Long-Term Contracting with Markovian Consumers

By MARCO BATTAGLINI*

To study how a firm can capitalize on a long-term customer relationship, we

characterize the optimal contract between a monopolist and a consumer whose
preferences follow a Markov process. The optimal contract is nonstationary and has
infinite 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 the
types and along any history, though convergence is history-dependent. In contrast,
as with constant types, the optimal contract can be renegotiation-proof, even with
highly persistent types. These properties provide insights into the optimal ownership
structure of the production technology. (JEL D23, D42, D82)

Advances in information processing and new eliminate the benefits for the seller. The existing
management strategies have made long-term, literature has studied this problem, focusing on
nonanonymous relations between buyers and those cases in which the consumer’s type is
sellers feasible in an increasing number of mar- constant over time.1 Here, it is well known that
kets. Many retailers can now store large data- the seller finds it optimal to offer the optimal
bases on consumers’ choices and utilize them static contract period after period. In a sense, the
for pricing decisions at a very low cost. In part seller commits not to use the information gath-
because of these new technologies, recent man- ered from the consumer’s choices.
agerial schools have stressed the importance of A model of long-term contracting that as-
capitalizing on long-term relations with custom- sumes constant types, however, clearly misses
ers (see, e.g., Louis V. Gerstner, Jr., 2002, and an important dimension of the problem. Con-
Jack Welch, 2001). When a long-term relation- sider the case of a monopolist selling to an
ship is nonanonymous and types are persistent, entrepreneur whose type depends on the num-
the seller can mitigate the problem of asymmet- ber of customers waiting for service. As is well
ric information by using consumers’ choices to known, under standard assumptions on the ar-
forecast future behavior. As a result, however, rival rate of customers, the type of this entre-
buyers are more reluctant to reveal private in- preneur follows a Markov process (see, e.g.,
formation that affects their consumption deci- Samuel Karlin and Howard M. Taylor, 1975).
sions: their strategic reaction may limit or even Or, to give another example, consider the case
of a company selling cellular telephones. These
contracts often last for years and it would not be
* Department of Economics, Princeton University, reasonable to assume that the telephone com-
001 Fisher Hall, Princeton, NJ 08544 (e-mail: pany, or the customer, does not take into ac-
mbattagl@princeton.edu). I gratefully acknowledge finan- count the likely, but uncertain, evolution of
cial support from the National Science Foundation (Grant
No. SES-0418150) and the hospitality of the Economics preferences (see, e.g., Eugenio J. Miravete,
Department at the Massachusetts Institute of Technology 2003, for evidence). In all these situations, the
for the academic year 2002–2003. I am grateful for helpful assumption that the consumer’s type is constant
comments to Pierpaolo Battigalli, Douglas Bernheim, Ste- is clearly not realistic. Even if types are very
phen Coate, Eddy Dekel, Avinash Dixit, Glenn Ellison, Jeff
Ely, Bengt Holmström, John Kennan, Alessandro Lizzeri,
persistent, it is reasonable to assume that they
Rohini Pande, Nicola Persico, William Rogerson, Ariel may vary over time and follow a stochastic
Rubinstein, Jean Tirole, Asher Wolinsky, seminar partici- process.
pants at Bocconi University, Cornell University, Harvard In this paper, we characterize the optimal
University, MIT, Northwestern University, Princeton Uni- contract offered in an infinitely repeated setting
versity, the Stanford Institute of Theoretical Economics,
UCLA, University College London, and University of
Wisconsin-Madison, as well as to three referees. Marek
1
Picia provided valuable research assistance. The related literature is discussed in Section I.
637
638 THE AMERICAN ECONOMIC REVIEW JUNE 2005

by a monopolist to a consumer whose prefer- rent, it explains why “old” customers should be
ences evolve following a Markov process. In treated more favorably than “new” customers.2
this case, even if types are highly persistent, In a stochastic environment, the incentives
the contract is very different from the contract for renegotiation are also very different. As
with constant types because the seller finds it shown in the received literature (see discussion
optimal to use information acquired along the below) when types are constant over time, the
interaction in a truly dynamic way. For this monopolist benefits from the ability to commit
reason, the characterization of the optimal con- to not renegotiating the contract, because the
tract when there is heterogeneity within and optimal contract is never time-consistent. With
across periods allows a new understanding of variable types, in contrast, this is not the case:
important aspects of a dynamic principal-agent indeed, even when types are highly correlated, a
relationship that previous models could not cap- simple and easily satisfied condition guarantees
ture—particularly, with regard to the memory renegotiation-proofness. Interestingly, when
and complexity of the contract, its efficiency, types are constant the optimal renegotiation-
and its robustness to renegotiation. Perhaps sur- proof contract always requires the agent to use
prisingly, it also provides insights into the op- sophisticated mixed strategies: with correlated
timal ownership structure of the production but stochastic types, the optimal renegotiation-
technology. proof contract has an equilibrium in pure strat-
As noted, when types are constant, the con- egies and simply requires the agent to report his
tract has no memory and the inefficiency of the type.
optimal static contract is repeated period after There is an intuitive argument which explains
period. With persistent but stochastic types, the dynamics of the distortions in the optimal
even in a simple stationary environment with contract and the efficiency result mentioned ear-
one period memory (i.e., a Markov process), the lier. Assume that the agent’s type can take two
contract is nonstationary and has infinite mem- values: high and low marginal valuation for the
ory; despite this, however, it can be represented good (respectively, ␪H and ␪L ). Consider Fig-
in a very economical way by a simple state ure 1, which shows the impact on profits at time
variable. Even if types are arbitrarily highly zero of a marginal increase ⌬q in the quantity
correlated and the discount factor is arbitrarily q(ht) offered to the consumer after a history ht.
small, the seller’s optimal offer converges over On the one hand, this change increases the
time to the efficient supply schedule along all surplus that can potentially be appropriated by
possible histories. The speed of convergence, the seller if history ht is realized (which is
however, is state contingent and occurs in a represented by the “thick arrow” on the left-
particular way, which extends a well-known hand panel in Figure 1).3 However, as in a static
property of the static model. On the one hand, in model, this increase in supply increases the rent
fact, we have a generalized no distortion at the that the principal must leave to the agent to
top (GNDT) principle: after any history, if the satisfy incentive compatibility. In every period,
agent reveals himself to have the highest possi- optimal supply is determined by this marginal
ble marginal valuation for the good, supply is cost–marginal benefit trade-off, and the dy-
set efficiently from that date onward in any namic properties of the contract are driven by its
infinite history that may follow. On the other evolution. To determine optimal supply, there-
hand, and more importantly, we have a novel fore, it is important to understand the impact
vanishing distortion at the bottom (VDB) prin- over expected rents of this change at time t.
ciple: even in the history in which the agent To this goal, consider the right panel of Fig-
always reveals to have the lowest marginal val- ure 1 and assume that the rent of the high type
uation for the good, the contract converges to
the efficient menu offer. One immediate impli-
cation of this result is that in the “steady state,” 2
See Igal Hendel and Alessandro Lizzeri (2003) and
or even after a few periods, the monopolist’s Georges Dionne and Neil A. Doherty (1994) for evidence of
supply schedule may be empirically indistin- this phenomenon.
3
Because consumption is always distorted below its first
guishable from the outcome of an efficient best level, starting from the seller’s optimum, a marginal
competitive market; moreover, since higher ef- increase in supply after history ht corresponds to an increase
ficiency is associated with a higher consumer in efficiency of the contract at that node.
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 639

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.

increases by ⌬Rt at time t. At time t ⫺ 1 the the agent’s expected rent is proportional to the
expected utility of the agent in the history im- “cumulative effect” of the difference in expec-
mediately preceding ht increases as well be- tations of the types: [Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L )]t⫺1.5
cause, although the agent is a low type at t ⫺ 1, Accordingly, the marginal cost–marginal bene-
he can become a high type in the following fit ratio at time t is proportional to
period, and then benefit from the increase in

冋 册
t⫺1
rent. Part of this extra expected rent can be Pr共␪H兩␪H 兲 ⫺ Pr共␪H兩␪L 兲
extracted by the seller at t ⫺ 1, but not all, since (1) .
Pr共␪L兩␪L 兲
incentive compatibility must be satisfied at that
time as well. At time t ⫺ 1, the high type cannot The dynamics of the optimal contract depends
receive less than what he would receive if he on the evolution of this cost-benefit ratio. When
chose the option designed for the low type. types are constant, the term in parentheses is
Even if the seller extracts all the expected in- exactly equal to one, so (1) is independent of t
crease in consumption of the low type with an and the distortion is constant: this explains why,
increase in price ⌬pt ⫺ 1 ⫽ Pr(␪H兩␪L )⌬Rt at t ⫺ 1, with constant types, it is optimal to offer the
the change in rent of the high type at t ⫺ 1, static contract repeatedly. When types are pos-
⌬Rt ⫺ 1, would be equal to Pr(␪H兩␪H)⌬Rt ⫺ itively but imperfectly correlated, even if types
⌬pt ⫺ 1, that is, (Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L ))⌬Rt,4 are highly persistent, optimal supply converges
which is positive if types are positively corre- to an efficient level along all histories as t 3 ⬁
lated. If the seller tries to extract this extra rent because (1) converges to zero.
at t ⫺ 1, then, repeating the same argument, she In general, any change in the contract at a
still must provide an increase in rent to the high time t has cascade effects on the expected rents
type at time t ⫺ 2 equal to ⌬Rt ⫺ 2 ⫽ in the previous periods. These effects depend
(Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L ))⌬Rt ⫺ 1, which can be not only on the transition probabilities, but also
written as (Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L ))2⌬Rt. Pro- on the structure of the constraints that are bind-
ceeding backward, we arrive at an increase in ing at the optimum. As time passes, these cas-
the rent left to the consumer at time 1 propor- cade effects become increasingly complicated
tional to (Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L ))t ⫺ 1 (see the because the number of histories grows exponen-
dashed arrows in the right panel of Figure 1). tially. A methodological contribution of this
While the marginal impact of the change in paper is in a novel characterization of the bind-
supply on expected surplus evaluated at time ing constraints by an inductive argument which
zero is proportional to the probability of the
history ht (i.e., ␮LPr(␪L兩␪L )t ⫺ 1), the impact on
5
The expected change in the agent’s rent at time one is
␮H[Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L )]t ⫺ 1⌬Rt, where ␮H is the proba-
4
The probability that a type i in period t becomes a type bility that the agent is a high type in the first period. The
j in t ⫹ 1 is denoted Pr(␪j兩␪i). constant ␮H, however, is irrelevant for our argument.
640 THE AMERICAN ECONOMIC REVIEW JUNE 2005

