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Princeton University - Economic Theory Center

Research Paper No. 46-2012

Optimal Dynamic Contracting

October 2012

Marco Battaglini

Rohit Lamba

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October 2012

Optimal Dynamic Contracting

Abstract

We study a simple dynamic Principal-Agent model in which the agents types are serially correlated.

In these models, the standard approach consists in first solving a relaxed version in which only local

incentive compatibility constraints are considered, and then in proving that the local constraints are

sufficient for implementability. We show that, with the exception of few notable examples highlighted in

the literature, this approach is not generally valid: even assuming standard regularity conditions, both

local and global incentive constraints are generally binding when serial correlation is sufficiently high. We

uncover a number of interesting features of the optimal contract that cannot be observed in the special

environments in which the standard approach works. Finally, we show that even in complex environments,

approximately optimal allocations can be easily characterized by focusing on a particular class of contracts

in which the allocation is forced to be monotonic.

Marco Battaglini

Department of Economics

Princeton University

Princeton NJ 08544

mbattagl@princeton.edu

Rohit Lamba

Department of Economics

Princeton University

Princeton NJ 08544

rlamba@Princeton.EDU

For useful comments and discussions we thank Dilip Abreu, Dirk Bergemann, Sylvain Chassang, StephenMorris, Wolfgang Pesendorfer, Roland Strausz, Balazs Szentes, Juuso Valimaki. Rohit Lamba is especially gratefulto Stephen Morris for guidance and encouragement.

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1 Introduction

Most contractual relationships have a dynamic nature, involving long term, non-anonymous inter-

action between a principal and an agent. Examples of these contractual relationships include the

cases of income taxation, of regulation, or of a monopolist repeatedly selling a non durable good

to a buyer. In these environments contracts can be made contingent on past realizations of the

agents type, allowing the principal to use the agents revealed preferences to screen future types

realizations. This may be particularly useful in limiting asymmetric information and agency

problems when the agents type is persistent over time.

Despite recent advances in contract theory, there is still a limited understanding about how

to use this information to design optimal screening contracts. Dynamic contracts are difficult to

study because they involve a large number of incentive compatibility constraints. The analysis

of optimal dynamic contracts has therefore been limited to economic environments in which a

form of the first-order approach can be applied: that is, in which the optimal contract can be

fully characterized using only the necessary conditions implied by the local incentive compatibility

constraints. While the first-order approach can be generally applied in static environments under

mild regularity assumptions, in dynamic environments local incentive compatibility constraints

have proven to be sufficient only in very special economic environments. 1

This leaves two questions open. First, what is the general applicability of the first-order

approach? In order to characterize the optimal contract in general environments, can we just

proceed by ignoring the global incentive constraints? Second, in environments in which the first-

order approach does not hold, what does the optimal contract look like? Are there phenomena

associated with dynamic contracts that we are ignoring by focusing on environments in which

solving the contract is easy? For all environments in which the optimal contract has been

characterized the first-order approach could be applied: this may suggest that under appropriate

regularity conditions the answer to the first question is affirmative, and the second question is

therefore irrelevant. In this paper we show that this is not generally correct. Even in the

simplest (and most natural) environments, both local and global constraint are typically binding

in dynamic environments. When global incentive constraints cannot be ignored, moreover, the

optimal contract shows features that are not apparent in environments in which the first-order

1 We will discuss the literature in greater details in the next subsection.

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approach is valid.

To make our point, we consider a simple Principal-Agent model in which a monopolist re-

peatedly sells a non durable good to a buyer. The marginal valuation of the buyer is his private

information, and it evolves over time according to a generalN-state Markov process. Higher types

are assumed to have higher marginal valuations and their associated conditional distribution on

future types first-order stochastically dominates the distribution of lower types.

We present three sets of results. We first explore the applicability of the first-order approach.

We show that if we ignore the global constraints, the necessary local incentive compatibility con-

straints allow us to state a dynamic envelope theorem through which we can express the agents

equilibrium rents just as a function of the expected allocation. The dynamic envelope theorem

in turn allows a simple characterization of the profit maximizing contract in the relaxed problem

that includes only local incentive constraints (first-order optimal contract, or FO-contract). We

also show that the envelope formula and a simple form of monotonicity of the allocation are suf-

ficient for implementability.2 Monotonicity requires that ifht ht, then q(ht) q(ht), whereq(ht) (resp., q(ht)) is the quantity allocated following a history ht (resp.,ht).3 Although thiscondition is only sufficient and quite strong, it is verified for virtually all environments in which

the optimal dynamic contract has been characterized in the existing literature.4 These results

directly extend the analysis in Battaglini [2005] who studied the same environment under the

assumption of only two types. This generalization allows us to put into a common framework

the key features of the environments in which the optimal contract has been characterized in the

literature.

We then show that the environments for which the dynamic envelope formula is sufficient to

characterize the optimal dynamic contract are quite special. In general, even in the simplest

examples, the allocation is not monotonic and local incentive constraints are not sufficient for

implementability. We proceed in two steps. To gain intuition on the structure of the optimal

contract, we start by proving this result in a simple case with three types and two periods where

the optimal contract can be fully solved in closed form. The characterization shows that the seller

2 An allocation is implementable if there exist transfers such that the contract is incentive compatible.

3 A history is a vector of reports ht = (ht1

,...,htt), so as usual ht ht ifhtj htj j t.

4 Less intuitive necessary and sufficient conditions, or stronger sufficient conditions can be easily stated, butthey have no practical interest.

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of persistence.

In the following, we proceed as follows. In Section 2 we present our simple benchmark model.

In section 3 we study the general problem. In Section 4 we characterize the optimal contract

in the case with 3 types and 2 periods. In Section 5 we study monotonic contracts. Section 6

concludes the analysis. We discuss the related literature in the next subsection.

1.1 Related literature

The first paper to present a general theory of dynamic contracts, and to state a dynamic version of

the so-called envelope formula, is Baron and Besanko [1984]. The envelope formula, an essential

result in static contract theory, is an implication of the local incentive compatibility constraints:

it allows us to express the agents equilibrium rent purely in terms of the final allocation. 6 In

static environments, under standard regularity conditions, the formula is sufficient for incentive

compatibility and therefore drastically simplifies the constraint set. In their paper, Baron and

Besanko [1984] state the formula in general terms and show it to be sufficient in two benchmark

cases: when types are constant over time, in which case the optimal dynamic contract corresponds

to a repetition of the static optimum; and when types realizations are independently distributed

over time, in which case the optimal contract is efficient starting from period 2. However, they do

not study the conditions under which the envelope formula is sufficient for incentive compatibility

in the general case with imperfectly correlated types.

Besanko [1985] and Battaglini [2005] are the first papers to present natural environments

in which the dynamic envelope theorem is sufficient and so the optimal contract can be fully

characterized even with an infinite horizon.7 Besanko [1985] studies a model in which the type

follows a first-order autoregressive process: the type at time t is t = t1+ t, where t is

the realization of an i.i.d random variable and (0, 1) is a parameter measuring persistence.

This environment is particularly simple to study because the stochastic process can be separated

in a deterministic part that induces persistence, and in a shock that is uncorrelated with the

agents type. Because the shock at t+ 1 is uncorrelated to the type at t, the agent at time t

does not have any private informational advantage on the stochastic part of his future type: the

6 See Stole [2001], Laffont and Martimort [2002], Milgrom (2004), and Bolton and Dewatripont [2005] for generaldiscussions of the envelope formula in the static case.

7 A number of other paper have studied dynamic contracts in two period environments. See for example Laffontand Tirole [1990].

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shock is therefore irrelevant for incentive compatibility. As shown by Besanko [1985], the optimal

contract in this environment is qualitatively similar to a fully deterministic one in the sense that

the equilibrium distortions are independent of the distribution of the shocks, t.8 Battaglini

[2005] considers a two types Markov process with correlated types. In this case the agent has

private information both on his current type and on the shape of future types distributions. The

distortions in the allocation are, thus, state contingent and depend on the entire history of the

contract. However, these distortions persist only along the lowest history in which the agents

valuation is low in every period and they decrease over time, converging to zero as time becomes

inifinitely large. The contract thus converges to efficiency along all histories.9 The model that

we study in the reminder of the paper is a direct generalization of this framework to the case with

N types.10

A recent literature has attempted to apply the first-order approach to general environments.11

The implicit assumption in this literature is that the first-order approach works if a sufficient

dose of regularity conditions is imposed on the model. Pavan, Segal and Toikka [2011] have

studied the necessary conditions characterized by local incentive compatibility in the form of a

dynamic envelope formula in quite general environments. These formulas have been used in

recent applied work to study a variety of principal agent models in which sufficient conditions

for the validity of the first-order approach have not been established. This literature typically

either assumes sufficiency without checking, or relies on numerical methods to verify a sample of

the global incentive constraints.12 The problem with this approach is that the verification of

all incentive compatibility constraint is impossible in an infinite horizon problem, and impractical

in non trivial finite problems, since the number of constraints grows exponentially with the types

and periods.

All the papers in the literature discussed above study environments in which the allocation

8 This model has b een used in a variety of applications: regulation (Besanko [1985]), managerial compensation(Garret and Pavan [2011]) and others (see Pavan, Segal and Toikka [2011]).

