and IRht(L) we can write

UH ht H j 0

jPrHH

PrLHjqeL

where remember qe(L) is the efficient quantitywhen the type is L If the agent reports himself tobe a low type on the contrary he will receive aninefficient quantity q(Lh) that is never strictlyhigher than the efficient level qe(L) therefore hiscontinuation value is not higher than U(H ht H)

39 The formal derivation of (2) is in the proof of Prop-osition 1 below


So U(H ht H) U(H ht L) 0 for any htand since types are positively correlated(Pr(HH) Pr(HL))U(H ht) 0 It followsthat

pH ht pL ht qH ht

qL ht L PrHLUH ht

which implies that ICht(L) is satisfied at ht

Step 2 the IRht(H) constraints By ICht

(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0

and therefore IRht(H) is satisfied as well

We can now prove Lemma 1 Assume that pq solves PII then by Claim 1 and 2 it mustalso solve PI Assume that p q solves PIthen by Claim 2 it must also solve P I

R sinceV(P I

R) V(PI) By Claim 1 there exists a psuch that p q solves PII and achieves thesame value as PI We conclude that q solves PIif and only if it solves PII

PROOF OF PROPOSITION 1Let us define ht

L L L L the historyalong which the agent always reports himself tobe a low type for t 1 periods Using the factthat ICht

(H) and IRht(L ) are equalities we can

formulate the utility of the high type at time 1as

UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0

This formula can be written as

(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L

Using the equality IRh1(L) we know that the low

type receives zero at time one It follows that PIIcan be represented as

B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L

where the first summation is the expected sur-plus generated by the supply contract (W(i j)is the expected social welfare generated by thecontract from period 2 if the realization in pe-riod 1 is j and the realization in period 2 is i)

We have two possible cases

Case 1 ht ht1 HthtL

t 1 Thefirst-order condition implies qt (ht) andthe contract is efficient

Case 2 ht htL

t 1 The first-order condi-tion with respect to a generic quantity offeredalong the lowest branch qt (Lht

L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1

which completes the characterization of the op-timal contract

PROOF OF PROPOSITION 4Starting in period t from any history ht

the expected first best surplus from time t on-ward is independent from t and equal toWt ( ) yenjt j tEt[1frasl2 j

2t ] Consider acontract c in which a fixed fee F W1(L ) ischarged in period 1 and then an efficient menuplan in which q( ) p( ) 1frasl2 2 for any 1 is offered This contract is clearly incentivecompatible and individually rational for any 1 moreover it is renegotiation-proof since it isefficient Therefore it is a feasible option in the


monopolistrsquos program even if the renegotiation-proofness constraint must be satisfied and mustyield an average profit not larger than the profit ofthe optimal contract This implies

(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1

1w where w( ) is the per-period Marshaliansurplus when the type is As 3 1 theright-hand side can be written as

t lim31

1

E t

1w 1 L E

t

1w where t is a finite integer Since (C5) mustholds for any t 1 and limt3( t) 0(because the process converges to a stationarydistribution) we have that lim31(1 )must be equal to lim31(1 ) E[yen1

1w()] This also implies that theagentrsquos average payoff is zero

PROOF OF PROPOSITION 5Since the optimal contract is efficient after

the agent reveals himself to be a high type themonopolist finds it optimal to offer to the con-sumer the same quantities that the consumerhimself would choose if he could directly con-trol supply The first part of the result thenfollows from the fact that all players have quasi-linear utilities and therefore they are indifferent

between paying or receiving a positive amountevery period or a large amount equal to thefuture expected payments at some period andzero afterward The second part follows fromthe fact that from (2) we know that along theldquolowest historyrdquo supply is distorted in the fu-ture with positive probability and the monopo-list cannot achieve these distortions withoutcontrol of the technology

PROOF OF PROPOSITION 6When IRht

(L ) is satisfied as equality for allht the low type receives zero expected utility inall periods Therefore we need only show thatthe average utility of the high type is nonde-creasing in time Using (2) and the incentive com-patibility constraint of the high type we can write

UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1

where U(H t Xt1) is the expected utility of ahigh type at time t given the state Xt1 Considernow two periods t and t t It is easy to showthat U(H t Xt1) U(H t Xt1) is propor-tional to X(L Xt1) X(L Xt1)[1 (Pr(HL)Pr(LL))](tt) which is nonnegativebecause Xt1 Xt1 and strictly positive if X(LXt1) 1 Therefore the average rent of theagent is nondecreasing in any history and strictlyincreasing in a nonempty subset of histories Itfollows that at time zero the expected averagerent starting from period t is strictly increasingin t

PROOF OF PROPOSITION 7Since the monopolistrsquos technology is sold as

soon as compatible with profit maximization itssale can occur only along a history in which theagent has always reported himself to be a lowtype Consider any such history ht The priceP(ht) paid for the technology by the high type isdetermined by the equation


(F6) WH Pht UH L ht

where U(i j ht) is the utility of a type ifrom declaring to be a type j after a historyht and W(i) is the expected first best sur-plus from time t if the type at t is i

40 SinceW(H) is clearly history independent theresult follows by the fact that supply is in-creasing over time and therefore U(H Lht) is increasing (see [B2])

PROOF OF PROPOSITION 8Consider the problem of ex post maximiza-

tion faced by the monopolist after a history htwith t 1 in which the agent has never reportedhimself to be a high type At this stage expectedprofits can be written as

(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL

where qi i H L is the sequence of quantitiesin the menus offered if the agent reports himselfto be a type i at t W(i qi) is the expectedsurplus generated in the contract if the agent isof type i and qi is offered and R(qL ) is theexpected rent of the high type starting from htwhich guarantees incentive compatibility (ByLemma 1 it depends only on qL as in [B2] andthe rent of the low type is zero) Indeed (G7) isa compact way to write (B3) when the posteriorprobability that the type is high starting from htis Pr(HL ) The monopolistrsquos ex post problem(Pex post) consists of maximizing (G7) under theadditional constraint that the expected rents ofthe agent are at least as high as the expectedrents starting from ht obtained keeping the orig-inal ex ante optimal contract It is howeveruseful to consider the program (P ex post) inwhich (G7) is maximized under the additionalconstraint that expected welfare is at least ashigh as the level achieved with the original ex

ante optimal contract which after ht is a con-stant that we denote W yeniHL Pr(iL )W(iqi) W We denote this constraint (G8) If weshow that the ex ante optimal quantities solveP ex post then they must also solve Pex post andbe renegotiation-proof The Lagrangian ofP ex post is

(G9) LP 1 LWH qH

WL qL] LRqL

where L is the ex post likelihood ratio[Pr(HL )Pr(LL )] and is the Lagrangianmultiplier associated with (G8) We proceed inthree simple steps

Step 1 0 Given that Pr(LL ) 1 HPr(HH) implies (4) j 0 if 0 thenthe solution of (G9) implies that all the quantitiesin qH are set efficiently and the quantities in qL aredistorted downward more than the solutions of theex ante optimal problem (the argument is the sameas in Section VI) But then the welfare constraint(G8) must be violated a contradiction

Step 2 Let deg be the ex ante likelihood ratio(HL ) We can show that [L(1 )] t1 Indeed it can be verified that the quan-tities following history ht in the optimal solutionof the ex ante problem maximize

(G10) LA WH qH

WL qL t 1RqL

If [L(1 )] t1 then the solution of(G9) would be less distorted than the solution of(G10) implying that the welfare constraint (G8)is not binding and so 0 a contradictionSimilarly we can prove that the reverse inequal-ity is not possible From [L(1 )] t1

we conclude that the solution of (G9) and (G10)coincide and the optimal ex ante contract isrenegotiation-proof

Step 3 Finally it is easy to see that ifPr(LL ) 1 HPr(HH) then there mustbe a finite t such that for t t then (4) issatisfied and the argument in steps 1 and 2 isvalid for any t t

40 To avoid confusions in what follows it is worth em-phasizing that U(j i ht) and U(j ht i) are differentobjects the first represents the case in which a type j reportsuntruthfully to be a type i after a history ht the secondrepresents the case in which a type j truthfully reports histype after a history ht1 ht i Indeed the secondexpression is equivalent to U(j j ht i)


