the strategic value of incomplete contracts for competing...

The Strategic Value of Incomplete Contracts for Competing Hierarchies�

David Martimorty Salvatore Piccoloz

June 22, 2006

Abstract

We provide a rationale for contractual incompleteness in a setting with competing manufacturer-

retailer hierarchies. In a class of simple manufacturers-retailers economies, characterized by adverse

selection and moral hazard, we show that principals dealing with (exclusive) agents who compete on a

downstream market, may prefer to leave contracts silent on some (potentially) veri�able performance

measures whenever certain other aspects of agents� activity remain noncontractible. Two e¤ects are

at play once one moves from a complete to an incomplete contracting. First, reducing the number of

screening instruments has a detrimental e¤ect on principals�pro�ts as it makes information revelation

more costly and drives up the information rents associated to informational asymmetry. Second, it may

provide principals with strategic power in that it allows to force competing hierarchies to behave in a more

friendly manner on the downstream market. In contrast with previous literature, conjecturing a positive

relationship between transaction costs and contracting incompleteness, it is demonstrated that equilibrium

contracts display forms of incompleteness whenever agency costs are small enough. Implications on the

use of vertical arrangements based on resale price maintenance in imperfectly competitive industries are

also discussed.

1 Introduction

The basic principal-agent paradigm predicts that an optimal contract must limit as much as possible agents�

discretion within agency relationships. According to this view, contracts have to be complete in the sense that

principals should pro�tably exploit all available screening and monitoring instruments in order to prevent

agents�misbehavior. Strikingly, however, seldom in real life contracts display such high degree of complexity.

Contracts appear rather simple and, more importantly, quite often fail to specify veri�able obligations of

contracting counterparts. Such examples of arm�s length relationships are widespread in business practices.

Manufacturers often delegate marketing and advertising activities to retailers; shareholders typically delegate

�We are grateful to Jim Dana, Matteo Galizi, Jakub Kastl, Massimo Marrelli, Alessandro Pavan, Bill Rogerson and MikeWhinston for useful discussions. Of course, all remaining mistakes are ours. Comments welcome.

yUniversity of Toulouse, France (IDEI-GREMAQ), [email protected] University and University of Salerno (DISES-CSEF) , [email protected].

1

to managers �rms� competitive and organizational strategies; lenders usually leave entrepreneurs free to

perform certain tasks that a¤ect the pro�tability of their ventures; insurance companies monitor only to a

limited extent the behavior of insurees, etc..

So far a few rationales behind this incompleteness puzzle have been provided by the contracting literature.

First, high transaction costs can limit individuals�capability to enforce complete contracts because many

relevant aspects of agents�performances are not veri�able at the time of enforcing contracting obligations.

Second, bounded rationality may prevent complete contracting as not all possible future contingencies can

be foreseen at the time contracts are designed. Nevertheless, these views clearly fail to explain why there

are circumstances under which sophisticated agents deliberately choose to write incomplete contracts even

though transaction costs are low.

In the current paper we address this issue by taking a traditional agency perspective. Speci�cally, we ar-

gue that once one moves from the isolated principal-agent set-up to games played by competing hierarchies,

the design of incomplete contracts can be rationalized by the interaction between agency costs associated to

alternative incentive schemes and contracting externalities. In a setting with competing hierarchies, encom-

passing both adverse selection and moral hazard, it is demonstrated that when agents impose externalities

on each other, and some aspects of the individuals�actions are nonveri�able, leaving contracts incomplete

or silent on certain (potentially) veri�able agents�performance measures may have a strategic value. In-

complete contracts emerge at equilibrium even when more complex arrangements can be enforced without

additional transaction costs, i.e., further enforcement and monitoring out�ows.

More precisely, we show that the equilibrium determinants of contracting incompleteness hinge one three

somewhat natural aspects of information asymmetries: (i) the way di¤erent screening instruments shape

agency costs; (ii) the type of externalities that bilateral negotiations between principal-agent pairs impose

on competing organizations; (iii) the e¤ectiveness of agency constraints on �rms�technological frontier.

To take an archetypical example of incomplete contracting, we consider manufacturers-retailers relation-

ships.1 Particular emphasis has indeed been stressed by the IO literature on the very incomplete nature of

the arrangements regulating the terms of trade between vertically related �rms. Not only manufacturers

often delegate marketing activities to retailers, but they also frequently give up vertical control by refusing

to impose contractual restraints that would reduce agency costs and, typically, would allow to improve upon

allocative e¢ ciency.2 We assume that two retailers (agents or dealers) selling di¤erentiated products com-

pete on a downstream market by setting quantities. Production of �nal outputs requires an essential raw

input which is supplied by a pair of exclusive, upstream suppliers (principals or manufacturers). Retailers

1The applied literature on this topic (Lafontaine and Slade, 1997, among many others) has recently argued that the empiricalevidence, fairly consistent across industries and �rms, quite often appears to be inconsistent with some aspects of the theoreticalpredictions based on agency theory.

2Clearly, our analysis can be also extended to many other environments where competition takes place between verticalhierarchies or where agents, dealing with exclusive principals, impose externalities on each other. For instance it is easy toverify that our results would also hold in a multiple regulators set-up where di¤erent authorities regulate economic sectorscharacterized by technological spill-overs among �rms; in the case of lenders �nancing competing entrepreneurs; or in aninsurance economy where risk-neutral principals share risk with individuals imposing consumption externalities on each other.

2

are selected from a very large population of agents, so that upstream �rms dictate the terms of trade in

the wholesale-retail relationship. Downstream demands are uncertain and only agents observe a pay-o¤

relevant signal which realizes before contracts are designed: an adverse selection problem. Moreover, an

unveri�able demand-enhancing activity (promotional expenditures and/or production of indivisible services)

is carried out by downstream agents: a moral hazard issue. Principals and agents are risk-neutral. Each

upstream �rm hires an agent before production occurs, but after uncertainty about demand is realized. Two

alternative wholesale-retail trade regimes are analyzed. Speci�cally, each principal can either commit to an

incomplete contract, referred to as a quantity �xing scheme (QF), or to a more sophisticated arrangement,

comparable to resale price maintenance (RPM). A QF arrangement is incomplete relative RPM in the sense

that, beyond �xing the quantity supplied to �nal consumers, it leaves the downstream �rm free to choose its

most preferred level of demand-enhancing activity. Instead, a RPM mechanism, besides �xing the quantity

supplied in the �nal market, also restrains the retail price charged to �nal consumers to indirectly control

the retailer�s non-market activity.3 A three-stage game is examined. In the �rst stage, before setting their

competitive strategies, upstream suppliers (simultaneously) make a public announcement on the type of

mechanism that they will enforce at the contracting stage. In the second stage, allocations are secretly ne-

gotiated after all players have observed the announcement made in the �rst stage. Finally, in the third stage,

product market competition takes place and payments are made according to the prespeci�ed contracting

rules.

Within this framework, we prove that the degree to which agents�(unveri�able) e¤ort a¤ects competing

organizations plays a crucial role in shaping the features of equilibrium contracts. Two opposite e¤ects

are at play once one moves from a complete (RPM) to an incomplete contract (QF) in this framework.

On the one hand, an incomplete contract leaves more information rents simply because it limits the set of

screening instruments. On the other hand, introducing a source of contracting incompleteness might provide

principals with strategic power in that it allows them to induce a more friendly behavior by rivals at the

market stage. Noteworthy, while the former e¤ect has been widely discussed in previous contributions, the

second e¤ect is novel in the principal-agent literature and fully driven by our focus on competition between

vertical hierarchies.

We provide simple conditions, related to the severity of asymmetric information problem, under which

incomplete contracting may endogenously emerge when agents impose either positive or negative externali-

ties on each other. The key argument is as follows, when demands for each good are independent, i.e., there

are no externalities between agents and so retailers are monopolists in their own markets, the enforcement

of a complete contract is bene�cial to principals because it reaches the best trade-o¤ between e¢ ciency and

3Of course, in real life suppliers can credibly commit to use retail price restrictions in several ways. First, one can imaginethat investments in speci�c monitoring technologies and/or delegation to third parties (such as intermediaries or experts) thetask of monitoring retail prices, have the goal of signaling in a credible way the use of retail price restrictions. Second, in adynamic perspective, suppliers can credibly induce their competitors to believe that retail price restrictions will be used throughreputation.

3

rent extraction.4 With competing hierarchies a new and opposite channel through which contracting incom-

pleteness can a¤ect principals�pro�ts comes at play. By leaving downstream agents free to set some aspects

of their performance, each principal can in�uence in her own interest the market behavior of competing

organizations under certain circumstances.

After having proved that incomplete contracts can be signed at equilibrium, we relate contracting incom-

pleteness to the nature of the externality between downstream retailers. In particular, when agents impose

negative (resp. positive) externalities on each other, that is goods are substitutes (resp. complements) and

e¤ort has a sel�sh (resp. cooperative) value, the game of contractual choices has multiple equilibria. In

a free-riding environment, that emerges when goods are substitutes and e¤ort has a sel�sh value, multiple

equilibria may emerge only if retail competition is not very intense, when this condition is not satis�ed only

complete contracts are signed at the equilibrium. We also show that when there are multiple equilibria

incomplete contracts are always Pareto superior (resp. inferior) relative to complete ones whenever e¤ort

has a cooperative (resp. sel�sh) value. A welfare analysis aimed at assessing the social desirability of retail

price restriction is also performed.

Section 2 relates our work to the literature on incomplete contracting. In Section 3, by introducing

a simple example, we motivate our approach arguing that when all aspects of agents� performance are

veri�able contracting incompleteness will never emerge at equilibrium. Section 4 presents an illustrative

example showing why, and under what conditions, contracting incompleteness may have a strategic value

whenever some aspects of agents�performances are nonveri�able. In Section 5, the model is extended to

a fully symmetric competing hierarchies framework. Finally, Section 6 discusses the extent to which our

analysis applies to more general environments. Section 7 concludes. All profs are relegated to an Appendix.

2 Related Literature

The common wisdom identi�es the major sources of contracting incompleteness in the presence of transaction

costs and bounded rationality. As argued by Mookerjee (2005), apart from the usual noncooperative incentive

and participation constraints, these approaches search for additional constraints on contracts that rationalize

simple real-world mechanisms. On the one hand, as suggested by the property rights approach (Grossman

and Hart, 1986, and Hart and Moore, 1990, among others), agents�inability to describe future (uncertain)

events might severely prevent them from signing complete contracts. On the other hand, even in the presence

of very sophisticated agents, enforcement and monitoring costs could limit the possibility of writing complete

contracts as well (see Maskin, 2002, Maskin and Tirole, 1999, and Segal, 1999, among others). Although

based on di¤erent presumptions, both approaches support the view that the standard principal-agent analysis

is unsatisfactory in building a convincing theoretical foundation for contracting incompleteness. The present

paper questions this view and it argues that, once one moves from the isolated principal-agent set-up to games

4Basically, each incentive feasible allocation implementable under incomplete contracting can always be replicated by acomplete contract.

4

played by competing hierarchies, contracting incompleteness can be more easily rationalized by looking at

how agency costs change with the nature of the contractual externalities between agents.

Few previous studies have addressed the issue of contracting incompleteness by taking a perspective

somewhat more in line with the principal-agent paradigm.