9 Application of this model include regulation (Battaglini [2007]), optimal taxation (Battaglini and Coate [2008]),non-durable goods monopoly with renegotiation (Maestri [2012]).

10 In Section 3.2 we show how the model with persistent Markov shocks and the model with an auto regressivecomponent can be easily combined and studied in the same framework.

11 Further, important work has been done to study the dynamic implementation of efficient mechanisms withmultiple agents by Athey and Segal [2007] and Bergemann and Valimaki [2010].

12 This approach has been adopted by Farhi and Werning [2011], Golosov, Troshkin and Tsyvinski [2011], Kapicka(2011), Zhang [2009] and others.

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is determined in every period, as is natural to assume in application such as nonlinear pricing

(where a good is sold to a customer in every period), or nonlinear taxation (where income is

produced in every period). A related literature has studied environments in which the agent

receives information gradually over time, but the allocation is determined only in the last period.

The seminal contribution here is Courty and Li [2000] who consider a monopolist model in which

the buyer first observes a signal about the distribution of his type, and then, before the good is

sold, observes his type. This literature studies the conditions under which it is optimal for the

seller to screen the agent sequentially. The fact that the allocation is determined only at the

end makes implementation simpler in these environments. As in the papers listed before, Courty

and Li [2000] focus on environments in which global incentive constraints can be ignored, showing

sufficient conditions under which this approach is valid.13

We finally note that our characterization of monotonic contracts fits into a recent literature

devoted to the study of approximately optimal mechanisms in environments in which fully optimal

mechanisms are hard to characterize (see Madarasz and Prat [2010], Chassang [2011] for recent

contributions and Hartline [2012] for a summary of the computer science approach). While parts

of this literature deal with more general environments than ours, our approach takes full advantage

of the dynamic structure of the framework we study; this allows us to obtain an approximately

optimal contract that guarantees incentive compatibility for all types at all histories.

2 Model

There are two players, a buyer and a seller. The buyer repeatedly buys a non-durable good from

the seller. He enjoys a per-period utility tq p for q units of the good bought at a price p.

In every period, the seller produces the good with a cost function c(q) = 12q2. The marginal

benefit t evolves over time according to a Markov process. There are N + 1 possible types,

= {0, 1,...,N}, with i i+1 = > 0 for any i = 0,...,N1. Let N = {0, 1, 2,...,N}

denote the set of all indices of types. The probability that state k is reached if the agent is in statei is given by f(k|i) =ik. LetFbe the conditional CDF, defined F(j|i) =

Njk=0

i(j+k). The

distribution of types conditional on being type i is denoted i = (i0, i1,...,iN). We assume that

13 Other important theoretical contributions in this literature on sequential screening are Eso and Szentes [2007]and Krahmer and Strausz [2011]. A related model of price discrimination with information revelation at theinterim stage is presented by Inderst and Ottaviani [2011] to study the market for financial advice.

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i has full support (soij >0 for anyi, j), and i first-order stochastically dominates j for any

iand any j > i: so, given that higher indices imply lower values, we have F(j|i)F(j|k) for

anyj andi k. In each period the consumer observes the realization of his own type; the seller,

in contrast, can only observe past allocations. At date 0 the seller has a prior = (0,...,N) on

the agents type; the prior has full support, so i >0 for any i. For future reference, note that

the efficient level of output is equal to qe(t) = t in all periods and after any history of types

realizations.

We assume that time t is discrete and the relationship between the buyer and the seller lasts

for T 2 periods. In period 1 the seller offers a supply contract to the buyer. The buyer can

reject the offer or accept it; in the latter case the buyer can walk away from the relationship at any

timet 1 if the expected continuation utility offered by the contract falls below the reservation

valueU= 0. In line with the standard model of price discrimination, the monopolist commits to

the contract that is offered. The common discount factor is (0, 1).

It is easy to show that in this environment a form of the revelation principle is valid, which

allows us to consider, without loss of generality, only contracts that in each period t depend on

the revealed type at time t and on the history of previous type revelations, i.e. the contract p, q

can be written asp, q=p

tht1

, q

tht1

Tt=1

, where ht1 andt are, respectively, the

public history up to period t 1 and the type revealed at time t.14 In general,ht can be defined

recursively as ht =

ht1, t

, h0 =. The set of possible histories at timet is denoted Ht (for

simplicity H=HT). Let t be the mapping that projects the first t elements of a vector. The

set of full histories that follow ht till time t is given by H(ht) = {hH|t(h) = ht}. A strategy

for a seller consists in offering a direct mechanism p, q as described above. The strategy of a

consumer is, at least potentially, contingent on a richer history htA=

ht1A , , t,t1, wheret is

the actual type every period andt is the revealed type. Note thath0A= . For a given contract,a strategy for the consumer, then, is simply a function that maps a history htA into a revealed

type: htAs(htA).

In the study of static models it is generally assumed that all types are served, i.e., each type

is offered a positive quantity. This assumption considerably simplifies the analysis because it

permits ignoring the non-negativity constraints in quantities. In our model, as in a static model,

14 When it does not create confusion, the subscript of signifies time period or a specific type depending on thecontext: so when we write i, we mean i,t. The extra notation would be superfluous in most of the paper.

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this property is guaranteed by the assumption that is sufficiently small. In the following we

will assume that this is the case, and so ignore the non-negativity constraints on qt ht1 .153 The dynamic envelope formula and optimality

In this section we characterize the sellers problem and we discuss the standard approach that has

been used in the literature to solve it- the so called first-order approach. The sellers problem

consists of choosing a contract p, q that maximizes profits under two sets of constraints: the

incentive compatibility constraints, that guarantee that agent i does not want to report being

a type j after any history ht; and the individual rationality constraints that guarantee that all

types receive at least their reservation utility U = 0 after any history ht

. Since the choice ofprices and quantities corresponds to the choices of utilities and quantities levels for the buyer,

this problem can be conveniently represented as a choice of utilities and quantities U, q =U

tht1 , qt ht1 Tt=1, whereUt ht1 is the expected utility of a type t after history

ht1.

The generic incentive compatibility constraint ICi,j(ht1) requires U(i|ht1)U(j; i|ht1),

whereU(j; i|ht1) the expected utility for a type i of reporting to be a type j at time t after

historyht1. This constraint can be easily rewritten in terms ofU, qas:

U(i|ht1) U(j; i|ht1) (1)

= U(j|ht1) +

Nk=0

(ik jk) U(k|ht1, j) + (j i)q(j|h

t1).

The individual rationality constraint for type i at history ht1, IRi(ht1), is a simple non nega-

tivity constraint:

U(i|ht1) 0. (2)

For future reference, we calllocal downward constraintsthe incentive constraints that are associated

with a deviation to a contiguous lower type (i.e. ICi,i+1(ht1)), and local upward constraints

the incentive constraints that are associated with a deviation to a contiguous higher type (i.e.

ICi+1,i(ht1)). We refer to all the other constraints as global. A contract that satisfies all the

incentive and rationality constraints is said to be implementable.

15 The exact condition that guarantees that the non-negativity constraints is non binding will be obvious in theanalysis since the optimal contract will be characterized in closed form.

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The monopolists problem constitutes maximizing expected surplus net of the buyers expected

equilibrium rents:

maxU,q

E [S(q)]

Ni=0

iU(i|h0)

s.t. IRi(ht1),I Ci,j(ht1) for any i, j and, ht1 Ht1, t

(3)This is a standard maximization problem of a concave function under a system of linear constraints.

AsTandNincrease, however, the number of variables and constraints becomes prohibitively large

making (3) analytically intractable.

The standard approach in the literature is to first study a relaxed problem in which only

individual rationality constraint of the lowest type and the local downward constraintsI Ci,i+1(ht

)are considered. The remaining constraints can be verified ex-post after the solution of the relaxed

problem has been characterized.

Definition 1. A contract is first-order optimal if and only if it maximizes profits under the

following constraints: ICi,i+1(ht) andI RN(ht), i N\{N}, ht Ht, t.

The interest in the FO-optimal contracts derives from the fact that in many environments they

coincide with optimal contracts. Under standard assumptions, the FO-optimal contract coincides

with the optimal contract in a static environment both with finite and continuous types (Stole

[2001]).16 This result has also been used in all papers that have extended the Principal-Agent

model to dynamic environments: the first-order autoregressive environment (Besanko [1985]) and

the Markov environment with two types (Battaglini [2005]). Perhaps more importantly, the applied

literature often focuses on FO-optimal contracts even in the absence of an explicit proof that local

incentive constraints are sufficient for implementability.17

It is easy to show that when we consider the relaxed problem with only local downward

constraints, the incentive compatibility constraints can be assumed to hold as equalities.18

This allows us to eliminate utilities from the optimization problem and drastically simplify the

constraint set. Define:

F(j |i ) = F(j |i ) F(j |i1 )

16 For the utility funtion we are assuming in this paper, the FO-optimal contract is optimal in a static environmentif the prior satisfies the monotone hazard rate condition. See Stole [2001] for discussion of these results.