allows a substantial simplification of the the optimal static menu is repeated in every
problem. period (see, e.g., Laffont and Tirole, 1993).
The particular features of the optimal contract With constant types the dynamics becomes
described above also have implications for the interesting only when other constraints are
optimal ownership structure of the monopolist’s binding, in particular when a renegotiation-
business. It is indeed interesting to ask why the proofness constraint must be satisfied. Seminal
monopolist keeps control of the production papers in this literature are Dewatripont (1989),
technology: after all, only the consumer benefits Oliver Hart and Tirole (1988), and Laffont and
directly from it and has information for its ef- Tirole (1990).7 In contrast to our findings with
ficient use. We show that the optimal contract variable types, a common result in this literature
can be interpreted as offering the high-type con- with constant types is that the ex ante optimal
sumer a call option to buy out the technology contract is never renegotiation-proof.
used by the monopolist. The sale of the tech- Kevin Roberts (1982) and Robert M.
nology, however, is state contingent and the Townsend (1982) are the first to present re-
monopolist tends to retain control more often peated principal-agent models with stochastic
than what would be socially optimal: by keep- types. In these frameworks, however, types are
ing the ownership rights, the monopolist can serially independent realizations, and therefore
control future rents of the high types and this incentives for present and future actions can
improves surplus extraction because types have easily be separated. Indeed, in this case, except
different expectations for the future. This in- for the first period, there is no asymmetric in-
sight seems relevant to understand the owner- formation between the principal and the agent
ship structure of a new technology. The initial because both share the same expectation for the
owner of a new technology generally has mo- future.8 David Baron and David Besanko
nopoly power on its use thanks to a patent and (1984) and Laffont and Tirole (1996) extend
must decide if it is more convenient to use the this research, presenting two period procure-
technology directly selling its products, or to ment models in which the type in the second
sell the patent. period is stochastic and correlated with the type
The paper is organized as follows. Section I in the first period. Because these models have
surveys the related literature. In Section II we only two periods, however, they cannot capture
describe the model. In Section III we character- such important aspects of the dynamics of the
ize the optimal contract and discuss its effi- optimal contract as its memory and complexity
ciency properties. Section IV discusses the after long histories, or its convergence to effi-
theory of property rights that follows from the ciency. Aldo Rustichini and Asher Wolinsky
characterization. Section V discusses the prop- (1995) characterize optimal pricing in a model
erties of the monetary payments in the optimal with infinite horizon and Markovian types as
contract. Section VI studies renegotiation- ours. However, in their model consumers are
proofness. Section VII presents concluding not strategic and ignore that future prices de-
comments. pend on their current actions; demand, more-
over, can assume two values, zero or one. None

I. Related Literature

As mentioned above, in dynamic models of 7

These papers study the optimal renegotiation-proof
price discrimination it is generally assumed that contract with constant types under different assumptions.
the agent’s type is constant over time.6 In this Hart and Tirole (1988) and Dewatripont (1989) present
models with many periods: the first paper assumes that
case we have a “false dynamics” in which the supply can have two values, zero or one; the second focuses
monopolist finds it optimal to commit to a con- on pure strategies and assumes some simplifications in the
tract in which past information is ignored and nature of the contractual agreement. Laffont and Tirole
(1990) solve a model in which supply can assume more than
two values, assuming two periods.
8
Because Townsend (1982) is specifically interested in
6
For excellent overviews of the literature on dynamic modelling risk sharing, he assumes that the principal is less
contracting, see Patrick Bolton and Mathias Dewatripont risk averse than the agent. In this case, even with i.i.d. types,
(2005) and Jean-Jacques Laffont and Jean Tirole (1993). the contract depends on the cumulated wealth of the agent.
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 641

of these papers with variable types considers We assume that the relationship between the
renegotiation-proofness.9 buyer and the seller is infinitely repeated and the
discount factor is ␦ 僆 (0, 1). In period 1 the seller
II. The Model offers a supply contract to the buyer. The buyer
can reject the offer or accept it; in the latter case
We consider a model with two parties, a the buyer can walk away from the relationship
buyer and a seller. The buyer repeatedly buys a at any time t ⱖ 1 if the expected continuation
nondurable good from the seller. He enjoys a utility offered by the contract falls below the
per-period utility ␪tq ⫺ p for q units of the good reservation value uគ ⫽ 0. In line with the stan-
bought at a price p. In every period, the seller dard model of price discrimination, the monop-
produces the good with a cost function c(q) ⫽ olist commits to the contract that is offered: in
1⁄2 q2. The marginal benefit ␪ evolves over time Section VI we relax this assumption, allowing
t
according to a Markov process. To focus on the the parties to renegotiate the contract.
dynamics of the contract, we consider the sim- It is easy to show that in the environment that
plest case in which each period the agent can we will study a form of the revelation principle
assume one of two types, ␪L, ␪H with ⌬␪ ⫽ is valid and allows us to consider without loss of
␪H ⫺ ␪L ⬎ 0. The probability that state l is generality only contracts that in each period t
reached if the agent is in state k is denoted depend on the revealed type at time t and on the
Pr(␪l兩␪k) 僆 (0, 1); the distribution of types con- history of previous type revelations. In this case
ditional on being a high (low) type is denoted the contract 具p, q典 can be written as 具p, q典 ⫽
␣H ⫽ (Pr(␪H兩␪H), Pr(␪L兩␪H)) (␣L ⫽ (Pr(␪H兩␪L ), (pt(␪ˆ 兩ht), qt(␪ˆ 兩ht))⬁
t⫽1, where ht and ␪ are, respec-
ˆ
Pr(␪L兩␪L ))). We assume that types are positively tively, the public history and the type revealed
correlated, i.e., Pr(␪H兩␪H) ⱖ Pr(␪H兩␪L ). How- at time t, and qt⵺ and pt⵺ are the quantities and
ever, we do not make assumptions on the degree prices conditional on the declaration and the
of correlation: indeed, an environment with history.11 In general, ht can be defined recur-
constant types can be seen as a limiting case of sively as ht :⫽ {␪ˆ t ⫺ 1, ht ⫺ 1}, h1 :⫽ A where
our model in which the probability that a type ␪ˆ t ⫺ 1 is the type revealed in period t ⫺ 1. The set
does not change converges to one. In each pe- of possible histories at time t is denoted Ht; the
riod the consumer observes the realization of his set of histories at time j following a history ht
own type; the seller, in contrast, cannot see it. (t ⱕ j) is denoted Hj(ht). A strategy for a seller
At date 0 the seller has a prior ␮ ⫽ (␮H, ␮L ) on consists of offering a direct mechanism 具p, q典 as
the agent’s type.10 For future reference, note described above. The strategy of a consumer is,
that the efficient level of output is equal to at least potentially, contingent on a richer his-
qe(␪t) ⫽ ␪t in all periods and after any history of tory hC t :⫽ {␪t ⫺ 1, ␪t, ht⫺1}, h1 :⫽ ␪1 because
ˆ C C

realizations of the types. the agent always knows his own type. For a
given contract, a strategy for the consumer,
then, is simply a function that maps a history hC t
into a revealed type: hC t 哫 b(ht ).
C
9
Dynamic environments with adverse selection and sto-
chastic types have recently been used to study models of In the study of static models it is often as-
leasing, insurance, and other applications. See Pascal sumed that all types are served, i.e., each type is
Courty and Hao Li (2000), and Hendel and Lizzeri (1999, offered a positive quantity, which is guaranteed
2003). John Kennan (2001) has studied a model with vari-
able types, but in which only short-term contracts with one by the assumption that ⌬␪ is not too large. The
period length can be offered. Battaglini and Stephen Coate same condition that guarantees this property in
(2003) apply the techniques of the present paper to charac- the static model also guarantees it in our dy-
terize the Pareto optimal frontier of taxation with correlated namic model; therefore, to simplify notation,
types.
10 we assume that this condition is verified in our
The fact that the agent’s type follows a Markov pro-
cess can be modelled in many natural ways. The agent may model.12 This assumption can easily be relaxed,
be a firm whose type depends on its list of customers
waiting for services, which according to the “inventory
model” follows a Markov process (see Karlin and Taylor, 11
Note, therefore, that p(␪兩h) is not the per-unit price
1975, §2.2.d). Or the agent’s type may depend on his paid after history {␪, h}, but the total monetary transfer at
investment opportunities: if these follow a branching pro- that history.
12
cess, then they are described by a Markov process (see The condition that guarantees that all types are served
Karlin and Taylor, 1975, §2.2.f). is ⌬␪ ⱕ (␮L/␮H)␪L. As we will see, the distortion introduced
642 THE AMERICAN ECONOMIC REVIEW JUNE 2005

but this would complicate notation with no gain 共IC h t 共 ␪ i 兲兲 q共 ␪ i 兩h t 兲 ␪ i ⫺ p共 ␪ i 兩h t 兲

in insight.
In the first part of the analysis we focus on the ⫹ ␦ E关U共 ␪ 兩h t , ␪ i 兲兩 ␪ t ⫽ ␪ i 兴
case with unilateral commitment in which the
monopolist can commit, but the consumer can ⱖ q共 ␪ j 兩h t 兲 ␪ i ⫺ p共 ␪ j 兩h t 兲
leave the relationship anytime. This assumption
seems the most appropriate in many markets.13 ⫹ ␦ E关U共 ␪ 兩h t , ␪ j 兲兩 ␪ t ⫽ ␪ i 兴
On the other hand, there are many situations in
which renegotiation is an important component @i ⫽ j, i, j 僆 H, L, where U(␪兩ht, ␪i) is the value
of the problem: in Section VI, we show that function of a type ␪ after a history {ht, ␪i}.
under general conditions the optimal contract is These constraints guarantee that type i does not
renegotiation-proof and therefore it can be ap- want to imitate type j after any history ht. And
plied to these environments too. the individual rationality constraint IRht(␪i) sim-
ply requires that the agent wants to participate
III. The Optimal Contract in the relationship each period: U(␪兩ht, ␪i) ⱖ 0
for any i and ht.
The monopolist’s optimal choice of contract The classic approach to characterize the so-
maximizes profits under the constraint that after lution to this problem in a static environment is
any history the consumer receives (at least) his in two steps. First, a simplified program, in
reservation utility and, also after any history, which the participation constraints of the high
there is no incentive to report a false type: type and the incentive compatibility constraints
of the low type are ignored, is considered (the
共PI 兲 max具p,q典␮H 关p共␪H兩h1 兲 ⫺ q2共␪H兩h1 兲/2 “relaxed problem”). Then it is shown that there
is no loss of generality in restricting attention to
⫹␦E 关⌸共␪兩h1 , ␪H 兲兩␪t ⫽ ␪H 兴] this case. In a static model, the remaining con-
straints of the relaxed problem are necessarily
⫹␮L 关p共␪L兩h1 兲 ⫺ q2共␪L兩h1 兲/2 binding at the optimal solution: this simplifies
the analysis because it allows us to substitute
⫹␦E 关⌸共␪兩h1 , ␪L兲兩␪t ⫽ ␪L兴] them directly in the objective function.
It is easy, however, to see that in a dynamic
s.t. ICht 共␪H兲, ICht 共␪L兲, IRht 共␪H兲, model this cannot be true. Given an optimal
contract, we can always add a “borrowing”
IRht 共␪L 兲 ᭙ ht agreement in which the monopolist receives a
payment at time t and pays it back in the fol-
where E[⌸(␪兩h1, ␪i)兩␪t ⫽ ␪i] i ⫽ H, L is the lowing periods. If the net present value of this
expected value function of the monopolist after transaction is zero, then neither the monopo-
history {h1, ␪i}. The incentive constraints list’s profit changes, nor any constraint, would
ICht(␪i) for i ⫽ H, L are described by: be violated, so the contract would remain opti-
mal: but the individual rationality constraints
need not remain binding after some histories.
More importantly, the incentive compatibility
by the monopolist is declining over time in all histories and,
in the first period, it is equal to the distortion of the static
constraints may also not be binding. In order to
model. Therefore if the monopolist serves all customers in provide incentives to the high type to reveal his
the static model, then she serves all customers after all private information, the monopolist may find it
histories in our dynamic model too. useful to use future payoffs instead of present
13
Discussing the life insurance market, Hendel and Liz- payoffs to screen the agent’s types. If this were
zeri observe that the term value contracts in the insurance
market which account for 37 percent of ordinary life insur- the case, there would be a history after which
ance, “... are unilateral: the insurance companies must re- the contract leaves to the high type more surplus
spect the terms of the contract for the duration, but the buyer than what a binding incentive compatibility
can look for better deals at any time. [...] These features fit constraint would imply.
a model of unilateral commitment.” (Hendel and Lizzeri,
2003, p. 302). Moreover, there is evidence that firms seem The following result generalizes the “binding
aware that the possibility to commit is important to win constraints” result of the static model, showing
exclusive long-term contracts. that in a dynamic setting, although constraints
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 643