Information Asymmetries, Dynamics and Renegotiation: Dewatripont and Maskin (1990, 1995)investigate the set of optimal contingencies on which an incentive contract should depend whenever rene-

gotiation is possible. They show that, within a repeated principal-agent set-up, where renegotiation issues

produce dynamic externalities, it might happen that a principal voluntarily limits the set of publicly ob-

servable screening devices in order to relax future renegotiation constraints. Similarly, Crémer (1995) shows

that in a repeated moral hazard setting, principals may voluntarily weaken the e¤ectiveness of their (ex

post) monitoring technology in order to make less likely renegotiation threats. Martimort (1999) and Olsen

and Torsvik (1993) show also how moving away from a centralized regulation by introducing competing

regulators, another form of incompleteness, may improve commitment in an intertemporal context. Rather

than looking at the e¤ects of incomplete contracts on renegotiation constraints with the same principal,

we focus on their strategic value and analyze the extent to which those arrangements weaken the market

behavior of agents�dealing with competing principals.

Bernheim and Whinston (1998) show that when some aspects of individuals�behavior are observable but

not veri�able, before playing a dynamic game, two individuals may pro�tably agree on designing contracts

which are silent on some potentially contractible aspects of the relationship. Bernheim and Whinston

analyze a class of complete information games where the possibility for players to sign (ex ante) binding

contracts allows to limit in a mutually bene�cial way the space of actions in their subsequent interactions.5

Two major aspects di¤erentiate our work by this paper. First, we analyze an incomplete information set-up

and incentive constraints arise here from the need for inducing information revelation whereas incentive

constraints are deduced from the self-enforceability of nonveri�able actions in their framework. Second,

di¤erently form them we examine a competing organizations set-up.

Information Asymmetries and Signaling: Aghion and Hermalin, (1990), Allen and Gale (1992) andSpier (1992) present a principal-agent model in which asymmetric information leads to contractual incom-

pleteness. With an informed principal, incomplete contracts may be pro�tably used to signal the principal�s

type when transactions costs are su¢ ciently high.

Ignorance in Vertical Contracting: Caillaud and Rey (1995) analyze the optimal information structureof producers with respect to their retailers. They show that ignorance on the retailer�s cost function might

create a strategic advantage that could outweigh the associated agency costs. While they rather focus on

the value of ignorance in environments where principals can chose whether to acquire (at no additional

costs) the relevant market information or, alternatively, to be (strategically) uninformed, in our set-up full

extraction is prevented by the moral hazard component. As we shall discuss, this assumption is key for

5 In this respect, their analysis seems much more related to Dewatripont and Maskin (1990, 1995) and Crémer (1995).

5

equilibria to display incomplete contracting.6 Finally, Kessler (1999) examines an agency model where the

agent�s information structure is endogenous and the possibility of remaining uninformed for the agent has

a positive strategic value. The crucial di¤erence between our paper and Kessler�s analysis is that, while

she analyzes the strategic value of ignorance from the agents�perspective, we rather focus on principals�

incentives to create contracting incompleteness.

Miscellaneous: Our analysis is also closely related to the literature on vertical restrains (D�Amato etal., 2005, Gal-Or, 1991, 1992, 1999, Martimort, 1996, Rey and Stiglitz, 1995 and Rey and Tirole, 1986).

The fundamental di¤erence between us and this literature is that while previous contributions have adopted

a complete contracting approach and mainly taken the set of control instruments as given, we endogenize

this set.7 From an organizational design viewpoint our results also provide a simple explanation for the

evidence showing that often, in real life, upstream manufacturers delegate non-market decisions, such as

advertising and marketing activities, to downstream retailers (see for instance La¤ontaine and Slade, 1997,

and Sheppard, 1993). In this respect, besides providing a theory of delegation based on information asym-

metries, we also contribute to assess the welfare properties of vertical arrangements based on vertical price

control (RPM) in imperfectly competitive industries. Speci�cally, normative implications advocating for the

lawfulness of vertical arrangements based on retail price control are drawn.

3 The Model

Environment: Consider a downstream industry where two retailers, R1 and R2, producing two symmet-

rically di¤erentiated products compete by setting quantities. The production of �nal output requires an

essential raw input which is supplied by a pair of exclusive upstream suppliers, S1 and S2. The inverse

market demand function facing the good produced by the retailer-i is uncertain and is de�ned by:8

pi(~�; ei; e�i; qi; q�i) = ~� + ei + �e�i � qi + �q�i;

where qi is the quantity produced of good-i, pi is the retail price level charged for this product, and ~� is a

common shock a¤ecting both demands.

6Relatedly, Kastl and Piccolo (2005) examine a dynamic game where vertical separation can be supported as an equilibriumstrategy in a in�nitely repeated game, provided that contracting externalities are intense enough and information asymmetriesare not particularly severe.

7An exception is Martimort and Piccolo (2006).8This demand system is generated by a representative consumer whosw preferences are:

V (q1; q2; I; �) =Xi=1;2;

ei(qi + �q�i) + �

Xi=1;2

qi

!� 1

2

Xi=1;2

q2i

!+ �q1q2 + I:

The sign of � determines whether produced goods are complements, independents or substitutes. Whereas the sign of �determines whether retailers impose positive (� � 0) or negative externalities (� < 0) on each other through their unveri�ableactivities.

6

This parameter is drawn on the compact support � � [�; �] according to the cumulative distribution

function F (�) having a (strictly) positive density, f(�), with jf 0(�)j being bounded. We assume also that thehazard rate h(�) = (1 � F (�))=f(�) is monotonically decreasing, _h(�) < 0 for all � 2 �. The realization of~� is private information of retailers at the time contracts are signed, and ei denotes an unveri�able activity

(e¤ort) performed by each retailer-i.

The e¤ort variable has two e¤ects on the (inverse) demands system. First, it enhances own consumers�

willingness to pay.9 Second, it may also in�uence the competitor�s inverse demand.

Following Che and Hausch (1999) we say that e¤orts have a cooperative (resp. sel�sh) nature if � �0 (resp. <). Throughout we shall assume that j�j � 1 in order to guarantee that own-e¤ort e¤ects are

larger than cross ones, i.e., @pi(:)=@ei � @pi(:)=@e�i.10 Furthermore, � denotes the measure of products�

di¤erentiation and it also satis�es j�j � 1 in order to guarantee that own-price e¤ects are larger than

cross-price ones in the direct demands�system, @qi(:)=@pi � @qi(:)=@p�i.

We shall focus either on the case where �� 0 or in that where � > 0 and � < 0 which captures a

free-riding set-up.11

Providing e¤ort is costly and we denote by (ei) the disutility function satisfying 0(ei) � 0; the Inadaconditions 0(0) = 0, limei!+1 0(ei) = +1, and 00(ei) > 1=2 for all ei 2 R+ in order to guarantee wellbehaved optimization programs.

Finally, as for production technologies, we assume that both upstream and downstream �rms produce

at constant marginal costs normalized to zero.

Most of our results will be derived by using Taylor expansions, hence they hold for �� = � � � small

enough. Alternatively, they also hold more generally whatever �� when (e) is quadratic since Taylor

expansions are then exact.

Mechanisms: We assume that the mechanisms ruling each hierarchy are private, i.e., cannot have anycommitment power. However, the choice of a contractual mode, i.e., whether quantity forcing QF or resale

price maintenance RPM is chosen, is itself observable.

Within this framework, we follow Myerson (1982) and Martimort (1996) to characterize the set of

incentive feasible allocations in each hierarchy. Indeed, for any output choice made by retailer-i, there is no

loss of generality in looking for Si�s best response to S�i�s contractual o¤er within the class of direct and

truthful mechanisms.

If a QF arrangement has been chosen, such mechanism is of the formnti(�̂i); qi(�̂i)

o�̂i2�

where �̂i is

retailer-i�s report on the demand parameter, and ti(�̂i) (resp. qi(�̂i)) the corresponding �xed-fee (resp.

9This assumption seems completely natural if one interprets e¤ort as being production of indivisible goods and/or investmentin advertising.10One could reasonably interpret the e¤ort as being production of indivisible services and/or advertising investments. When

goods are substitutes ei a¤ects negatively pj as it may have a business stealing e¤ect, i.e., � < 0. Whereas, when goods displayforms of complementarities or network e¤ects it is natural to assume that ei has also a positive impact on pj , i.e., � > 0.11 In fact, assuming that produced goods are complements, � > 0, and that e¤orts create negative externalities, � < 0, seems

unreasonable, thus it can be ruled out.

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output) paid (resp. supplied) to (resp. from) the supplier Si. If RPM is preferred, an incentive mechanism

is of formnti(�̂i); qi(�̂i); pi(�̂i)

o�̂i2�

where pi(�̂i) denotes the retail price of good-i following report �̂i.

A RPM arrangement is more complete relative to QF because this latter contract leaves unspeci�ed

the retail market price. Therefore, it is reasonable to think of a QF arrangement as being equivalent to a

vertically decentralized organizational structure or to an arm�s length relationship. Under this contractual

scheme, the upstream manufacturer does not have enough instruments to monitor the promotional e¤ort

level exerted by the retailer. Instead, RPM will be seen to replicate the constrained vertical integration

outcome, since by dictating the retail price and the quantity sold to the retailer, the upstream manufacturer

is able to control directly the retailer�s e¤ort level, although there is no possibility of full extraction.

Timing, Strategies and Equilibrium Concept: Firms play a three-stage game whose sequence of eventsis as follows:

1. Each supplier Si either publicly commits to verify the (ex post) realization of the retail price in market-i, together with Ri�s sales level or, alternatively, she might give up price control and use only sales as a

screening device.12

2. Uncertainty about demand realizes and only R1 and R2 observe it.

3. Each supplier Si secretly o¤ers a menu of contracts to his own retailer Ri. If the contract is accepted,Ri reports a message �̂i 2 � to Si about the realized demand state. E¤ort is exerted, product market

competition takes place and, �nally, payments are made after veri�able actions have been observed. If the

o¤er is turned down, Si and Ri enjoy their outside options which are normalized to zero, and R�i acts as a

monopolist in the downstream market.

The equilibrium concept we use is Perfect Bayesian Equilibrium with the added re�nement that, provided

Ri receives any unexpected o¤er from Si he still believes that R�i will produce the same quantity. We denote

by G the three stages game of contractual choices cum mechanisms o¤ers and market interactions.

Complete Information Benchmark: Under complete information the two kinds of contracts QF andRPM implement the same allocation. Both mechanism can emerge in an equilibrium of G. Let us denoteby fp�(�); q�(�); e�(�)g the solution to the following equation:

(1) � + (1 + �)�(q�(�))� (2� �)q�(�) = 0,

with �(:) = 0(:)�1, q�(�) = 0(e�(�)) and p�(�) = q�(�).

Proposition 1 Assume �� is small enough and 00(e) > maxn12 ;1+�2��

ofor all e 2 R+. Then the following

properties are satis�ed:

12 In practice suppliers can credibly commit to use retail price restrictions in several ways. First, one can imagine that(observable) investments in speci�c monitoring technologies may induce competitors to believe that retail price restrictions willbe enforced. Second, in a dynamic perspective, suppliers may well achieve the same goal through reputation e¤ects.

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� E¤orts, outputs and �xed fees are the same under both contractual regimes: qP (�) = qQ(�) = q�(�),

eQ(�) = eP (�) = e�(�), and tP (�) = tQ(�) for all �;

� E¤orts and outputs are increasing in �, i.e., _e�(�) � 0 and _q�(�) � 0 for all �.13

This result is similar to Katz (1991) arguing that the use of agents in games of competing hierarchies

does not a¤ect the equilibrium outcomes if there exists a contract which perfectly internalizes the vertical

externality between the principals and their agents. We shall see below that this result drastically changes

under asymmetric information. Furthermore, it should be noticed that, under complete information, both

types of contracts produce the same external e¤ect on third parties, i.e., they generate the same consumers�

surplus and total welfare. As we prove, this property no longer holds once asymmetric information is

introduced in the model, this result will thus motivate our welfare analysis.