17 See the Section 1.1 for a discussion of this literature.

18 The details of the statements made in this section are formally proven in the appendix.

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Recalling that H(ht) is the set of histories follow ht, and representing by hk the kth element of

hisotryh, we have the following characterization of the agents utility as a function of only q:19

Lemma 1. In correspondence to a FO-optimal contract we have:

U(i|ht1) U(i+1|ht1)

= q(i+1|h

t1)

+

hH(ht1,i+1)

>t

tk=t+1

F

hk

hk1 q(h|h1). (4)for anyi N\{N}, ht Ht1 andt = 1,...,T.

Lemma 1 presents a straightforward dynamic extension of the envelope formula introduced by

Myerson [1981]. This can be seen by takingto zero, in which case the second term in the righthand side vanishes and (4) coincides with the classic static formula. A continuous type version

of the formula is presented in Baron and Besanko [1984] for the case in which T = 3 and by

Besanko [1985] for an infinite horizon model with first-order autoregressive types in which shocks

have independent realizations. Battaglini [2005] states the formula for a Markov process with two

states: (4) is a direct, but more involved, extension of this result for the case with|| 2. Pavan,

Segal and Toikka [2011] present a general version of the formula for a continuous type space and

other stochastic processes.

Lemma 1 allows us to express the utility vector only in terms ofq. Define:

U(i|ht1; q) =

Nij=1

q(i+j|ht1)

+ hH(ht1,i+j)

>t

tk=t+1

F

hk

hk1 q(h|h1)

(5)

for any i < N, andU(N|ht1; q) = 0. The following result derives immediately from (4):

Corollary 1. In correspondence to a FO-optimal contract, we haveU(i|ht1) = U(i|h

t1; q)

for anyi N, ht1 Ht1, t.

Note, that this result allows us to express U(i|h0

) solely as a function ofq. The FO-optimalcontract can now be characterized as the solution of the following program:

maxq

E [S(q)]

Ni=0

iU(i|h

0; q)

(6)

19 To interpret (4) note that given a history h= (h1, ..., hs),his the realization of the type at time s. It

follows that q (h|h1) is the quantity at time when the realized history ish1.

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This problem can be solved to obtain the closed form solutions presented in Proposition 1. Let

D(ht) be equal to 1 at t = 1, and for t >1, define:

D(ht) =

0 ifht=0 for any t

t1=1

F(ht+1|h

t)

f(ht+1|ht)

else

From the first-order conditions of (6) we obtain:20

Proposition 1. In correspondence to the FO-optimal contract we have:

q

i|ht1

=i

1

Nk=jk

jD(ht1, i) i N, ht Ht1, t (7)

wherej= i for t=1 andj =ht11 for t>1.

The FO-optimal contract has very distinctive characteristics. As soon as the type becomes0,

the highest type, the contract becomes efficient in all following periods, a phenomenon that has

been called the Generalized No-Distortion at the Top. For any other history, the quantities are

distorted below and the distortion isj1k=0 k/i

D(ht1, i): this formula is complicated

because the wedge is state contingent and it depends on the entire history. The distortion,

however, converges to zero as the serial correlation of types converges to zero.

Given the (relatively) simple characterization of Proposition 1, the key question is: when is

it with loss of generality to focus on the first-order approach? Substituting (7) for the relevant

allocation, (1) and (2) are the necessary and sufficient conditions on the primitives for the imple-

mentability ofq, but they are not very practical to check. A much simpler sufficient condition for

the validity of the first-order approach is the following. Letq(ht) =q

htt|ht1

be an allocation

after history ht, and let ht ht ifhtj htj j t. We have:Definition 2. An allocation is monotonic ifq(ht) q(ht) for anyht ht.The following result was used in Battaglini [2005] to establish the sufficiency of (5) for the case

||= 2. The generalization to the case withNtypes is immediate:21

20 Note that, in the following expression, D(ht1, i) corresponds to D(ht) for ht =

ht1, i

.

21 Indeed in Battaglini [2005] it is shown that a weaker monotonicity condition than the one in Proposition 2

is actually sufficient. The condition requires thatN

k=0ikU

kht1, i

is non decreasing in i (see Step

1 of Claim 2 in Battaglini [2005]). This condition is implied by the monotonicity of the allocation as defined inDefinition 2. Analogous monotonicity results applicable to other stochastic processes have b een presented by Pavan,Segal and Toikka [2011], see Theorem 4.

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Proposition 2. The envelope formula (5) and monotonicity of the FO-optimal contract are

sufficient for implementability.

Proposition 2 directly parallels the well known results in static environments that show that

local incentive compatibility (i.e. the envelope formula) and monotonicity of the allocation are

necessary and sufficient for implementability. The result is however weaker for two reasons: first

the monotonicity condition is stronger than in a static environment, since it involves all histories

following a report; second, the result is only sufficient.22 The problem with Proposition 2 is that

it is useful only to the extent that it is easy to apply. The are a number of applications in which

the FO-optimal contract is monotonic, but the number is limited.

Example 1. Battaglini [2005] has considered a model identical to the model presented above,

but with||= 2: a high type and low type. In this case, the validity of the first-order approach

is established as follows. Note, that the FO-optimal contract is always distorted below and as

soon as the type is high, the contract becomes efficient. With two types distortions are present

only in the history in which the type is always low. This history corresponds to the lowest history

according to the order introduced above. It follows that the contract is monotonic according

to definition 1, and so the FO-optimal contract is optimal.

Example 2. Besanko [1985] has considered a case in which t = t1+ t, where t is the

realization of an i.i.d. random variable and (0, 1) is a parameter measuring persistence. The

Markovian framework developed above can be easily adapted to generalize Besankos environment.

We present a two period model to drive home the point in a simple fashion. The type in the first

period 1 is drawn from some prior with support 0,...N,where k k+1 = . The second

period type 2 is determined by the following stochastic process.

2 =1 + (1 )2,where 1 is the realization at t = 1 and

2 is the random realization of a Markov process with

withN+ 1 states, the same support as 1, and transition probabilities f(j|i) =ij (where ij

is the probability of2 =j given1 =i). Note that measure the level of autocorrelation. Ifwe assume ij =kj for any i ,j, k, then we have i.i.d. shocks and we are in a model isomorphic

to Besanko [1985].

22 This sufficient condition is stronger than what it is needed. The point here is to state the simplest sufficientcondition that allows us to apply the first order condition to all the environments in which it is known to work.

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When we consider only local incentive constraints, it is easy to show that they must hold as

equalities. In period 2, we have

U(k|i) = U(k+1|i) + (1 ) q(k+1|i),

whereU(k|i) and q(k|i) are respectively the second period utility and quantity of the agent

who has typeiin the first period and andi+(1 )kin the second. Without loss of generality

we can setU(N|i) = 0, so we haveU(k|i) = (1 ) Nl=k+1 q(l|i). Similarly, in the first

period, we have

U(i) = U(i+1) +

q(i+1|h

0) + N

k=0ikq(k|i+1)

+Nk=0

ik (i+1)k

U(k|i+1) (8)

The expected utility of the agent with type i is equal to the utility of type i+1 plus an in-

formational rent. The informational rent can be decomposed into two parts. First, we have a

deterministic part q(i+1|h0) + Nk=0

ikq(k|i+1): the realization of type at time 1 affects

rents att = 1 (i.e., q(i+1|h0)), but it effects rents at t = 2 as well, in a way that is proportional

to the degree of autocorrelation (i.e., Nk=0

ikq(k|i+1)). Second, we have the stochastic

part

Nk=0 ik (i+1)kU(k|i+1). This term depends only on the fact that a typei and a typei + 1 have different expectation on the probabilities of future types. Besanko [1985] assumes that

the shocks are i.i.d., so thatik= (i+1)k. In this case, the stochastic term of the information rent

is zero. The distortions are then exclusively deterministic. The first order optimal quantities are

given byq(i|h0) = i1N

k=i ki

,in period 1, andq(k|i) = i+ (1 )k 1N

k=i ki

in period 2. It is immediate to observe that, under monotone hazard rate on the priors, quanti-

ties are monotonic in the sense of Definition 1. It follows from Proposition 2 that the first order

approach works and these quantities describe the optimal contract. Following the same steps we

can generalize the analysis to Tperiods as in Besanko [1985], assuming t

= t1

+ (1 )twheret has the same i.i.d. distribution as2. In this case it is easy to verify that we have:

q(k|ht1) = j+ (1 )k

t1 1

Nk=i ki

wherej is the realization at time t 1 (i.e. ht1t1) andi is the realization at time 1 (i.e. h

t11 ).

Note that j+ (1 )k is the first best efficient quantity: the optimal contract is characterized

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by deterministic distortions that are independent of the Markov process governing the evolution

of types.23

Example 3. When the shock is i.i.d. it is easy to extend the analysis of Example 2 to a case with

k-order autoregression: t =kj=1

jtj+ (1 0)t with k

j=0j = 1. The example can also

be extended to a case in which t =l(t1, qt1) + ( 1 )t, wherel(t1, qt1) is a function oft1, the types up to t 1, and ofqt1, the allocation up to t 1. To see this point, note that

at timet the termskj=1

jtj or l(t1, qt1) are just constants for all types, so they do not

have any effect on incentives to reveal the true type. Another simple variation is to assume that

t =l1(t1, qt1)+(1 )l2(t1, qt1)

t, whereli(

t1, qt1)i = 1, 2 are functions oft1, the

types up to t 1, and ofqt1

, the allocation up to t 1. Both of these extensions of Besankosmodel have been suggested by Pavan, Segal and Toikka [2011]. The key assumption here is that

the shock is independent of the agents type.

