Optimal Collusion in Oligopoly Supergames:

Marginal Costs Matter

Etienne Billette de Villemeur∗, Laurent Flochel∗∗, Bruno Versaevel�

December 31, 2004

∗ Corresponding author. University of Toulouse 1, IDEI & GREMAQ, 21 allee de Brienne,

31000 Toulouse France (Etienne.devillemeur@univ-tlse1.fr)

∗∗ University of Lyon 2 & GATE, 93 chemin des Mouilles, BP 167 69131 Ecully Cedex France

(ßochel@gate.cnrs.fr)

�Ecole de Management de Lyon & GATE, 23 avenue Guy de Collongue, 69134 Ecully Cedex

France (versaevel@em-lyon.com)

1

Optimal Collusion in Oligopoly Supergames:

Marginal Costs Matter

Abstract

In a standard oligopoly supergame with identical Þrms, a necessary condition on the level of

marginal costs is derived for optimal collusion to be sustainable, either in prices or in quantities,

for any degree of product differentiation, and any number of Þrms. Only in the Cournot version

of the model, and with at most three Þrms, the level of marginal costs plays no role in the

sustainability of collusion. Otherwise, for any degree of differentiation and any size of industry,

the marginal cost must be higher than a threshold value for an optimal single-period punishment

to be obtained. When parameter values are such that no optimal single-period punishment can

be implemented, a multi-period punishment can always be constructed that leads Þrms to

sustain collusion at the proÞt-maximizing price or quantity level. In that case, punishments

last longer when the level of marginal costs decreases. By contrast, the length of the punishment

phase does not impact the minimum discount factor for which collusion can be implementable.

More differentiation and/or concentration unambiguously facilitate collusion. When Þrms use

prices, the optimal punishment phase lasts longer than when they use quantities, and the

boundary condition on the discount factor is more stringent.

JEL ClassiÞcation: C72; D43; L13.

Keywords: Collusion ; Product differentiation ; Industry concentration ; Cournot ; Bertrand.

1 Introduction

Antitrust authorities are interested in relying on simple guidelines in order to investigate the

behavior of Þrms. This motivates the understanding of the impact of product differentiation and

industry concentration on the sustainability of collusive agreements in oligopolistic industries.

In a legal context in which collusive agreements cannot be contractually enforced, it is well

known that each Þrm may Þnd it proÞtable to deviate from a tacitly collusive price or quantity

that maximizes industry proÞts. This may not be the case however if a deviation can be

sufficiently and credibly �punished� via lower industry prices or larger quantities in a subsequent

period of time. In order to identify structural conditions for the sustainability of collusive

strategies, the theoretical literature has constructed a class of dynamic models usually referred

to as supergames. These models feature a repeated market game in which Þrms maximize a

ßow of discounted individual proÞts by non-cooperatively choosing a price or a quantity over

an inÞnite number of periods.

One stream of that literature follows Friedman (1971) by considering trigger strategies (com-

monly referred to as �grim� strategies) which call for reversion to the one-shot stage game Nash

equilibrium forever when a deviation is detected in a previous period. A general result is that

collusion can be sustained if the discount factor is above a threshold value. In a duopoly model

with horizontal differentiation, Deneckere (1983, 1984) Þnds that tacitly collusive agreements

in prices are less easily sustained than in quantities unless goods are highly substitutable. This

is sensitive to the two-Þrm assumption, since in an extension of the same model Majerus (1988)

demonstrates that, with three Þrms or more, a collusion in prices is less easily sustained than

in quantities for all degrees of substitutability. Other studies examine the relationship between

alternative formalizations of product differentiation and the sustainability of collusion. They in-

clude papers by Chang (1991) and Ross (1992), who conÞrm in two spatial competition models

3

that collusion is more easily sustained when products are increasingly differentiated.1

A weakness that is common to all models of collusion with trigger strategies (i.e. pun-

ishments of inÞnite duration) is that they rule out the possibility of modulating the level of

punishments. More precisely, by assuming that Þrms revert to the Nash equilibrium of the

one-shot stage game in all periods that follow a deviation, they put an upper bound on the

severity of punishments.

This problem is tackled by a second stream of the literature, which reconsiders the impact

of various model speciÞcations on the sustainability of collusion with a stick-and-carrot penal

code. In the latter mechanism, if a Þrm deviates from the collusive strategy, all Þrms play a

punishment strategy before returning to the collusive price or quantity. In a repeated Cournot

(i.e., quantity-setting) oligopoly model, with identical sellers of a homogenous good and con-

stant marginal costs, Abreu (1986) characterizes the most severe punishments as �optimal� by

proving that they lead Þrms to sustain the most proÞtable collusive strategy. A dual interpre-

tation is to consider the minimum discount factor that sustains a given collusive outcome. Also

in a repeated Cournot model, Wernerfelt (1989) uses a stick-and-carrot framework to Þnd that

more product differentiation may render collusion less sustainable when the number of Þrms is

relatively large.2 In a repeated Bertrand (i.e., price-setting) duopoly model, with spatial differ-

entiation, and constant marginal costs normalized to zero, Hackner (1996) demonstrates that a

symmetric stick-and-carrot structure is an optimal price path, and conÞrms that differentiation

tends to facilitate collusive agreements. It is also demonstrated that, when the punishment

price is constrained to be non-negative, a prolonged price war can sustain the same collusive

1With vertical differentiation, Hackner (1994) Þnds that more differentiation unambiguously implies more

sustainable collusion in a Bertrand setup, with a trigger penal code. In the present paper we concentrate on

horizontal differentiation exclusively.2Although of interest, this ambiguous result is derived from demand assumptions (adapted from Deneckere,

1983) which are not standard (on this see Osterdal, 2003, pp. 54-55).

4

strategy as in the unrestricted case. With two Þrms and constant marginal costs again, but

with another speciÞcation of the horizontal differentiation assumption, Lambertini and Sasaki

(2002) Þnd a qualitatively similar relationship between product substitutability and collusion

sustainability.3 In addition, for all degrees of product differentiation, they show that collusion is

less easily sustainable in Bertrand than in Cournot. This is obtained in a setup where quantities

are constrained to be non-negative but prices may fall below zero.4

In this paper, we show that the level of constant marginal costs matters for collusion sus-

tainability. Our starting point is the intuitive proposition by Lambertini and Sasaki (2001) that

collusion in a repeated oligopoly game with high marginal costs may be more easily sustainable

than with lower marginal costs, all other things remaining equal. The underlying idea is that,

if prices are constrained to be positive, the ability to punish a deviation by charging prices

below marginal costs is limited when these costs are close to zero. We unveil a condition on

the level of marginal costs that is necessary for collusion to be sustainable, either in prices or in

quantities, in a stick-and-carrot setup. This is done with standard demand speciÞcations, for

any degree of product differentiation, and any number of Þrms.

The introduction of positive marginal costs, together with a non-negativity constraint on

prices as in Hackner (1996), leads to several new results. We Þnd that only in the Cournot

version of the model, and with at most three Þrms, the level of marginal costs plays no role in the

3Lambertini and Sasaki (2002) also consider complements. Under non-negative quantity constraints, they

Þnd that longer (i.e., multi-period) punishments are required only when products are close complements and

the quantity is the strategic variable.4Beyond the degree of product differentiation and the number of Þrms, other studies focus on the role of

various cost and demand speciÞcations. For example, Rothschild (1999) demonstrates that collusion sustainabil-

ity depends on Þrms� relative efficiencies. Other examples include Lambertini (1996), Tyagi (1999), and Collie

(2004), who investigate the relationship between assumptions on the form of the demand function and Þrms�

ability to collude in a repeated games with trigger strategies.

5

sustainability of collusion. Otherwise, for any degree of differentiation and any number of Þrms,

the marginal cost must be higher than a threshold value. This comes in addition to the usual

lower bound condition on the discount factor. High marginal costs can thus be included in the

list of factors that facilitate collusion. When parameter values are such that no optimal single-

period punishment can be implemented, we Þnd that a longer � i.e., multi-period � punishment

can always be constructed that leads Þrms to sustain collusion. More precisely, the length of the

punishment phase is demonstrated to be a substitute for the level of marginal costs. By contrast,

it does not impact on the critical threshold value of the discount factor that allow to sustain

collusion in multi-period punishment scheme. A comparative statics analysis establishes that

more differentiation and/or concentration in the industry relax the sustainability conditions on

the marginal cost and the discount factor, and thereby facilitate collusion. Eventually, when

Þrms use prices as opposed to quantities, we Þnd that the optimal punishment phase lasts

longer, and the boundary condition on the discount factor is more stringent. In that sense,

collusion is less easily sustainable in Bertrand than in Cournot.

The remainder of the paper is organized as follows. Section 2 describes the model. In Section

3, we investigate the role of the level of marginal costs on the existence of an optimal single-

period punishment, either in prices or in quantities. In Section 4, we consider situations in

which collusion is not sustainable with a single-period punishment. Conditions of sustainability

of optimal collusion for the Bertrand and Cournot cases are compared in Section 5. Final

remarks appear in Section 6. Computations are in the appendix.