need not bind in every optimal scheme, there is increase in payments at time t. If the value of the
no loss of generality in assuming that con- high type’s outside option goes down, then his
straints in the relaxed problem are satisfied as equilibrium rent goes down as well. Expected
equalities. Let us define PII as the program in profits, therefore, would be larger after the change
which expected profits are maximized, assum- in the contract and all constraints would be re-
ing that the incentive constraints of the high spected: but this is not possible if the contract is
type and the participation constraints of the low optimal, so we have a contradiction. After a his-
type hold as equalities after any history, and no tory h2 ⫽ ␪H we proceed in a similar way: in this
other constraint is assumed. We say that a sup- case profits remain constant after the change in
ply schedule q*t (␪兩ht) is a solution of a given prices, so the constraint needs not necessarily be
program if there exists a payment schedule binding at the optimum, but it can be reduced to an
p*t (␪兩ht) such that the menu {q*t (␪兩ht), p*t (␪兩ht)} equality without loss. The argument for the par-
is a solution of the program. ticipation constraints is analogous.
It is important to point out that Lemma 1 does
LEMMA 1: The supply schedule q*t (␪兩ht) not claim that any solution 具p, q典 of a relaxed
solves PI if and only if it solves PII. problem in which the incentive constraint of the
low type and the participation constraint of the
The result that the constraints may be assumed high type are ignored is a solution of PI. In PII
to hold as equalities without loss may be intu- we assume that the constraints are satisfied as
itively explained in a two-period version of the equalities, so it is not just a relaxed version of
model (the complete argument, presented in the PI. Indeed, such a claim would not be true:
Appendix, is by induction on t). Assume that at some solutions of the relaxed problem would
time t ⫽ 2 the incentive compatibility constraint of imply future rents for the high type that would
the high type is not binding after a history h2 ⫽ ␪L. violate the incentive compatibility constraint of
Consider this change in the contract: reduce the the low type after some histories. However, if
extra rent at t ⫽ 2 and reduce the price paid by the 具p, q典 solves the relaxed problem, then there
low type at t ⫽ 1 so that his participation con- exists a p⬘ such that 具p⬘, q典 solves PI; and if 具p,
straint is satisfied as an equality after the change. q典 solves PI then there exists a p⬙ such that 具p⬙,
The rent of the high type at time 1 depends on his q典 solves PII and, because of this, solves the
outside option (the utility obtained by reporting relaxed problem as well.
himself untruthfully to be a low type), so it is We can now focus the simpler problem with
affected by both these changes. Even if the net equality constraints PII; from the first-order
change in payments has a neutral effect on the low conditions, we obtain:
type’s expected utility, however, it will reduce the
outside option of the high type: because the high PROPOSITION 1: At any time t, the optimal
type is more optimistic about the future realization contract is characterized by the supply
of his type, the reduction in future rents if he function:
reports his type untruthfully will be larger than the

冦
␪H if ␪ ⫽ ␪ H

冋 册
t⫺1
␮ H Pr共␪H兩␪H 兲 ⫺ Pr共␪H兩␪L 兲
(2) q *t 共 ␪ 兩h t 兲 ⫽ ␪ L ⫺ ⌬ ␪ if ␪ ⫽ ␪L and ht ⫽ hLt
␮L Pr共␪L兩␪L 兲
␪L if ␪ ⫽ ␪L and ht 僆 Ht⶿hLt

where hLt :⫽ {␪L, ␪L, ... ␪L}, the history along induce different menus in the optimal mecha-
which the agent always reports himself to be a nism.14 This fact, however, does not imply that
low type in the first t ⫺ 1 periods.
14
From (2) we can see that the optimal contract Consider two histories that differ only in the first
realization of types, the first being high, the second being
is nonstationary and has unbounded memory: low, and which have low realizations in any period follow-
for any T ⬎ 0, we can always find two histories ing date two. If these histories are longer than a positive
that are identical for the last T periods but that parameter T, say they have T ⫹ 1 length, then they coincide
644 THE AMERICAN ECONOMIC REVIEW JUNE 2005

the contract has a complicated structure. From the distortion, therefore, does not depend on
Proposition 1 we can see that the only thing that consumption smoothing, but it is a necessary
matters for the contract is whether we are on the feature of dynamic price discrimination. In a
lower branch or not. Since this depends only on dynamic environment, the principal has more
the current type, and if in the previous periods freedom to redistribute distortions over time
the agent reported himself to be a high type, the and states in order to screen the agent’s types.
state can be described by a simple 0-1 variable Propositions 1 and 2 characterize the optimal
which can be defined recursively: way to redistribute the distortion, proving that
the principal finds it optimal to introduce dis-
(3) X t ⫽ X共 ␪ t , X t ⫺ 1 兲 tortions even in the far future, potentially for an
unbounded number of periods. This is perhaps
⫽ 0 再
1 if Xt ⫺ 1 ⫽ 1 and ␪t ⫽ ␪L
else
surprising since the agent’s taste follows a
Markov process and therefore the relevant eco-
nomic environment has a memory of only one
for t ⱖ 1, and X0 ⫽ 1. This variable starts with period.15
value one and remains one if the agent persists We now turn to the particular pattern in
in reporting a low type; once the agent has which distortions are introduced. In Sections III
reported himself to be a high type, the state A and III B we discuss the dynamics of distor-
switches to zero and remains constant forever. tions and the asymptotic properties of the con-
Let us define ⌳ ⫽ [Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L )/ tract as ␦ 3 1. In Section III C, we discuss the
Pr(␪L兩␪L )]; we have: key assumptions of the model.

PROPOSITION 2: The optimal solution is a A. Efficiency: The GNDT and VDB Principles
function of time and the 0-1 state variable de-
scribed by (3): q*t (␪t, Xt ⫺ 1) ⫽ ␪t ⫺ ⌬␪ (␮H/ In order to interpret (2), it is useful to com-
␮L ) X(␪t, Xt ⫺ 1)⌳t ⫺ 1. pare it with the benchmark with constant types.
In this case, there are only two possible histo-
The length of the memory of the optimal ries: either the agent is always a high type, in
contract is a central issue in the literature on which case the contract is efficient; or the agent
dynamic moral hazard (see William P. Roger- is always a low type, and the contract is dis-
son, 1985), but it has not been studied in ad- torted below the efficient level in all periods by
verse selection models, because when the a constant ⌬␪ (␮H/␮L ).
agent’s type is perfectly constant we know that When types follow a Markov process, the
the contract is also constant over time and in- contract instantly becomes efficient as soon as
dependent of past histories, so it has no mem- the agent reports himself to be a high type: but
ory. In the moral hazard literature, the memory now efficiency “invades” the histories in which
of the contract is a direct consequence of the the agent subsequently reports himself to be a
agent’s risk aversion. With risk aversion, it is low type. This is the generalized no distortion
optimal not only to smooth consumption over at the top (GNDT) principle. Its intuition is the
states of the world, as in the static moral hazard following. Distortions are introduced only to
framework, but also to smooth consumption extract more surplus from higher types; there-
over periods. To this end, the contract must fore there is no reason to distort the quantity
keep track of the past realizations of the agent’s offered to the highest type. After any history ht
income. In the model presented above, how- the rent that must be paid to a high type to
ever, the agent is risk neutral; the persistence of reveal himself is independent of the quantities
that 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 monopolist
offers an efficient contract in the first history, i.e., regardless
15
of the realizations in the following periods, the quantity As we discuss in greater detail in Section IV, the fact
offered is efficient in any period following the first. Not so that the seller wants to distort supply for a unlimited and
for the second history. Therefore, even if there is no intrin- state-contingent number of periods has important implica-
sic economic reason in the environment to offer different tions for the allocation of the property rights of the produc-
menus at date T ⫹ 1, the contracts are different. tion technology.
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 645