The examples presented above show that the first-order approach can be extended to study

quite complex dynamic environments. All the examples, however, can be reconducted to two

basic assumptions. The environment studied in Besanko [1985] allows for many possible types

(infact a continuum), but with the stochastic shocks being uncorrelated to the agents type. In

this environment the shocks are irrelevant for the equilibrium distortions, which are independent

of the history of realized types (except for the first). The environment studied in Battaglini

[2005] allows the conditional distributions of the types to depend on the type, but limits the

analysis to two types only. In this case the optimal contract is history dependent. These

two environments have a common feature that explains why it has been impossible to find other

natural environments in which the first-order approach works: they limit, or completely neutralize

the multidimensionality of the dynamic screening problem. When there areNtypes, an agents

type is typically at least anN1 dimensional object: the type includes the shape of the conditional

distribution function, in the model described above. Limiting the analysis to these environments,

however, is with loss of generalityand itartificiallylimits the range of economic phenomena that

naturally arises in dynamic screening problems, as the example in Section 4.1 will clearly illustrate.

23 Besanko [1985] assumses the intial type to be determinstic, so ther are no priors in the model. The distortions,

therefore, are just functions of- distortion in period t being t1.

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4 Global constraints and optimality: impossibility results

In this section we discuss the conditions under which the first-order approach cannot be used,

and explore the structure of the optimal contract in these situations. We first present a simple

example with 3 types and 2 periods that can be fully solved. This example allows us to analyze

the complexities of an optimal dynamic contract when there are more than two types and the

shocks are not independent from the types. The example shows that global constraints cannot in

general be ignored and, even when they can be ignored, the envelope formula (4) is not sufficient to

characterize the optimal contract because additional monotonicity constraints need to be satisfied.

It also shows that, contrary to what happens when the first-order approach can be adopted, the

seller finds it optimal to strategically pool the types to extract higher rents: pooling is dynamic and

history dependent. We then generalize the results of the example by showing that the first-order

approach always fails when the time horizon is sufficiently important.

4.1 A fully characterized optimal contract

In this section we analyze a three types and two periods model. In particular, the set of types is

{H, M, L} where H> M> L withH M= = M L. The transition probabilities

between period one and two are described by f(|) = and f(|) = 12 for any ,

{H, M, L}, =

, with > 1/3. In this model, therefore, measures the persistence of the

types. Conditional on not being the same type, there is an equal probability of being any of the

other two types. An appealing feature of this simple transition matrix is that it ranks the types

according to first-order scholastic dominance, as requested in the model described in the previous

sections.

The first-order approach cannot generally be used in this setting. To characterize the optimal

contract we therefore focus on aweakly relaxed programthat constitutes problem (3) with||= 3

andT= 2 once we replace the constraint set with the following constraints:

IRL, ICHM, ICML , ICHL , (9)

ICHM(M), ICML(M), ICLM(M), ICHM(L), ICML(L), ICLM(L)

where IRLis the individual rationality constraint of type L at t= 0,ICi,jis incentive compatibility

constraint requiring that typei doesnt want to misreport being a type jin period 1, and ICi,j(k) is

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Figure 1: The dashed arrows are the constraints in the WR-problem that are ignored in the firstorder approach.

the incentive compatibility constraint requiring that typei doesnt want to misreport being a type

j in period 2, after the agent reports to be a type k in period 1. With respect to (3), this problem

has two key differences. First, now we are ignoring all the individual rationality constraints of the

lowest type in period 2 and incentive compatibility constraints after history H. Second, and most

importantly, we are adding three new constraints: the global downward constraintI CHL,and the

local upward constraints ICLM(M), ICLM(L) in period 2. The constraint set of the problem is

illustrated in the relevant history tree in Figure 1. In the following we will refer to this program

as theWR-program.

Since this is a three type and two period model we simplify notation. LetUi be the expected

utility of type i in the first period and ui(h) be the expected utility of type i after history h in

the second period. Note that since the second period is the terminal period, the expected utility

and stage utility are the same. Similarly, we defineqi andqi(h) to be the first and second period

allocations respectively. The following lemma allows to simplify the constraint set:24

Lemma 2. In the WR-program, constraintsIRL, ICHM, I CML bind at the optimum.

We can now use the equalities implied by Lemma 2, to reduce the number of free variables

24 When only the usual local downward incentive compatibility constraints are considered, the following result isimmediate. If, for example I CHMwere not binding, the principal could simply raise the price a type His paying.In a WR-problem the proof of the result is complicated by the additional constraints: reducing type Hs rent att= 0 may conflict with I CHL .

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in the optimization problem. In particular we can eliminate the period 1 utility vectors. Define

HM(i) = uH(i) uM(i) and ML(i) = uM(i) uL(i) for i = M, L. The variablekl(i) is the

net utility of reporting to be type k rather than a type l after history i. Using this notation, we

can rewrite the WR-programas a maximization problem in which the control variables are the

quantities q and second period marginal utilities :

max,q

i=H,M,L

i

iqi

12q

2i +

k=H,M,L

(k |i )

kqk(i) 12qk(i)

2

H

qM+ 312 HM(M)

(H+ M)

qL+

312

ML(L)

(10)

subject to

[] : qM+ 3 1

2 HM(M) qL+

3 1

2 HM(L)

[HM(M)] : HM(M)qM(M) | [HM(L)] : HM(L)qM(L)

[ML(M)] : ML(M) qL(M) | [ML(L)] : ML(L) qL(L)

[LM(M)] : ML(M)qM(M) | [LM(L)] : ML(L) qM(L)

where the variables in the square brackets on the left are the Lagrange multipliers associated with

the constraints. Program (10) is a standard maximization problem, but it is complicated by a still

significantly large number of constraints. The first constraint is global: it prevents the high type

from reporting to be a low type. All the remaining constraints are local, associated to deviations to

a contiguous type. The difference between (10) and the problem of the first-order approach (6) is

the global constraintI CHL and the presence of the upward constraints I CLM(M) andI CLM(L).

The upward constraintsICLM(M) andICLM(L) are essentiallymonotonicity conditionsrequiring

qM(h) qL(h) forh =M,L.25 We cannot ignore any of these three constraints. Moreover now

we cannot assume without loss of generality that all local downward incentive constraints are

binding att = 2: so the envelope formula (4) cannot be directly applied. This is why we still have

utilities in the objective function. We have:

Lemma 3. A contract is optimal if and only if it solves the WR-program.

25 To see this note that given I CML (h), qM(h) qL(h) if and only if ICLM(h) is satisfied.

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Figure 2: Fully characterized contract

We can therefore focus on problem (10). The analysis can be divided into two cases: first the

case in which the global constraint can be ignored and so it is sufficient to look at local constraints,

i.e. = 0; second, the case in which the global constraint is binding, i.e. > 0.

The following result characterizes the necessary and sufficient condition for = 0. For a given

L and, the environment is fully described by two parameters, M, , and therefore it can be

represented in the two dimensional box (M, ) E(L) = (0, 1 L) (1/3, 1).26 In the rest

of the analysis we will fix L andand study how the equilibrium changes as we change M, .

This approach is without loss of generality and it allows for simpler statements (and a graphical

representation) of the relevant cases. We have:

Lemma 4. There exists a threshold () such that the global incentive constraintICHL can be

ignored if and only ifM ().

The reason why the global constraint may bind in this WR-programis that in correspondence

with the FO-optimal contract, the expected rent of an agent in the second period is not generally

monotonic in the type revealed in the first. To see this point, consider the global incentive

26 The thresholds defined below do not depend on the types.

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constraint I CHL. BecauseI CHM is binding, the rent of a high typeUH is equal to the expected

utility that a type Hreceives in equilibrium by reporting to be a type M, that isU(M; H),where

U(j; H) is the expected utility that a type Hreceives in equilibrium by reporting to be a type

j. The global constraintI CHL can therefore be written as U(M; H) U(L; H), or:

j=M,L

qj+ 3 1

2

j=M,L

qj(M)2qL+ 3 1

2

j=M,L

qj(L)

Each side of the inequality has two components: a static term,j=M,L

qj, on the left hand

side, and 2qL, on the right hand side; and a dynamic term,j=M,L

312 qj(M), on the left

hand side and312

j=M,L qj(L), on the right hand side.

It can be verified that the static term on the right hand side is always lower than the staticrent on the left hand side since qL qM.27 The same, however, is not true for the dynamic terms

becausej=M,L qj(L) may be strictly larger than

j=M,L qj(M). For example, we always have

qM(M) qM(L). Whenj=M,L qj(L) >

j=M,L qj(M), ICHL may become binding. Why

does the seller find it optimal to offer a lower quantity to an M type after history M than after

historyL (i.e.,qM(M) qM(L))? In static contracts, the intuition behindqMqL is relatively

simple. The quantities chosen after historyL determine the informational rent of type Mand of

type H28 ; the quantities chosen after M determine only the informational rent of type H: this

is why in a static environment, we should expect that distorting quantities after L should have alarge impact on total expected rents in the first period, and so be a better choice for the seller.

