On the profitability of production perturbations in a dynamic natural resource oligopoly

Journal of Economic Dynamics & Control 27 (2003) 1237–1252www.elsevier.com/locate/econbase

On the pro%tability of production perturbations ina dynamic natural resource oligopoly

Hassan Benchekrouna, G-erard Gaudetb; ∗aDepartment of Economics, Florida Atlantic University, USA

bD�epartement de sciences �economiques and Centre de recherche et d�eveloppement en �economique,Universit�e de Montr�eal, C.P. 6128, Succursale Centre-ville, Montr�eal,

Canada H3C 3J7

Accepted 1 December 2001

Abstract

Static oligopoly analysis predicts that if a single %rm in Cournot equilibrium were to beforced to marginally contract its production, its pro%ts would fall. On the other hand, if all the%rms were simultaneously forced to reduce their production, thus moving the industry towardsmonopoly output, each %rm’s pro%t would rise. We show that these very intuitive results maynot hold in a dynamic oligopoly, such as a nonrenewable natural resource oligopoly, wherethe exogenous constraint would take the form of a contraction of the %rm’s output path oversome %xed interval of time: there are situations where a %rm will gain from being the lone%rm constrained in this way and cases where each %rm will lose if all the %rms in the industryare so constrained, thus exactly reversing the conclusions obtained from purely static analysis.? 2002 Elsevier Science B.V. All rights reserved.

Keywords: Production perturbations; Dynamic oligopoly; Di8erential games; Nonrenewable resources

1. Introduction

Consider an industry composed of N %rms in a symmetric static Cournot equilibriumand assume the goods are substitutes in demand and strategic substitutes. Supposethis equilibrium is perturbed by an exogenously imposed marginal contraction of theindividual production of a subset of S of those %rms, while the remaining N − S%rms are allowed to simultaneously adjust their production optimally in response tothis perturbation. The question then arises as to what will be the e8ect of such anexogenously imposed contraction on the pro%ts of the %rms in the designated subset.

∗ Corresponding author.E-mail addresses: [email protected] (G. Gaudet).

0165-1889/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S0165 -1889(02)00023 -4

1238 H. Benchekroun, G. Gaudet / Journal of Economic Dynamics & Control 27 (2003) 1237–1252

Clearly, if the subset is composed of all the %rms (S =N ), then the pro%ts of each ofthe %rms in the subset will rise, since the resulting exogenously supported reduction inindustry output is consistent with what a perfectly collusive industry would choose todo. At the other extreme, if the subset is composed of only one %rm (S = 1), then itspro%ts will necessarily fall, since if that lone %rm could credibly commit to a changein its output away from the Cournot equilibrium output, in the fashion of a Stackelbergleader, than it would choose to expand its output, not contract it. This is because, withsubstitutes in demand and strategic substitutes, when the lone %rm decreases its output,all the other %rms optimally react by increasing their own output, hence decreasingprice. For all intermediate cases, Gaudet and Salant (1991) have derived a necessaryand suFcient condition on the minimum number of %rms that must be included in thesubset in order for their pro%ts to rise as a result of the exogneous contraction. Thatnumber generally depends on the curvature of the demand and cost functions, as wellas on the total number of %rms in the industry. 1

In this paper, we reconsider those results for the case of a dynamic output gamewhere there exists a constraint on the cumulative actions of each player. This is thecase for an oligopoly where each %rm owns a %xed stock of a nonrenewable naturalresource and competes in quantity with the other %rms. The cumulative output of eachresource extracting %rm is constrained by its %xed initial stock and the evolution ofthe stocks over time is described by a di8erential equation which makes the game adi8erential game. The marginal contraction of output being imposed to the %rms in thedesignated subset is modeled as a perturbation of the production path of each of the%rms in the subset in a neighborhood of its initial equilibrium path during some %xedinterval of time. All the %rms outside the subset are allowed to react optimally to thisperturbation of the initial equilibrium.