6

2 The Model

Identical Þrms in N = {1, . . . , n} maximize intertemporal proÞts by simultaneously and non-cooperatively choosing either quantities (Cournot supergame) or prices (Bertrand supergame)

in an inÞnitely repeated game over t = 1, 2, ...,∞. The discount factor δ = 1/(1+ r), where r isthe single period interest rate, is common to all Þrms. Each Þrm i incurs a constant marginal

(unit) cost c ∈ [0, 1] to supply a non-negative quantity qi. Over all periods, each consumer hasthe same identical utility function, adapted from Hackner (2000), of the form

U(q, I) =nXi=1

qi − 12

nXi=1

q2i + 2γXi6=j

qiqj

+ I, (1)

which is quadratic in the consumption of q-products and linear in the consumption of the com-

posite I-good (i.e., the numeraire).5 The parameter γ ∈ (0, 1) measures product substitutabilityas perceived by consumers. If γ → 0, the demand for the different product varieties are inde-

pendent and each Þrm has monopolistic market power, while if γ → 1, the products are perfect

substitutes. Consumers maximize utility subject to the budget constraintPpiqi+I ≤ m, where

m denotes income, pi is the non-negative price of product i, and the price of the composite

good is normalized to one. By symmetry, we notePj 6=i qj = (n− 1)qj .

Bertrand In the price-setting version of the model, from (1) the demand function for Þrm

i writes

qi(pi, pj) =

1− pi if pi ≤ �pi(pj)

11+γ(n−1)

³1− 1+γ(n−2)

1−γ pi +γ(n−1)1−γ pj

´if �pi(pj) ≤ pi ≤ �pi(pj)

0 if pi ≥ �pi(pj)

, (2)

5In Hackner (2000), quantities qi are multiplied by a parameter ai, that is a measure the distinctive quality

of each variety i. Here we exclude vertical product differentiation by assuming that ai = 1, all i ∈ N .

7

and the demand for each other symmetric Þrm j writes

qj(pi, pj) =

0 if pi ≤ �pi(pj)

11+γ(n−1)

³1− 1

1−γ pj +γ1−γ pi

´if �pi(pj) ≤ pi ≤ �pi(pj)

1−pj1+γ(n−2) if pi ≥ �pi(pj)

, (3)

with �pi(pj) =1γ [pj − (1− γ)] and �pi(pj) =

11+γ(n−2) (1− γ + γ (n− 1) pj). The piecewise

deÞnition of demand functions is a consequence of the non-negativity constraint we impose

on quantities. Remark however that the two functions (24) and (25) are continuous in their

respective arguments. This implies that proÞt functions are also continuous. One can check

that they are also concave in the relevant decision variable.

Cournot In the quantity-setting version of the model, from (1) Þrm i�s inverse demand

function in each period is

pi(qi, qj) = max{0, 1− qi − γ(n− 1)qj}, (4)

all qj ≥ 0, j 6= i, and the inverse demand for each other symmetric Þrm j is deÞned by

pj(qi, qj) = max {0, 1− γqi − (1 + γ(n− 2))qj} . (5)

Here again, it is straightforward to check that a Þrm�s proÞt function is continuous and the

associated maximization problem is convex.

Penal Codes Both in the Bertrand and Cournot versions of the model, a strategy proÞle

is a set of available actions a, including a collusive price (quantity), which yields joint proÞt

maximization, and a punishment price (quantity), which leads to low per-Þrm proÞts. In the

stage game, Þrms� strategies in prices (quantities) may differ from period to period. An action

path {at}∞t=1 is deÞned as an inÞnite sequence of n-dimensional vector prices (quantities), as

8

chosen by each Þrm in each period. We assume all Þrms initially follow a collusive path by

choosing the collusive action aM . The collusive price and quantity are given by pM = (1 + c) /2

and qM = (1− c) / [2 (1 + γ(n− 1))] respectively. If the collusive strategies are played by allÞrms in all periods, each Þrm earns a sum of proÞts ΠM = πM/ (1− δ), where πM refers to

symmetric single-period collusive proÞts. All Þrms have a short-run incentive to deviate, that

is to lower (increase) its own price (quantity) in order to increase individual proÞts at every

other Þrm�s expense. If such a deviation by one Þrm i in N is detected in period t, all Þrms

switch to the punishment action aP in period t+ 1.

As standard in the literature, we consider two alternative penal codes, namely the �trigger�

penal code and the �stick-and-carrot� penal code. We are interested in investigating the con-

ditions on marginal costs, product differentiation, and size of the industry, in which collusion

is sustainable in a stick-and-carrot setup. We compare them with conditions on the parameter

values for which Þrms collude in equilibrium in the trigger model, we thereby use as a bench-

mark. Both in the trigger and stick-and-carrot cases, we have a collusive equilibrium of the

supergame if all Þrms Þnd it proÞtable to adopt the collusive action whenever the action path

calls for them to do so, and to implement the penal code whenever the action path calls for

them to do so.

¥ Following Friedman (1971), in the trigger penal code, all Þrms revert forever to the stage

game Nash equilibrium in the period that follows any deviation from the collusive strategy. In

this case, each Þrm earns a discounted ßow of proÞts Πdi = πdi + δπ

N/ (1− δ), all i, where πdirefers to a Þrm i�s single-period deviation proÞts, and πN refers to symmetric single period

Nash equilibrium proÞts. For the collusive strategy to be sustainable, the discounted ßow

of proÞts from adhering to the collusive strategy forever must exceed the discounted ßow of

9

proÞts obtained in the case of deviation, i.e. ΠM ≥ Πdi . It follows that collusion is sustainablewith a trigger penal code if and only if the discount factor is higher than a threshold level

δtrigger =¡πdi − πM

¢/¡πdi − πN

¢. Remark that this result holds true for all (γ, c) ∈ [0, 1)2 and

all n ≥ 2.

¥ In a single-period stick-and-carrot penal code a la Abreu (1986) also, all Þrms initially follow

a collusive path by choosing the collusive action aM . If a deviation by any Þrm occurs, all

Þrms switch to the punishment action aP in the next period (the stick). After one period of

punishment, if any deviation from aP is detected, the punishment phase restarts, otherwise all

Þrms resume the collusive behavior by adopting aM forever (the carrot).

The need for a punishment is rooted in the fact that each individual Þrm, assuming that all

other Þrms play the collusive action, has a short-run incentive to lower (increase) its own price

(quantity) to increase individual proÞts at every other Þrm�s expense. Then the choice of a low

(high) punishment price (quantity) aP in the next period renders free-riding less attractive. As

standard in the penal code literature, we say punishments are optimal if, for a given discount

parameter δ, they lead Þrms to obtain the highest level of sustainable collusive proÞts. As

established by Abreu (1986, p. 192), these optimal punishments are also the most severe.

Following Lambertini and Sasaki (2002), we adopt an equivalent interpretation according to

which a penal code is optimal if, given a collusive action aM , it requires the lowest critical

threshold in the discount factor in order to sustain the collusive equilibrium path. Intuitively,

this is because the lower the discount factor, the smaller the present value of future collusive

proÞts, and the higher the incentives to deviate.

10

3 Collusion Sustainability

In this section, we investigate the role of the level of marginal costs on the existence of an optimal

single-period punishment, either in prices or in quantities, in the stick-and-carrot mechanism.

As in the existing literature, we also examine the effects of the degree of product differentiation

and of the number of Þrms.

For each player to have no incentive to deviate, a deviation must be followed by a punishment

that leads the discounted ßow of proÞts to be less than the actualized stream of collusive

equilibrium proÞts. Moreover, for the punishment to be a credible threat, one should verify

that Þrms do implement the punishment strategy. This occurs if individual gains to deviate

from the punishment phase are smaller than the loss incurred by prolonging the punishment

by one more period. The strategy proÞle {aM , aP } is a subgame perfect equilibrium of the

supergame if and only if the two conditions are satisÞed, as made formal in the system of

incentive constraints

πdi (aM )− π(aM ) ≤ δ (π(aM )− π(aP )) , (6)

πdi (aP )− π(aP ) ≤ δ (π(aM )− π(aP )) , (7)

where π(a) denotes each Þrm�s stage proÞt when all Þrms play a, and πdi (a) is Þrm i�s proÞt from

a one-shot best deviation from the action a selected by all Þrms j 6= i, all i, with a = aM , aP .The Þrst condition says that the incentive for the initial deviation must be smaller than what is

lost due to the punishment phase. The second condition says that the incentive to deviate from

the punishment phase must be smaller than the loss incurred by prolonging the punishment by

one more period. We look for an optimal punishment action a∗P and the threshold level of the

discount factor δK such that, if a∗P is prescribed and δ ≥ δK , then Þrms can sustain collusion

11

at aM . The subscript K = B,C refers to the Bertrand or Cournot versions of the model,

respectively.