Dynamic contracts, however, are different. A reduction in qM(M) induces a reduction of

uH(M), the rent of a high type after M. If this reduction is equal to, say, the expected

reduction in the rent of type H in the first period is : the expectation is multiplied by

since even after a deviation, a high type in the first period is still highly likely to remain a high

type in the second. Similarly, a reduction (say ) in qM(L) induces a reduction in uH(L) and

the expected rent of type M, equal to (1 ) . Note that the expected change in rent is

now proportional to 1 rather than . Thus, distortions in histories with constant types(i.e. for exampleh = M, M) have a greater impact on rents at t = 0 and they are a much more

potent screening device than distortions after histories with non constant types (i.e. for example

h= L, M). Non-monotonic allocations, therefore, should not be seen as a phenomenon induced

27 Explicit solutions for the optimal quantities for all feasible parameters are presented in Table 1 in the appendix.

28 As in static contracts, type Hreceives the informational rent of type Mplus an additional rent.

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by lack of regularity. As we show in this and the next subsection, they are infact a prominent

featureof the optimal solution in dynamic screening problems.

Within the two regions defined by (), the particular shape of the optimal contract depends

on the remaining set of binding constraints. The following proposition describe how the optimal

contract looks like for M (), when the global constraint an be ignored:

Proposition 3. AssumeM (). There is a threshold0(M),such that:

- Case A1. If < 0(M), the optimal contract is fully separating and first-order optimal.

- Case A2. If 0(M), the optimal contract is fully separating after all histories except

h1 =M. After this history types M and L are pooled: qM(M) =qL(M).

Regions A1 and A2 are illustrated in Figure 2 in a simple parametric example, where the

threshold 0(M) is represented by a dashed line.29 In region A1, the envelope formula is

sufficient to characterize the optimal contract. It is interesting to note, that in this region,

however, the optimal contract is not monotonic. This is a feature that, as discussed in Section 3, is

never true whenN= 2, or when the shocks are serially uncorrelated as in Besankos autoregressive

model. In addition, Case A2 in Proposition 3 shows that even if we can ignore the global incentive

compatibility constraints, the optimal contract cannot be characterized by simply relying on the

envelope formula (4). Under standard regularity conditions, in static environments the contract

that is optimal under only local constraints is monotonic. This is not necessarily the case in

a dynamic environment. While in this case the missing binding constraint set can be easily

identified, in more general environments withNtypes andTperiods, it is likely to be hard to pin

down which incentive constraints are binding. In these cases the envelope formula alone is not

sufficient to characterize the optimal contract.

WhenM< ()boththe global constraintICHL andthe local constraintsICHMand ICML

are simultaneously binding in the first period. There are three relevant cases. The following result

characterizes the optimal contract in these situations:

Proposition 4. There exists a threshold() such that:

- Case B1&B2. AssumeM[(), ()). The optimal contract is fully separating at

t= 1. There exists a threshold0(), such that the optimal contract is fully separating at

29 In Figure 2 we assume L = 0.25 and = 0.95.

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t = 2 as well ifM > 0() (case B1). IfM 0(), typesM and L are pooled after

historyM : qM(M) =qL(M) (case B2).

- Case B3. If M < (), the optimal contract pools typesM and L in the first period:

qM = qL. In the second period, after history H, the contract is separating and efficient.

After histories M andL, typesM andL are pooled across both histories: qM(j) = qL(j)

andqj(M) = qj(L) forj = M, L.

Propositions 3 and 4 provide a full characterization of the optimal contract. Table 1, presented

in the appendix, describes the closed form solution of the optimal quantities for all parameters

regions. It is interesting to note that although in regionB 1 the global constraint is binding, we

have full separation of the types. In region B2, the global constraint is binding and we have

separation in the first period, but not in the second; in region B3, we have pooling ofM andL

both in the first and second periods.

An important lesson of this example is the significance of pooling in dynamic contracts, a

phenomenon that is not apparent in environments that are solvable with the simple envelope

formula. Pooling may occur only in period 2 (regionsA2 andB2) or both in period 1 and period

2 (region B3). Regions A2 and B2 are interesting because they involve strategic separation of

types in period 1 followed by history dependent pooling in period 2, we term this dynamic pooling.

Finally, note that in dynamic contracts, unline static, global incentive constraints can bind

even types are fully separated. In casesB1 and B2, first period quantites are separated at the

opitmum even though the first period global incentve constraint I CHL binds.

4.2 General impossibility results

The results of the previous section show that even in a very simple example, the envelope theorem

and local incentive constraints are not sufficient to characterize the optimal contract. In this

section we generalize the results by showing that with N types and T periods, the first order

approach is not valid if the time horizon and persistence are sufficienty important. To make

this point, we continue to assume that types persistence is parameterized by a single parameter

, such that = ii. Now, however, besides allowing for any || 2 and T 2, we put no

restrictions on the off-diagonal probabilitiesij (i=j), that can be written asij =ij(1 ),

whereik are generic constants withk=i

ik = 1 i. We say that the first-order approach fails

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as 1 if there is a such that for > at least one of the omitted global constraints fails

to be satisfied when the allocation is given by the first-order optimal contract (7).

Proposition 5: There exist a < 1, and aT 2 such that for any > andT T the

first-order approach fails as 1.

Proposition 5 shows that the first-order approach fails precisely when it seems more important:

when the future horizon is important (many future periods, non myopic agent) and when types

are sufficiently persistent. Note that the proposition above puts no restriction whatsoever on the

priors. The fact that we require the future horizon to be sufficiently important for the result to

hold should not be surprising: when types have low levels of persistence, then the dynamic nature

of the environment has less bite since the agent has limited information about the future when

signing the contract. Corollary 2 presented below shows that a highT andare not necessary

for the FOA to fail. Consider environments that satisfy the following weak assumption:

Assumption 1. 1N1NN1

(N2)N(N1)N

>1.

This assumption is weak because it is natural to assume that the lowest two types do not have

a large mass when N is large. Note that1N1NN1

ifN1, N 0 as N increases. In

this case, if the ratio (N2)N/(N1)N is finite, then Assumption 1 is verified for N sufficiently

large. An example that satisfies Assumption 1 is the case with a uniform priori = 1/(N+ 1),

and a transition matrix ii = , ij = (1 ) /N for i = j as in the model of Section 4.1. We

have:

Corollary 2 Suppose Assumption 1 holds. Then, there exists (0, 1)such that the first-order

approach fails for all >0 andT2.30

The table in figure 3 provides additional support to the results of Proposition 5 and Corollary

2 by computing the lower bound forN= 4 (5 types) andT= 3, and a variety of configurations

of the other parameters. In the table it is assumed that the types follow a bell shaped transition

matrix in which the probability that type i becomes a type j declines with the distance between

i and j :

i,j= 1

1 + |i j| (11)

30 Note that the FOA does not fail when T= 1, but it may fail with T >1 for any arbitrarily small. Indeed,whenT= 1 there is no second period in which the incentive compatibility constraints can fail; when T >1 we canalways find a future t Tin which some necessary incentive constraint fails.

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Figure 3: Failure of FOA for ||= 5, T= 3.

where is a parameter regulating the curvature of the transition probabilities and =i=j

11+|ij|

is a constant chosen to have the probabilities sum to one. The left hand panel shows examples

in which = 0, and so the off-diagonal probabilities are uniform as in the model of Section 4.1;

the right hand panel assumes = 1.

The fact that global constraints are binding when types are highly correlated may seem surpris-

ing since it is well known that when types are constant (and so perfectly correlated) the optimal

contract is just a repetition of the optimal static contract (Baron and Besanko [1984]). As per-

sistence converges to one, shouldnt we expect the optimal contract to converge to the optimal

contract when persistence is one? If this were the case, then local incentive constraints would be

sufficient. Proposition 5 and Corollary 2 show that there is a failure of lower-hemicontinuity of

the optimal contract with respect to types persistence. The intuition is very simple. As per-

sistence converges to one, the FO-optimal contract does converge to the optimal static contract

on the main history, that is the history in which types remain constant. Outside this history,

however, the contract is generally not monotonic, even for arbitrarily high levels of persistence.

Implementability fails precisely because of this non monotonicity, rendering the FO-optimal con-

tract not incentive compatible. So, then, why is optimal to have repetition of the static optimal

contract when persistence is perfect? The reason is thatany allocation is optimal (including a

repetition of the static contract) after an history that has probability exactlyequal to zero. How-

ever, even an arbitrarily small probability to reach the non-constant histories makes the repetition

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of the static contract suboptimal and demands non-montonic allocations as part of the optimum.

5 Ironing, implementability and optimality

The results of the previous sections make clear that in order to solve for an optimal contract, the

principal cannot generally use the first-order approach and limit the analysis to local incentive

compatibility constraints. Without the first-order approach, we have no systematic way to simplify

the constraint set: potentially, all constraints could be binding. This may make the analysis

extremely complicated even from a numerical point of view. What does the optimal contract

looks like in general environments with large T andN? What kind of advice can we give to a

seller who needs to design an optimal contract? In this section we show that there is a classof contracts that is relatively easy to characterize, and that induces a minimal loss (if any) on

the principals payoff precisely when the first-order approach fails, that is, when the agents types

are highly persistent. This class consists of contracts that are monotonic as in Definition 2.