REFERENCES

Baron David P and Besanko David ldquoRegulationand Information in a Continuing Relation-shiprdquo Information Economics and Policy1984 1(3) pp 267ndash302

Battaglini Marco and Coate Stephen ldquoParetoEfficient Income Taxation with StochasticAbilitiesrdquo National Bureau of Economic Re-search Inc NBER Working Papers No10119 2003

Battaglini Marco ldquoOptimality and Renegotia-tion in Dynamic Contractingrdquo Centre forEconomic Policy Research CEPR Discus-sion Paper No 5014 2005

Bolton Patrick and Dewatripont Mathias Con-tract theory Cambridge MA MIT Press2005

Bulow Jeremy I ldquoDurable-Goods Monopo-listsrdquo Journal of Political Economy 198290(2) pp 314ndash32

Courty Pascal and Li Hao ldquoSequential Screen-ingrdquo Review of Economic Studies 200067(4) pp 697ndash717

Dewatripont Mathias ldquoRenegotiation and Infor-mation Revelation over Time The Case ofOptimal Labor Contractsrdquo Quarterly Journalof Economics 1989 104(3) pp 589ndash619

Dionne Georges and Doherty Neil A ldquoAdverseSelection Commitment and RenegotiationExtension to and Evidence from InsuranceMarketsrdquo Journal of Political Economy1994 102(2) pp 209ndash35

Gerstner Louis V Jr Who says elephants canrsquotdance New York Harper Collins 2002

Harris Milton and Holmstrom Bengt ldquoA Theoryof Wage Dynamicsrdquo Review of EconomicStudies 1982 49(3) pp 315ndash33

Hart Oliver D and Tirole Jean ldquoContract Re-negotiation and Coasian Dynamicsrdquo Reviewof Economic Studies 1988 55(4) pp 509ndash40

Hendel Igal and Lizzeri Alessandro ldquoAdverseSelection in Durable Goods Marketsrdquo Amer-ican Economic Review 1999 89(5) pp1097ndash1115

Hendel Igal and Lizzeri Alessandro ldquoThe Roleof Commitment in Dynamic Contracts Evi-dence from Life Insurancerdquo Quarterly Jour-nal of Economics 2003 118(1) pp 299ndash327

Karlin Samuel and Taylor Howard M A firstcourse in stochastic process San Diego Ac-ademic Press 1975

Kennan John ldquoRepeated Bargaining with Per-sistent Private Informationrdquo Review of Eco-nomic Studies 2001 68(4) pp 719ndash55

Laffont Jean-Jacques and Tirole Jean ldquoAdverseSelection and Renegotiation in Procure-mentrdquo Review of Economic Studies 199057(4) pp 597ndash625

Laffont Jean-Jacques and Tirole Jean A theoryof incentives in procurement and regulationCambridge MA MIT Press 1993

Laffont Jean-Jacques and Tirole Jean ldquoPollu-tion Permits and Compliance StrategiesrdquoJournal of Public Economics 1996 62(1ndash2)pp 85ndash125

Miravete Eugenio J ldquoChoosing the Wrong Call-ing Plan Ignorance and Learningrdquo Ameri-can Economic Review 2003 93(1) pp 297ndash310

Rey Patrick and Salanie Bernard ldquoLong-TermShort-Term and Renegotiation On the Valueof Commitment in Contractingrdquo Economet-rica 1990 58(3) pp 597ndash619

Rey Patrick and Salanie Bernard ldquoOn theValue of Commitment with Asymmetric In-formationrdquo Econometrica 1996 64(6) pp1395ndash1414

Roberts Kevin ldquoLong-Term Contractsrdquo Un-published Paper 1982

Rochet Jean-Charles ldquoA Necessary and Suffi-cient Condition for Rationalizability in aQuasi-Linear Contextrdquo Journal of Mathe-matical Economics 1987 16(2) pp 191ndash200

Rogerson William P ldquoRepeated Moral HazardrdquoEconometrica 1985 53(1) pp 69ndash76

Rustichini Aldo and Wolinsky Asher ldquoLearningabout Variable Demand in the Long RunrdquoJournal of Economic Dynamics and Control1995 19(5ndash7) pp 1283ndash92