Characterization of an optimal punishment The optimal punishment action a∗P and the

threshold level of the discount factor δK are such that (6) and (7) hold with equality.6 To see

that, assume Þrst that none of the two inequalities is binding for a = a∗P and δ = δK . Since both

expressions on the right-hand side of the inequality sign are continuous in δ and monotonically

decreasing when the discount parameter is decreasing, there exists a value �δ < δK for which

the system still holds true, a contradiction. Assume now that exactly one inequality is binding.

Recalling that proÞt functions πdi (.) and π(.) are continuous in Þrms� strategies, by changing

slightly the punishment action from a∗P to �aP one can relax the binding constraint and still let

the other inequality hold. Now with δ = δK and �aP , both constraints (6) and (7) are slack,

contradicting the previously established claim. As a result both constraints must be binding

for a = a∗P and δ = δK . It follows that

δK =πdi (aM )− π(aM )π(aM )− π(aP ) . (8)

A Þnal condition is that Þrms have incentives to participate in the game after deviation also.

This occurs when the individual rationality constraint is satisÞed (Lambson, 1987), that is

π(a∗P ) +∞Xτ=1

(δK)τ πM ≥ 0. (9)

In words, the losses incurred by a Þrm in the punishment period must not overbalance the dis-

counted stream of collusive proÞts earned in the following periods. We Þnd that this constraint

is always satisÞed for the speciÞcations of the model (on this see the Appendix, sections 7.1.1.

and 7.1.2).

When Þrms detect a deviation from the collusive strategy aM , they all switch to the pun-

6This is a restatement of Abreu (1986, Theorem 15) in the new context of the present model speciÞcations.

12

ishment phase before returning to collusive proÞts. As a change in parameters (γ, n, c) impacts

each Þrm�s deviation proÞts and collusive proÞts (i.e., πdi (aM ), πdi (aP ), and π(aM )), from (6)

and (7) it also impacts the optimal punishment action a∗P and thereby the lowest discount fac-

tor δK . The following proposition gives conditions on collusion sustainability under an optimal

single-period punishment.

Proposition 1 Optimal collusion is implementable with a single-period punishment if and only

if c ≥ max{0, cK} and δ ≥ δK , where K = B,C. If K = C and n ≤ 3 then cK = 0 for all γ, n,otherwise cK > 0. In all cases δK ≤ δtrigger.

Proof See Appendix 7.1. ¥

There are several results in the proposition. As standard in the literature, we obtain that

the single-period punishment proÞts must be sufficiently valued by a deviating Þrm. This means

that the discount factor must be sufficiently close to 1, i.e. δ ≥ δK . Another result is that thelevel of marginal costs matters for collusion sustainability.

For an intuitive interpretation of this claim, recall Þrst that, in all models of tacit collusion,

a necessary condition for the sustainability of agreements is the existence of a sufficiently severe

punishment mechanism, as formalized in (6). This holds when the discount factor is high

enough and punishment proÞts are low enough. More precisely, for a given discount factor, the

higher the deviation payoffs, and the lower the collusive proÞts, the tougher the punishment

must be for collusion to remain sustainable. In a trigger penal code, a deviation implies that

Þrms stop colluding and revert to the one-shot stage game Nash equilibrium forever. A stick-

and-carrot setup authorizes a shorter and more severe punishment phase that may lead Þrms to

earn negative proÞts for some time without violating the individual rationality constraint (9).

The severity of punishment depends on the extend to which Þrms can charge a non-negative

13

price sufficiently below marginal costs. This leads to identify the following fact: the higher the

level of c, the larger the range [0, c] of below-cost non-negative prices that Þrms can possibly

charge in a punishment phase.

Secondly, note that a more severe punishment is likely to render deviation from the pun-

ishment path more proÞtable. This is captured in the present stick-and-carrot penal code, by

a second necessary condition displayed in (7), which requires that the punishment is enforced

when a previous-period deviation is observed. When punishment proÞts are negative (which

holds true with below marginal cost pricing), the deviator can be better off by interrupting

production temporarily. This occurs when goods are relatively close substitutes and the mar-

ginal cost is sufficiently high.7 We thus have another effect that may potentially countervail

the former one: the higher the marginal cost c, the more severe the maximal punishment, and

the higher the incentive to deviate from a severe punishment path.

Proposition 1 establishes that the latter effect is always dominated by the former one. This

leads to the conclusion that cmust lie above a threshold value cK for collusion to be sustainable.

Remark that this constraint on c is ineffective in the quantity-setting version of the model with

two or three Þrms. This implies that results obtained in the literature with a duopoly and a

constant marginal cost normalized to zero, all other speciÞcations remaining equal, are robust

to the introduction of a positive marginal cost. Otherwise, the constraint on c must be satisÞed

for a single-period punishment to be optimal. This result contrasts sharply with trigger penal

code models, in which one can easily check that the sustainability of collusion is not directly

connected to the level of marginal costs (at least in the linear cost setup).

7When goods are differentiated enough and/or the marginal cost is very low, a Þrm can earn positive proÞts

by deviating from the punishment strategy (on this see Appendices 7.1.1 and 7.1.2).

14

Comparative statics We now turn to the impact of a change in the differentiation parameter

γ or in the number of Þrms n on the threshold values cK and δK .

Proposition 2 For K = B,C and for all γ, n :

(i)∂cK∂γ ≥ 0, ∂δK

∂γ > 0;

(ii)∂cK∂n ≥ 0, ∂δK

∂n > 0.

Proof The sign of derivatives can be obtained by direct computation of expressions given in the

Lemmas 1 and 2 of Appendix 7.1. ¥

This proposition establishes that the threshold values cK and δK evolve in the same direction

when parameter values change. We Þnd that an increase in product differentiation (i.e., a

reduction in γ) facilitates collusive agreements, as it relaxes the constraints on c and δ. This

extends to the present model speciÞcations a previous result unveiled by Chang (1991) with a

trigger penal code, and by Hackner (1996) and Lambertini and Sasaki (2002) in a stick-and-

carrot framework with two Þrms. The driving force behind this claim is that a rise in product

substitutability renders a deviation from the collusive or punishment path individually more

proÞtable. This effect dominates all other changes in the expression of δK as displayed in (8).

This contrasts with the fact that, with a trigger penal code, δtrigger is not monotonic in γ. An

interesting and novel result relates to the level of marginal costs. When product substitutability

increases, individual deviation proÞts rise, but simultaneously punishment proÞts are lowered.

We Þnd that the magnitude of the former effect dominates the latter. In consequence, an

increase in product substitutability makes a deviation from the collusive path more likely. This

results in the strengthening of the constraint on c, that is Proposition 2(i). An increase in

the number of Þrms has similar effects, as claimed in Proposition 2(ii). Again, this is because

an individual deviation becomes most proÞtable, both with price and quantity competition,

15

whereas the reduction in punishment proÞts is relatively limited.

4 Multi-Period Punishments

In this section we focus on situations in which optimal collusion is not sustainable with a

single-period penal code. This occurs when parameter values are such that one of the two

constraints on c and δ we described in Proposition 1 is violated. To do that, we Þrst introduce

a multiple-period extension to our model, before comparing optimal symmetric punishments

with the trigger penal code equilibrium benchmark.

As in Lambertini and Sasaki (2002), consider a multi-period stick-and-carrot penal code

in which, if any deviation from aM by any Þrm is detected, all Þrms switch to a l-period

punishment phase (the stick) during which they play aP,k, with k = 1, . . . , l. In any period of

punishment, if a deviation from the punishment action by a Þrm is detected, the punishment

phase restarts for l more periods, after which all Þrms revert to the initial collusive path aM

forever (the carrot).

For Þrms not to be incentivized to deviate from the collusive path, a deviation should lead

them to earn a discounted ßow of proÞts which is less than their discounted ßow of collusive

equilibrium proÞts. This writes

πdi (aM ) +l−1Xk=1

δkπ(aP,k) + δlπ(aP,l) +

∞Xτ=l+1

δτπ(aM ) ≤∞Xτ=0

δτπ(aM ), (10)

where π(a) denotes each Þrm�s stage symmetric proÞts when all Þrms charge a, and πdi (a) is

Þrm i�s proÞts from a one-shot best deviation from the action a, as selected by all Þrms j 6= i, alli, j. For some parameter values, which satisfy conditions described in the previous section, this

inequality may hold with a single-period punishment, that is l = 1. However, a longer duration

of punishment is needed when parameter values are such that the severity of a single-period

16

punishment is not sufficient to deter deviation, in which case l > 1. In the latter situation,

we consider punishment phases in which the l− 1 Þrst periods of punishment are the toughestpossible. The most severe action, we denote by aP , is such that prices are driven down to zero.

This occurs when, in the Bertrand version of the model, all Þrms charge pP= 0, while in the

Cournot version all Þrms expand their output so that pi(qP , qP ) = 0, all i. Then in the l-th

Þnal period of punishment, Þrms play an optimal strategy aP,l. Remark that, for all l ≥ 1,

when marginal costs are positive Þrms may earn negative proÞts if they play the punishment

action aP,l.