he receives the same utility as if he falsely stant over time. Indeed, the case with constant
reported himself to be a low type; therefore, types is not asymptotically efficient because
only the quantities that follow such a history {1 ⫺ [Pr(␪L兩␪H)/Pr(␪L兩␪L )]}t ⫺ 1 is exactly one,
affect his rents. This implies that the monopolist and therefore, independent from t.
is residual claimant on the surplus generated on Clearly, as the persistence of types converges
histories after a high type report, and therefore to one, we have that [Pr(␪L兩␪H)/Pr(␪L兩␪L )] 3 0.
the quantities that follow such histories are cho- Not surprisingly, this implies that ceteribus pa-
sen efficiently. In our dynamic framework, this ribus the contract converges in every period to
simple principle has strong implications be- the optimal static contract as Pr(␪H兩␪H) and
cause it forces the contract to be efficient not Pr(␪L兩␪L ) converge to one. There are, however,
only in the first period in which the agent truth- two important observations. First, convergence
fully reveals himself to be a high type, but also to efficiency appears to be relatively fast even if
in all the following periods. types are highly correlated.16 Second, as we
A distortion persists on the lowest branch discuss below, the results with fixed and sto-
of the history tree (i.e., when the agent al- chastic types are very different when ␦ 3 1,
ways declares to be a low type). By a sim- regardless of the level of persistence of the
ple manipulation of the formula in Proposition types.
2, the distortion can be written as ␪L ⫺ q*t(␪L兩ht)
⫽ ⌬ ␪ ( ␮ H / ␮ L ) X( ␪ t , X t ⫺ 1 ){1 ⫺ [Pr( ␪ L 兩 ␪ H )/ B. Distribution of Surplus with Large
Pr( ␪ L 兩 ␪ L )]} t ⫺ 1 , since the efficient level of Discount Factors
output with a low type is ␪L. Given that types
are positively correlated, we have [Pr(␪L兩␪H)/ All the results presented above are valid for
Pr(␪L兩␪L )] 僆 (0, 1) and it follows that any ␦ 僆 (0, 1); if we assume that ␦ 3 1, even
limt3⬁q*t(␪兩ht) ⫽ ␪ ⫽ qe(␪ ), which proves: stronger results emerge. In this case we can
easily bound the inefficiency and determine the
PROPOSITION 3: For any discount factor distribution of surplus between the seller and
␦ 僆 (0, 1), the optimal contract converges over the buyer.17
time to an efficient contract along any possible With constant types, the average utility of the
history. consumer is bounded away from zero and inde-
pendent of ␦; the average payoff of the monop-
This is the vanishing distortion at the bottom olist is equal to the profit that would be achieved
(VDB) principle. The monopolist introduces a in a static model and independent of ␦ as well.
distortion along the “lowest” history because However, even an arbitrarily small reduction in
this minimizes the cost of screening the agent’s the persistence of the types has a very high
types: however, even this distortion converges impact on surplus and payoffs when the dis-
to zero as t 3 ⬁. count factor is high.
The optimal distortion simply equalizes the
marginal cost of a decrease in supply (in terms
of reduced surplus generated in the relationship)
16
and its marginal benefit (in terms of reduced Assume, for example, that types are ex ante equally
rent to be paid to the high type). With constant likely, and the types are very much correlated (for example,
the type is persistent 80 percent of the time). Then the
types, after any history ht in which the agent expected inefficiency of the contract after ten periods will
declares to be a low type, the marginal benefit be 0.03779⌬␪; the expected inefficiency after 50 periods
of increasing surplus with a higher q(ht) is in- will be 5.0517 ⫻ 10⫺12⌬␪.
17
dependent of the length of the history: it is In this comparative statics exercise we change the
proportional only to ␮L, because once the type discount factor, keeping the transition probabilities con-
stant. Another interesting exercise would be to modify the
is low in the first period then it is low forever. level of persistence of the types, or the frequency of their
Similarly, the marginal cost of an increase in changes. Increasing the frequency of changes would rein-
q(ht) is proportional to ␮H, the probability that force the effects of an increase in ␦. But when we simulta-
the high type receives the increase in the rent. neously increase types persistence and the discount factor,
the result depends on which of the two (persistence and ␦)
Since, therefore, the marginal cost/marginal converges faster to one. The case considered in the paper, in
benefit ratio is time-independent, it is not sur- which only ␦ 3 1, corresponds to the case in which the
prising that the optimal distortion is also con- discount factor converges more quickly than persistence.
646 THE AMERICAN ECONOMIC REVIEW JUNE 2005

PROPOSITION 4: When types are imperfectly that types are positively correlated.19 When this
correlated, even if correlation is not positive, is the case, a “high” type has not only a higher
then as ␦ 3 1 the average profit of the monop- marginal valuation for the good today, but also
olist converges to the first-best level of surplus a higher expected valuation for a contract in the
and the average utility of the consumer con- future. Without this assumption, a type would
verges to zero, regardless of the renegotiation- be “high” or “low” depending on which of these
proofness constraint. two components of utility prevails. Along with
the rest of the literature on dynamic contract-
When the discount factor is high, it does not ing,20 we also assume that at any point in time
matter what happens in the first T periods, for T the type ␪t can assume one of two values. When
finite. However, because we are working with a there are n possible values, the conditional dis-
Markov process, in the long run the distribution tribution of future realization of ␪t is a n ⫺ 1
of types converges to a stationary distribution dimensional vector, so the characteristics of
which is independent from the initial value. each agent are n ⫺ 1 dimensional. In this case,
This implies that at time one the ability of the besides the problem of dynamic screening,
consumer to predict his own type realizations in we would have an additional problem of multi-
the far future is almost as good as the seller’s dimensional screening. As is well known, in this
ability. For this reason, when ␦ is high the case types are not “naturally” ordered, and the
monopolist can separate the agents paying only set of constraints that are binding can be more
a minuscule rent to the higher type.18 complicated. The environment studied above
has the advantage of separating the study of
C. Discussion the dynamics of the contract from the study
of the multidimensionality of the types, which
Before presenting further results, we now dis- is a conceptually distinct problem, and there-
cuss the assumptions of the model, emphasizing fore provides a better understanding of the
the issues that are still open for future research. dynamics.21
In particular, we focus on the stochastic process, A related issue regards the transition proba-
the utility function, cash constraints, and the bilities in the stochastic process. Clearly, many
time horizon. different assumptions can be made regarding
As noted, any change of the contract at time these probabilities. In this paper, we have con-
t has a “cascade” effect on expected utilities in sidered the case in which the transition proba-
the previous histories. These effects depend bilities do not change over time. This, however,
both on the structure of the constraints that are is not essential for the characterization: indeed
binding at the optimum, and on the transition even if the degree of positive correlation
probabilities, which determine the conditional changes over time (but remains positive), we
expectation of the consumer at each history would be able to perform the same simplifica-
node. This is the reason the properties of the tion of the incentive constraints as in
stochastic process are important in the charac- Lemma 1.22 Another assumption of the model is
terization. A key assumption of the model is that the transition probabilities between types

19
This assumption is used in the characterization of the
optimal contract, but not in Proposition 4.
20
See, e.g., Hart and Tirole (1988), Laffont and Tirole
18
It is worthwhile to point out the differences between (1990), and Rustichini and Wolinsky (1995).
21
this result and the results in Proposition 1 and 2 because the At the cost of higher complexities, the model can be
logic of their proofs is different. The proof of Proposition 4 extended to multidimensional types. Indeed, even in a mul-
does not require the assumption that types are positively tidimensional environment, types can be “endogenously”
correlated. For this reason, Proposition 4 is stronger than the ordered to simplify the set of incentive constraints (see, for
result that would have followed from taking the limit in the example, Jean-Charles Rochet, 1987).
formula of Proposition 1 as ␦ 3 1. However, while Prop- 22
In this case the optimal contract would not depend on
osition 1 characterizes the optimal contract for any ␦, Prop- the likelihood ratio ⌳ raised to the t as in Proposition 2, but
osition 4 is only a limit result. Even in the limit case in on the multiplication of the changing likelihood ratios along
which ␦ 3 1, Proposition 4 shows that the contract con- the lowest history. For this extension, we would also need to
verges to an efficient contract in probability, but it is silent continue assuming that a high type remains more likely to
on the behavior of the contract in any single history. be high in the future.
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 647

are all positive, although they may be small: this bound which is generally very small. For exam-
precludes a case in which there is a type i which ple, the upper bound on monetary transfers
will never become some other type j.23 An depends (among other variables) on the persis-
interesting extension of the model could be to tence of agents’ types: as persistence converges
consider a process with more than two types and to one, the per-period payments converge to the
a form of long-term heterogeneity in which same payments as in the static model. If we
different transition probabilities correspond to assume that cash constraints are satisfied in the
each initial type. A systematic analysis of the standard static version of the model, then when
properties of the stochastic process and the ex- types are sufficiently persistent (as perhaps rea-
tension to the case of dynamic screening with sonable to assume when consumption is fre-
multidimensional types is left for future quent), cash constraints would not be binding in
research. our dynamic model either.
Regarding the utility and the cost function, Finally, we turn to the time horizon. Besides
the results can easily be extended to the case in a direct theoretical interest, the analysis of a
which the cost function is a generic convex stationary model with infinite periods is useful
function and utility is a generic function u(␪t, q), for two reasons. First, with this assumption we
provided that the usual single crossing condi- can study long-term behavior and convergence
tion is assumed.24 A relevant assumption, how- of the contract, which would be impossible in
ever, is that the utility is quasilinear (as a two-period model. It is also instrumental,
generally assumed in the literature on nonlinear however, in the study of price dynamics. For
pricing). When the utility function is not quasi- example, we will show that the transfer price of
linear, we have an additional issue of consump- the monopolist’s technology is declining over
tion smoothing over time. In this case, too, the time. Since the model is stationary, the true
analysis of the quasilinear case allows us to value of the technology is constant and identical
separate the dynamic screening problem from in any period. Therefore, this decline in price
the conceptually different problem of consump- arises purely for strategic reasons: in a nonsta-
tion shooting.25 tionary model with finite periods we would not
As standard in the literature, we do not im- be able to separate the strategic effect from the
pose cash constraints on the consumer’s natural decline in value due to the shorter hori-
choice.26 At the cost of additional complica- zon. It is, however, easy to show that our char-
tions, it would be a simple exercise to incorpo- acterization would be valid even in a model
rate these constraints in our model. Under with T periods.
plausible assumptions, however, these con-
straints would be irrelevant for the analysis. In IV. Property Rights
the model, monetary transfers can always be
bounded above in all periods by a finite upper Before presenting results on the monetary
transfers, it is useful to discuss property rights,
since their allocation typically (although not
23
This environment, however, can be approximated necessarily) influences the flow of monetary
since transition probabilities can be arbitrarily small.
24
transfers. Up to this point, we have assumed that
This requires the cross derivative u␪q to be positive. the monopolist has the right to decide the quan-
When the utility and cost functions are generic functions,
however, the first-order conditions do not necessarily yield tity supplied in every period. Instead of selling
closed form solutions. All the results, however, continue to output on a period-by-period basis, however,
hold in this more general environment (see Battaglini and the monopolist may decide to sell the property
Coate, 2003, for details). rights of her exclusive technology to the con-
25
The results, however, are robust to changes in the sumer. Only the consumer benefits directly
degree of risk aversion. In Battaglini and Coate (2003) we
show that when risk aversion is below a critical value, the from the technology and has information for its
characterization with risk aversion is the same as the char- efficient use. It is therefore natural to expect that
acterization without risk aversion and a small change in risk the property rights are ultimately acquired by
aversion would imply only a small change in the contract. the agent who has a superior valuation of its
26
Cash constraints limit the maximum amount of per-
period monetary transfers between the principal and the future use. The decision to transfer property
agent. Clearly, such constraints can be incorporated in both rights, however, depends on the history of the
a dynamic model and a static model. agent’s types:
648 THE AMERICAN ECONOMIC REVIEW JUNE 2005