Townsend Robert M ldquoOptimal MultiperiodContracts and the Gain from Enduring Rela-tionships under Private Informationrdquo Journalof Political Economy 1982 90(6) pp 1166ndash86

Welch Jack Jack Straight from the gut NewYork Warner Books Inc 2001


by a monopolist to a consumer whose prefer-ences evolve following a Markov process Inthis case even if types are highly persistentthe contract is very different from the contractwith constant types because the seller finds itoptimal to use information acquired along theinteraction in a truly dynamic way For thisreason the characterization of the optimal con-tract when there is heterogeneity within andacross periods allows a new understanding ofimportant aspects of a dynamic principal-agentrelationship that previous models could not cap-turemdashparticularly with regard to the memoryand complexity of the contract its efficiencyand its robustness to renegotiation Perhaps sur-prisingly it also provides insights into the op-timal ownership structure of the productiontechnology

As noted when types are constant the con-tract has no memory and the inefficiency of theoptimal static contract is repeated period afterperiod With persistent but stochastic typeseven in a simple stationary environment withone period memory (ie a Markov process) thecontract is nonstationary and has infinite mem-ory despite this however it can be representedin a very economical way by a simple statevariable Even if types are arbitrarily highlycorrelated and the discount factor is arbitrarilysmall the sellerrsquos optimal offer converges overtime to the efficient supply schedule along allpossible histories The speed of convergencehowever is state contingent and occurs in aparticular way which extends a well-knownproperty of the static model On the one hand infact we have a generalized no distortion at thetop (GNDT) principle after any history if theagent reveals himself to have the highest possi-ble marginal valuation for the good supply isset efficiently from that date onward in anyinfinite history that may follow On the otherhand and more importantly we have a novelvanishing distortion at the bottom (VDB) prin-ciple even in the history in which the agentalways reveals to have the lowest marginal val-uation for the good the contract converges tothe efficient menu offer One immediate impli-cation of this result is that in the ldquosteady staterdquoor even after a few periods the monopolistrsquossupply schedule may be empirically indistin-guishable from the outcome of an efficientcompetitive market moreover since higher ef-ficiency is associated with a higher consumer

rent it explains why ldquooldrdquo customers should betreated more favorably than ldquonewrdquo customers2

In a stochastic environment the incentivesfor renegotiation are also very different Asshown in the received literature (see discussionbelow) when types are constant over time themonopolist benefits from the ability to committo not renegotiating the contract because theoptimal contract is never time-consistent Withvariable types in contrast this is not the caseindeed even when types are highly correlated asimple and easily satisfied condition guaranteesrenegotiation-proofness Interestingly whentypes are constant the optimal renegotiation-proof contract always requires the agent to usesophisticated mixed strategies with correlatedbut stochastic types the optimal renegotiation-proof contract has an equilibrium in pure strat-egies and simply requires the agent to report histype

There is an intuitive argument which explainsthe dynamics of the distortions in the optimalcontract and the efficiency result mentioned ear-lier Assume that the agentrsquos type can take twovalues high and low marginal valuation for thegood (respectively H and L ) Consider Fig-ure 1 which shows the impact on profits at timezero of a marginal increase q in the quantityq(ht) offered to the consumer after a history htOn the one hand this change increases thesurplus that can potentially be appropriated bythe seller if history ht is realized (which isrepresented by the ldquothick arrowrdquo on the left-hand panel in Figure 1)3 However as in a staticmodel this increase in supply increases the rentthat the principal must leave to the agent tosatisfy incentive compatibility In every periodoptimal supply is determined by this marginalcostndashmarginal benefit trade-off and the dy-namic properties of the contract are driven by itsevolution To determine optimal supply there-fore it is important to understand the impactover expected rents of this change at time t

To this goal consider the right panel of Fig-ure 1 and assume that the rent of the high type

2 See Igal Hendel and Alessandro Lizzeri (2003) andGeorges Dionne and Neil A Doherty (1994) for evidence ofthis phenomenon

3 Because consumption is always distorted below its firstbest level starting from the sellerrsquos optimum a marginalincrease in supply after history ht corresponds to an increasein efficiency of the contract at that node