When a Þrm deviates from the collusive strategy, a second incentive constraint must be

satisÞed so that Þrms implement the penal code. In words, the individual gain to deviate from

the punishment phase must be smaller than the loss incurred by prolonging the punishment

phase of l more periods. Formally, this writes

s−1Xk=1

δkπ (aP ) + δsπdi (a

sP ) +

s+l−1Xk=s+1

δkπ (aP ) + δs+lπ (aP,l) +

∞Xτ=s+l+1

δτπM

≤l−1Xk=1

δkπ(aP ) + δlπ(aP,l) +

∞Xτ=l+1

δτπM ,

for any period s in which a Þrm deviates from the penal code, with 1 ≤ s ≤ l (we adopt the

convention thatPs−1k=1 δ

kπ (aP ) = 0 if s = 1).

The strategy proÞle {aM , aP , aP,l} is a subgame perfect equilibrium of the supergame if andonly if (10) and (11) are satisÞed. There are l constraints in (11). The only one that will be

binding is for s = 1, since the incentive to deviate from the penal code is higher in the Þrst

period of punishment than in any subsequent period. The system of inequalities can thus be

rewritten as

πdi (aM )− π(aM ) ≤ δl (π(aM )− π(aP,l)) +¡δ(1− δl−1)/(1− δ)¢ (π(aM )− π(aP )) ,

πdi (aP )− π(aP ) ≤ δl (π(aM )− π(aP,l)) + δl−1 (π(aP,l)− π(aP )) .(12)

17

The optimal punishment action a∗P,l and the threshold level of the discount factor δK,l

are such that (12) holds with equality (the proof proceeds as in the single-period case, by

observing that for any l > 1, both expressions on the right-hand side of the inequality sign are

monotonically decreasing when δ decreases). A reorganization of terms leads to

δK,l =πdi (aM )− π(aM )πdi (aM )− πdi (aP )

, all l > 1, (13)

In this multi-period penal code, the individual rationality constraint for a deviation not to end

the game writes

l−1Xk=1

δkπ(aP ) + δlπ(a∗P,l) +

∞Xτ=l+1

δτπM ≥ 0. (14)

Remark that a shift to a longer punishment phase always results in lower intertemporal pun-

ishment proÞts. This is because π(aP,l) ≥ π(aP ) and π(aM ) ≥ π(aP,l+1). In words, for a givenperiod, the proÞts obtained in a short (i.e., l-period) punishment phase are bounded from below

by the proÞts earned in a longer (l+1-period) phase. The following proposition gives conditions

on collusion sustainability under an optimal multi-period punishment.

Proposition 3 Optimal collusion is implementable with a l-period punishment if and only if

max{0, cK,l} ≤ c ≤ max{0, cK,l−1} and δ ≥ δK,l(c), where K = B,C and l > 1. The threshold

value cK,l is strictly decreasing when l increases, but δK,l(c) is constant in l. Moreover, for all c,

there always exists a length l of punishment phase such that optimal collusion is implementable

if δ ≥ δK,l(c), with δK,l(c) ≤ δtrigger.

Proof Recall that tacit collusion is implementable if and only if (i) there exists a sufficiently

severe punishment mechanism, (ii) there are no incentive to deviate from this punishment, and (iii)

there are no incentives to drop out from the game, i.e. the individual rationality constraint (14) holds

18

true. The Þrst two conditions stated in (12) can be rewritten as

Xl (δ, c) ≤ π(aP,l, c) ≤ Yl (δ, c) , (15)

where the argument c highlights the fact that all expressions depend on the value of the marginal cost

while the lower-bound Xl (δ, c) and the upper-bound Yl (δ, c) of the last-period punishment proÞts

π(aP,l, c) are given by:

Xl (δ, c) ≡ 1

δl−11

1− δ¡πdi (aP , c)− π (aP , c)

¢− δ

1− δ π (aM , c) +1

1− δ π (aP , c) ,(16)

Yl (δ, c) ≡ π(aM , c)− 1

δl¡πdi (aM , c)− π (aM , c)

¢+

1

δl−11− δl−11− δ (π (aM , c)− π (aP , c)) .

(17)

Condition (14) can also be rewritten as Zl (δ, c) ≥ 0, where Zl increases when δ increases. Clearly,since an increase in δ relax the three constraints, if collusion is implementable with a discount factor

δK,l it is implementable for any discount factor δ ≥ δK,l. It is also plain that the lower bound of thediscount factor δK,l depends a priori on the marginal cost c since the inequality (16) must be binding

for a = a∗P,l and δ = δK,l and both Xl (δ, c) and Yl (δ, c) depends on c. In contrast, the impact of

c on collusion sustainability is less straightforward. Since Zl (δ, c) is decreasing in c, an increase in

the marginal costs makes the third constraint more stringent. Moreover, the last-period punishment

proÞts π(aP,l, c) also depend on c.

Note that, while a longer length of punishment results in lower intertemporal proÞts, the proÞts in

the last period of the punishment phase increase as the length l of the punishment phase increases. To

see that, observe that πdi (aP ) > π(aP ) implies that the lower bound of the last period punishment

proÞts Xl (δ) strictly increases with l. Since the inequality (16) must be binding for a = a∗P,l and

δ = δK,l and the critical discount factor δK,l as deÞned by equation (13) does not change with

19

l, it follows that π(a∗P,l+1) > π(a∗P,l). Note also that obvious feasibility constraints impose that

π(aP ) ≤ π(a∗P,l) ≤ π(aM ), all l.The threshold level of marginal cost cK,l is the lowest marginal cost that allow to implement

collusion by the means of a punishment period of length l. It happens to be also the highest marginal cost

that allow to implement collusion by the means of a punishment period of length l+1. It corresponds

indeed to the level of marginal costs such that both feasibility constraints π(aP ) ≤ π(a∗P,l) and

π(a∗P,l+1) ≤ π(aM ) are binding. Formally

π¡aP , cK,l

¢= −cK,lq (aP ) = Xl

¡δK,l, cK,l

¢= Yl

¡δK,l, cK,l

¢,

π¡aM , cK,l

¢= Xl+1

¡δK,l, cK,l

¢= Xl+1

¡δK,l+1, cK,l

¢.

By deÞnition, collusion implementation by the means of a l-period punishment with a∗P,l = aP is a

tantamount to collusion implementation by the means of a l+1-period punishment with a∗P,l+1 = aM .

The get the result that collusion can be implemented by the means of a l-period punishment for any

cK,l ≤ c ≤ cK,l−1, it is thus sufficient to show that, if collusion can be implemented with last periodpunishment proÞts π(aP ) < π(a

∗P,l) < π(aM ), then collusion can also be implemented by the means

of a l-period punishment for a strictly smaller and a strictly larger marginal cost c. Both Xl and Yl

depends continuously on δ and c. Moreover the difference Yl −Xl is monotonic, strictly increasing inδ. It follows that it is always possible to haveXl (δ, c) ≤ Yl (δ,c), possibly by adjusting δ, both after anincrease or a decrease in c. The only limit to this changes in the marginal cost follow from the ability to

match the value Xl¡δK,l (c) ,c

¢= Yl

¡δK,l (c) , c

¢with punishment proÞts π(aP,l, c). One can show

however that the difference π(aP,l, c)−Xl (δ,c) is continuous and increasing aP,l and continuous anddecreasing in c. Indeed,

∂ [π(aP,l, c)−Xl (δ, c)]∂c

=1

δl−11

1− δ¡qdi (aP )− q(aP )

¢− δ

1− δ q(aM )+1

1− δ q(aP )−q(aP,l) < 0.

To see that, observe that since q(aP,l) is decreasing with aP,l it is sufficient to show that the inequality

20

holds for aP,l = aM . Since q(aP ) > qdi (aP ), it is also sufficient to focus on the case l = 2. Thus the

monotonicity of π(aP,l, c)−Xl (δ, c) in c will follow from

δ <q(aP )− qdi (aP )q(aP )− q(aM )

.

As a result π(aP,l, c) − Xl (δ, c) is always decreasing in c since qdi (aP ) ≤ q(aM ).It follows that, if

collusion can be implemented by the means of a l-period punishment for level c of marginal costs, it

can also be implemented for a strictly higher (resp. lower) c as long as aP,l can be increased (resp.

decreased). One also get that cK,l < cK,l−1.

To sum up, if collusion can be implemented by using a l-period punishment for some (δ, c) , one

can deÞne δK,l+1 (c) = δK,l (c) and cK,l+1 < cK,l so that collusion can also be implemented for

any δ ≥ δK,l (c) and cK,l+1 ≤ c ≤ cK,l by using a l + 1-period punishment. The proof proceeds

by induction. Appendix 7.2 focus on the 2-period case and establishes that collusion can indeed be

sustained for δ ≥ δK,2 (c) and cK,2 ≤ c ≤ cK . In particular, the threshold value δK,l (c) = δK,2 (c)is exhibited.