PROPOSITION 5: Without loss of generality, equal to Rown ⫽ 1⁄2 ⌬␪ (␪H ⫹ ␪L ). This rent is
the optimal contract offers a call option to buy higher than the minimal rent that would guar-
out the firm to the agent as soon as he reveals antee truthful revelation: the incentive compat-
himself to be a high type. However, the monop- ibility constraint requires only a rent equal to
olist never finds it optimal to sell the firm to an RIC ⫽ ⌬␪␪L ⬍ Rown. Imagine now that the
agent who has always revealed himself to be a monopolist, after the agent reveals himself to be
low type. a low type, keeps the ownership in order to
reduce the rent of the high type at t ⫽ 2, instead
The first part of this result should not be of selling the firm. Assume, in particular, that
surprising. After the agent reveals himself to be instead of selling the good at cost in the second
a high type, there is no residual asymmetric period, she sells to the high type ␪H units at
information. At this stage, and before the real- price 1⁄2 ␪H
2
⫹ ␧, i.e., she reduces the extra rent
ization of the type in the following period, we of the high type by ␧ in case in period 1 the
should expect no reason for the monopolist to agent declares to be a low type. For ␧ small, the
keep the ownership of the technology.27 The contract remains incentive compatible in the
interesting observation, however, is in the sec- second period. In order to satisfy the constraints
ond part of the proposition: after a history in at t ⫽ 1, suppose that the monopolist reduces
which the agent has never revealed he is a high the price paid by the low type by ␦ Pr(␪H兩␪L )␧
type, the monopolist finds it strictly suboptimal dollars. The low type’s incentives in period 1
to sell the firm and prefers to introduce a dis- are unchanged: if he reports himself to be a low
tortion in the value of the firm, not only in the type, he receives ␦ Pr(␪H兩␪L )␧ dollars more in
period in which the type is revealed, but also in t ⫽ 1 and he expects to receive ␦ Pr(␪H兩␪L )␧ less
the subsequent periods.28 Indeed, as we dis- at t ⫽ 2; moreover, the contract does not change
cussed in Section III A, the distortion is intro- if the agent chooses to report himself to be a
duced to extract surplus from the high type. high type. Consider now the impact of this
This suggests that it is natural to observe a change on the incentive compatibility constraint
distortion in the period in which the agent re- of the high type at t ⫽ 1. If the high type
veals his type. But this does not explain why the deviates and reports himself to be a low type, he
monopolist still wants to introduce a distortion receives ␦ Pr(␪H兩␪L )␧ more, the same as the low
in the following periods: given that the agent type since this is paid “in cash” at t ⫽ 1 with a
has revealed his low type, there is no asymmet- reduced price. However, the expected loss for
ric information anymore in this case either. This the high type is ␦ Pr(␪H兩␪H)␧ because he is more
characteristic of the optimal contract depends optimistic than the low type about the future.
on the dynamic nature of the incentive con- Since ␦[Pr(␪H兩␪L ) ⫺ Pr(␪H兩␪H)]␧ is negative,
straint and it is instructive to see why it is true. this implies that the outside option of the high
Consider a simple two-period example. As- type, i.e., the utility of reporting untruthfully,
sume that after the declaration in period 1 the has a lower value and the monopolist can induce
monopolist sells the firm to the agent irrespec- truthful revelation by leaving a lower rent to the
tive of the type. In the second period the agent high type at t ⫽ 1. The monopolist, therefore,
would receive all the surplus, i.e., 1⁄2 ␪H
2
if he is prefers to keep strict ownership of the firm:
a high type and ⁄2 ␪L if he is a low type. This
1 2
ownership enables control of the rent of the
implies that the high type receives a rent, i.e., an agent in the second period, and this control is
extra payoff with respect to the lower type, important to extract surplus in the sale of the
technology to the high type in the first period.
27
The characterization of the optimal contract in
Note that this result is different from the classical (2) goes beyond this observation. In our infinite
results by Jeremy Bulow (1982) concerning the trade-off
between the sale and the rental of a durable good. In this and stationary environment, in fact, the monop-
literature, in fact, if a durable good is sold, then the quantity olist finds it optimal to reduce the efficiency of
remains constant in the following periods; in our frame- the firm for potentially infinite periods, until she
work, instead, the firm is selling the technology to produce hears a “high-type” report. Moreover, as we will
the good, and the future quantities depend on the realized
type. prove in Section V, the dynamics of the transfer
28
I am grateful to Bengt Holmström and Asher Wolin- price of the technology will be dictated by the
sky who have independently suggested this point. dynamics of the optimal inefficiency in supply.
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 649

V. The Dynamics of Monetary Payments incentive structure of the model. We can there-
fore ask what is the dynamics of prices and,
As mentioned above, two payment schedules more importantly, the dynamics of the consum-
with the same present value can give the same er’s utility in the optimal supply contract.
incentives to an agent; therefore the prices There is one particular case in which the
charged in the optimal contract are not uniquely monopolist receives anticipated payments from
identified. Indeed, although it is true that we can the consumer, which has special significance
assume without loss of generality that the opti- from an empirical and theoretical reason: the
mal contract keeps the lowest type at his reser- contract discussed in Section IV, in which the
vation utility in any period, we can construct monopolist, as soon as compatible with profit
equilibrium contracts that do not have this fea- maximization, sells the firm to the consumer
ture: an example is the contract in which the who reports himself to be a high type. We call
monopolist sells the technology to the agent. In this arrangement the sale-of-the-firm contract.
general, when we have many periods, we can In this case, too, the monopolist can add on top
find optimal contracts in which the monopolist of a sale-of-the-firm contract a lending contract,
receives a large payment at some date t and she as defined above, in which she borrows more
commits to pay it back in installments. The money than the value of the firm and repays the
installments can, in principle, follow any time extra amount over time. Since we are not inter-
pattern. In this section, we focus on two types of ested in this case, we assume without loss in
optimal monetary transfers that seem more in- generality that the IRht(␪L ) constraint is satisfied
teresting from a theoretical and empirical point as equality in all periods. Again, if this condi-
of view. tion is satisfied, the sale-of-the-firm contract is
For any contract in which the monopolist uniquely determined. The interesting question
borrows money and repays it in an arbitrary in this case is the dynamics of the strike price of
time pattern, we can distinguish two parts: a the call option on the technology.30
supply contract in which the relevant ICht(␪H) Regarding the supply contract, we have:
and IRht(␪L ) constraints are satisfied as equali-
ties, and a residual lending contract, in which PROPOSITION 6: In the optimal supply con-
the monopolist borrows some amount of money tract, the average per-period utility of the agent
and pays it back over time to the agent. A starting from any date t is nondecreasing in t in
reason why the supply contract is more inter- all possible histories and strictly increasing in
esting than other contracts is that if we assume some history; therefore, the expectation at time
that the monopolist is even slightly more patient zero of the average rent of the agent from date
than the agent, she would never find it optimal t is strictly increasing in t.
to ask the agent to anticipate payments for fu-
ture supply (as occurs when the technology is Recent empirical work has highlighted that in
sold to the agent), and therefore all the con- some important markets long-term contracts are
straints would be binding after all histories.29 front-loaded: prices are initially high and de-
The lending contract can take (almost) any form cline over time.31 A consequence of this effect,
because the monopolist can commit to repay it therefore, is that the expected utility of a con-
according to any time pattern. The supply con- sumer from continuing to remain a monopo-
tract, however, is uniquely determined by the list’s customer increases over time. Dionne and
Doherty (1994) explain this phenomenon as a
consequence of the possibility to renegotiate
29
If the monopolist is more patient than the agent, then contracts over time. Building on Milton Harris
the incentive compatibility for the low type would be bind-
ing in all periods and the monopolist would not find it
and Holmström (1982), Hendel and Lizzeri
optimal to lend money to the agent because the agent would
not be able to commit to repay the debt. Note that the
30
monopolist’s objective function is continuous in her dis- I am grateful to William Rogerson who suggested this
count factor and the constraints do not depend on it: there- point.
31
fore, an infinitesimal reduction in the monopolist’s discount This effect has been shown with California auto in-
factor would have only an infinitesimal impact on the opti- surance data by Dionne and Doherty (1994), and more
mal quantities qt(␪兩ht), but would eliminate equilibria in recently by Hendel and Lizzeri (2003) in the life insurance
which the monopolist borrows money. market.
650 THE AMERICAN ECONOMIC REVIEW JUNE 2005

(2000) present a model with no asymmetric option of the high type (i.e., the value of report-
information, but in which both the principal and ing untruthfully). This outside option changes
the agent learn over time from a public signal over time because the contract becomes increas-
the type of the agent: front-loading is therefore ingly efficient along the “lowest” history, and
a consequence of reclassification risk. Our the improvements in the contract benefit the
model suggests a new explanation for this phe- high type more than the low type. The higher
nomenon in which front-loading is precisely a efficiency of the contract, in fact, increases the
consequence of the commitment power of agent’s utility in the event in which he turns into
the seller.32 Indeed, in the optimal contract, a high type, and an agent who is a high type
even without imposing renegotiation con- today has a higher probability of being a high
straints, she finds it optimal to promise an effi- type tomorrow. For this reason, the price for the
cient contract to the agent if he reports himself service that the low type is willing to pay in-
to be a high type, or to provide a contract with creases more slowly than the increase in utility
decreasing inefficiency. Because of this, she of a deviation for a high type. For this reason,
must commit to pay a rent to the high type that the outside option of the high type increases
increases over time because the higher the effi- over time. This implies that the only way for the
ciency of the contract, the more expensive it is monopolist to induce a truthful revelation is to
to separate the agents’ types. reduce the strike price of the call option on the
We now turn to sale-of-the-firm contracts. As property rights of the firm.
we discussed in Section IV, the monopolist
finds it optimal to sell her technology only if the VI. Renegotiation-Proofness
agent reveals himself to be a high type. It is
therefore natural to look at the evolution of the So far we have assumed that the monopolist
call price of the option to buy the technology can commit to a contractual offer. We discussed
along the history in which it can be exercised this point above, arguing that this is the most
(i.e., when the agent always reveals himself to appropriate assumption in many environments,
be a low type). This question is interesting in particular when the monopolist is serving
because, given the stationary structure of the many consumers and is interested in maintain-
model, the present value of the firm along this ing her reputation, or when renegotiation costs
history is constant. Remember that the model are larger than the benefits. There are situations,
has infinite periods, and because the preferences however, in which the seller cannot commit not
of the consumer follow a Markov process, the to renegotiating the contract after some histo-
value of the firm depends only on the current ries. The received literature has shown that if
state of the consumer’s type.33 This fact may types are constant, the optimal contract is never
suggest that the price of the firm is constant over renegotiation-proof. Perhaps surprisingly, given
time. We have, however: a condition that is easily satisfied, this is no
longer true when the agent’s type follows a
PROPOSITION 7: In the optimal sale-of-the- stochastic process.34 We say that a contract is
firm contract, the strike price of the call option renegotiation-proof if after no history ht there is
to buy out the technology is strictly declining a new contract starting in period t that the con-
over time. sumer would accept in exchange for the original
contract, and that is strictly superior for the
What really matters in the determination of monopolist. This definition is standard in the
the transfer price of the firm is the outside
34
This result should not be confused with the findings in
32
In our model, we assume that the seller has monopoly Patrick Rey and Bernard Salanie (1990, 1996). Beside the
power. Clearly, this assumption should be taken into con- fact that they consider a different model with moral hazard
sideration when using the model to interpret evidence in and constant types, these authors show that a renegotiation-
markets with more competitive environments. proof contract can be implemented by a chain of short-term
33
In an equilibrium of a direct mechanism, the consumer contracts (two-period contracts in which the principal can
would reveal his type truthfully; therefore the firm would be commit). They do not characterize the renegotiation-proof
sold only to high types. This implies that whenever the firm contract and they do not prove that the ex ante optimal
is sold, the expected present value of the firm is constant, contract is renegotiation-proof (which indeed would not be
irrespective of the period in which it is sold. true in their models with constant types).
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 651