4 Preliminary Remarks

Before presenting the formal analysis it is worthwhile explaining why the issue of contracting incompleteness

must be addressed in an asymmetric information framework. To this end we shall use a simple example

of vertical contracting under asymmetric information. The centerpiece of the argument rests on the idea

that, when some aspects of agents�performance are nonveri�able, the interaction between agency costs,

second-best outcomes and the presence of non-market externalities that competing agents impose on each

other may provide principals with strategic power.

Consider the simple case where while R1 must deal with S1 to receive inputs, whereas R2 and S2 are

vertically integrated and produce as a unique entity, labeled S2-R2. Furthermore, assume that providing

e¤ort is too costly for both retailers, i.e., (e) = +1 for all e 2 R+, so that each (inverse) demand functionis de�ned by pi(~�; qi; qj) = ~� � qi + �q�i for i = 1; 2.

To begin with, observe that when S1 monitors both retail price and sales he can fully extract the

information rent of his retailer and infer (correctly at the equilibrium) the production of S2-R2.14

Lemma 2 When S1 monitors both sales and retail price no rents are left to R1.

At an equilibrium of the game outcomes must solve the following type-contingent system of �rst-order

conditions equalizing marginal revenues to marginal costs within each pair:

(2) � � 2qPi (�) + �qP�i(�) = 0 for i = 1; 2:

13Since demands are increasing in � it is completely natural to focus on cases where output is also increasing in �.14This mechanism is similar to Gal-Or (1991).

9

This yields a symmetric equilibrium output:

qPi (�) =�

2� � for i = 1; 2.

Consider now the optimal contract when S1 no longer controls the retail price of his retailer. The

upstream supplier does not have enough instruments to achieve full extraction. The underlying idea is

standard, high-demand types have an incentive to mimic low-demand ones, and S1 has to grant information

rents in order to induce information revelation. To trade-o¤ optimally allocative e¢ ciency and rent extrac-

tion, the supplier must also distort downward the output relative to the complete information level. In this

context, as we have argued above, the Revelation Principle applies in S1-R1 hierarchy (keeping as �xed the

production of R2). Denoting by U1(�) retailer R1�s information rent, we have:

U1(�) = p1(�; q1(�); q2(�))q1(�)� t1(�) = max�̂2�

np1(�; q1(�̂); q2(�))q1(�̂)� t1(�̂)

o:

Using standard techniques developed in Martimort (1996), we obtain the following �rst- and second-order

local conditions for incentive compatibility:

_U1 (�) = (1 + � _q2(�))q1(�); (IC1)

(1 + � _q2(�)) _q1 (�) � 0; (IC2)

which, together with the agent�s participation constraint

U1(�) � 0 8 � 2 �; (PC)

de�ne the set of incentive feasible allocations in S1-R1 hierarchy.

Expressing the �xed-fee as a function of U1(�) and q1(�), S1�s problem, denoted thereafter by (P), canbe rewritten as:

(P) : maxfq1(:);U1(:)g

Z �

�fp1(�; q1(�); q2(�))q1(�)� U1(�)g f(�)d�

subject to (IC1); (IC2) and (PC):

A contractual externality similar to the competing-contracts e¤ect studied in Martimort (1996) is at

play in this environment. Speci�cally, information rents have now a di¤erent structure relative to those

arising in a standard isolated principal-agent framework. In fact, the revelation of a good demand state

provides two pieces of information to S1. First, it reveals that consumers�willingness to pay is large. Second,

10

since � a¤ects symmetrically demands, it also provides information on the market behavior of the opponent

hierarchy, which (in equilibrium) will produce more in better demand states. In particular, information

rents now are also a¤ected by a measure of the competitiveness, � _q2(�), indicating how aggressively the

integrated structure S2-R2 behaves at the market stage. The steeper q2(�) is the lower is its level at any

given �. This implies in turn that, when _q2(�) increases, the integrated pair S2-R2 behaves less (resp. more)

aggressively at the market stage so as to make S1 less (resp. more) willing to grant information rents if

goods are substitutes (resp. complements).

By using standard techniques, and assuming that information rents are increasing in � so that (PC)

binds only at �, we solve a relaxed program (P 0), neglecting (IC2) in (P),

(P 0) : maxfq1(:)g

Z �

�fp1(�; q1(�); q2(�))q1(�)� h(�)(1 + � _q2(�))q1(�)g dF (�):

Optimizing pointwise yields the following �rst-order condition which is both necessary and su¢ cient

(given our assumptions ensuring concavity of the objective) for optimality:

(3) � � 2qQ1 (�) + �qQ2 (�)� h(�)(1 + � _q

Q2 (�)) = 0:

At a best-response to what the integrated structure produces, S1 equalizes her virtual marginal revenues

to zero at each �. Under asymmetric information, everything happens as if the demand parameter � is now

replaced by a lower virtual demand parameter, namely ��h(�)(1+ � _qQ2 (�)), which captures also the extentof competitive pressure on the downstream market.

On the other hand, S2-R2 chooses its output level so as to maximize pro�t under complete information,

i.e., � � 2qQ2 (�) + �qQ1 (�) = 0. From equation (3) one can show that in a Nash equilibrium of the market

subgame R1�s output, qQ1 (�), must be a solution to the following di¤erential equation:

(4) _qQ1 (�) =(2 + �)(� � (2� �)qQ1 (�)� h(�))

�2h(�);

with the boundary condition qQ1 (�) = qP1 (�).

Let �!1 (�) be S1�s state contingent pro�ts under a given mechanism ! 2 fQF;RPMg, and denote by�!1 = E~�[�

!1 (�)] its expectation. Formally we have:

�Q1 (�) = qQ1 (�)(� + �qQ2 (�)� q

Q1 (�)� h(�)(1 + �q

Q2 (�)))

and,

�R1 (�) = qR1 (�)(� + �qR2 (�)� qR1 (�))

The next Proposition summarizes the results for this simple case.

11

Proposition 3 The following properties are satis�ed at each � 2 [�1; 1]:

� qQ1 (�) � qP1 (�) for all � (with equality only at �);

� S1 strictly prefers to control the retail price; i.e., �P1 > �Q1 .

When the retail price is not monitored, the upstream manufacturer chooses the quantity schedule so

as to equalize her virtual marginal revenues to zero. By doing so, output is reduced below the complete

information level for rent extraction reasons. On the contrary, under vertical price control, S1 can extract

R1�s private information at no costs, so S1 can always replicate any allocation implemented when sales are

the only screening instrument by means of vertical price control, but of course this is never optimal.

The lesson of this simple example is that restricting the set of screening instruments used to control the

agents�behavior, i.e., not controlling the retail price in the present setting, is never worthwhile when the

supplier can achieve the vertically integrated structure �rst-best pro�t under complete contracting.

Of course, when, besides produced output, the agent does not perform any other unveri�able task a¤ect-

ing both his own principal�s payo¤ and the competing organization objective function, a complete contract

can always replicate every allocation obtained with an incomplete one, hence contracting incompleteness

has no commitment value. To properly address the question one should thus consider environments where

information asymmetries prevent upstream suppliers to achieve full extraction even under complete con-

tracting. The rest of the analysis goes in this direction and, as we shall see in the next sections, in such a

framework the interaction between contracting externalities created by (extra) unveri�able tasks performed

by agents, information rents and second-best outcomes associated to information asymmetries may give a

commitment value to an incomplete contract.

5 An Illustrative Example

We start our analysis with a simple example considering again a fully integrated �rm competing with a

vertically separated supplier-retailer hierarchy. Once the argument highlighted by this simple example will

be clear, it will be easier to understand the properties of our complete model, which entails symmetric

competing hierarchies. To further simplify the analysis, we also assume that S2-R2 does not exert any e¤ort

(formally this amounts to assume 2(e2) = +1 for all e2 2 R+).The system of (inverse) market demands is thus de�ned by:

p1(~�; e1; q1; q2) = ~� + e1 � q1 + �q2 and p2(~�; e1; q2; q1) = ~� + �e1 � q2 + �q1:

Observe that none of the contractual regimes allows S1 to fully extract R1�s information rents since even

when the retail price can be contracted upon, S1 cannot disentangle the impact of the demand parameter

� and the retailer�s e¤ort on the residual demand the latter faces. The underlying idea is as follows: the

12

possibility for the retailer to claim that large sales are due to a high e¤ort level, whereas they result instead

from a high demand, induces the upstream manufacturer to give up some information rent to the high

demand retailer (high �) in order to induce truthtelling. As a result, the second-best allocation will be

characterized by a downward distortion of both quantity and e¤ort supplied by the retailer when he faces

low demand states. This information rent, of course, depends on the chosen contractual mode.

As we shall clearly point out below, the choice of the contractual mode has two opposite e¤ects on S1�s

pro�ts whenever R1�s e¤ort and output have the same impact on S2-R2�s (inverse) demand, i.e., sign(�) =

sign(�). On the one hand, committing to an incomplete contract may lead the upstream supplier to grant

more information rents relative to RPM because a screening instrument is given up: an agency e¤ect. On

the other hand, an incomplete contract may have a strategic value relative to a complete one in that it

enables S1 to force S2-R2 to behave in a more friendly manner at the market stage: a strategic e¤ect.

Indeed, besides creating a vertical externality between S1 and R1, an incomplete contract also generates an

horizontal externality a¤ecting the competitive behavior of the rival pair S2-R2. The tension between those

two e¤ects will depend upon the severity of the agency problem. More precisely, when information rents are

small enough, the agency e¤ect will be dominated by the strategic one. In a free riding set-up, i.e., � > 0

and � < 0, however, the oversupply of e¤ort provided by R1 under QF makes its competitor more aggressive

at the market stage so as to make the strategic e¤ect reinforcing the agency one. An incomplete contract is

thus always dominated by the complete one.

Below we solve the game in two steps. First, we characterize the market allocation under both contractual

regimes. Then, the equilibrium contract will be derived by using a backward induction argument.

S2-R2�s Program: To begin with, let us brie�y analyze the program solved by S2-R2. For each realization

of ~� and any incentive feasible allocation fe1(�); q1(�)g�2� implemented by S1, the vertically integrated

structure S2-R2 solves:

(5) maxq22R+

p2(�; e1(�); q2; q1(�))q2:

The corresponding necessary and su¢ cient condition yields the following reaction function at each �:

(6) q2(�) =� + �q1(�) + �e1(�)

2;

From (6) one can infer that S1 has an incentive to choose strategically the contractual mode to soften S2-

R2�s behavior at the competitive stage. In fact, the contractual regime maximizing the term f�e1(�) + �q1(�)gin equation (6) will be the most preferred one either when goods are substitutes or when they are comple-

ments, everything else being kept constant.15

Complete Contracting: Let us �rst consider a RPM arrangement. In this case S1 can contract also the

15Obviously, this e¤ect must be traded o¤ with the e¤ect of contracting incompleteness on information rents, which may wellgo in the opposite direction of decreasing S1�s pro�ts.

13

retail market price besides the quantity supplied by the downstream �rm to �nal consumers. The e¤ort

level is then indirectly �xed as a function of � through the inverse demand, i.e., e1 = p1+ q1��q2� �. RPMis less �exible than QF just because when the retailer faces a retail price target, she is indirectly forced to

choose the e¤ort level.16

Let us de�ne retailer R1�s information rent as:

U1(�) = p1(�)q1(�)� (p1(�) + q1(�)� �q2(�)� �)� t1(�):

By de�nition of incentive compatibility, we have:

U1(�) = max�̂2�

np1(�̂)q1(�̂)� (p1(�̂) + q1(�̂)� �q2(�)� �)� t1(�̂)

o:

This yields the following �rst- and second-order local conditions for incentive compatibility:

_U1 (�) = (1 + � _q2(�)) 0 (e1(�)) ; (IC1)

(1 + � _q2(�))(1 + _e1(�) + � _q2(�)) � 0; (IC2):

Then, (IC1) and (IC2) together with the retailer�s participation constraint

U1(�) � 0 8 � 2 �; (PC)

de�ne the set of incentive feasible allocations in S1-R1 hierarchy for a �xed output schedule q2(:) chosen by

the rival pair S2-R2. S1�s problem, denoted thereafter by (PP ), is to design a menu of contracts to maximizethe expected franchise fee he receives from R1 subject to the participation and incentive compatibility

constraints, together with the additional restriction required by the retail price target.