We show that the necessary and suFcient condition on the number of %rms derivedin Gaudet and Salant (1991) for the static case is in fact neither necessary nor suFcientin that context. As a result, we can show that even forcing a single %rm to marginallycontract its production path in a neighborhood of equilibrium over a %xed interval oftime can result in an increase in its pro%ts. This in fact can happen when any numberof %rms are subject to the exogenous output contraction and no matter what is thetotal number of %rms in the industry. 2 At the extreme, if all the %rms in the industryare subject to the exogenously supported output contraction, then, paradoxically, pro%tsmay fall. The reason for these results is that the contraction a8ects not only the outputduring the period of contraction, but a8ects also the stocks that remain at the end ofthe period of contraction. This in turn a8ects the equilibrium output paths and values

1 Perfectly symmetrical results can of course be derived for a marginal expansion rather than contraction:when pro%ts rise for a contraction they would fall for an expansion, and vice versa. See Gaudet and Salant(1991).

2 It may seem that if imposing on a %rm some feasible contraction of its equilibrium path can leave itbetter o8, the %rm should have voluntarily contracted its path initially in the same fashion. That reasoningneglects the fact that had it done so, the other %rms would have responded by modifying their own paths,with the result that it would itself then want to deviate from this new path. In other words, the new pathcannot be sustained as an equilibrium unless supported by some exogenous constraint, as we have assumed.Hence the importance of the contraction we consider being exogenously imposed.

H. Benchekroun, G. Gaudet / Journal of Economic Dynamics & Control 27 (2003) 1237–1252 1239

of the game that is played from that time on. We derive a necessary and suFcientcondition for the pro%t of the %rms in the exogenously constrained subset to increasein this context as a result of being imposed a marginal contraction of their outputpaths over a %xed interval of time. The condition of Gaudet and Salant (1991) on thenumber of %rms in the subset is shown to obtain only in a limiting case where thedynamic game in fact reduces to the equivalent of a sequence of static games.

We will consider both an open-loop and a closed-loop production game. In theopen-loop game, the %rms commit to an entire production path at the beginning of thegame. Each %rm chooses its own extraction path taking as given the whole extractionpaths of its competitors. Hence each %rm knows that the choice of its own path exertsno strategic inGuence on the extraction paths of its competitors, since the rivals’ ex-traction rate is strictly a function of time. In the closed-loop game, on the other hand,each %rm chooses an extraction rule taking as given its competitors extraction rules.These extraction rules specify for each instant an extraction rate for each level of itsown stock and those of its competitors. Thus, contrary to the open-loop case, each%rm is aware of the impact of its own resource stock on the production decisions ofthe other %rms at each instant and knows that, through this e8ect, it can inGuence theextraction rate of its rivals.

It may seem paradoxical that the phenomenon we describe can occur in an open-loopgame but not in a static game, since open-loop games and static games are formallyidentical from an information structure point of view. It should be noted however thatthe open-loop production game we describe has properties which cannot be capturedin a purely static game such as the conventional Cournot production game referred toabove in the %rst paragraph, where the controls belong to the space of real numbers.Furthermore, the open-loop production game allows types of perturbations in a neigh-borhood of equilibrium that cannot even be considered in such a static framework.Indeed, a nonrenewable resource extraction game is necessarily dynamic in nature,since resource stocks (the state variables) are being irreversibly changed as a resultof decisions to extract. This dependance of the current state on past actions cannot becaptured in the conventional static game. The nonrenewable resource extraction gamemay however be formulated as a di8erential game with open-loop information, whichcan be viewed as a static game where the controls belong to the in%nite-dimensionalspace of functions. The rates of production (the controls) being functions of time, itbecomes pertinent to ask what is the e8ect of a perturbation in a neighborhood of equi-librium of the production paths of some of the players over some 8xed subinterval oftime, a question which does not even arise in a static game where controls must belongto the space of real numbers. Such a perturbation has the e8ect not only of changingthe rate of production during the interval in question, but of modifying the vector ofstocks remaining at the date at which the subinterval ends. Hence the possibility thatthe qualitative conclusions as to the e8ect on the values of the game for the playersbe di8erent.

The exogenous output contraction to which the analysis applies may be one that isthe direct result of some outside intervention, such as a temporary strike, for instance. Itmay also be one that is supported indirectly by some policy decisions, such as taxes orquotas, which are expected only to be in e8ect for some predetermined length of time.


The analysis applies as well—albeit with the direction of the e8ects being reversed—to policy variables, such as subsidies, that support an output expansion rather than acontraction. The approach can also be useful to analyze, in some cases, the e8ects onpro%ts of shifts in the output paths due to decisions to merge or to cartelize all or partof the industry. The usefulness of the methodology itself is not limited to nonrenewablenatural resource industries, although the qualitative results that follow from it may bedi8erent in other cases. It could be usefully applied to analyze the e8ect of similaroutside interventions in other dynamic oligopoly situations—such as when learning bydoing is present—where the current state depends on past actions.