FIGURE 2. THE RENEGOTIATION-PROOFNESS CONDITION

literature and natural: when a contract is plating a change only in the quantity offered
renegotiation-proof, either the monopolist or the after the agent reports a type ␪ t ⫹ j after a
consumer would reject a revision of the initial history X t ⫹ j⫺1 , keeping constant all other
agreement. quantities. The complete argument clearly
needs to consider a change on the entire se-
PROPOSITION 8: The optimal contract is quence of contingent menus: this is a more
renegotiation-proof if Pr(␪L兩␪L ) ⱕ 1 ⫺ sophisticated problem (solved in the Appen-
␮HPr(␪H兩␪H). Moreover, if this condition is not dix), but this example provides a useful intu-
satisfied, there exists a t ⬍ ⬁ such that the ition. First, note that a contract is renegotiated
contract is not renegotiation-proof only in the only by a contract that is Pareto superior;
first t periods. otherwise either the seller or the buyer would
not accept the change. Since welfare is
Figure 2 represents the condition of Proposi- strictly concave in the quantity supplied, and
tion 7 in an example in which there is a 30- the ex ante optimal quantity at time t (i.e.,
percent initial probability that the agent is a high q*( ␪ t ⫹ j , X t ⫹ j⫺1 )) is not larger than the effi-
type: the contract is renegotiation-proof for any cient output q e ( ␪ t ⫹ j ), it must be that welfare
point below the straight “thick line.” As can be is strictly increasing in the interval [0,
seen from the figure, the set of parameters for q*( ␪ t ⫹ j , X t ⫹ j⫺1 )] (see Figure 3). This im-
which the contract is renegotiation-proof covers plies that the new output q⬘( ␪ t ⫹ j , X t ⫹ j⫺1 )
most of the set of feasible parameters. (Because prescribed by a renegotiated contract must be
types are positively correlated, Pr(␪L兩␪L ) and strictly larger than q*( ␪ t ⫹ j , X t ⫹ j⫺1 ).
Pr(␪H兩␪H) are both larger than 1⁄2 .)35 Consider now a history h t . If the agent has
The intuition of this result can be seen from previously reported himself to be a high type,
Figure 3.36 Assume, for the sake of illustra- then the contract is efficient and not renego-
tion, that at time t the monopolist is contem- tiable; assume therefore that the agent has
always reported himself to be a low type. If
after h t the monopolist could choose any
35
If we measure the persistence of types by ␥ ⫽ other continuation contract, the quantity sup-
max{Pr(␪L兩␪L ), Pr(␪H兩␪H)}, an immediate implication of
Proposition 8 is that, given any initial prior, there is an
plied after a history h t ⫹ j following or equal to
upper bound ␥(␮H) on persistence such that for ␥ ⱕ ␥(␮H) h t in the new contract would be q R ( ␪ t ⫹ j ,
the optimal contract is renegotiation-proof (area below the
semicircle in Figure 2).
36
The two concave functions in the figure represent the
profits and welfare generated in period t ⫹ j after history the efficient contract, and qR(␪t ⫹ j , Xt ⫹ j⫺1) is the contract
xt ⫹ j⫺1: q*(␪t ⫹ j , Xt ⫹ j⫺1) is the optimal contract, qe(␪t ⫹ j) is that is ex post optimal after history ht.
652 THE AMERICAN ECONOMIC REVIEW JUNE 2005

FIGURE 3. THE EX POST MAXIMIZATION PROBLEM AFTER A HISTORY ht

X t ⫹ j⫺1 ) ⫽ ␪t ⫹ j ⫺ ⌬␪[Pr(␪H兩␪L )/Pr(␪L兩␪L )]X but follow a stochastic process, even if the
(␪t⫹j , Xj⫺1)⌳j⫺1. This formula is identical to the correlation level is very high (as in Fig-
formula of the ex ante optimal contract in Prop- ure 2), consumers do not need to use mixed
osition 2, except that instead of the prior ␮H, we strategies in equilibrium, but simply truthfully
use the appropriate posterior after ht, Pr(␪H兩␪L ), report their type. The conflict between optimal-
and the state Xt ⫹ j is started afresh at time t. ity and renegotiation-proofness, and the sophis-
Comparing qR(␪t ⫹ j , Xt ⫹ j⫺1) with the contract tication of equilibrium strategies necessary to
in Proposition 2, it is easy to verify that the ex guarantee the latter property, therefore, are im-
ante optimal q*(␪t ⫹ j , Xt ⫹ j⫺1) is larger than the plications of the assumption that types are con-
ex post optimal level q R ( ␪ t ⫹ j , X t ⫹ j⫺1 ) if stant or very highly correlated.37

␮ H t ⫹ j⫺1 Pr共␪H兩␪L 兲 j⫺1 VII. Conclusion

(4) ⌳ ⱕ ⌳ ,
␮L Pr共␪L兩␪L 兲 This paper shows that a long-term contractual
relationship in which the type of the buyer is
which is always satisfied if Pr( ␪ L 兩 ␪ L ) ⱕ 1 ⫺ constant over time is qualitatively different
␮ H Pr( ␪ H 兩 ␪ H ) since t ⬎ 1. (Obviously, the from a contractual relationship in which the
contract can be renegotiated only starting type follows a Markov process, even if the types
from the second period.) Because the profit are highly persistent. While in the first case the
function is also strictly concave, this implies contract is constant, in the second the contract is
that when (4) holds, any quantity larger than truly dynamic and converges to the efficient
q*( ␪ t ⫹ j , X t ⫹ j⫺1 ) reduces expected profits at contract. Even if the environment has only one-
h t . But then any change that would be ac- period memory and risk-neutral agents, the op-
cepted at h t by the customer would necessar- timal contract is not stationary and has
ily reduce profits, implying that the quantity unbounded memory. The structure of the opti-
q*( ␪ t ⫹ j , X t ⫹ j⫺1 ) would not be renegotiated mal contract, however, is remarkably simple. In
at any time t. analogy with the static model, we have a stron-
When types are constant, the optimal ger version of the generalized no distortion at
renegotiation-proof contract requires the con- the top principle, which implies that the entire
sumer to play sophisticated mixed strategies, state-contingent contract becomes forever effi-
and this may appear unrealistic. These strategies cient as soon as the agent reports himself to be
are necessary to guarantee that, after any pos-
sible history, the monopolist’s posterior be- 37
A complete analysis of the mixed strategy equilibrium
liefs are such that there are no ex post Pareto in the optimal renegotiation-proof contract with variable
superior contracts. The result presented above, types when (4) is not satisfied is presented in Battaglini
however, shows that when types are correlated (2005).
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 653

a high type. In our dynamic setting, however, straints satisfied as equalities as well, while leav-
we also have a novel vanishing distortion at the ing the incentive constraints untouched and
bottom principle which clearly could not be keeping the value of the program constant.
appreciated in a static model.
With constant types there always is a conflict Step 1: We show the result by induction. Let t
between optimality and renegotiation-proofness, be a finite integer. Assume that for any solution
and the latter property is guaranteed only if 具p, q典 of P RI and any t⬘ ⱕ t, there is a pt⬘ such
consumers use sophisticated mixed strategies. that 具pt⬘, q典 is also a solution of P RI ; and all
With stochastic types, in contrast, even if there incentive compatibility constraints are satisfied
is high persistence, the optimal contract is as equalities up to period t⬘, the value of the
renegotiation-proof for natural specifications of objective function is unchanged, and pt⬘ is iden-
the parameters. Consumers, moreover, adopt tical to p in any period j ⬎ t⬘: pt⬘(␪; hj) ⫽ p(␪;
simple pure strategies. hj) for j ⬎ t⬘. This step is clearly satisfied for t ⫽
The dynamic theory of contracting presented 1, since in this period the incentive compatibil-
in the paper also provides insights into the own- ity constraint is necessarily binding at the opti-
ership structure of the monopolist’s exclusive mum. We now show that if pt⬘ exists @t⬘ ⱕ t,
technology and contributes to explaining some then there must be a price vector pt ⫹ 1 such that
empirical findings. The monopolist may find it all the incentive compatibility constraints are
optimal to keep the ownership of the technology satisfied as equalities up to period t ⫹ 1, pt ⫹ 1 is
even when it would be inefficient in order to identical to p in any period j ⬎ t ⫹ 1, and the
control the agents’ future rents and therefore value of the objective function is unchanged.
maximize rent extraction. This inefficient reten- Given the induction step, assume without loss
tion of property rights may potentially last for of generality that the incentive constraints are
an arbitrarily large number of periods, but the satisfied as equalities for any j ⱕ t. There are
allocation of property rights will be efficient two cases to consider. Assume first that at pe-
with probability one in the long term. riod t ⫹ 1, after a history ht ⫹1 ⫽ {ht, HL}, the
high type receives an expected continuation
APPENDIX utility equal to the utility he would receive if he
declares to be a low type plus a constant ␧ ⬎ 0.
PROOF OF LEMMA 1: Modify the contract so that the new prices after
For a generic maximization program P, we histories {ht ⫹1, ␪H} and {ht ⫹1, ␪L} are re-
define V(P), the value of the objective function spectively pt⫹1(␪H; ht⫹1) ⫽ p(␪H; ht⫹1) ⫹ ␧ and
at the optimum. Let us also define P RI the p(␪L; ht ⫹1) ⫽ p(␪L; ht ⫹1);38 simultaneously,
program in which expected profit is maximized reduce the price after history {ht , ␪L} so that
only under the incentive compatibility con- pt ⫹ 1(␪L; ht) ⫽ p(␪L; ht) ⫺ ␦ Pr(␪H兩␪L )␧. We call
straints of the high type and the individual ra- this new price vector p̂t ⫹ 1. This change would
tionality constraints of the low type: ICht(␪H), not reduce the monopolist’s expected profit, it
IRht(␪L ) @ht. We proceed in two steps. would not violate IRht(␪L ) in any period, it
would not violate the incentive compatibility
CLAIM 1: constraints for histories following ht, and it
If 具p, q典 solves P RI , then there exists a p⬘ such would satisfy ICht⫹1(␪H) as equality. Consider
that 具p⬘, q典 satisfies IRht(␪L ) and ICht(␪H) as now the ICht(␪H) constraint at ht. The utility of a
equality @ht, and achieves the same value as 具p, high type that is truthful U(␪H; ht) is unchanged;
q典: V(P RI ) ⫽ V(PII). if the high type reports himself to be a low type,
however, he would receive:
PROOF:
Given a solution 具p, q典, we first show that the (A1) U共 ␪ H ; h t ⫺ 1 兲 ⫹ ␦ 共Pr共␪H兩␪L 兲
price vector can be modified to guarantee that
all the incentive constraints are satisfied as equal- ⫺ Pr共␪H兩␪H 兲兲␧ ⱕ U共␪H ; ht ⫺ 1 兲
ities without reducing the value of the program;
then we show that the resulting contract with 38
Remember that p(␪; ht) is the price charged after
incentive constraints satisfied as equalities can be history ht if the agent declared to be a type ␪, so it is the
modified to make the individual rationality con- price charged after a history {ht, ␪}.
654 THE AMERICAN ECONOMIC REVIEW JUNE 2005