Expressing the �xed fee as a function of the retailer�s revenue and information rent we can thus write

(PP ) as:

(PP ) : maxfU1(:);q1(:); e1(:)g

Z �

�f(� + e1(�)� �q2(�)� q1(�))q1(�)� (e1(�))� U1(�)g f(�)d�;

subject to (IC1), (IC2) and (PC):

We will �rst assume and check ex post that (1 + � _q2(�)) � 0 for all �. Then U1(�) is increasing and

16 Indeed, under retail price restrictions the upstream producer has full control of all available instruments. See also Blair andLewis (1994) and Martimort and Piccolo (2006).

14

(PC) binds only at �, i.e., the lowest demand parameter. We obtain immediately:

U1(�) =

Z �

�(1 + � _q2(x))

0(e1(x))dx:

Integrating by parts into the expression of the maximand of (PP ) and neglecting (IC2) (that will bechecked ex post also) yields a simpli�ed optimization problem (PP 0):

(PP 0) : maxfq1(:); e1(:)g

Z �

�

�(� + e1(�)� q1(�) + �q2(�))q1(�)� (e1(�))� h(�)(1 + �q2(�)) 0(e1(�))

f(�)d�:

At a best response to the schedule q2(:) implemented by the competing pair S2-R2, the production and

e¤ort in S1-R1 hierarchy are respectively given by the following �rst-order conditions:17

(7) q1(�) = p1(�) = � + e1(�) + �q2(�)� q1(�);

(8) q1(�) = 0(e1(�)) + h(�)(1 + � _q2(�)) 00(e1(�)):

Notice that the dichotomy result (La¤ont and Tirole, 1993, Ch. 3) holds in this setting, that is since

under RPM the only screening variable is e¤ort, which is downward distorted relative to the complete

information level, output is produced according to the e¢ cient rule that would prevail under complete

information.

By using (6) together with (7) and (8), one can also check that the allocation�eP1 (�); q

P1 (�)

�2� solves

the following system of di¤erential equations:

(9) _qP1 (�) =2(qP1 (�)� 0(eP1 (�))) + h(�) 00(eP1 (�))(2 + �(1 + � _eP1 (�)))

�2h(�) 00(eP1 (�));

and,

(10) _eP1 (�) =(4� �2) _qP1 (�)� (2 + �)

2 + ��;

with the boundary conditions qP1 (�) = q�1(�) and eP1 (�) = e�1(�).

Simple algebra shows that at an equilibrium the complete information allocation, fe�1(�); q�1(�)g�2�, isnow de�ned by the following equations:

(2 + �)� + (2 + ��)�(q�1(�))� (4� �2)q�1(�) = 0;17Given concavity of the objective, these conditions are also su¢ cient.

15

and

q�1(�) = 0(e�1(�)):

Next Proposition characterizes some useful features of the equilibrium allocation under RPM.

Proposition 4 Assume �� is small enough and 00(e) > maxn12 ;

2+��4�2�(�+�)

ofor all e 2 R+. Then the

following properties hold:

� eP1 (�) � e�1(�) and qP1 (�) � q�1(�) for all � (with equality holding only at �)

� (IC2) always holds.

These are standard properties of adverse selection models, so we shall not discuss them (see La¤ont and

Martimort, 2002).18

Incomplete Contracting: Under QF, S1 does not observe the retail price, but she still observes thequantity supplied by R1 on the retail market. A QF mechanism can thus be viewed as an incomplete

contract relative to RPM since the upstream producer voluntarily gives up one of the possible screening

instruments.

Let us now rede�ne retailer R1�s information rent under a QF regime as:

U1(�) = �t1(�) + maxe12R+

f(� + e1 � q1(�) + �q2(�))q1(�)� (e1)g :

By de�nition of incentive compatibility, we have:

U(�) = max�̂2�

��t1(�̂) + max

e12R+

n(� + e1 � q1(�̂) + �q2(�))q1(�̂)� (e1)

o�:

From which we obtain the following �rst- and second-order local conditions for incentive compatibility:

(11) _U1(�) = (1 + � _q2(�))q1(�);

(12) (1 + � _q2(�)) _q2(�) � 0;

to which we must also add the participation constraint:

(13) U(�) � 0 8 � 2 �:18One can easily show that under some regularity conditions these properties also hold when �� is arbitrarily chosen.

16

This leads to state S1�s contracting problem, denoted thereafter as (PQ), as:

(PQ) : maxfU1(:);q1(:)g

Z �

�f(� + �(q1(�))� q1(�) + �q2(�))q1(�)� U1(�)g f(�)d�:

subject to (11), (12) and (13).

For any given quantity schedule speci�ed by the direct revelation mechanism QF, R1 gains �exibility

under a quantity-�xing arrangement in the sense that the e¤ort level is chosen to command more information

rents than it would be e¢ cient from the manufacturer�s viewpoint. More speci�cally, while choosing the

optimal e¤ort level, the retailer does not internalize the impact of his e¤ort on the information rent given

up by the upstream manufacturer. QF introduces a vertical externality between the manufacturer and his

retailer. As rents and e¤ort are positively related via quantity, it will be thus pro�table to oversupply e¤ort

relative to RPM everything else being kept equal.

We will again �rst assume and check ex post that (1 + � _q2(�)) � 0 for all �. Then U1(�) is increasingand (13) binds at � only. We obtain immediately:

U1(�) =

Z �

�(1 + � _q2(x))q1(x)dx:

Integrating by parts the above expression and inserting into the maximand of (PQ) yields the expressionof the relaxed program (PQ0):

(PQ0) : maxfq1(:)g

Z �

�f(� + �(q1(�))� q1(�) + �q2(�))q1(�)� (� (q1(�)))� h(�)(1 + � _q2(�))q1(�)g f(�)d�:

At a best-response to the schedule q2(:) implemented by the S2-R2 pair we get:

(14) � + �(qQ1 (�))� 2qQ1 (�) + �q

Q2 (�)� h(�)(1 + � _q

Q2 (�)) = 0:

Di¤erentiating equation (6) yields 2 _qQ2 (�) = 1 + � _qQ1 (�) + � _eQ1 (�), using this condition together with

_eQ1 (�) = �0(qQ1 (�)) _qQ1 (�), one can immediately show that q

Q1 (�) solves the the following di¤erential equation:

(15) _qQ1 (�) =(2 + �)(� � h(�)) + (2 + ��)�(qQ1 (�))� q

Q1 (�)(4� �2)

�h(�)(��0(qQ1 (�)) + �);

with the boundary condition qQ1 (�) = q�1(�).

Under the QF regime the dichotomy result no longer holds. In fact, since S1 gives up a screening

instrument by not controlling the retail price, the only screening device is output which must be downward

distorted relative to the complete information level for rent extraction reasons. This point will be key to

17

prove that an incomplete contract might be preferred to a complete one.

Next Proposition characterizes some useful features of the equilibrium allocation under QF.

Proposition 5 Assume �� is small enough and 00(e) > maxn12 ;1+��2��2

ofor all e 2 R+. Then the following

properties hold:

� eQ1 (�) � e�1(�) and qQ1 (�) � q�1(�) for all � (with equality only at �);

� 1 + � _qQ2 (�) � 0 for all � 2 �:

Choice of the Contractual mode: We now turn to determine whether RPM or QF is chosen at equi-

librium. As a preliminary result we state the next Proposition which provides a useful description of how

outputs and e¤orts are ordered under both regimes. This result will be key for our equilibrium characteri-

zation.


2+��4�2�(�+�) ;

1+��2��2

ofor all e 2 R+. Then

the following properties hold:

� eQ1 (�) � eP1 (�) for all � with equality holding only at �;

� qQ2 (�) � qP2 (�) (resp. <) if and only if � � 0 (resp. <) for all � with equality only at � when � 6= 0;

� qQ1 (�) � qP1 (�) (resp. <) if and only if �� 0 (resp. <) for all � with equality only at � when �� 6= 0.

The economic interpretation of the above result is simple. When a QF contract is enforced, the upstream

manufacturer gives up the control of the promotional activities (e¤ort) supplied by the downstream retailer.

This decentralized organizational mode, crucially, allows the downstream �rm to increase his information

rent by playing on his e¤ort choice. More speci�cally, as e¤ort a¤ects positively quantity,19 R1 �nds it

pro�table to supply more e¤ort (relative to RPM) in order to enjoy more rents.

From equation (6), the output produced by the integrated structure S2-R2 is directly a¤ected by the

e¤ort level. Since eQ1 (�) � eP1 (�) for all �, an incomplete contract provides S1 with strategic power in that it

shifts�s S2-R2�s reaction function. That is q2(:) increases (resp. diminishes) when one moves from RPM to

QF whenever � � 0 (resp. <). Besides introducing a vertical externality between S1 and R1, an incompletecontract also creates an horizontal externality between the competing hierarchies.

By moving from RPM to QF, three di¤erent e¤ects are at play simultaneously. First, for any given

output level, R1 will exert more e¤ort under QF relative to RPM. Under QF, the agent is residual claimant

for the full impact of his e¤ort on enhancing demand. This e¤ect raises e¤ort and thus R1�s output: a

demand-enhancing e¤ect. Second, since sales is the only screening instrument under QF, one needs to

19Remember that under QF the downstream retailer chooses his e¤ort according to the e¢ cient rule q1 = 0(e1).

18

distort it downward for rent extraction reasons: a rent extraction e¤ect. Third, because of the horizontal

externality, the output of the S2-R2 pair is shifted upward (resp. downward) when goods are substitutes

(resp. complements): a strategic e¤ect.

When the agents� e¤ort choices do not create externalities, � = 0, or goods are independent, � = 0,

the strategic e¤ect is absent. Therefore the present set-up displays the same features as the sequential

monopolies model studied in Martimort and Piccolo (2006). In particular, the demand enhancing and

the rent extraction e¤ects exactly compensate each other in the cutting-edge case where 000(e1) = 0, so

qQ1 (�) = qP1 (�) for all �.

As we have argued above, though, when products are di¤erentiated and the e¤ort choice creates con-

tracting externalities, the strategic e¤ect reinforces when �� > 0. In this case QF increases e¤ort and so

it moves q2(:) in a more suitable manner relative to RPM. In fact, since e¤ort in�uences negatively (resp.

positively) the output produced by the competing �rm when goods are substitutes (resp. complements),

and QF commands an e¤ort larger than that produced under RPM, the result follows immediately since

a QF contract always forces S2-R2 to behave in a more friendly manner (relative to RPM) on the retail

market.

When instead � < 0 and � > 0, the strategic e¤ect makes S2-R2 more aggressive at the market stage

since the consumers�willingness to pay increases and q2(:) increases, which in turn lowers q1(:) as quantities

are strategic substitutes.

Now, let �P1 = E~�[�P1 (�)] and �

Q1 = E~�[�

Q1 (�)] de�ne S1�s pro�ts under RPM and QF, respectively.

Next proposition shows that incomplete contracts may have a commitment value under certain conditions

related to the severity of the agency problem and to the presence of externalities between agents.