It remains to demonstrate that a trigger penal code can be replicated by a stick-and-carrot penal

code. Observe that, for any particular set of values taken by each of the structural parameters (c, n, γ),

it is always possible to construct a multi-period stick-and-carrot penal code which associates to each

Þrm a stream of proÞts which, evaluated in the Þrst period of punishment, is exactly equal to the

actualized sum of proÞts earned in the punishment phase of the trigger penal code. In our notation,

this replication argument is equivalent to saying that there exists a pair (l, aP,l) such that

l−1Xk=1

δkπ(aP ) + δlπ(aP,l) +

∞Xτ=l+1

δτπM| {z }≡ΠS&C(l,aP,l)

=∞Xτ=1

δτπN| {z }≡Πtrigger

, (18)

where the subscript S&C stands for �stick-and-carrot�, aP is the most severe action (i.e., either a price

equal to zero or a quantity which drives each Þrm�s price down to zero), and l ≥ 1. To see that, it is

21

sufficient to Þnd an integer l such that ΠS&C (l + 1, aM ) = ΠS&C (l, aP ) ≤ Πtrigger ≤ ΠS&C (l, aM )holds true. By continuity of ΠS&C (l, aP,l) in aP,l, there exist a last period punishment value such

that ΠS&C (l, aP,l) = Πtrigger. Rewrite the expression on the left-hand side of the equality side in

(18) for aP,l = aM :

ΠS&C (l, aM ) =δ(1− δl−1)1− δ π(aP ) + δ

lπM +δl+1

1− δ πM . (19)

Recalling that π(aP ) ≤ 0 (by construction) together with πM > 0 makes it clear that ΠS&C¡l, aM

¢is

decreasing when l increases. Moreover, as πM > πN (by deÞnition), it follows thatΠS&C¡l = 1, aM

¢=

(δ/ (1− δ))πM > Πtrigger and liml→+∞ΠS&C¡l, aM

¢= δ/(1− δ)π(aP ) < 0 < Πtrigger. There-

fore there exists a value for l such that ΠS&C (l + 1, aM ) ≤ Πtrigger ≤ ΠS&C (l, aM ) hence a pair

(l, aP,l) such that (18) holds true. ¥

There are several results in this proposition. Firstly, the constraint on the marginal cost

c we identiÞed in Proposition 1 for the single-period punishment case Þnds a counterpart in

the multi-period setup. Secondly, the latter constraint on c is relaxed by increasing the length

of the punishment phase. In other words, a longer duration of punishment is a �substitute�

to a high marginal cost. In contrast, the constraint on the discount factor cannot be relaxed

by the increase in l since δK,l depends on c (as well as n and γ) but not on the length of the

punishment phase l. The last result of Proposition 3 consists in showing that the constraint on

marginal costs can be fully relaxed by sufficiently lengthening the punishment phase. In other

words, if δ ≥ δK,l, there exists a punishment code that allow to implement collusion for any levelof the marginal costs c. This echoes an existing result by Hackner (1996), who demonstrates

that an optimal stick-and-carrot symmetric punishment a∗P always exists (Proposition 4, p.

624) with two Þrms, constant marginal costs normalized to zero, and in a spatial differentiation

framework. Proposition 3 extends this result to the general case, that is with n ≥ 2 Þrms,

constant marginal costs c in [0, 1), and for any value taken by the exogenous differentiation

22

parameter γ in [0, 1).

Corollary 4 δK,l ≥ δK , all l > 1 , K = B,C.

Proof Since δK,l does not depend on l, it is sufficient to show that δK,l=2 ≥ δK . This is doneby comparing values of δK,l=2 and δK as given in Appendices 7.1 and 7.2. ¥

To illustrate, we now consider the simplest possible case of multiple-period punishment, that

is l = 2, to obtain the following result.

Remark 1 For K = B ,C and for all γ, n :

If K = C and n ≤ 4 then cK,l=2 = 0 for all γ, n; otherwise cK,l=2 > 0. Moreover∂cK,l=2∂γ > 0,

∂δK,l=2∂γ > 0,

∂cK,l=2

∂n > 0, and∂δK,l=2∂n > 0.

It remains to compare the role of the nature of competition on the sustainability of collusion.

We tackle this next.

5 Price vs. Quantity

We now compare the threshold values cK and δK , as deÞned in Section 3, in the Bertrand

(K = B) and Cournot (K = C) versions of the model.

Proposition 5 For all γ, n : cB > cC and δB > δC .

This proposition establishes that collusion is less easily sustainable in a price-setting oligopoly

than in a quantity-setting one. This means Þrst that, in a Bertrand setup, a longer punishment

than in the Cournot case may be required. Indeed, since cB > cC , there exist parameters (c, n, γ)

which are such that collusion is not sustainable with a single-period punishment in Bertrand,

while a single-period punishment is sufficient in Cournot. From Proposition 3, we know there

23

exists l such that cC,l ≤ c, hence optimal collusion with be implementable in Bertrand with

a multi-period punishment. Moreover, the constraint on the discount factor is more stringent

when Þrms use prices as opposed to quantities. This claim holds for any degree of product

substitutability, and for all numbers of Þrms.

This can be made intuitive by observing that the incentives to deviate in the Bertrand case

are higher than in the Cournot setup. This is because a deviating Þrm can capture the whole

demand in a price-setting oligopoly. This does not apply in a quantity-setting oligopoly, because

a unilateral expansion in some Þrm�s output cannot eliminate its rivals� demand. By contrast,

the range of punishment possibilities [−cq, πM ) is the same in the Bertrand and Cournot cases,since we use symmetric punishments. In both versions of the model, the severity of punishments

thus depends only on the level of marginal costs.

To illustrate the multi-period case, we now focus on two-period punishment phases. For the

sake of clarity, denote by cK,l=1 and δK,l=1 the threshold values deÞned in the single-period

framework.

Remark 2 For all γ, n : cB,l > cC,l; and for all c, γ, n : δB,l > δC,l0 , l ≥ l0, with l, l0 = 1, 2.

This remark establishes that the results offered in Proposition 4 are still valid in a two-

period context. It can be easily extended to more than two periods. This is because the most

severe punishments π(aP ) = −cq inßicted in the Þrst l − 1 periods are the same in Bertrandand Cournot, with aP = p

P= 0 in the price-setting case, and pi(aP , aP ) = pi(qP , qP ) = 0

in the quantity version. The difference between the Bertrand and Cournot cases is therefore

fully captured by the difference in the unilateral deviation proÞts πdi (aP,l) and the optimal

punishment proÞts π(aP,l) in the last period l, as in the single-period framework.

24

6 Conclusion

We have presented an oligopoly supergame model with differentiated goods and positive mar-

ginal costs, under Bertrand or Cournot behavior. In this setup, we are able to derive quite

general results which are policy relevant. They conÞrm that antitrust authorities should con-

centrate investigations on industries in which Þrms are few and sell highly differentiated goods.

On top of that, we argue that the level of marginal costs should be added to the list of struc-

tural factors that facilitate collusion. Another interesting Þnding is that a longer punishment

phase is a substitute for a low level of marginal costs. However, real-world circumstances may

limit the duration of below cost price wars. In this case, a low level of marginal costs would

constitute an impediment to tacitly collusive agreements.

7 Appendix

In this appendix, we determine conditions of collusion sustainability by computing the threshold

values of c and δ for all parameter values.

7.1 Single-period punishment

The optimal punishment action aP and the minimum threshold δK are deÞned by

πdi (aP )− π (aP ) = πdi (aM )− π (aM ) (20)

and

δK =πdi (aM )− π (aM )π (aM )− π (aP ) . (21)

We check that the individual rationality constraint in the punishment phase is satisÞed by

verifying that the value of discounted proÞts from the punishment period onwards is strictly

25

positive, that is

π(aP ) +δ

1− δ π(aM ) ≥ 0. (22)

Since ∂∂δ

³π(aP ) +

δ1−δπ(aM )

´> 0, it will be sufficient to check in the following that

π(aP ) +δK

1− δKπ(aM ) ≥ 0 (23)

to conclude that the individual rationality constraint is satisÞed for all δ > δK .

Both in the Bertrand and in the Cournot cases, the collusive proÞts, price and individual

quantities are respectively given by πMi = (1− c)2 /4 (1 + γ (n− 1)) , pM = (1 + c) /2 and

qM = (1− c) /2 (1 + γ(n− 1)) .

7.1.1 Bertrand case

We start by recalling the Bertrand related content of Proposition 1, and by displaying threshold

values of c and δ. Then we turn to the proof.