where the inequality follows from the fact that CLAIM 2:

types are positively correlated. It follows that Any solution of PII satisfies all the constraints
IChj(␪H) are satisfied for any j ⱕ t, and 具p̂t ⫹ 1, q典 of PI, and V(PRI ) ⫽ V(PI).
is a solution of P RI . By the induction step we
can find a new price vector pt ⫹ 1 which is such PROOF:
that the incentive compatibility constraints are Since V(PRI ) ⫽ V(PII), we need only show
satisfied as equalities in all periods j ⱕ t, and that, in correspondence to the solution of PII,
that is identical to p̂t ⫹ 1 for periods j ⬎ t (which after any history ht the low type does not want
is identical to p in periods l ⬎ t ⫹ 1). Since this to imitate the high type (i.e., the ICht(␪L ) con-
change does not affect prices at t ⫹ 1 and straint is satisfied) and the high type receives at
following periods, in correspondence to pt ⫹ 1 least his reservation value (i.e., the IRht(␪H) con-
the incentive compatibility is satisfied as equal- straint is satisfied). This guarantees that
ity @t ⱕ t ⫹ 1, prices are equal to the prices in V(PI) ⱖ V(P RI ) and hence the result.
p for periods j ⬎ t ⫹ 1, and the value function
is equal to the original. Step 1: the ICht(␪L ) constraints. Note that by
Assume now that at period t ⫹ 1, after some ICht(␪H) and IRht(␪L ), after any history ht:
history ht ⫽ {ht ⫺ 1, ␪H}, the high type receives
a utility equal to the utility he would receive if p共 ␪ H ; h t 兲 ⫺ p共 ␪ L ; h t 兲 ⫽
he declares to be a low type plus a constant ␧ ⬎
0. Modify the contract so that the new prices 共q共 ␪ H ; h t 兲 ⫺ q共 ␪ L ; h t 兲兲 ␪ L
after histories {ht, ␪H} and {ht, ␪L} are respec-
tively pt ⫹ 1(␪H; ht) ⫽ p(␪H; ht) ⫹ ␧ and pt ⫹ 1(␪L; ⫹ ␦ Pr共␪H兩␪L 兲⌬U共␪H , ht 兲
ht) ⫽ p(␪L; ht); simultaneously, reduce prices
after history {ht, ␪H} so that pt ⫹ 1(␪H; ht ⫺ 1) ⫽ ⫹ 共q共 ␪ H ; h t 兲 ⫺ q共 ␪ L ; h t 兲兲⌬ ␪
p(␪H; ht ⫺ 1) ⫺ ␦ Pr(␪H兩␪H)␧. This new con-
tract pt ⫹ 1 would leave all the constraints of the ⫹ ␦ 共Pr共␪H兩␪H 兲 ⫺ Pr共␪H兩␪L 兲兲⌬U共␪H , ht 兲
relaxed problem satisfied @ht and the incentive
constraint satisfied as equalities in the first t ⫹ where p(␪i; ht)i ⫽ H, L is the price charged after
1 periods. And it would not reduce profits. the agent declares to be a type i and ⌬U(␪H, ht) ⫽
U(␪H; ht, ␪H) ⫺ U(␪H; ht, ␪L ), the difference
Step 2: By the previous step we can assume between the rent of a high type after a ␪H and a ␪L
without loss of generality that all the incentive declaration (the continuation value of a low type is
compatibility constraints are satisfied as equal- zero in PII). As can be seen from (2), in corre-
ities. We now show that the individual rational- spondence to the solution of PII, 39 after an agent
ity constraints can also be reduced to equalities. declares to be a high type an efficient contract is
It can be verified that the individual rationality offered in the optimal solution of the relaxed prob-
constraint must be binding at t ⫽ 1. Again, we lem; so using ICht(␪H) and IRht(␪L ) we can write:
prove the result for the remaining periods by
冘 ␦ 共Pr共␪ 兩␪ 兲
⬁
induction. Assume that in all periods j ⱕ t, U共 ␪ H ; h t , ␪ H 兲 ⫽ ⌬ ␪ j
H H
IRhj(␪L ) holds as equality and that the expected j⫽0
utility of a low type agent after history ht ⫹ 1 is
␬ ⬎ 0. Consider an increase by ␬ of the prices ⫺ Pr共␪L兩␪H 兲兲jqe共␪L 兲
charged in the period t ⫹ 1, pt ⫹ 1(␪; ht, ␪i) ⫽
p(␪; ht, ␪i) ⫹ ␬ @␪; and a reduction of the price where, remember, qe(␪L ) is the efficient quantity
at time t so that pt ⫹ 1(␪; ht ⫺ 1) ⫽ p(␪; ht ⫺ 1) ⫺ when the type is ␪L. If the agent reports himself to
␦␬ @␪. Clearly, this change would not violate be a low type, on the contrary, he will receive an
the constraints of P RI , would leave the incentive inefficient quantity q*(␪L兩h) that is never strictly
compatibility constraints untouched, and would higher than the efficient level qe(␪L ): therefore his
satisfy all the individual rationality constraint as continuation value is not higher than U(␪H; ht, ␪H).
equality up to period t ⫹ 1. Profit would remain
unchanged as well.
39
The formal derivation of (2) is in the proof of Prop-
We now prove: osition 1 below.
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 655

So U(␪H; ht, ␪H) ⫺ U(␪H; ht, ␪L ) ⱖ 0 for any ht; Using the equality IRh1(␪L ) we know that the low
and, since types are positively correlated, type receives zero at time one. It follows that PII
(Pr(␪H兩␪H) ⫺ Pr(␪H兩␪L ))⌬U(␪H, ht) ⱖ 0. It follows can be represented as:

冘␮
that:
共B3兲 E⌸共␪兩h1 兲 ⫽ i
p共 ␪ H ; h t 兲 ⫺ p共 ␪ L ; h t 兲 ⱖ 共q共 ␪ H ; h t 兲 i ⫽ H,L

⫺ q共 ␪ L ; h t 兲兲 ␪ L ⫹ ␦ Pr共␪H兩␪L 兲⌬U共␪H , ht 兲
冋
⫻ q共 ␪ i 兲 ␪ i ⫺
q共 ␪ i 兲 2
2 冉
W共␪H ; ␪i 兲
⫹ ␦ ␣ i W共␪ ; ␪ 兲
L i
冊册
which implies that ICht(␪L ) is satisfied at ht.

冘 ␦ 共Pr共␪ 兩␪ 兲
⬁
Step 2: the IRht(␪H) constraints. By ICht(␪H) ⫺ ␮ H⌬ ␪ j
H H
and IRht(␪L ) we have: j⫽0

q共 ␪ H ; h t 兲 ␪ H ⫺ p共 ␪ H ; h t 兲 ⫺ Pr共␪H兩␪L 兲兲 jq*j ⫹ 1 共␪L兩hjL⫹ 1兲

⫹ ␦ Pr共␪H兩␪H 兲U共␪H ; ht , ␪H 兲 ⱖ ␦共Pr共␪H兩␪H 兲 where the first summation is the expected sur-
plus generated by the supply contract (W(␪i; ␪j)
⫺ Pr共␪L兩␪L 兲兲U共␪H ; ht , ␪L 兲 ⬎ 0 is the expected social welfare generated by the
contract from period 2 if the realization in pe-
and therefore IRht(␪H) is satisfied as well. riod 1 is ␪j and the realization in period 2 is ␪i.)

We can now prove Lemma 1. Assume that 具p, We have two possible cases:
q典 solves PII; then, by Claim 1 and 2 it must
also solve PI. Assume that 具p, q典 solves PI; Case 1: ht ⫽ {ht ⫺ 1, ␪} 僆 Ht⶿hLt @t ⱖ 1. The
then, by Claim 2 it must also solve P RI , since first-order condition implies q*t (␪兩ht) ⫽ ␪, and
V(P RI ) ⫽ V(PI). By Claim 1 there exists a p⬘ the contract is efficient.
such that 具p⬘, q典 solves PII and achieves the
same value as PI. We conclude that q solves PI Case 2: ht ⫽ hLt @t ⱖ 1. The first-order condi-
if and only if it solves PII. tion with respect to a generic quantity offered
along the lowest branch q*t (␪L兩hLt ) implies that:
PROOF OF PROPOSITION 1:
Let us define hLt :⫽ {␪L, ␪L, ... ␪L}, the history (B4) q *t 共 ␪ L 兩h Lt 兲 ⫽ ␪ L
along which the agent always reports himself to
be a low type for t ⫺ 1 periods. Using the fact
冋 册
t⫺1
␮ H Pr共␪H兩␪H 兲 ⫺ Pr共␪H兩␪L 兲
that ICht(␪H) and IRht(␪L ) are equalities, we can ⫺ ⌬␪ ,
␮L Pr共␪L兩␪L 兲
formulate the utility of the high type at time 1
as:
which completes the characterization of the op-
timal contract.
U共␪H ; h1 兲 ⫽ ⌬␪q*1 共␪L兩h1 兲 ⫹ ␦共␣H ⫺ ␣L 兲