2+��4�2�(�+�) ;

1+��2��2

ofor all e 2 R+. Then

the following properties hold:

� �Q1 > �P1 whenever �� > 0;

� �Q1 � �P1 whenever �� 0.

Of course, once one moves from a complete to an incomplete contract, information rents increase because

a screening instrument is given up and information revelation becomes more costly, an agency cost e¤ect.

When goods are independent or e¤ort does not create externalities, this e¤ect drives the upstream supplier

to always prefer RPM. When goods are di¤erentiated and there are contracting externalities such that both

e¤ort and output by R1 a¤ect in the same direction S2-R2�s demand and make it behave in a more friendly

manner, the strategic e¤ect may outweigh excessive agency costs. In fact, as an incomplete contract allows

S1 to force S2-R2 to behave in a more friendly manner at the market stage, it raises the supplier�s pro�ts

by increasing e¤ort and so the expected transfer that can be extracted from R1.

19

6 Symmetric Hierarchies

We now extend the example presented in the previous section to a more symmetric model where each

downstream retailer deals with an exclusive upstream supplier who remains asymmetrically informed on

market demand. So the retailer Ri faces the following inverse demand function:

pi(~�; ei; e�i; qi; q�i) = ~� + ei + �e�i � qi + �q�i

on the downstream market. Again, two e¤ects are key in determining the equilibrium of the game G. First,an incomplete contract commands more information rents. Second, for any given mechanism ruling the

competing hierarchies, contractual incompleteness may create a strategic e¤ect in�uencing rivals�market

behavior. Speci�cally, the vertical externality that a QF mechanism creates within each vertical hierar-

chy is translated horizontally on the competing organization. Increasing retailers�e¤ort, indeed, provides

a bene�cial e¤ect on a supplier�s pro�ts to the extent that it weakens the competitive stance of the op-

posing hierarchy on the downstream market. Of course, since we now examine a full model of competing

organizations, multiple equilibria of the game of contracting mode choices can emerge.

Furthermore, as our analysis is also related to the practice of vertical restraints, which have been for a

long time at heart of antitrust and regulation concerns, we shall also discuss some welfare properties of the

equilibrium contracts and relate them to the use of retail price restrictions in vertical contracting.

Complete Contracting: With this hierarchy framework it is straightforward to describe the set of incentivefeasible allocations within Si-Ri pair with the following �rst- and second-order local conditions for incentive

compatibility:

(16) _Ui (�) = (1 + � _e�i(�) + � _q�i(�)) 0 (ei(�)) ;

(17) (1 + � _e�i(�) + � _q�i(�))(1 + _ei(�) + � _e�i(�) + � _q�i) � 0;

and the participation constraint:

(18) Ui(�) � 0; 8 � 2 �:

Equipped wit this characterization, we may de�ne Si�s problem as:

(PPi ) : maxfUi(:);qi(:);ei(:)g

Z �

�f(� + ei(�) + �e�i(�)� qi(�) + �q�i(�))qi(�)� (ei(�))� Ui(�)g f (�) d�;

subject to (16), (17) and (18).

20

Assuming that the quantity (1 + � _e�i(�) + � _q�i(�)) remains positive, Ui(�) is increasing and the partic-

ipation constraint (18) binds only at �. This leads to the following equation:

Ui(�) =

Z �

�(1 + � _e�i(x) + � _q�i(x))

0 (ei(x)) dx:

Inserting Ui(�) into the maximand of (PPi ) and integrating by parts yields the following relaxed program(PP 0i ):

(PP 0i ) : maxfqi(:);ei(:)g

Z �

�

�pi(�; ei(�); e�i(�); qi(�); q�i(�))qi(�)� (ei(�))� h(�)(1 + � _e�i(�) + � _q�i(�)) 0 (ei(�))

f (�) d�:

Pointwise optimization with respect to qi(:) and ei(:) respectively lead to the following �rst-order condi-

tions:

(19) qi(�) = pi (�) = � + ei(�) + �ej(�)� qi(�) + �qj(�);

(20) qi (�) = 0(ei(�)) + h(�)(1 + � _ej(�) + � _qj(�)) 00 (ei(�)) = 0:

Let�qP (�); eP (�)

�2� be the symmetric allocation obtained when both suppliers use complete contracts.

Next Proposition characterizes some useful features of this allocation.

Proposition 8 Assume �� is small enough, �+� < 1 and 00(e) > maxn12 ;1+�2�� ;

1+�2(1��)

ofor all e 2 R+:

Then the following properties hold:

� eP (�) � e�(�) and qP (�) � q�(�) for all � (with equality only at �);

� Information rents are increasing in � and the local-second-order condition (17) holds.

Incomplete Contracting: As before, under this contractual regime the upstream supplier gives up the

retail price as a screening instrument and types�revelation must be elicited only through sales. Proceeding

as before, the retailer Ri�s information rent under a QF regime can be rewritten as:

Ui(�) = �ti(�) + maxei2R+

f(� + ei + �e�i(�)� qi(�) + �qj(�))qi(�)� (ei)g :

By incentive compatibility, we immediately obtain the following local �rst- and second-order conditions:

(21) _Ui(�) = (1 + � _e�i(�) + � _q�i(�))qi(�);

(22) (1 + � _e�i(�) + � _q�i(�)) _qi(�) � 0:

21

Incentive feasible allocations must also satisfy the usual participation constraint:

(23) Ui(�) � 0; 8 � 2 �:

Taking into account that Ri�s e¤ort satis�es ei(�) = �(qi(�)), Si�s problem (PQi ) can be rewritten as:

(PQi ) : maxfUi(:);qi(:)g

Z �

�f(� + �(qi(�)) + �e�i(�)� qi(�) + �q�i(�))qi(�)� (�(qi(�)))� Ui(�)g f (�) d�;

subject to (21), (22) and (23).

Assuming that (1 + � _e�i(�) + � _q�i(�)) � 0 (a condition to be also checked ex post), Ui(�) is increasingand thus (23) binds at � only. Hence we get:

Ui(�) =

Z �

�(1 + � _e�i(x) + � _q�i(x))qi(x)dx:

Inserting Ui(�) into the maximand of (PQi ) and integrating by parts yields the new expression of the

relaxed program (PQ0i ):

(PQ0i ) : maxfqi(:)g

Z �

�fpi(�; ei(�); e�i(�); qi(�); q�i(�))qi(�)� (�(qi(�)))� h(�)(1 + � _e�i(�) + � _q�i(�))qi(�)g f (�) d�:

Optimizing pointwise yields the following �rst-order condition:

(24) � + �(qi(�))� 2qi(�) + �ej(�) + �qj(�)� h(�)(1 + � _ej(�) + � _qj(�)) = 0;

together with the e¤ort optimality condition:

(25) ei(�) = �(qi(�)):

Let�qQ(�); eQ(�)

�2� denote the output-e¤ort pair when both suppliers o¤er an incomplete contract.

Next Proposition characterizes some useful features of this allocation.

Proposition 9 Assume �� is small enough and 00(e) > maxn12 ;1+�2�� ;

1+2�2(1��)

ofor all e 2 R+. Then the


� eQ(�) � e�(�) and qQ(�) � q�(�) for all � (with equality only at �);

� Information rents are increasing and (22) holds.

22

6.1 Equilibrium Characterization

We now turn to characterize the pure strategy equilibrium choice of contracting mode. As we show, besides

the extra agency costs associated to an incomplete contract which are kept small in the limit of small

uncertainty, two key e¤ects are at play once one moves from a complete to an incomplete contract in this

symmetric hierarchies framework. As we have argued above, an incomplete contract makes a retailer willing

to exert more e¤ort relative to a complete mechanism since the agent is residual claimant of the full impact

of his non-market activities (e¤ort) on demand enhancing when retail price is not monitored. This in turn

might a¤ect: (i) the e¤ort-output choice of the competing retailer via the �demand� driven horizontal

externality: a direct strategic e¤ect; and (ii) it e¤ects own reaction function and in turn the output-e¤ort

pair of the competing hierarchy: an indirect strategic e¤ect. Depending upon the nature of downstream

externalities, i.e., cooperative versus sel�sh e¤ort and complements versus substitutes goods, those e¤ects

might well go in opposite directions.

The next result extends Proposition 7 to a symmetric hierarchies environment and summarizes our

results.

Proposition 10 Assume �� is small enough, �1 < � + � < 1 and that 00 is large enough. Then the


� If �� > 0 the game G displays multiple (symmetric) equilibria in the contractual choice.

� If �� 0 there are multiple equilibria when j�j � minn1; �

00

o. When min

n1; �

00

o= �

00the unique

equilibrium entails complete contracts if � 00� j�j � min

n1; �(2

00+1)2 00

o. No equilibria in pure strategy

exist for j�j > �(2 00+1)2 00

whenever minn1; �(2

00+1)2 00

o= �(2 00+1)

2 00.

The intuition underlying this result is again driven by the di¤erent impact that the two kinds of mech-

anisms have on retailers�e¤ort choices.

Let us begin by explaining why incomplete contracts can be part of a PBE of G: Assume then, withoutloss of generality, that an incomplete contract rules the hierarchy S�i-R�i. We must consider three cases.

� If e¤orts have a cooperative value (� > 0) and goods are complements (� > 0), then when Si moves

from a complete to an incomplete contract Ri�s e¤ort increases, this has a direct positive e¤ect on R�i�s

reaction function which shifts upward, thus making Si better o¤ as goods are complements. Moreover,

a larger ei(�) has also a positive e¤ect on Ri�s demand and so it increases q�i(�) via a higher qi(�). Of

course, both e¤ects are bene�cial to Si who prefers an incomplete contract to a complete one in the

limit of small uncertainty.

� Assume now that e¤orts have a sel�sh nature (� < 0) and goods are substitutes (� < 0). In this caseincreasing Ri�s e¤ort through the choice of an incomplete contract lowers R�i�s reaction function and,

23

as a consequence, it relax retail competition. Moreover, it also increases qi(�) so to further reduce

q�i(�). Both e¤ects again go in the same direction of increasing Si�s pro�ts, and incomplete contracts

remain an equilibrium of G.

� Finally, in the free-riding case, i.e., � > 0 and � � 0, when Si moves from a complete to an incomplete

contract the increased e¤ort by Ri has a direct positive e¤ect on R�i�s reaction function which shifts

upward thus making R�i more aggressive at the market stage. However, a larger q�i(�) calls for a

higher e�i(�), which has a positive e¤ect on Ri�s demand. Of course, the latter e¤ect dominates the

former one when competition is not very intense, so that incomplete contract remain an equilibrium

of G. The opposite is true when j�j is large enough.

Let us now explain why complete contracts are also an equilibrium of G. Assume then that a completecontract rules the hierarchy S�i-R�i. We must look at Si�s incentive to play a complete contract (relative

to an incomplete one). Again, three cases must be discussed.

� When � > 0 and � > 0, moving from an incomplete to a complete contract reduces Ri�s incentive to

exert e¤ort due to the dichotomy result, this has a direct positive e¤ect on R�i�s e¤ort through the

RPM condition e�i(�) = 2q�i(�)��qi(�)��ei(�)� �, everything else being kept constant. Hence Ri�sreaction function shifts upward. Moreover, as e�i(�) increases q�i(�) increases too, thus making Sibetter o¤ since goods are complements. Both e¤ects go in the direction of increasing Si�s pro�ts, as a

consequence, complete contracts are part of an equilibrium for small demand uncertainty.

� When � < 0 and � < 0, the same kind of argument drives the result. The only di¤erence being thatthe e¤ort reduction driven by the choice of a complete contract now pushes e�i(�) down.