In static environments the envelope formula plus monotonicity are necessary and sufficient for a

contract to be implementable: if we ignore the monotonicity constraint, then the contractmust

be ironed out to make it monotonic, otherwise implementability fails (see Myerson [1981]). In a

dynamic environment monotonicity is not necessary: it follows that if we impose monotonicity in

the sellers problem, we guarantee implementability even if we ignore the global constraints, butwe may obtain a suboptimal contract. The main result of this session is that as types persistence

converge to one, the optimal monotonic contract converges in probability to the optimal contract,

and so the loss from focusing on this class of contracts converges to zero.

DefineM as the set of monotonic contracts:

M =

q

q(i|ht1)q(i+1|ht1), i < N, andq(i|ht1) q(i|ht1),i= 1,...,N, ht1 andht1

ht1

. (12)

where, as before, ht

ht ifhtj htj j t. It follows immediately from Proposition 2 that theoptimal monotonic contract can be characterized by solving the following program:

maxqM

E [S(q)]

Ni=0

iU(i, h

0; q)

(13)

whereU(i, h0; q) is given by the envelope formula (5). Problem (13), moreover, is sufficiently

tractable to allow a partial characterization of the properties of its solution.

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Figure 4:

Proposition 6. In the optimal monotonic allocation, q

tht1 tfor anytandht1. More-

over, for any arbitrarily small1, 2> 0 there exists aT such thatPr

qt ht1 t> 12 for any environment withTT periods.

The first part of the proposition establishes that analogous to the static model, the optimal

monotonic contract is uniformly downward distorted. The second part of the proposition states

that the contract converges to an efficient contract in probability.

What about the loss in profits for high levels of persistence, owing to the deployment of the sub-

optimal monotonic contracts? Letq = {q(ht)}htHbe an allocation and letq= {q (ht)}htH

be the optimal allocation. Let I be the identity matrix that describes the transition matrix

when types are perfectly correlated. As types become perfectly persistent, we must have that

the transition matrix converges to I, i.e. I. We say that q converges in probability to the

optimal allocation as types become perfectly persistent if limIPr (|q(ht) q (ht)| ) = 0

for any >0.

Proposition 7. The optimal monotonic contract converges in probability to the optimal allocation

as types become perfectly persistent.

A major implication of above proposition is that, the sellers profit associated with the op-

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timal monotonic contract converges to the profit level in the optimal contract as types become

increasingly persistent.

The table in Figure 4 illustrates the loss of profits associated with the optimal monotonic

contract in an example with 3 periods and 3 types and the Markov matrix used in section 4.1.

The loss is expressed as a percentage of the profit in the optimal contract. As can be seen, the

approximation is quite good for all cases, with a loss of profit that is never higher than 0.06%.

The table also provides the loss that would be incurred by ignoring the dynamic nature of the

problem and repeatedly offering a state uncontingent optimal static contract: the loss can be

higher than 10% of the optimal profits, even in this trivial example with only 3 periods. Naturally

larger losses should be expected with longer horizons.

It is interesting to note the inverse-U relationship between losses and the level of persistence.

As persistence increases, losses increase, peak and then come down again. The reason is simple. At

= 1/3, the model is akin to the iid shock framework, where we know that the optimal contract

is monotonic. At the other extreme, = 1, the optimal contract constitutes repititon of the static

opitmum which too is monotonic. As we increase , the distortions vary and the probability

of non-constant histories decreases. Thus, the loss in using monotonic contracts increases with

the non-monotonicities only to be supressed in probability by the increasing weight of constant

histories along which the optimal monotonic allocation converges to the optimal allocation.

The results of this section may be useful in applied work. As mentioned in the introduction,

many works in the applied literature postulate that the first-order approach works. The risk is

that the contracts thus characterized are not incentive compatible. Further in the most natural

environments, this risk can not be fully resolved by numerical methods.. To the extent that it is

not possible to check al lthe incentive compatibility constraints, studying the optimal monotonic

contract may be a more robust option, since it guarantees implementability and it is equal to the

true optimal contract with high probability when types are highly persistent.

6 Conclusion

In this paper we have studied a simple Principal-Agent model in which the agents type is private

information and follows a Markov process. We have presented three sets of results. First,

following the standard approach in the literature, we have studied the optimal contract when only

local incentive constraints are considered. We have shown that the agents equilibrium rents can

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be represented purely as a function of the allocation through a dynamic version of the so called

envelope formula. Moreover, as in static model, the envelope formula and a natural monotonicity

condition on the allocation guarantee that the contract is implementable. Although this condition

is only sufficient and quite strong, it is verified for virtually all the natural environments in which

the optimal dynamic contract has been characterized in the existing literature.

Second, and most importantly, we have shown that the environments for which the dynamic

envelope formula is sufficient to characterize the optimal dynamic contract are quite special.

In general, even in the simplest examples, the allocation is not monotonic and local incentive

constraints are not sufficient for implementability. We have first proved this result in a simple

case with three types and two periods where the optimal contract can be fully solved in closed

form. The characterization shows that the optimal contract is characterized by what we call

dynamic pooling: strategic, state contingent treatment of types in which types may be initially

separated, but then be pooled conditioned on particular histories. We have then generalized the

analysis showing that with N types and T periods the applicability of the first-order approach

is even more limited: under quite weak conditions it fails when the time horizon is sufficiently

important. The fact that the envelope formula is not sufficient to characterize implementability

suggests that a new approach is required to study dynamic contracts in general environments.

Finally, we have shown that some insights on the optimal contract in general environments

with many types and periods can be gained by studying a simple class of suboptimal contracts:

monotonic contracts, in which non monotonicities in the allocation are ironed out. The appeal

of this class is that the optimal monotonic contract converges in probability to the optimal contract

as the persistence of types converges to one, that is precisely when the first-order approach tends

to fail.

The analysis suggests a number of important research questions. The characterization of the

optimal contract with three types and two periods suggests that state dependent pooling of types

plays an important role in dynamic screening. The example suggests a number of features that

are natural to expect to hold in more general environments as well. The analysis in section 5,

moreover, suggests that even when it is not possible to fully characterize the optimal contract,

useful insights can be gained by studying contracts that are approximately optimal. We leave

the further development of these idea for future research.

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References

Athey S. and I. Segal (2007), An Efficient Dynamic Mechanism, mimeo.Baron, D. and D. Besanko (1984), Regulation and Information in a Continuing Relation-ship, Information Economics and Policy, 1(3):267-302.

Battaglini, M. (2005), Long-term contracting with Markovian consumers. American Eco-nomic Review. 95, 637658.

Battaglini, M. (2007). Optimality and Renegotiation in Dynamic Contracting, Gamesand Economic Behavior, 60 (2), 213-246.

Battaglini M. and S. Coate (2008). Pareto Efficient Income Taxation with Stochastic Abil-ities, Journal of Public Economics, 92:3-4, pp. 844-68.

Bergemann D. and J. Valimaki (2010). The Dynamic Pivot Mechanism, Econometrica,

78(2), 771-789.

Besanko, David (1985). Multiperiod Contracts between prinipal and Agent with Adverseselection,Economic Letters, 17, 33-37.

Bolton, P. and M. Dewatripont (2005). Contract Theory. The MIT Press, Cambridge, MA.

Chassang, S. (2011), Calibrating Incentive Contracts, working paper.

Courty, P. and H. Li (2000), Sequential Screening, Review of Economic Studies, 67(4):697-718.

Eso, P and B. Szentes (2007), Optimal Information Disclosure in Auctions and the HandicapAuction, Review of Economic Studies, 74(3), 705-731.

Farhi, E. and I. Werning (2011), Insurance and Taxation over the Life Cycle, workingpaper.

Garrett, D. and A. Pavan (2011), Managerial Turnover in a Changing World, , workingpaper.

Golosov, M., M. Troshkin, and A. Tsyvinski, (2011), Optimal Dynamic Taxes, workingpaper.

Hartline, J. (2012), Approximation in Mechanism Design, American Economic ReviewP&P.

Inderst R. and M. Ottaviani (2011), How (not) to pay for advice: A framework for consumerfinancial protection, mimeo.

Kapicka, M., (2011), Information Economies with Persistent Shocks: A First-Order Ap-proach, working paper.

Krahmer, D. and R. Strausz (2011), The Benefits of Sequential Screening working paper.

Laffont, J.J. and D. Martimort (2002),The Theory of Incentives,Princeton University Press,Princeton, NJ.

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Laffont J.J. and J. Tirole (1990): Adverse Selection and Renegotiation in Procurement.Review of Economic Studies, 1990, 57(4), pp. 597625.

Laffont, Jean-Jacques and J. Tirole (1993), A theory of incentives in procurement and reg-ulation. Cambridge, MA: MIT Press, 1993.

Madarasz, K. and A. Prat (2010), Screening with an Approximate Type Space, workingpaper.