冉 冉 冊冊
PROOF OF PROPOSITION 4:
⌬␪q*3 共␪L兩h3L兲 ⫹ ...
⌬␪q*2 共␪L兩h2L兲 ⫹ ␦共␣H ⫺ ␣L 兲 Starting in period t from any history {ht, ␪},
⫻ 0 .
the expected first best surplus from time t on-
0
ward is independent from t and equal to
W*t (␪ ) ⫽ ¥jⱖt ␦ j ⫺ tEt[1⁄2 ␪2j 兩␪t ⫽ ␪]. Consider a
This formula can be written as contract c in which a fixed fee F ⫽ W*1(␪L ) is
charged in period 1 and then an efficient menu

冘 ␦ 共Pr共␪ 兩␪ 兲 plan in which q␶(␪ ) ⫽ ␪, p␶(␪ ) ⫽ 1⁄2 ␪2 for any

⬁

(B2) U共 ␪ H ; h 1 兲 ⫽ ⌬ ␪ j
H H ␶ ⱖ 1 is offered. This contract is clearly incentive
j⫽0 compatible and individually rational for any ␶ ⱖ
1; moreover, it is renegotiation-proof since it is
⫺ Pr共␪H兩␪L 兲兲 jq*j ⫹ 1 共␪L兩hjL⫹ 1兲. efficient. Therefore it is a feasible option in the
656 THE AMERICAN ECONOMIC REVIEW JUNE 2005

monopolist’s program, even if the renegotiation- between paying or receiving a positive amount
proofness constraint must be satisfied, and must every period, or a large amount equal to the
yield an average profit not larger than the profit of future expected payments at some period and
the optimal contract ⌸*. This implies zero afterward. The second part follows from
the fact that from (2) we know that along the
(C5) 共1 ⫺ ␦ 兲⌸* ⫺ “lowest history,” supply is distorted in the fu-
ture with positive probability, and the monopo-

冋冘 册
⬁ list cannot achieve these distortions without
control of the technology.
共1 ⫺ ␦ 兲 E ␦ ␶ ⫺ 1 w*共 ␪ ␶ 兲 ⱖ
␶⫽1
PROOF OF PROPOSITION 6:

冋冘 册
When IRht(␪L ) is satisfied as equality for all

共1 ⫺ ␦ 兲 E
⬁

␶⫽1
冏
␦ ␶ ⫺ 1 w*共 ␪ ␶ 兲 ␪ 1 ⫽ ␪ L ⫺
ht, the low type receives zero expected utility in
all periods. Therefore we need only show that
the average utility of the high type is nonde-

冋冘 册
creasing in time. Using (2) and the incentive com-
⬁ patibility constraint of the high type we can write:
共1 ⫺ ␦ 兲 E ␦ ␶ ⫺ 1 w*共 ␪ ␶ 兲
␶⫽1 U共 ␪ H ; t, X t ⫺ 1 兲

where w*( ␪ ) is the per-period Marshalian

冘 ␦ 共Pr共␪ 兩␪ 兲 ⫺ Pr共␪ 兩␪ 兲兲
⬁
surplus when the type is ␪. As ␦ 3 1, the ⫽ ⌬␪ j
H H H L
j

right-hand side can be written as j⫽0

⍀共 ␪ ; t̂兲 ⫽ lim共1 ⫺ ␦ 兲
␦ 31 冋
䡠 ␪L ⫺ ⌬␪X共␪L , Xt ⫹ j ⫺ 1 兲
␮H
␮L

再
⫻ E 冋冘 ␦ˆ
␶ ⱖt
␶⫺1
冏
w*共 ␪ ␶ 兲 ␪ 1 ⫽ ␪ L 册 冉
⫻ 1⫺
Pr共␪H兩␪L 兲
Pr共␪L兩␪L 兲 冊 册 t ⫹ j⫺1

⫺E 冋冘 ˆ
␶ ⱖt
␦ ␶ ⫺ 1 w*共 ␪ ␶ 兲 册冎 where U(␪H; t, Xt⫺1) is the expected utility of a
high type at time t given the state Xt⫺1. Consider
now two periods: t and t⬘ ⬍ t. It is easy to show
that U(␪H; t, Xt⫺1) ⫺ U(␪H; t⬘, Xt⬘⫺1) is propor-
where t̂ is a finite integer. Since (C5) must tional to X(␪L, Xt⬘⫺1) ⫺ X(␪L, Xt⫺1)[1 ⫺
holds for any t̂ ⱖ 1 and limt̂3⬁ ⍀( ␪ ; t̂) ⫽ 0 (Pr(␪H兩␪L )/Pr(␪L兩␪L ))](t⫺t⬘), which is nonnegative
(because the process converges to a stationary because Xt⫺1 ⱕ Xt⬘⫺1 and strictly positive if X(␪L,
distribution), we have that lim␦31(1 ⫺ ␦)⌸* Xt⬘⫺1) ⫽ 1. Therefore, the average rent of the
must be equal to lim␦31(1 ⫺ ␦ ) E[¥ ␶⬁⫽1 agent is nondecreasing in any history and strictly
␦ ␶ ⫺ 1 w*( ␪ ␶ )]. This also implies that the increasing in a nonempty subset of histories. It
agent’s average payoff is zero. follows that, at time zero, the expected average
rent starting from period t is strictly increasing
in t.
PROOF OF PROPOSITION 5:
Since the optimal contract is efficient after PROOF OF PROPOSITION 7:
the agent reveals himself to be a high type, the Since the monopolist’s technology is sold as
monopolist finds it optimal to offer to the con- soon as compatible with profit maximization, its
sumer the same quantities that the consumer sale can occur only along a history in which the
himself would choose if he could directly con- agent has always reported himself to be a low
trol supply. The first part of the result then type. Consider any such history ht. The price
follows from the fact that all players have quasi- P(ht) paid for the technology by the high type is
linear utilities and therefore they are indifferent determined by the equation
VOL. 95 NO. 3 BATTAGLINI: LONG-TERM CONTRACTING WITH MARKOVIAN CONSUMERS 657

(F6) W*共 ␪ H 兲 ⫺ P共h t 兲 ⫽ U共 ␪ H , ␪ L ; h t 兲 ante optimal contract, which, after ht, is a con-
stant that we denote W*: ¥i ⫽ H,L Pr(␪i兩␪L )W(␪i ,
where U( ␪ i , ␪ j ; h t ) is the utility of a type ␪ i qi) ⱖ W*. We denote this constraint (G8). If we
from declaring to be a type ␪ j after a history show that the ex ante optimal quantities solve
h t ; and W*( ␪ i ) is the expected first best sur- P *ex post, then they must also solve Pex post and
plus from time t if the type at t is ␪ i . 40 Since be renegotiation-proof. The Lagrangian of
W*( ␪ H ) is clearly history independent, the P *ex post is:
result follows by the fact that supply is in-
creasing over time and therefore U( ␪ H , ␪ L ;
h t ) is increasing (see [B2]). (G9) LP ⫽ 共1 ⫹ ␶兲关␭L W共␪H , qH 兲

PROOF OF PROPOSITION 8: ⫹ W共␪L , qL 兲] ⫺ ␭L R共qL 兲

Consider the problem of ex post maximiza-
tion faced by the monopolist after a history ht where ␭L is the ex post likelihood ratio
with t ⬎ 1 in which the agent has never reported [Pr(␪H兩␪L )/Pr(␪L兩␪L )], and ␶ is the Lagrangian
himself to be a high type. At this stage, expected multiplier associated with (G8). We proceed in
profits can be written as: three simple steps:

(G7) E关⌸共␪兩ht 兲兩ht 兴 Step 1: ␶ ⬎ 0. Given that Pr(␪L兩␪L ) ⱕ 1 ⫺

␮HPr(␪H兩␪H) implies (4) @j ⱖ 0, if ␶ ⫽ 0 then
⫽ Pr共␪H兩␪L 兲关W共␪H , qH 兲 ⫺ R共qL 兲兴 the solution of (G9) implies that all the quantities
in qH are set efficiently and the quantities in qL are
⫹ Pr共␪L兩␪L 兲W共␪L , qL 兲 distorted downward more than the solutions of the
ex ante optimal problem (the argument is the same
where qi i ⫽ H, L is the sequence of quantities as in Section VI). But then the welfare constraint
in the menus offered if the agent reports himself (G8) must be violated, a contradiction.
to be a type i at t; W(␪i , qi) is the expected
surplus generated in the contract if the agent is Step 2: Let ␭° be the ex ante likelihood ratio
of type i and qi is offered; and R(qL ) is the (␮H/␮L ). We can show that [␭L/(1 ⫹ ␶)] ⫽
expected rent of the high type starting from ht ␭⬚⌳t ⫺ 1. Indeed, it can be verified that the quan-
which guarantees incentive compatibility. (By tities following history ht in the optimal solution
Lemma 1 it depends only on qL as in [B2], and of the ex ante problem maximize:
the rent of the low type is zero.) Indeed (G7) is
a compact way to write (B3) when the posterior
(G10) LA ⫽ ␭⬚W共␪H , qH 兲
probability that the type is high starting from ht
is Pr(␪H兩␪L ). The monopolist’s ex post problem
⫹ W共␪L , qL 兲 ⫺ ␭⬚⌳t ⫺ 1R共qL 兲
(Pex post) consists of maximizing (G7) under the
additional constraint that the expected rents of
the agent are at least as high as the expected If [␭L/(1 ⫹ ␶)] ⬍ ␭⬚⌳t ⫺ 1, then the solution of
rents starting from ht obtained keeping the orig- (G9) would be less distorted than the solution of
inal, ex ante optimal contract. It is, however, (G10), implying that the welfare constraint (G8)
useful to consider the program (P *ex post) in is not binding and so ␶ ⫽ 0, a contradiction.
which (G7) is maximized under the additional Similarly we can prove that the reverse inequal-
constraint that expected welfare is at least as ity is not possible. From [␭L/(1 ⫹ ␶)] ⫽ ␭⬚⌳t ⫺ 1
high as the level achieved with the original ex we conclude that the solution of (G9) and (G10)
coincide and the optimal ex ante contract is
renegotiation-proof.
40
To avoid confusions in what follows, it is worth em-
phasizing that U(␪j , ␪i; ht) and U(␪j; ht̂, ␪i) are different Step 3. Finally, it is easy to see that if
objects: the first represents the case in which a type j reports Pr(␪L兩␪L ) ⬎ 1 ⫺ ␮HPr(␪H兩␪H), then there must
untruthfully to be a type i after a history ht; the second
represents the case in which a type j truthfully reports his
be a finite t̃ such that for t ⬎ t̃, then (4) is
type after a history ht̂ ⫹ 1 ⫽ {ht̂, ␪i}. Indeed the second satisfied, and the argument in steps 1 and 2 is
expression is equivalent to U(␪j , ␪j; ht̂, ␪i). valid for any t ⬎ t̃.
658 THE AMERICAN ECONOMIC REVIEW JUNE 2005

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