4





(1) PrHH PrHL

PrLL t 1




















II The Model





C t1 t ht1C h1


C















E h1 Ht H13]

LpLh1 q2Lh12

E h1 Lt L13]

st IChtH ICht

L IRhtH

IRhtL ht



IChti qiht i piht

EUht i t i 13

qjht i pjht

EUht j t i 13
















(2) qt ht H if H

L H

LPrHH PrHL

PrLL t 1

if L and ht htL

L if L and ht HthtL








(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else



























C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES






























4





(1) PrHH PrHL

PrLL t 1




















II The Model





C t1 t ht1C h1


C















E h1 Ht H13]

LpLh1 q2Lh12

E h1 Lt L13]

st IChtH ICht

L IRhtH

IRhtL ht



IChti qiht i piht

EUht i t i 13

qjht i pjht

EUht j t i 13
















(2) qt ht H if H

L H

LPrHH PrHL

PrLL t 1

if L and ht htL

L if L and ht HthtL








(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else



























C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES









































II The Model





C t1 t ht1C h1


C















E h1 Ht H13]

LpLh1 q2Lh12

E h1 Lt L13]

st IChtH ICht

L IRhtH

IRhtL ht



IChti qiht i piht

EUht i t i 13

qjht i pjht

EUht j t i 13
















(2) qt ht H if H

L H

LPrHH PrHL

PrLL t 1

if L and ht htL

L if L and ht HthtL








(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else



























C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES



































E h1 Ht H13]

LpLh1 q2Lh12

E h1 Lt L13]

st IChtH ICht

L IRhtH

IRhtL ht



IChti qiht i piht

EUht i t i 13

qjht i pjht

EUht j t i 13
















(2) qt ht H if H

L H

LPrHH PrHL

PrLL t 1

if L and ht htL

L if L and ht HthtL








(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else



























C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES




































(2) qt ht H if H

L H

LPrHH PrHL

PrLL t 1

if L and ht htL

L if L and ht HthtL








(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else



























C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES






























(3) Xt Xt Xt 1

1 if Xt 1 1 and t L

0 else



























C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES












































C Discussion
















IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES


































IV Property Rights




















(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES










































(H)and IRht




































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES




















































(4)H

Lt j1

PrHL

PrLLj1




VII Conclusion








APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES
































APPENDIX




(H)IRht




(L ) and ICht(H) as


R) V(PII)










(A1) UH ht 1 PrHL

PrHH UH ht 1











We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES





































We now prove


R) V(PI)

PROOFSince V(PI






ICht(H) and IRht


pH ht pL ht

qH ht qL ht L

PrHLUH ht

qH ht qL ht

PrHH PrHLUH ht




UH ht H j 0

jPrHH

PrLHjqeL





pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES






























pH ht pL ht qH ht

qL ht L PrHLUH ht



(H)and IRht

(L ) we have

qH ht H pH ht

PrHHUH ht H PrHH

PrLLUH ht L 0



R sinceV(P I






UH h1 q1Lh1 H L

q2Lh2L H Lq3Lh3

L

0 0


(B2) UH h1 j 0

jPrHH

PrHL jqj 1Lhj 1L



B3 Eh1 i HL

i

qi i qi

2

2 iWH i

WL i H

j 0

jPrHH

PrHL jqj 1Lhj 1L



Case 1 ht ht1 HthtL


Case 2 ht htL


L) implies that

(B4) qt LhtL L

H

LPrHH PrHL

PrLL t 1







(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES






























(C5) 1

1 E 1

1w

1 E 1

1w 1 L

1 E 1


t lim31

1

E t

1w 1 L E

t








UH t Xt 1

j 0

jPrHH PrHL j

L XL Xt j1H

L

1 PrHL

PrLLt j1





(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES





























(F6) WH Pht UH L ht





(G7) E htht13

PrHLWH qH RqL13

PrLLWL qL



(G9) LP 1 LWH qH

WL qL] LRqL




(G10) LA WH qH

WL qL t 1RqL






REFERENCES





























REFERENCES





























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