� Finally, in the free-riding case the lower incentive to exert e¤ort for Ri under a complete contract has adirect positive e¤ect on R�i�s e¤ort through the RPM condition e�i(�) = 2q�i(�)��qi(�)��ei(�)��.This shifts upward Ri�s reaction function: a positive direct e¤ect. However, since e�i(�) increases,

then q�i(�) increases too, thus making R�i more aggressive at the market stage and, in turn, Si worse

o¤. When retail competition is soft enough the former e¤ect dominates the latter one, hence complete

contracts remain part of an equilibrium of G in the limit of small uncertainty.

Since the game G admits multiple equilibria, in the next Proposition we order these equilibria accordingto a Pareto criterion from the suppliers�viewpoint. As we will see the key element will be the nature of

e¤ort externalities.

Let �!;! denote the suppliers�pro�t in a symmetric choice of contractual modes where in both hierarchies

the mechanism ! 2 fQF;RPMg is chosen.

24

Proposition 11 Assume �� is small enough, �1 < �+� < 1 and that 00 is large enough. When the game

G displays multiple equilibria in the contracting choice, then �Q;Q > �P;P (resp. �) if � > 0 (resp. �).

Incomplete contracts are thus mutually bene�cial to suppliers (relative to complete ones) only when

e¤ort spillovers are positive or e¤ort has a cooperative value. Indeed, in the limit of small uncertainty,

as retailers are residual claimants for the full impact of their e¤ort choices under incomplete contracts,

consumers�willingness to pay increases under incomplete contracting relative to complete contracts, thus

improving upon productive e¢ ciency.

6.2 Welfare Analysis

This section investigates the welfare e¤ects of contracting incompleteness on consumers� surplus. As we

have considered a simple manufacturer-retailer economy, the analysis developed in previous sections can be

used to assess the impact of retail price restrictions on third parties such as consumers.20 The question of

whether this type of contracts are welfare detrimental has indeed been at the heart of a controversial debate

in the antitrust and regulation literature for a long time.

Next Proposition extends the welfare results provided in Martimort and Piccolo (2006) within a isolated

manufacturer-retailer economy to a competitive environment.

Proposition 12 Assume �� is small enough, �1 < �+� < 1 and that 00 is large enough. Then incomplete

contracts are detrimental (resp. bene�cial) to consumers if � � 0 (resp. �).

Interestingly, the result relates the welfare desirability of vertical contracting based on retail price re-

strictions to the nature of non-market activities performed by retailers, i.e., cooperative versus sel�sh e¤ort.

Since the implementation of complete contracts forces �rms to be more aggressive at the market stage when

e¤ort has a cooperative value, contracts based only upon sales must be bene�cial to consumers relative

to more complex arrangements aimed at controlling retail price, as the former ones stimulate productive

e¢ ciency. The converse is obviously true when e¤ort has a sel�sh nature. An interesting corollary of this

result is that suppliers�and consumers�preferences are aligned with respect to the equilibrium contractual

mode.

7 Concluding Remarks

The paper has provided a simple rationale for incomplete contracting. By taking a traditional agency per-

spective we have showed that, once one moves from the isolated principal-agent set-up to games where

vertical organizations impose externalities on each other, incomplete contracting may emerge in equilibrium

whenever agency costs are su¢ ciently small and some aspects of the agents�performance are unveri�able.

20Much scholars have indeed advocated that the sole role of Antitrust policies should be to promote consumers�surplus. SeeBork (1978, Chapter 2, pp. 51) for instance.

25

Strikingly, in a set-up where transaction or retailing costs are endogenously created by asymmetric informa-

tion, our predictions are at the odds with the common wisdom postulating a negative relationship between

transaction costs and degree of contracting incompleteness.

From a normative perspective, we provide conditions under which the welfare loss that double mark-

ups in�ict on consumers is minimized either by vertical arrangements imposing retail price restrictions

or those controlling only sales depending on whether non-market activities performed by retailers have

a cooperative or sel�sh value. Our results show that RPM enhances (resp. reduces) consumers� surplus

relative to QF whenever these kind of activities (e¤ort) produce negative (resp. positive) spillovers on

downstream demands. In light of these results, the analysis reveals that the concerns about social desirability

of RPM, typically addressed by antitrust authorities, are justi�ed only under certain conditions related to

the underlying economic environment.

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27

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8 Appendix

8.1 Proof of Proposition 1

It is obvious and thus omitted. �

8.2 Proof of Lemma 2

To prove the claim it su¢ ces to verify that vertical price control allows S1 to learn (ex post) the realized

demand state. So, suppose that R1 lies at the revelation stage and announces a message �̂ 6= �, S1 then

supplies q1(�̂) units of inputs at the �xed fee t(�̂) and expects to observe (ex post) a retail price level

28

p1(�̂) = (2+�)(�̂+�q1(�̂)). However, since S2-R2 produces at q2(�) = (�+�q1(�̂))=2, ex post S1 will observe

p1(�̂; �) = (2 + �)(� + �q1(�̂)) 6= p1(�̂) for each �̂ 6= �. R1�s deviation will then be immediately detected. �


As a preliminary step we show that in a properly de�ned neighborhood of �, throughout denoted B(�), itmust be qQ1 (�) � qP1 (�) with equality holding only at �. That q

Q1 (�) = qP1 (�) = q�(�) is obvious. Notice

then, that for all � 2 B(�), a �rst-order Taylor expansion of qQ1 (�) around � yields:

qQ1 (�) � qP1 (�)� _qQ1 (�)(� � �);

where _qQ1 (�) can be obtained by a simple application of l�Hopital�s rule to the di¤erential equation (4):

lim�!�

_qQ1 (�) =2 + �

�2lim�!�

1� (2� �) _qQ1 (�)� _h(�)_h(�)

,

Using the fact that _h(�) = �1 and rearranging terms in the above equation we have _qQ1 (�) = (2+ �)(2��2)�1 � _qP1 (�) = (2� �)�1. Therefore, for all � 2 B(�), it must be:

_qQ1 (�) = (2 + �)(2� �2)�1 � (2� �)�1 = _qP1 (�);

which in turn implies qQ1 (�) � qP1 (�) for all � 2 B(�) since qQ1 (�) = qP1 (�). Now, let Bc(�) =

�� 2 �j� =2 B(�)

,

we show that the result also holds globally for all � 2 Bc(�). Suppose, indeed, that there exists a �� 2 Bc(�)such that qQ1 (�

�) = qP1 (��), and without loss of generality, consider that �� is the lowest of such values. By

using equation (4) one can easily verify that _qQ1 (��) = �(2 + �)��2 < 0 < _qP1 (�

�) = (2 � �)�1. Hence, in

a properly de�ned neighborhood of ��, say B(��), one must have qQ1 (�) � qP (�) (resp. <) if and only if

� � �� (resp. >). This yields a contradiction with the de�nition of ��.

We shall now prove the following results: Information rents are increasing in �, the local second-order

condition (IC2) holds, program (P 0) has a unique solution and global incentive compatibility holds. As apreliminary step we prove that qQ1 (�) � qm1 (�) for all �, where q

m1 (�) satis�es:

(26) � � (2� �)qm1 (�)� h(�) = 0:

To begin with, we show that the result holds in a properly de�ned neighborhood of �, say Bm(�).Di¤erentiating equation (26) with respect to � we have _qm1 (�) = 2(2 � �)�1 � (2 � �)(2 � �2)�1 = _qQ1 (�).

Therefore, since qm1 (�) = qQ1 (�) = qP1 (�), it must be qm1 (�) � qQ1 (�) for all � 2 Bm(�). Moreover, one can also

show that the result remains true for all � 2�� 2 �j� =2 Bm(�)

. Indeed, suppose that there exists a �� 2 �

such that qm1 (��) = qQ1 (�

��), and without loss of generality assume that �� is the lowest of such values. In

this case, from equation (4) one immediately gets _qQ1 (��) = 0. Since _qm1 (�) = (2 � �)�1(1 � _h(�)) > 0, we

29

have _qm1 (��) > _qQ1 (�

��) and qm1 (�) < qQ1 (�) for � � ��, a contradiction.

By using the same logic one can immediately show that (IC2) holds too. Also, uniqueness follows directly

by linearity of pi(:) and strict concavity of (P 0).For global incentive compatibility constraints, let UQ1 (�; �̂) de�ne R1�s pro�ts evaluated at the allocationn

qQ1 (�̂); tQ1 (�̂)

o, that is when the true demand state is � and the message sent to S1 is �̂ 6= �. To show

that global incentive compatibility constraints hold, we must have �(�; �̂) � UQ1 (�; �) � UQ1 (�; �̂) > 0 for

each pair (�; �̂) 2 �2. Assume then � > �̂ without loss of generality, simple algebraic manipulations allow to

rewrite �(:) as:

�(�; �̂) =

Z �

�̂

n� _tQ1 (s) + (� � q

Q1 (s) + �q

Q2 (�)) _q

Q1 (s)� _qQ1 (s)q

Q1 (s)

ods:

By using (IC1) and substituting for _tQ1 (s) = (s � qQ1 (s) + �qQ2 (s)) _q

Q1 (s) � qQ1 (s) _q

Q1 (s) into the above

equation one obtains:

�(�; �̂) =

Z �

�̂2 _qQ1 (s)

�Z �

s(1 + � _qQ2 (x))dx

�ds:

Since we have already proved that _qQ1 (�) > 0 and (1 + � _qQ2 (�)) > 0 for all �, it follows that �(�; �̂) > 0,

which concludes the proof.

Finally, from the �rst-order conditions (2) and (3) one has �P1 = E~�[qP1 (�)]

2 and �Q1 = E~�[qQ1 (�)]

2. Then

�P1 > �Q1 follows immediately from qQ1 (�) � qP1 (�) for all � 2 �. �


Observe that as we are considering �� small, we can use the following �rst-order Taylor expansions in

describing output and e¤ort: qP1 (�) � q�1(�)� _qP1 (�)(� � �) and eP1 (�) � e�1(�)� _eP1 (�)(� � �) for all �.To show that _qP (�) � 0, simple application of l�Hopital�s rule yields:

lim�!�

_qP1 (�) = lim�!�

2( _qP1 (�)� 00(eP1 (�)) _eP1 (�))� (2 + �(1 + � _eP1 (�)))( _h(�) 00(eP1 (�))� h(�) 000(eP1 (�)) _eP1 (�))�2�_h(�) 00(eP1 (�)) + h(�)

000(eP1 (�)) _eP1 (�)

� ;

which, from (10) together with _h(�) = �1 and h(�) = 0, yields:

(27) _qP1 (�) =2 00(2 + �)

2 00 (2� � (� + �))� �� 2 :

Therefore, monotonicity follows if the following restriction holds:

00(e) >2 + ��

2(2� � (� + �)) 8 e 2 R+;

30

where (�; �) 2�(�; �) 2 [�1; 1]2 : 2� � (� + �) 6= 0

.

By using the de�nition of q�1(�) we get:

_q�1(�) = 00(2 + �)

00 (4� �2)� �� 2 :

Since qP1 (�)� q�1(�) � ( _q�1(�)� _qP1 (�))(� � �) for all �, we get:

qP1 (�)� _q�1(�) =(2 + ��)

�2 00 � 1

�(2 + �) 00

(2 00 (2� � (� + �))� �� 2)( 00 (4� �2)� �� 2)

which immediately implies _q�1(�) < _qP1 (�) since 00(e) > max

n12 ;

2+��2(2��(�+�))

ofor all e 2 R+, hence qP1 (�) �

q�1(�) for all � with equality only at �.