Maestri, L. (2011), Dynamic Contracting under Adverse Selection and Renegotiation, working paper.

Milgrom, P. (2004), Putting Auction theory to Work, Cambridge University Press.

Myerson R. (1981), Optimal Auction Design, Mathematics of Operations Research.

Pavan, A., I. Segal, and J. Toikka (2011), Dynamic mechanism design: Incentive compati-

bility, profit maximization and information disclosure.working paper.

Stole, Lars (2001), Lectures on the Theory of Contracts and Organizations, mimeo, TheUniversity of Chicago.

Zhang Yuzhe (2009), Dynamic Contracting with Persistent Shocks, Journal of EconomicTheory, 144, 635-675.

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7 Appendix

7.1 Proof of Lemma 1 and Corollary 1

We first show that all the constraints in the relaxed problem can be assumed to hold as equalities.

Lemma A1. In a FO-relaxed problem: IRN(ht1) can be assumed to hold as equality for all

ht1 Ht1; ICi,i+1(ht1) can be assumed to hold as an equality for all ht1 Ht1 and

i= 0, 1,...,N 1.

Proof. The proof of this result is in the on line appendix. The appendix is available for

download at http://www.princeton.edu/mbattagl/OptDyn appendix.pdf.

We can now prove Lemma 1 and Corollary 1 together. We shall proceed by (backward)induction ont. Note that at t = T, Lemma A1 implies:

U(N|hT1) = 0 and U(i|h

T1) = Nil=1

q(i+l|hT1)i N 1. (14)

Similarly, for t =T 1, we have for i N 1:

U(i|hT2) = q(i+1|h

T2) + U(i+1|hT2) +

Nk=0

(ik (i+1)k)U(k|hT2, i+1)

=Ni

n=1 q(i+n|hT2) + N

k=0((i+n1)k (i+n)k)U(k|hT2, i+n)= Nin=1

q(i+n|h

T2) + N1k=0

((i+n1)k (i+n)k)Nkl=1

q(k+l|hT2, i+n)

Now, it is easy to verify that:

N1k=0

((i+n1)k (i+n)k)Nkl=1

q(k+l|hT2, i+n) =

Nk=1

k1l=0

(i+n1)lk1l=0

(i+n)l

q(k|h

T2, i+n)

=Nk=1

F(k |i+n )q(k|hT2, i+n)

It follows that we can write:

U(i|hT2) =

Nin=1

q(i+n|h

T2) + Nk=1

F(k |i+n )q(k|hT2, i+n)

(15)

= Nin=1

q(i+n|hT2) + hH(hT2,i+n)

>T1

tk=t+1

F

hk

hk1 q(h|h1)

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It is easy to see that (14) and (15) prove the statement in Corollary 1 and in Lemma 1 respectively

fort = T andt = T 1. We therefore conclude that our hypothesis holds fort T 1. Next,

suppose it holds for t + 1 where t T 2. We want to show that it holds for t. We have,

U(i|ht1) = q

i+1|h

t1

+ U

i+1|ht1

+ Nk=0

(ik (i+1)k)U(k|ht1, i+1) (16)

=Nin=1

q

i+n|h

t1

+ Nk=0

((i+n1)k (i+n)k)U

k|ht1, i+n

= Nin=1

q

i+n|ht1

+ N1k=0

((i+n1)k (i+n)k)Nkm=1

q

k+m|ht1, i+n

+

hH(ht1,i+n,k+m) >t+1(t+1)

k=t+2Fhk hk1 qh|h1,where the third equality follows from the induction hypothesis. Now,

N1k=0

((i+n1)k (i+n)k)Nkm=1

q(k+m|ht1, i+n) =

Nk=1

F(k |i+n )q(k|ht1, i+n),

(17)

and,

N1k=0

((i+n1)k(i+n)k)Nkm=1

hH(ht1,i+n,k+m)

>t+1

(t+1)k=t+2

F

hk

hk1 qh|h1

=N1k=0

((i+n1)k (i+n)k)Nkm=1

Qk+m,

where,

Ql=

hH(ht1,i+n,l)

>t+1

(t+1)k=t+2

F

hk

hk1 qh|h1 .With some algebra we can write:

N1k=0

((i+n1)k (i+n)k)Nkm=1

Qk+m= Nk=1

F(k|i+n) Qk (18)

Combining (17) and (18) we obtain:

Nk=1

F(k|i+n) q(k|ht1, i+n) + Qk (19)= Nk=1

F(k|i+n)

q(k|ht1, i+n)+ hH(ht1,i+n,k)

>t+1

(t+1)k=t+2

F

hk

hk1 qh|h1

=

hH(ht1,i+n)

>t

tk=t+1

F

hk

hk1 qh|h1

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Combining (16) and (19), we obtain:

Ui|ht1= Nin=1

qi+n|ht1 + hH(ht1,i+n)

>t

tk=t+1

Fhk hk1 q(h|h1) .This proves Corollary 1. SubtractingU(i+1|h

t1) and dividing by from the above ex-

pression gives us Lemma 1.

7.2 Proof of Proposition 2

Recall that U(kht1, i ) = U(k ht1, i ) U(k ht1, i+1 ). We start with a useful lem-

mata.

Lemma A2. Ifq(i|ht1) andU(k ht1, i ) are non increasing ini for anyht1, then (5)implies that local upward incentive compatibility constraints are satisfied.

Proof. The proof of this result is in the on line appendix.

Lemma A3. If q(i|ht1) andU(kht1, i ) are non increasing in i for any ht1 and (5)

holds, then the local incentive compatibility constraints imply the global incentive compatibility

constraints.

Proof. The proof of this result is in the on line appendix.

Given the lemmas presented above, Proposition 2 is proven if we establish that when the

allocation is monotonic as defined in Definition 2, then q(i|ht1) and U(kht1, i ) are non

increasing in i for any ht1. The fact thatq(i|ht1) is non increasing in i for any ht1 is an

immediate consequence of the monotonicity. The fact that U(kht1, i ) is non increasing in

i for any ht1 is established by the following result.

Lemma A4. If the allocation is monotonic, thenU(kht1, i )is non increasing in i for any

ht1.

Proof. Note first thatU(N ht1, i ) = U(N ht1, i+1 ) = 0, so U(N ht1, i ) = 0. ByLemma 1, we have:

U(N1|ht1, i)/= q(N|h

t1, i) +

hH(ht1,i,N1)

>t+1

t1k=t+2

F

hk

hk1 q(h|h1).(20)

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It is useful to write this expression with a different notation. LetHt(i) be set of realizations

of lengthT tthat start with the first element equal to i (we denote this the typical element of

Ht(i), so th1 = i). A possible history of lengtht following ht with (t + 1)-th element equal

to i (ht+1= i) is then h

={ht,t ht} for thHt(i) (by convention we write ht =

ht,t h0

).

We can then write:

U(N1ht1, i )/ = q(N|ht1, i) +

thHt(N1)

>t+1

t1

l=t+2

F(thl |thl1 )

q(th|ht1, i,t ht1)

(21)Similarly we can write:

U(N1ht1, i+1 )/ = q(N|ht1, i+1) + (22)

thHt(N1)

>t+1

t1

l=t+2

F(thl |thl1 )

q(th|ht1, i+1,t ht1)

(23)Therefore we have:

U(N1 ht1, i )/

= q(N|ht1, i) q(N|h

t1, i+1)

+

thHt(N1)

>t+1

t1

l=t+2

F(thl |thl1 )

q(th|ht1, i,t ht1)q(th|ht1, i+1,t ht1)

Note that by monotonicity, we must have q(N|ht1, i) q(N|ht1, i+1) 0 and

q(th|ht1, i,t h

t1) q(th|ht1, i+1,t h

t1) 0

It follows that U(N1 ht1, i ) U(N

ht1, i ). Assume now that U(j ht1, i ) is

monotonic in j forj m. We prove the result by induction proving that U(m1 ht1, i )U(m

ht1, i ). Applying Lemma 1 and using the notation developed above, we have:

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U(m1 ht1, i )/= U(m

ht1, i )/+ q(m|ht1, i) q(m|ht1, i+1)+

thHt(m1)

>t+1

t1

l=t+2

F(thl |thl1 )

q(th|ht1, i,t ht1)q(th|ht1, i+1,t ht1)

And so by monotonicity of the allocation we have U(m1

ht1, i ) U(m ht1, i ). 7.3 Proof of Lemma 2

First, we prove a useful lemma that will be invoked in the proof of Lemma 2.

Lemma A5. The optimal solution satisfies: qL L andqM(L) M.

Proof. The proof of this result is in the on line appendix.

Now, we show thatI RLbinds. Suppose not. DecreaseUH, UM, ULby the same small amount.

The first period incentive compatibility constraints continue to hold and the second period con-

straints are unaffected. This increases the profit of the monopolist without disturbing any of the

constraints, giving us a contradiction. Thus, UL = 0. Next, we show that I CML binds. Suppose

not. DecreaseUM by. Then, all the constraints are satisfied and we increase the monopolists

profit, giving us a contradiction. Using these two binding constraints we can eliminate UL and

UM from the maximization problem. In particular, I CHMcan now be written as

UH (qM+ qL) + 3 1

2 [(uH(M) uM(M))+ (uM(L) uL(L))]

Also,I CHL is given by

UH2qL+ 3 1

2 [uH(L) uL(L)]

First, note that at least one of ICHM and ICHL must bind. If not, then we can decrease

UHand increase the monopolists profit. Suppose I CHMdoes not bind. Then, I CHL must bind.