Lemma 6 Collusion can be sustained by a single-period punishment price if and only if c ≥ cBand δ ≥ δB, where:

cB =

c0B if 0 ≤ γ < min½2((n−3)+(n−1)

√2)

2n+n2−7 , n−3+√n2−1

3n−5

¾c1B if min

½2((n−3)+(n−1)

√2)

2n+n2−7 , n−3+√n2−1

3n−5

¾≤ γ

≤ max½n−3+√n2−1

3n−5 ;(n−3)(n2−2n+5)+(n2−1)

√(n2−2n+5)

2(12n−7n2+2n3−11)

¾c2B if γ ≥ max

½n−3+√n2−1

3n−5 ;(n−3)(n2−2n+5)+(n2−1)

√(n2−2n+5)

2(12n−7n2+2n3−11)

¾with

c1B =

�c1B for n = 2

�c1B for n > 3

26

and

δB =

116

(γ(n−3)+2)2(1−γ)(γ(n−2)+1) if 0 ≤ γ ≤ min{n−3+

√n2−1

3n−5 , γ1} and min{0, c0B} ≤ c ≤ 1 for all n

γ2(2n−3)+γ(3−n)−1(2γ−1)(1+nγ−γ) if max

nn−3+√n2−1

3n−5 ; γ2

o≤ γ ≤ 1 and c2B ≤ c ≤ 1 for all n

((2γ−1)(1+γ)−γ2)(γ−2)2(γ2−2)(γ2−4γ+2)+4γ2

√(1−γ)(γ+γ2−1) if

n−3+√n2−13n−5 ≤ γ ≤ γ2 and �c1B ≤ c ≤ 1 for n = 2

γ2 (n−1)2(γ(n−3)+2)2 if γ1 ≤ γ ≤ n−3+√n2−1

3n−5 and �c1B ≤ c ≤ f (γ) for n > 3

where γ1 and γ2 are expressions of n, and f (γ) = (1− γ) / (1 + γ (n− 2)) .

Proof: The demand function is given by

qi(pi, pj) =

1− pi if pi ≤ �pi(pj)

11+γ(n−1)

³1− 1+γ(n−2)

1−γ pi +γ(n−1)1−γ pj

´if �pi(pj) ≤ pi ≤ �pi(pj)

0 if pi ≥ �pi(pj)

,

(24)

and demand for each other symmetric Þrm j is deÞned by

qj(pi, pj) =

0 if pi ≤ �pi(pj)

11+γ(n−1)

³1− 1

1−γ pj +γ1−γ pi

´if �pi(pj) ≤ pi ≤ �pi(pj)

1−pj1+γ(n−2) if pi ≥ �pi(pj)

, (25)

with �pi(pj) =1γ [pj − (1− γ)] and �pi(pj) = 1

1+γ(n−2) (1− γ + γ (n− 1) pj).Firm i�s proÞt writes πi(pi, pj) = (pi − c) qi(pi, pj), with c < 1. This proÞt function is locally

27

concave in each market structure. One can Þnd that the best deviation proÞt for Þrm i writes

πdi (p) =

0 if p ≤ pj

[(1−γ)(1−c)+γ(n−1)(p−c)]24(1+γ(n−1))(1+γ(n−2))(1−γ) if p ≤ p ≤ pj(1−pj)γ2 ((p− c)− (1− γ) (1− c)) if p ≤ p ≤ pj¡1−c2

¢2if p ≥ pj

(26)

where pj− c ≡ − (1−γ)(1−c)

γ(n−1) , pj − c ≡ (1−γ)(1−c)(2+γ(2n−3))2(1−γ)+γ(n−1)(2−γ) , pj − c ≡ (1−c)(2−γ)

2 . Note that

pj> 0⇔ c >

(1− γ)(1 + γ (n− 2)) ≡ f(γ).

We now compute the best deviation to the collusive price. Since pj< pM < pj and pM ≤

pj ⇔ γ ≤ ¡n− 3 +√n2 − 1¢ / (3n− 5) < 1, when all other Þrms charge pM , Þrm i�s maximum

deviation proÞt is given by

πdi (pM ) =

(1−c)2[2+γ(n−3)]2

16(1+γ(n−1))(1+γ(n−2))(1−γ) if γ ≤ n−3+√n2−13n−5 ,³

1−c2γ

´2(2γ − 1) if γ ≥ n−3+√n2−1

3n−5 .

(27)

In order to solve (20), we distinguish several cases that depend on the form of πdi (pP ) and

πdi (pM ).

� Case 1. πdi (pP ) = 0 if 0 ≤ pP ≤ pj and c > f(γ)

The left-hand side term of (20) is a convex parabolic function of p that takes value 0 if either

p = c or p = 1. Since pj< c < 1, we know that it is monotone decreasing for all p ≤ p

j.

Case 1.1. γ ≤ n−3+√n2−13n−5

We Þnd:

(i) πdi (0)− π (0) > (=)¡πdi (pM )− π (pM )

¢⇔ c > (=) �c1B with

�c1B ≡ (17−10n+n2)γ2−8(3−n)γ+8−4(2+γ(n−3))√(1−γ)(1+γ(n−2))

γ2(n−1)2 .

28

Remark that �c1B is a monotone increasing function of γ, with limγ→1�c1B = 1, and that for

γ = n−3+√n2−13n−5 the difference �c1B − f (γ) is negative if n = 2, nil if n = 3 and positive if n > 3.

(ii) πdi

³pj

´− π

³pj

´< (=)

¡πdi (pM )− π (pM )

¢⇔ γ > (=)γ1 with

γ1 ≡2¡(n− 3) + (n− 1)√2¢

2n+ n2 − 7 ,

with γ1 > (=)(n−3+

√n2−1)

3n−5 ⇔ n < (=) 3. We also Þnd that �c1B = f (γ) if γ = γ1.

We conclude that (20) admits a unique solution pP ∈ (0, pj) iff n > 3, γ1 ≤ γ ≤ n−3+√n2−13n−5

and c ≥ �c1B. The threshold value of the discount factor is given by

δ1B ≡γ2 (n− 1)2

(γ(n− 3) + 2)2 ∈ (0, 1).

From (23) , the individual rationality constraint is satisÞed since one can easily check that

π(pP ) +δ1Bπ(pM )

1−δ1B= 0.

Case 1.2. γ ≥ n−3+√n2−13n−5

We Þnd:

(i) πdi (0)− π (0) > πdi (pM )− π (pM )⇔ c > c2B with

c2B ≡(2n− 1) γ2 + (3− n) γ − 1− 2γp(2γ − 1) (1 + γ (n− 1))

(2γ − 1) (1 + γ (n− 1))− γ2 ,

which is a monotone increasing function of γ, with limγ→1c2B = (

√n − 1)/(√n + 1) < 1. For

γ = n−3+√n2−13n−5 we check that �c1B = c2B and c

2B − f (γ) is negative if n = 2, nil if n = 3 and

positive if n > 3.

(ii) πdi

³pj

´− π

³pj

´<¡πdi (pM )− π (pM )

¢⇔ γ > γ2 with

γ2 ≡³(n− 3) ¡n2 − 2n+ 5¢+ ¡n2 − 1¢p(n2 − 2n+ 5)´

2 (2n3 − 7n2 + 12n− 11)

and γ2 > (=)(n−3+

√n2−1)

3n−5 ⇔ n < (=) 3 and c2B = f (γ) if γ = γ2.

29

We conclude that (20) admits a unique solution pP ∈ (0, pj) if γ ≥ max{n−3+√n2−1

3n−5 ; γ2}and c ≥ c2B. The discount factor threshold is given by

δ2B ≡γ2 (2n− 3) + γ (3− n)− 1(2γ − 1) (1 + γ(n− 1)) ∈ (0, 1).

From (23) , the individual rationality constraint is satisÞed since π(pP ) +δ2Bπ(pM )

1−δ2B = 0.

� Case 2 : πdi (pP ) = [(1−γ)(1−c)+γ(n−1)(pj−c)]24(1+γ(n−1))(1+γ(n−2))(1−γ) if sup{0, pj} ≤ pP ≤ pj

The lower bound of pP is sup{0, pj} since pP must be non-negative. The left-hand side termof (20) is a convex parabolic function of pP which takes value 0 for pP =

1+c−γ(1−c(n−2))2+γ(n−3) ≡ ÿp

only. As pj< ÿp < pj , this function is monotone decreasing on [pj , ÿp) and monotone non-

decreasing on [ÿp, pj ].

Case 2.1. γ ≤ n−3+√n2−13n−5

The only admissible solution to (20) is pP − c = 12 (1− c) 2−γ(n+1)2+γ(n−3) , with pP < ÿp < pj for all

parameter values. We Þnd pP > pj ⇔ γ < γ1 and pP > 0⇔ c > c0B with

c0B ≡−2 + γ(n+ 1)2 + γ(3n− 5) .

We check that (i) −2+γ(n+1)2+γ(3n−5) > 0⇔ γ > 2(n+1) <

n−3+√n2−13n−5 and (ii) 2

(n+1) < γ1, all n.

We conclude that there is a solution pP such that pj < pP < pj for all γ < min{γ1, n−3+√n2−1

3n−5 }and c ≥ c0B. We also Þnd

δ3B ≡1

16

(γ(n− 3) + 2)2(1− γ) (γ(n− 2) + 1) ∈ (0, 1).

From (23) ,the individual rationality constraint is satisÞed since π(pP ) +δ3Bπ(pM )

1−δ3B > 0 for all

γ < min{γ1, n−3+√n2−1

3n−5 }.