By using the same kind of argument, we have:

_eP1 (�) =

�2 00 + 1

�(2 + �)

2 00 (2� � (� + �))� �� 2 :

Then, since 00 _q�1(�) = _e�1(�), one gets:

_eP1 (�)� _e�1(�) =2 (2 + �) 00

(2 00 (2� � (� + �))� �� 2)( 00 (4� �2)� �� 2) ;

implying _e�1(�) < _eP1 (�) since 00(e) > max

n12 ;

2+��2(2��(�+�))

ofor all e 2 R+ and, as a consequence, eP1 (�) �

e�1(�) for all �.

Notice also that program (PP 0) displays interior solutions whenever �� is su¢ ciently small. Moreover,from the above results one can show that information rents are increasing in �. In fact, simple algebra yields:

0 < _UP1 (�) =(2 + �)

�2 00 � 1

� 0(eP1 (�))

2 00 (2� � (� + �))� �� 2 .

The same kind of argument allows to show that the monotonicity condition holds, i.e., 2 _qP1 (�) (1 +

� _qP2 (�)) > 0 for all �. Moreover, uniqueness simply follows from linearity of pi(�; e1; qi; qj) for i = 1; 2 and

strictly concavity of (PP 0).Finally, let UP1 (�; �̂) be R1�s pro�ts at

npP1 (�̂); q

P1 (�̂); t

P1 (�̂)

o, namely when the true retailer�s type

is � and the message sent to the manufacturer is �̂ 6= �. To show that global incentive compatibil-

ity constraints hold we must have �P (�; �̂) � UP1 (�; �) � UP1 (�; �̂) > 0 for each pair (�; �̂) 2 �2. As-

sume then � > �̂ without loss of generality, simple algebraic manipulations allow to rewrite �P (:) as

�P (�; �̂) =R ��̂

�_tP1 (s)� 2 0(eP1 (s; �)) _qP1 (s)

ds, where eP1 (s; �) � 2qP1 (s) � �qP2 (�) � �. By using (IC1) and

31

substituting for _tP1 (s) � 2 0(eP1 (s)) _qP1 (s) into �P (:), one obtains:

�P (�; �̂) = 2

Z �

�̂_qP1 (s)

�Z �

s 00(e(s; x))(1 + � _qP2 (x))dx

�ds:

Then, since _qP1 (�) > 0 whenever 00(e) > max

n12 ;

2+��2(2��(�+�))

ofor all e 2 R+, it follows that �P (�; �̂) > 0

for all � and �̂, which concludes the proof. �


Again, for �� small enough we consider qQ1 (�) � q�1(�)� _qQ1 (�)(�� ) for all �. From (15), a straightforward

application of l�Hopital�s rule yields:

lim�!�

_qQ1 (�) = lim�!�

(2 + �)(1� _h(�)) + (2 + ��)�0(qQ1 (�)) _qQ1 (�)� _qQ1 (�)(4� �2)

��_h(�)(��0(qQ1 (�)) + �) + �h(�)�

00(qQ1 (�)) _qQ1 (�)

� ;

hence, using _h(�) = �1 and h(�) = 0, we get:

_qQ1 (�) =(2 + �) 00

00 (2� �2)� �� 1 ;

where monotonicity follows, i.e., _qQ1 (�) � 0, if the following restriction holds:

00(e) >1 + ��

2� �2 8 e 2 R+:

In the limit of small uncertainty, i.e., �� small, it also follows that qQ1 (�)�q�1(�) � ( _q�1(�)� _qQ1 (�))(��)

for all �, and as a consequence:

_qQ1 (�)� _q�1(�) =(2 + �)(2 00 � 1) 00

( 00 (2� �2)� �� 1)( 00 (4� �2)� �� 2) ;

which immediately implies _q�1(�) < _qQ1 (�) for all 00(e) > max

n12 ;1+��2��2

ofor all e 2 R+. Hence qQ1 (�) � q�1(�)

for all � with equality only at �. By using the same kind of argument one also has eQ1 (�) � e�1(�)� _eQ1 (�)(��).

Hence:

_eQ1 (�) = �0(q�1(�)) _qQ1 (�) > �0(q�1(�)) _q

�1(�) = _e�1(�);

which directly implies eQ1 (�) � e�1(�) for all �.

That program (PQ0) displays interior solutions whenever �� is small enough is obvious. Hence, for qQ1 (�)

32

being interior, information rents are increasing in � since 00(e) > maxn12 ;1+��2��2

ofor all e 2 R+ implies:

0 < _UQ1 (�) =qQ1 (�)

�2 00 � 1

�(2 + �)

2( 00 (2� �2)� �� 1) :

Since _qQ1 (�) > 0, one can check that the monotonicity condition also holds, i.e., (1 + � _qQ2 (�)) _qQ1 (�) > 0.

Moreover, uniqueness simply follows from linearity of p1(�; e1; q1; q2) and strictly concavity of (PQ).Finally, let UQ1 (�; �̂) de�ne the retailer�s pro�ts evaluated at

nqQ1 (�̂); t

Q1 (�̂)

o, when the true demand

realization is � and the message sent to S1 is �̂ 6= �. To show that global incentive compatibility constraints

hold we must have �Q(�; �̂) � UQ1 (�; �)�UQ1 (�; �̂) > 0 for each pair (�; �̂) 2 �2. Assume then � > �̂ without

loss of generality, simple algebraic manipulations allow to rewrite �Q(:) as:

�Q(�; �̂) =

Z �

�̂

n� _tQ1 (s) + _qQ1 (s)p1(�; e

Q1 (s); q

Q1 (s); q

Q2 (�))� q

Q1 (s) _q

Q1 (s)

ods:

By using IC1 and substituting for _tQ(s) � _qQ1 (s)p1(s; eQ1 (s); q

Q1 (s); q

Q2 (s)) � qQ1 (s) _q

Q1 (s) into the above

equation, we get:

�Q(�; �̂) =

Z �

�̂_qQ1 (s)

�Z �

s(1 + � _qQ2 (x))dx

�ds.

Then, since we have proved above that (1 + � _qQ2 (�)) > 0 when 00(e) > maxn12 ;1+��2��2

ofor all e 2 R+,

one has �Q(�; �̂) > 0, which concludes the proof. �


First, observe that eQ1 (�)� eP1 (�) � ( _eP1 (�)� _eQ1 (�))(�� ) for all �. Using the de�nition of _eP1 (�) and _e

Q1 (�)

we get:

_eP1 (�)� _eQ1 (�) =(2 00 � 1)(2 + �)( 00(2� �2)� 1)

(2 00 (2� � (� + �))� �� 2)( 00 (2� �2)� �� 1) ;

which immediately implies _eP1 (�) > _eQ1 (�) as we have assumed 00(e) > max

n12 ;

2+��2(2��(�+�)) ;

1+��2��2

ofor all

e 2 R+. Hence, eQ(�) � eP (�) for all � with equality only at �.

By using the same kind of argument we have qQ2 (�)� qP2 (�) � ( _qP2 (�)� _qQ2 (�))(� � �) for all �. Hence:

(28) _qP2 (�)� _qQ2 (�) =� (2 + �)

�2 00 � 1

�22(2 00 (2� � (� + �))� �� 2)( 00 (2� �2)� �� 1) ;

which immediately yields the result since 00(e) > maxn12 ;

2+��2(2��(�+�)) ;

1+��2��2

ofor all e 2 R+.

33

Similarly, we have:

_qP1 (�)� _qQ1 (�) =(��)

�2 00 � 1

�(2 + �) 00

(2 00 (2� � (� + �))� �� 2)( 00 (2� �2)� �� 1) ;

yielding sign(qQ1 (�)� qP1 (�)) = sign (��) since 00(e) > maxn12 ;

2+��2(2��(�+�)) ;

1+��2��2

ofor all e 2 R+: �


First, observe that in the limit of small uncertainty the type-contingent pro�t of the upstream supplier

under both contractual regimes can be obtained by using a second-order Taylor expansion of (virtual)

pro�ts around �:

(29) �!1 (�) � �!1 (�)� _�!1 (�)(� � �) +1

2��!1 (�)(� � �)2 for each ! 2 fQF;RPMg :

Observe that when �� is su¢ ciently small one can also approximate the distribution of ~� with a uniform,

i.e., ~� � U [�], we then have E~�[� � �] � ��=2 and E~�[� � �]2 � ��2=3. Taking expectations on both sidesof (29) we get:

(30) �Q1 ��P1 ��_�P1 (�)� _�Q1 (�)

� ��2+��Q1 (�)� ��P1 (�)

� ��26;

We need then to compute each term in (30). First, consider program (PP 0), di¤erentiating with respectto � and using the Envelope Theorem we have:

_�P1 (�) = 0(eP1 (�))(1 + � _qP (�))(1� _h(�)) + h(�) 00(eP1 (�))(1 + � _qP (�))2:

From h(�) = 0 and _h(�) = �1, taking limits for � ! � on both sides it follows _�P1 (�) = 2q�1(�)(1+� _q

P2 (�)).

Using the �rst-order condition (8) in _�P1 (�), di¤erentiating again with respect to �, we have ��P1 (�) =

( _qP1 (�) + 00 _eP1 (�))(1 + � _q

P2 (�)).

Consider now program (PQ0), again di¤erentiating with respect to � and using again the EnvelopeTheorem we have:

_�Q1 (�) = (1 + � _qQ2 (�))q

Q1 (�)� _h(�)(1 + � _q

Q2 (�))q

Q1 (�);

taking limits for � ! � on both sides and using _h(�) = �1 we have _�Q1 (�) = 2q�1(�)(1 + � _qQ2 (�)). By using

the same kind of argument, we get ��Q1 (�) = ( _qQ1 (�) +

00 _eQ1 (�))(1 + � _qQ2 (�)).

Substituting _�!1 (�) and ��!1 (�); for ! 2 fQF;RPMg, into �

Q1 ��P1 , it follows:

�Q1 ��P1 � �q�(�)�_qP2 (�)� _qQ2 (�)

�� (��P1 (�)� ��Q1 (�))

��2

6:

34

Taking �� small enough and substituting (28) into the above equation we have:

�Q1 ��P1 �(��)��q�(�)

�2 00 � 1

�(2 + �) 00

(2 00 (2� � (� + �))� �� 2)( 00 (2� �2)� �� 1) ;

which immediately yields the result since �Q1 ��P1 > 0 (resp <) whenever �� > 0 (resp. �).To conclude the proof we must consider the case �� = 0. The sign of �Q1 � �P1 is now obtained by

looking at the second order term of the Taylor approximation (30):

�Q1 ��P1 � (( _qQ1 (�) +

00 _eQ1 (�))(1 + � _qQ2 (�))� ( _qP1 (�) + 00 _eP1 (�))(1 + � _qP2 (�)))

��2

6:

First assume � = 0, in this case _qP1 (�) = _qQ1 (�) and _qP2 (�) = _qQ2 (�), then we have:

�Q1 ��P1 � �(1 + � _qQ2 (�))( _e

P1 (�)� _eQ1 (�))

��2

6;

which yields the result since _eP1 (�) > _eQ1 (�). The same kind of argument allows to show that �Q1 � �P1 �

� 00( _eP1 (�)� _eQ1 (�))��2=6 < 0 when � = 0, and thus concludes concludes the proof. �


We follow the same technique used in the proof of Propositions 4 and 5. First, notice that the allocation�qP (�); eP (�)

�2� solves the following system of di¤erential equations:

_qP (�)(2� �)� 1� _eP (�)(1 + �) = 0;

qP (�)� 0(eP (�))� h(�)(1 + � _eP (�) + � _qP (�)) 00(eP (�)) = 0;

with boundary conditions qP (�) = q�(�) and eP (�) = e�(�). As we have assumed �� small enough, from

a �rst-order Taylor expansion of qP (:) around � we have qP (�) � q�(�) � _qP (�)(� � �). From a simple

application of l�Hopital�s rule we have:

(31) _qP (�) =2 00

2 00 (1� �� )� (1 + �) and _eP (�) =2 00 � 1

2 00 (1� �� )� (1 + �) :

Having assumed � + � < 1, monotonicity is insured if the following restriction holds:

00(e) >1 + �

2(1� �� ) 8 e 2 R+;

35

Now, it is easy to show that in the complete information benchmark we have:

_q�(�) = 00

00(2� �)� (1 + �) ;

with _q�(�) > 0 since 00(e) > (1 + �)=(2� �) for all e 2 R+.Hence, as qP (�) = q�(�), it follows that _qP (�) > _q�(�) must imply qP (�) � q�(�) for all � < �. Simple

algebra then yields:

_qP (�)� _q�(�) =

�2 00 � 1

�(1 + �) 00

( 00(2� �)� (1 + �))(2 00 (1� �� )� (1 + �)) ;

which implies the result since 00(e) > 1=2 for all e 2 R+. By using the same kind of argument one showsthat eP (�) � e�(�) for all � � �. Finally, notice that for �� small enough program e¤orts and outputs will

be positive. The rest of the proof is omitted as it follows Martimort (1996). �


By de�nition the allocation�qQ(�); eQ(�)

�2� solves the following system of di¤erential equations:

� + (1 + �)eQ(�)� qQ(�)(2 + �)� h(�)(1 + � _eQ(�) + � _qQ(�)) = 0;

_eQ(�) = �0(qQ(�)) _qQ(�);

with boundary conditions qR(�) = q�(�) and eR(�) = e�(�). Again, as we have assumed �� small, l�Hopital�s

rule yields:

(32) _qQ(�) =2 00

2 00(1� �)� (1 + 2�) and _eQ(�) =2

2 00(1� �)� (1 + 2�) .

which directly implies monotonicity if:

00(e) >1 + 2�

2(1� �) 8 e 2 R+ :

Simple algebra then yields:

_qQ(�)� _q�(�) =

�2 00 � 1

� 00�

2 00(1� �)� (1 + 2�)� � 00(2� �)� (1 + �)

� ;that immediately implies qQ(�) � q�(�) for all � with equality holding only at �. The same kind of argument

allows to show that eQ(�) � e�(�) for all � with equality holding only at �. Finally, (PQ0) has interiorsolutions whenever �� is small enough. The rest of the proof is omitted as it follows Martimort (1996). �

36


To begin with, we need to characterize allocations in an asymmetric play of G. So, suppose that Si choosesa complete contract while S�i plays an incomplete one. By de�nition the allocations

nqP;Qi (�); eP;Q(�)

o�2�

andnqQ;P�i (�); e

Q;P�i (�)

o�2�

solve the following system of di¤erential equations:

(33) qP;Qi (�)� 0(eP;Qi (�))� h(�)(1 + � _eQ;P�i (�) + � _qQ;P�i (�)))

00(eP;Qi (�)) = 0;

(34) eP;Qi (�) = 2qP;Qi (�)� �eQ;P�i (�)� �qQ;P�i (�)� �;

(35) � + eQ;P�i (�)� 2qQ;P�i (�) + �e

P;Qi (�) + �qP;Qi (�)� h(�)(1 + � _eP;Qi (�) + � _qP;Qi (�)) = 0;

(36) eQ;P�i (�) = �(qQ;P�i (�)):

with boundary conditions qQ;P�i (�) = qP;Qi (�) = q�(�) and eP;Qi (�) = eQ;P�i (�) = e�(�).

Di¤erentiating (36) it follows _eQ;P�i (�) = �0(qQ;P�i (�)) _qQ;P�i (�), then plugging _e

Q;P�i (�) into (33)-(35) and

linearizing the reduced system around the point��; q�(�); e�(�)

�we have:

(37) 2 _qP;Qi (�)� 1� _eP;Qi (�)� �_qQ;P�i (�)

00� � _qQ;P�i (�) = 0;

(38) _qP;Qi (�)� 00 _eP;Qi (�) + (1 +�

00_qQ;P�i (�) + � _q

Q;P�i (�))

00 = 0;

(39) 1 +1

00_qQ;P�i (�)� 2 _q

Q;P�i (�) + � _e

P;Qi (�) + � _qP;Qi (�) + (1 + � _eP;Qi (�) + � _qP;Qi (�)) = 0:

One can check that the solution of the linearized system (37)-(39) yields:

_eP;Qi (�) =�(1� 2�) + 2 00 (2� + �) + 4( 00)2 (1 + �)

a+ b 00 + c( 00)2;

_qP;Qi (�) =�2 00 (1� 2�) + 4( 00)2 (1 + �)

a+ b 00 + c( 00)2;

and

_qQ;P�i (�) =�2 00 (1� �) + ( 00)2 (1 + � + �)

a+ b 00 + c( 00)2:

37

Moreover, _eQ;P�i (�) = �0(q�(�)) _qQ;P (�) implies:

_eQ;P�i (�) =1

00

��2 00 (1� �) + ( 00)2 (1 + � + �)

a+ b 00 + c( 00)2

�;

where we have de�ned a = (1� 2�), b = �2�3��+ 2�2 + 2

�and c = 4

�1� �2 � ��

�. With c > 0 since we

have assumed 1 > �+�. Notice when 00 is su¢ ciently large and �1 < �+� < 1, monotonicity follows under

all possible contractual regimes. That is, _q!;!0(�) � 0 for all pairs (!; !0) 2 fQF;RPMg2 and � 2 � if:

00(e) > max

�1� �

1 + � + �;1 + 2�

2(1� �) ;1 + �

2(1� �� ) ; 001;

002

�; 8 e 2 R+

where,

001 = max� 00 2 R+j � (1� 2�) + 2 00 (2� + �) + 4( 00)2 (1 + �) = 0

;

and

002 = max� 00 2 R+ja+ b 00 + c( 00)2 = 0

:

Equipped with this characterization we can conclude the proof. Again, since �� is small enough, from a

second-order Taylor approximation around � the Si�s type-contingent pro�t when he chooses a mechanism

! while S�i chooses !0 can be written as:

�!;!0

i (�) � �!;!0

i (�)� _�!;!0

i (�)(� � �) + 12��!;!

0

i (�)(� � �)2:

Then, using E~�[� � �] � ��=2 and E~�[� � �]2 � ��2=3 it follows that �!;!0

i = E~�[�!;!0

i (�)] �E~�[�

!0;!0

i (�)] = �!0;!0

i (resp. <) if:

(40) �( _�!;!0

i (�)� _�!0;!0

i (�))��

2+ (��!;!

0

i (�)� ��!0;!0

i (�))��2

6� 0 (resp. <):

We �rst show that incomplete contracts are part of a PBE of G if one of the following conditions

holds: (i) �� > 0; or (ii) j�j < �= 00 when � � 0 and � � 0. Since players are symmetric, we only need

to show that �Q;Qi � �P;Qi . Consider then program (PQ0i ), di¤erentiating with respect to � and usingthe Envelope Theorem one can easily show that _�Q;Qi (�) = 2q�(�)(1 + � _eQ(�) + � _qQ(�)) and ��Q;Qi (�) =

( _qQ(�)+ 00 _eQ(�))(1+� _eQ(�)+� _qQ(�)). By using the same kind of argument, di¤erentiating program (PP 0i )with respect to � and taking limits for � ! � one also gets _�P;Qi (�) = 2q�(�)(1 + � _eQ;P�i (�) + � _qQ;P�i (�)) and��P;Qi (�) = ( _qP;Qi (�) + 00 _eP;Qi (�))(1 + � _eQ;P�i (�) + � _qQ;P�i (�)). Substituting in (40) and taking �� small we

have:

�Q;Qi ��P;Qi � q�(�)��( _eP;Q�i (�)� _eQ(�)) + �( _qP;Q�i (�)� _qQ(�))

��;

38

using the solutions of the linearized system of equations (37)-(39) together with _qQ(�) and _eQ(�), we have:

�Q;Qi ��P;Qi �2�� + � 00

��q�(�)

�2 00 � 1

�2�a+ b 00 + c( 00)2

� �2 00 (1� �� )� (1 + �)

� ;which yields immediately the result when � + � 00 6= 0.

Assume now � + � 00 = 0, for �� small the sign of �Q;Qi � �P;Qi is given by the second-order terms of

equation (40):

lim�!��= 00

��Q;Qi ��P;Qi

�� 2

6lim

�!��= 00

��Q;Qi (�)� ��P;Qi (�)

�= � 00��

2

6< 0;

the same result holds when � = 0.

We now show that complete contracts are part of a PBE of G if one of the following conditions holds:(i) �� > 0, or (ii) j�j � �(2 00 + 1)=2 00 when � � 0 and � � 0. Consider now the following di¤erence

�P;Pi ��Q;Pi , for �� small the same argument used above allows to obtain:

�P;Pi ��Q;Pi � q�(�)��( _eP;Q�i (�)� _eP (�)) + �( _qP;Q�i (�)� _qP (�))

��:

After simple algebra we have:

�P;Pi ��Q;Pi �� + 2 00(� + �)

��q�(�)

�2 00 � 1

�2�a+ b 00 + c( 00)2

� �2 00 (1� �� )� (1 + �)

� ;yielding the result for � + 2 00(� + �) 6= 0.

Consider now the case � + 2 00(� + �) = 0, for �� small the sign of �P;Pi � �Q;Pi is given by the

second-order terms of equation (40):

lim�!�� (2

00+1)2 00

��P;Pi ��Q;Pi

�� 2

6lim

�!�� (2 00+1)2 00

��P;Pi (�)� ��Q;Pi (�)

�= 00

��2

6> 0;

it is easy to check that the same result holds when � = 0 and �+2 00(�+ �) 6= 0. This concludes the proof.�


Again, as we have assumed �� small:

�Q;Q ��P;P � q�(�)��( _eP (�)� _eQ(�)) + �( _qP (�)� _qQ(�))

��:

39

From equations (31) and (32) we have:

�Q;Q ��P;P ��q�(�)

�2 00 � 1

�2(2 00 (1� �� )� (1 + �))(2 00(1� �)� (1 + 2�)) ;

which directly proves that �Q;Q > �P;P (resp. <) if � > 0 (resp. < 0).

When � = 0, the sign of �Q;Q ��P;P is given by the second-order terms of (40):

lim�!0

��Q;Q ��P;P

�� 2

6lim�!0

��Q;Q(�)� ��P;Pi (�)

�= ��

2

6

�2 00 � 1

�2 00�

1� 2 00(1� �)�2 < 0;

which concludes the proof. �


Since we have considered a simple representative consumer economy where, for any given wealth level w,

demands are derived as solution of the program,

max(q1;q2;I)

8<:V (q1; q2; I; �) : Xi=1;2

piqi + I � w

9=; ;

with:

V (q1; q2; I; �) =X

i;j=1;2;i6=jei(qi + �qj) + �

Xi=1;2

qi �1

2

0@Xi=1;2

q2i � 2�q1q2

1A+ I:Hence, as we focus only on what happens to consumers in the symmetric equilibria of contracting choices

we have:

V (qQ(�); qQ(�); I; �)� V (qP (�); qP (�); I; �) = 1

2

Xi=1;2

(qQ(�)� qP (�))(qQ(�) + qP (�)):

As we have assumed �� small enough, the result follows from:

_qQ(�)� _qP (�) = � 2� 00(2 00 � 1)�2 00(1� �)� (1 + 2�)

� �2 00 (1� �� )� (1 + �)

� ;which directly yields sign

�qQ(�)� qP (�)

�= sign(�). �

40

the strategic value of incomplete contracts for competing...

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