Thus, we can eliminate UH from the maximiztion problem. In particular, ICHM can now be

written as

qL+ 3 1

2 [uH(L) uM(L)] qM+

3 1

2 [uH(M) uM(M)]

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Second, we claim that if ICML and ICHL bind and ICHM does not bind, then ICHM(L)

binds. SupposeuH(L) uM(L)> qM(L). DecreaseuH(L) by, thereby, increasing the profit

of the monopolist without disturbing any of the remaining constraints, giving us a contradiction.

Thus,I CHM(L) must bind.

Rewriting , using I CHM(M) and the binding I CHM(L) we get

qL+ 3 1

2 qM(L) qM+

3 1

2 M

Since ICHMdoes not bind, it is easy to see that qM = M. In Lemma A5, we showed that

qL L (and thus qL < M), and qM(L) M. These clearly contradict the above inequality.

Thus, we must have that I CHM binds.

7.4 Proof of Lemma 3

We prove the lemma as follows. LetU = U(ht) be the vector of expected utilities, mapping an

historyht to the corresponding agents expected utility. First, we construct a vector of utilities U

using the solution of the WR-problem,, q. We then show that the solutionU, qsatisfies all

the constraints of the sellers profit maximization problem and it maximizes profits. We proceed

in two steps:

Step 1. We setuL(M), uL(L),uL(H) all equal to zero. We also define:

uM(M) = ML(M), uM(L) = ML(L), uM(H) = qL(H)

uH(M) = ML(M) + HM(M), uH(L) =ML(L) + HM(L), uH(H) = (qL(H) + qM(H))

Since I RL,I CML andI CHMhold as an equality, we must have:

UL = 0,

UM = qL+ 3 1

2 ML(L), and

UH = UM+ qM+

3 1

2 HM(M).

Step 2. We now show thatU, qsatisfies all the constraints of the profit maximizing problem.

By construction it is immediate that U, q satisfies all the constraints in the WR-problem. It

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reamains to be shown that it also satisfies the other constraints,

IRH, IRM, ICMH, ICLM, ICLH, (24)

ICHM(H), ICML(H), IRL(H), IRL(M), IRL(L)

ICMH(H), ICLM(H), ICLH(H), ICHL(H)), ICMH(M),

ICLH(M), ICHL(M), ICMH(L), ICLH(L), ICHL(L).

First, we show that I RM is satisfied. FromI CML we have

UM=UL+ qL+ 3 1

2 [uM(L) uL(L)]

= qL+

3 1

2 [uM(L) uL(L)] [UsingI RL]

qL+ 3 1

2 qL(L)>0 [Using I CML(L)]

Similarly, we can show that IRH is satisfied. To prove the remaining constraints we need the

following properties of the solution of the WR-problem.

Lemma A6. For all parameter configurations, in the WR solution we have: 1. qi(H) =i fori =

M,L,H, qM(M)< M, qL(M) L, andqL(M) qL(L) 2. HM(M) = qM(M), ML(M) =

qL(M), HM(L) = qM(L); 3. quantities att = 2are nondecreasing in type after any history;

4. qHqMqL.

Proof. The proof of this result is in the on line appendix.

Consider the first period constraints. To show thatI CLMholds it is sufficient to prove:

0 = UL LqM+

uL(L) +

1

2 uM(M) +

1

2 uH(M)

(25)

= UM qM 3 1

2 uL(M)

= UM qM 3 1

2 qL(M)

Since UM= qL+ 312 qL(L), (25) can be written as:

qM+ 3 1

2 qL(M) qL+

3 1

2 qL(L)

The fact that tis inequality is satisfied follows from Point 1 and 4 in Lemma A6. (In the following,

when we mention a point, we refer to the points of Lemma A6.)

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Next, we show that I CMHholds. FromI CHMwe have:

UH=UM+ qM+ 3 12 [uH(M) uM(M)]

Thus,

UM=UH qM 3 1

2 [uH(M) uM(M)]

=UH qH 3 1

2 [uH(H) uM(H)]

+ (qH qM) + 3 1

2 [(uH(H) uM(H)) (uH(M) uM(M))]

> UH qH 3 1

2 [uH(H) uM(H)] .

The last inequality follows from the observation that:

uH(H) uM(H)qM(H) = M>qM(M) = uH(M) uM(M), (26)

where the first inequality follows from the definition ofui(H), the first equality and the second

inequality follow from Point 1. From (26) and the fact thatqH > qM (Point 4), it follows that

ICMHholds. We now turn to I CLH. Using I CLMfirst and thenI CMH, we have:

UL UM qM 3 1

2 [uM(M) uL(M)]

UH qH 3 12

[uH(H) uM(H)] qM 3 12

[uM(M) uL(M)]

=UH 2qH 3 1

2 [uH(H) uL(H)]

+ (qH qM) + 3 1

2 [(uM(H) uL(H)) (uM(M) uL(M))]

> UH 2qH 3 1

2 [uH(H) uL(H)] ,

The last inequality follows from the observation that:

uM(H) uL(H) qL(H) = L qL(M) = uM(M) uL(M), (27)

where the first inequality follows from the definition ofui(H), the first equality and the second

inequality follow from Point 1. From (27) and qH> qM(Point 4), it follows that I CLH holds.

Consider now the second period constraints. The constraints IRL(M), IRL(L) IRL(H),

ICML(H), and ICHM(H)) follow immediately by the definition of the utilities at t = 2. The

proof thatU, qsolves the sellers problem is therefore completed if we prove that it satisfies the

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constraints in the last two lines of (24). This result follows from the fact that the local downward

incentive constraints are satisfied in period 2 and quantities are weakly monotonic after any history

(Point 3). Finally, to see that the contract is optimal, we note that it maximizes expected profits

in the less restricted WR-problem, so it must be optimal in the sellers problem. Note moreover

that since the original problem is concave in q this is in fact the unique solution (in quantities).

7.5 Proof of Lemma 4

For the reminder of the proof, it is useful to state the first order conditions of the WR-problem.

It is easy to see that the Htype always gets the efficient quantity. After historyH, moreover,

quantities are always efficient, implying: qH=qH(M) = qH(L) = HandqH(H) = H, qM(H) =

M, qL(H) = L. The remaining first-order conditions are given by:

[qM] : M(M qM) H+ = 0

[qL] : L (L qL) (H+ M) = 0

[qM(M)] : M (M qM(M)) HM(M)+ LM(M)= 0

[qL(M)] : M1

2 (L qL(M)) ML(M)= 0

[qM(L)] : L1

2 (M qM(L)) HM(L)+ LM(L)= 0

[qL(L)] : L (L qL(L)) ML(L)= 0

[HM(M)] : H3 1

2 +

3 1

2 + HM(M) = 0

[ML(M)] : ML(M) LM(M) = 0

[HM(L)] : 3 1

2 + HM(L) = 0

[ML(L)] : (H+ M) 3 1

2 + ML(L) LM(L) = 0

We can now proceed with the proof. In the reminderof this section, we firs characterize the

optimal allocation assuming = 0. We then derive the conditions under which the assumption

of = 0 is admissible.

Assuming = 0, we have

qM=MHM

and qL= LH+ M

L. (28)

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Clearly, = 0 implies HM(L) = 0. Also, it is easy to show that LM(L) = 0, else qM(L)> M,

which contradicts lemma A5. We therefore haveML(L) = (H+ M) 31

2

, and the solution

after history L is given by:

qM(L) = M and qL(L) =LH+ M

L

3 1

2 . (29)

Next, note that we must have HM(M) = H312 and ML(M) = LM(M). We have two

possible cases:

Case 1 (Region A1). ML(M) = LM(M) = 0. In this case:

qM(M) = MH

M

3 1

2

and qL(M) = L (30)

For this to be a solution, we must have M HM

312 L, so 0(M) where

0(M) = H

3H 2M.

We conclude that for 0(M) the solution is given by qH=H, qH(j) =H, qj(H) =j for

allj = H,M,Lin addition to (28)-(30).

Case 2 (Region A2). ML(M) =LM(M)> 0. Then, qM(M) andqL(M) are both equal to

a constant q. From the first order condition with respect to qM(M) andqL(M) we have:

qM(M) = qL(M) = 2

1 + M+

1

1 + L

HM

3 1

1 + . (31)

We conclude that for > 0(M) the solution is given by qH=H, qH(j) =H, qj(H) =j for

allj = H,M,L, (28)-(29) and (31).

To characterize the necessary and sufficient condition for = 0, we need to verify that given

the solution defined above,I CHL is satisfied. Plugging in the values of Case 1, we obtain:

MHM

+ 3 1

2 M

HM

3 1

2

L

H+ ML

+ 3 1

2 M,

(32)

that is,

ML(1 L)

1 +

312

21 + L

1 +

312

2 =1() (33)

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Plugging in the val