30

Case 2.2. γ ≥ n−3+√n2−13n−5

The only admissible solution pP to (20) is such that pP < pj for all parameters, pP ≥ pj ⇔[γ ≤ γ2 and c ≥ f(γ)] and pP ≥ 0⇔ c ≥ �c1B, with

�c1B ≡γ(γ − 1) +p(1− γ) (γ + γ2 − 1)

γ +p(1− γ) (γ + γ2 − 1) .

We check that �c1B ≤ f(γ) whenever n−3+√n2−13n−5 ≤ γ ≤ γ2, that �c1B = f(γ) if γ = γ2 and that

�c1B = c0B if γ =

n−3+√n2−13n−5 . We conclude that there exists a solution pP such that pj < pP < pj

whenever n−3+√n2−1

3n−5 ≤ γ ≤ γ2 and c ≥ �c1B, which is a non empty case only for n = 2. Now forn = 2, we Þnd

δ4B ≡(2γ − 1) (γ + 1)− γ2

(γ2 − 2) (γ2 − 4γ + 2) + 4γ2p(1− γ) (γ2 + γ − 1) (γ − 2)2 ∈ (0, 1).From (23) , the individual rationality constraint is satisÞed since π(pP ) +

π(pM )δ4B

1−δ4B > 0.

� Case 3 : πdi (pP ) = (1−pj)γ2 ((pj − c)− (1− γ) (1− c)) if pj ≤ pP ≤ pj

The left-hand side of (20) is increasing when p increases, all p ∈ [pj , pj ].

Case 3.1. γ ≤ n−3+√n2−13n−5

We use πdi¡pj¢ − π ¡pj¢ > πdi (pM ) − π (pM ) ⇔ γ < n−3+√n2−1

3n−5 to conclude that (20) has

no solution in the range [pj , pj ] and for γ ≤ n−3+√n2−13n−5 .

Case 3.2. γ ≥ n−3+√n2−13n−5

There is no admissible solution to (20) in the range [pj , pj ] and for γ ≥ n−3+√n2−13n−5 .

7.1.2 Cournot case

Let us start by recalling the Cournot related content of Proposition 1 before expressing the

threshold values of c and δ.

31

Lemma 7 An optimal single-period punishment quantity qp exists in any Cournot supergame

if δ ≥ δC with γ ∈ (0, 1) for n = 2, 3, or c ≥ cC and n > 3, where

cC =

c0C =

γ(n−1)−23γ(n−1)+2 if

2n−1 ≤ γ ≤ min

n21+

√2

n−1 , 1o

c1C =

³(2+γ(n−1))

√(1+γn−γ)−2(1+γ(n−1))

´³2(1+γ(n−1))+(2+γ(n−1))

√(1+γn−γ)

´ if minn21+√2n−1 , 1o≤ γ.

and

δC =

116(2+γ(n−1))21+γ(n−1) if n < 6 or if n ≥ 6 and 2

n−1 ≤ γ ≤ minn21+

√2

n−1 , 1o

³(2+γ(n−1))

√(1+γn−γ)−2(1+γ(n−1))

´³2(1+γ(n−1))+(2+γ(n−1))

√(1+γn−γ)

´ if n ≥ 6 and minn21+√2n−1 , 1o≤ γ.

Proof: The inverse demand function for Þrm i and all other symmetric Þrms j are given

by

pi(qi, qj) = max{0, 1− qi − γ(n− 1)qj}, all qj ≥ 0, j 6= i,

pj(qi, qj) = max{0, 1− γqi − (1 + γ(n− 2))qj}. (28)

Symmetric proÞts write

π(q) =

(1− q (1 + γ(n− 1))− c)q if q ≤ 1(1+γ(n−1))

−cq if q ≥ 1(1+γ(n−1))

, (29)

One can easily check that the one-shot best deviation proÞt is

πdi (q) =

14 [1− c− γ (n− 1) q]2 if q ≤ 1−c

γ(n−1) ;

0 otherwise.

(30)

Since qM < 1−cγ(n−1) for all parameter values, π

di (qM ) =

(1−c)216

(γ(n−1)+2)2(1+γ(n−1))2 . Now, the optimal

punishment strategy qP is a solution in q of

πdi (q)− π (q) = πdi (qM )− π (qM ) . (31)

There are two cases:

32

� Case 1 : πdi (q) = 14 (1− c− γ (n− 1)) qj)2 if q ≤ 1−c

γ(n−1)

Solution to (31) is qP =12 (1− c) 3γ(n−1)+2

(2+γ(n−1))(1+γ(n−1)) > 0 which is admissible if and only

if qP ≤ 1−cγ(n−1) ⇔ γ ∈ £0,min©2 ¡1 +√2¢ / (n− 1) , 1ª¢, where min©2 ¡1 +√2¢ / (n− 1) , 1ª =

1⇔ n < 6. The price pi(qP , qP ) ≥ 0 iff c ≥ c0C with

c0C ≡γ (n− 1)− 23γ(n− 1) + 2 .

This frontier c0C is upward sloping and intersects c = 0 for γ = 2n−1 . Note that we need

2n−1 < 1 ⇔ n > 3 for the frontier to be above zero. Now we compute the threshold value of δ

to Þnd

δ1C =1

16

(2 + γ (n− 1))21 + γ (n− 1) < 1.

From (23) , the individual rationality constraint is satisÞed since π(qP ) +δ1Cπ(qM )

1−δ1C > 0 ⇔ γ ∈µ2(3−2

√3)

(n−1) ,2(3+2

√3)

(n−1)

¶which is always true since γ ≤ 21+

√2

n−1 < 23+2

√3

n−1 , all n.

� Case 2 : πdi (q) = 0 if q > 1−cγ(n−1)

The only admissible solution to (31) is qP =(1−c)(2+γ(n−1))

³2+√1+γ(n−1)

´4(1+γ(n−1))2 if and only if

γ ∈h21+

√2

n−1 , 1´and n ≥ 6. We Þnd that qp ≤ (1− c) /γ (n− 1) iff c > c1C with

c1C ≡(2 + γ(n− 1))p(1 + γ(n− 1))− 2 (1 + γ(n− 1))2 (1 + γ(n− 1)) + (2 + γ(n− 1))p(1 + γ(n− 1)) .

This frontier c1C is upward sloping and we check that c1C = c

0C for γ = 2

1+√2

n−1 . Now we compute

the threshold value of δ to Þnd:

δ2C =

µγ (n− 1)

(2 + γ (n− 1))¶2< 1.

From (23) , the individual rationality constraint is satisÞed since π(qP ) +δ2Cπ(qM )

1−δ2C = 0.

33

7.2 Two-period punishment

We solve the two-period punishment case which occurs when collusion cannot be sustained by

using a single-period punishment, that is if c is below a threshold cK . We examine in turn the

Bertrand and Cournot cases.

7.2.1 Bertrand

During the Þrst period of punishment, Þrms inßict the most severe punishment, that is pP = 0.

We solve in pP,2 and δ the system

πdi (pM )− π(pM ) = δ2 (π(pM )− π(pP,2)) + δ (π(pM )− π(0)) ,

πdi (0)− π(0) = δ2 (π(pM )− π(pP,2)) + δ (π(pP,2)− π(0)) , (32)

with

πdi (0) =

[(1−γ)(1−c)+γ(n−1)(−c)]2

4(1+γ(n−1))(1+γ(n−2))(1−γ) if 0 ≤ c ≤ f (γ) ,0 if c ≥ f (γ) ,

πdi (pM ) =

(1−c)2[2+γ(n−3)]2

16(1+γ(n−1))(1+γ(n−2))(1−γ) if γ ≤ n−3+√n2−13n−5 ,³

1−c2γ

´2(2γ − 1) if γ ≥ n−3+√n2−1

3n−5 .

and

π(pM ) =(1− c)2

4 (1 + γ (n− 1)) and π(0) =−c

1 + γ (n− 1) .

We start by recalling a partial result (that appears in Proposition 2) before displaying the

thresholds values of c and δ.

Lemma 8 For n ≥ 2, an optimal two-period punishment penal code exists for cB ≥ c ≥

34

cB,l=2 (γ) and δ > δB,l=2 where:

cB,l=2 =

0 if 2n+1 ≤ γ ≤ γ4 and n = 2

c0B,l=2 if γ5 ≤ γ ≤ min©¡n− 3 +√n2 − 1¢ / (3n− 5) , γ6ª and n ≥ 3

ec0B,l=2 if γ6 ≤ γ ≤ ¡n− 3 +√n2 − 1¢ / (3n− 5) and n > 4c1B,l=2 if max

©γ4,¡n− 3 +√n2 − 1¢ / (3n− 5)ª ≤ γ ≤ γ3

c2B,l=2 if max©γ3,¡n− 3 +√n2 − 1¢ / (3n− 5)ª ≤ γ ≤ 1

and for n = 2:

δB,l=2 =

γ(c−1)2(c+1)(γ(c−3)+4(1−4)) if

2n+1 ≤ γ ≤ n−3+√n2−1

3n−5

(γ2+γ−1)(c−1)2(−1+γ)(c−1)2+γ(−2(c−1)2+(2c(c−1)+γ)γ2) if

n−3+√n2−13n−5 ≤ γ ≤ γ3

γ+γ2−1(2γ−1)(1+γ) if γ3 ≤ γ ≤ 1

and for n ≥ 3:

δB,l=2 =

−(c−1)2(n−1)γ(c+1)(γn(3c−1)−4(1−c)+γ(5−7c)) if

2n+1 ≤ γ ≤ min

nn−3+√n2−1

3n−5 , γ6

oand

min{0, c0B,l=2} ≤ c ≤ min{c0B, f (γ)}

γ2(n−1)2(2+γ(n−3))2 if γ1 ≤ γ ≤ n−3+√n2−1

3n−5 and max{f (γ) , �c0B,l=2} ≤ c ≤ �c1B

γn(2γ−1)−3γ(γ−1)−1(2γ−1)(1+γ(n−1)) if n−3+

√n2−1

3n−5 ≤ γ ≤ 1 and max{f (γ) , c2B,l=2} ≤ c ≤ c2B

(c−1)2(γ−1)(γ2(2n−3)+γ(3−n)−1)(1+γ(n−2))D if n−3+

√n2−1

3n−5 ≤ γ ≤ γ3 andc1B,l=2 ≤ c ≤ f (γ) and n = 3, 4

with D = 1+¡¡2(1− 2c) + 3c2¢n2 − 2 ¡3− 7c+ 5c2¢n+ 5− 12c+ 8c2¢ γ4 + (−3 (c− 1)2 n2 +

(15c− 13) (c− 1)n−14+30c−16c2)γ3+((1−c)2n2−9(c−1)2n+14(c−1)2)γ2+2(c−1)2(n−

35

3)γ + c2 − 2c. We verify that δB,l=2 is continuous over the range of γ.

Proof: There are two cases:

� Case 1. πdi (0) = [(1−γ)(1−c)+γ(n−1)(−c)]24(1+γ(n−1))(1+γ(n−2))(1−γ) if 0 ≤ c ≤ f (γ)

We have to distinguish two subcases according to the optimal deviation to the collusive

price.

Case 1.1. πdi (pM ) =(1−c)2[2+γ(n−3)]2

16(1+γ(n−1))(1+γ(n−2))(1−γ) if γ ≤ n−3+√n2−13n−5

Only one solution to (32) is below the collusive price and is positive if c0B,l=2 ≤ c ≤ c0B withc0B,l=2 = 0 for all γ if n = 2 and

c0B,l=2 ≡γ(n−1)((19−14n+3n2)γ2−(24−8n)γ+8)−4(2+γ(n−3))2

√(1−γ)(1+γ(n−2))

(−97+131n−59n2+9n3)γ3+16(n−2)(2n−5)γ2−8(11−5n)γ+16

for n ≥ 3, all γ. We also obtain

δ0B,l=2 =− (c− 1)2 (n− 1) γ

(c+ 1) ((n(3c− 1)− 7c+ 5) γ − 4(1− c)) .

Case 1.2. πdi (pM ) =³1−c2γ

´2(2γ − 1) if γ ≥ n−3+√n2−1

3n−5

System (32) has only one admissible solution pP . Now for n = 2, pP has four roots in c.

Two of them are negative and two are positive, we denote by c0 and c00, with 0 < c0 < c1B < c00.

We show that c0 = 0 for γ = γ4. Since pP < 0 ⇔ γ > γ4 for c = 0, we conclude that

pP > 0 ⇔ c > max {c0, 0} and we deÞne c1B,l=2 ≡ c0. For n = 3, 4, since we have to consider

only γ > γ4, we know that pP < 0 for c = 0. We also Þnd that pP > 0 for c = f(γ) and for

n−3+√n2−13n−5 ≤ γ < γ3. Eventually, for n ≥ 5, (32) has no solution in pP since c1B,l=2 > f(γ) for

all γ ≥ n−3+√n2−13n−5 . We also obtain

δ1B,l=2 =(c− 1)2 (γ − 1) ¡(2n− 3) γ2 + (3− n) γ − 1¢ (1 + γ (n− 2))

D,

36

where D is as expressed above.

� Case 2. πdi (0) = 0 if (1−γ)(1+γ(n−2)) ≡ f (γ) ≤ c ≤ cB

Again we distinguish two subcases.

Case 2.1. πdi (pM ) =(1−c)2[2+γ(n−3)]2

16(1+γ(n−1))(1+γ(n−2))(1−γ) if γ ≤ n−3+√n2−13n−5

Remark that this subcase cannot occur if n = 2, 3. Moreover, for n = 4, we have c1B,l=2 <

f (γ) and collusion can always be sustained with a two-period punishment. For n > 4, only one

solution to (32) is below the collusive price and is positive if c > �c0B,l=2 with

�c0B,l=2 ≡ A−4(2+γ(n−3))2√2(1−γ)(1+γ(n−2))((5−4n+n2)γ2+(2n−6)γ+2)

γ4(n−1)4 ,

where A =¡161− 212n+ 102n2 − 20n3 + n4¢ γ4+16 (n− 3)3 γ3+48 (n− 3)2 γ2+64(n−3)γ+

32.

Let γ6 be deÞned implicitly by �c0B,l=2 = c

0B,l=2 = f (γ6). We also obtain

δ2B,l=2 =γ2 (n− 1)2

(2 + γ (n− 3))2 .

Case 2.2. πdi (pM ) =³1−c2γ

´2(2γ − 1) if γ ≥ n−3+√n2−1

3n−5

Only one solution to (32) is below the collusive price and is positive if c > c2B,l=2 with

c2B,l=2 =B−2γ(2γ−1)(γn−γ+1)

√(4n−5)γ2+(−2n+6)γ−2

((2n−3)γ2+(−n+3)γ−1)2

where B =¡−1− 4n+ 4n2¢ γ4−2 (2n− 1) (n− 3) γ3+¡11− 10n+ n2¢ γ2−(6− 2n) γ+1.We

also obtain

δ3B,l=2 =γn (2γ − 1)− 3γ (γ − 1)− 1(2γ − 1) (1 + γ (n− 1)) .

We also show that cB2 > cB,l=22 (γ) on the range

³max

nγ2,

n−3+√n2−13n−5

o, 1ifor all n ≥ 2.

37

7.2.2 Cournot

We solve in qP,2 and δ the following system

πdi (qM )− π(qM ) = δ2 (π(qM )− π(qP,2)) + δ³π(qM )− π(qP )

´, (33)

πdi (qP )− π(qP ) = δ2 (π(qM )− π(qP,2)) + δ³π(qP,2)− π(qP )

´, (34)

where qPis the Þrst-period punishment quantity which is such that pi

³qP, qP

´= 0 which

yields to π(qP) = −cq

P. We have

πdi (qP ) =

14

³1− c− γ (n− 1) q

P

´2if q

P≤ 1−c

γ(n−1) ⇔ c < 11+γ(n−1) ,

0 otherwise.

The two-period case requires that c < cC(γ). For n = 4, 5, πdi (qP ) = 0 is impossible since

11+γ(n−1) > c

0C . For n ≥ 6, since 1

1+γ(n−1) < c1C for γ = 1, there exists a ùγ such that

11+γ(n−1) <

cC ⇔ γ > ùγ.

There are two cases:

� Case 1. πdi (qP ) = 14

³1− c− γ (n− 1) q

P

´2if 0 < c < min

n1

1+γ(n−1) , cCo

Only one solution to (33) is above the collusive quantity and is below qPif c > c0C,l=2 with

c0C,l=2 ≡γ(n−1)((3(1−n)2)γ2+(8(n−1))γ+8)−4(2+γ(n−1))2

√(1+γ(n−1))

9(n−1)3γ3+32(1−n)2γ2−40(1−n)γ+16 ,

which is below cC for all γ and n > 4. Moreover c0C,l=2 intersects1

1+γ(n−1) for n > 9. We Þnd

also

δ1C,l=2 =γ (1− c)2 (n− 1)

(1 + c) ((n(1 + 3c)− 1 + 3c) γ + 4− 4c) .

� Case 2. πdi (qP ) = 0 if 11+γ(n−1) ≤ c ≤ cC which is possible for n ≥ 6 only.

38

Here again, there is only one solution to (33) above the collusive quantity. This second-

period punishment is below qPif c > c1C,l=2 with

c1C,l=2 =E−4(2+γ(n−1))2

q2(1+γ(n−1))((1−n)2γ2+2(n−1)γ+2)

γ4(n−1)4 ,

where E = (1− n)4 γ4 − 16 (1− n)3 γ3 + 48 (1− n)2 γ2 − 64(1− n)γ + 32. We also obtain

δ2C,l=2 =γ2 (n− 1)2

(γ(n− 1) + 2)2 .

We check that c1C,l=2 ≤ cC for all γ and all n > 4, and that c1C,l=2 > 11+γ(n−1) only for n ≥ 9.

39

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41