After describing the model in Section 2, we present results for the open-loop gamein Section 3. The reason for considering the open-loop game before going on to theclosed-loop game is twofold. First, it makes clear that our conclusions do not rest onthe fact that the individual %rm can inGuence its rivals’ rates of production by changingthe time path of its own resource stock. Second, it will make it easier to isolate anddescribe the e8ects of the exogenous contraction which can be attributed speci%cally tothe closed-loop strategies. In the Section 4, we show how the analysis can be extendedto the closed-loop game. Section 5 o8ers a brief conclusion.

2. The model

Consider a nonrenewable natural resource industry composed of N identical %rms,each one owning at date t = 0 identical resource stocks. 3 Let ui(t) denote the rate ofextraction of %rm i at time t, xi(t) its remaining stock at time t and Ti its terminaldate. For all t¿Ti and all i; ui(t) ≡ 0. We will denote, respectively, by x(t) =(x1(t); x2(t); : : : ; xN (t)); u(t)=(u1(t); u2(t); : : : ; uN (t)) and T =(T1; T2; : : : TN ) the vectorsof stocks, of extraction rates and of terminal dates, and by i(u(t)) %rm i’s pro%tsat time t. Ji(u) will denote the present value of those pro%ts over the interval [0; Ti].Thus,

Ji(u) =∫ Ti

0i(u(t))e−rt dt:

We assume i(u(t)) to be twice continuously di8erentiable and strictly concave withrespect to ui(t). We also assume that resources i and j are, at each date t, substitutes indemand (@i=@uj ¡ 0) and strategic substitutes, which means that %rm i’s best responseto an aggressive action from %rm j (an increase of uj(t)) at time t is to reduce itsown production at that same date, i.e., dui(t)=duj(t)¡ 0. 4 Given the assumptions justmade on i(u(t)), this is equivalent to assuming @2i=@ui@uj ¡ 0.

In the open-loop game, %rm i’s problem is to determine, at time t = 0, the ratesui(t) at which it will be extracting the resource at each instant t, taking as given itscompetitors extraction paths. It can commit at time t = 0 to this extraction path. In

3 We make the assumptions of identical %rms and identical initial stocks because this signi%cantly simpli%esthe derivations by allowing us to restrict attention to symmetrical equilibria. There is no loss in insight indoing so. Similar results can be derived with nonidentical %rms and initial stocks.

4 See Bulow et al. (1985).


the absence of any exogenous constraints on its extraction path, %rm i chooses its bestresponse to solve the following problem:

max{ui(t)};Ti

Ji(u)

s:t: xi(t) = −ui(t) (1)

and

xi(0) = x0; xi(Ti)¿ 0: (2)

An open-loop equilibrium is a vector of extraction paths that solves the above prob-lem simultaneously for all i = 1; 2; : : : ; N . Denote this equilibrium vector u∗(x(0)),where x(0) is the vector of initial endowments. Note that the vector x(0) has scalarelements xi(0) = x0, which represent the identical individual initial stocks. The equi-librium present value of %rm i is therefore also a function of x0 and will be writtenVi(x(0)) ≡ Ji(u∗(x(0))).

In the closed-loop game, each %rm’s problem is to determine at the initial datea strategy which takes the form of a decision rule based on time and the vector ofremaining stocks at the moment the decision is to be made. We thus limit our attentionto Markov strategies, in the sense that the decision rule depends only on the currentstocks and not on their entire history. Firm i chooses its strategy taking as given thestrategies of the N − 1 other %rms.

Denote %rm i’s decision rule by ui(t; x(t)). Then its problem can be stated as thatof choosing ui(t; x(t)) in order to solve

max{ui(t;x)};Ti

∫ Ti

0i(u(t; x))e−rt dt

s:t: xi = −ui(t; x)

and

xi(0) = x0; xi(Ti)¿ 0

taking as given the decision rules of the N − 1 other %rms. The strategy chosen by%rm i therefore constitutes a best response to the set of strategies of the N − 1 other%rms. A closed-loop, or Markov perfect, equilibrium is a set of N such strategies, onefor each %rm, each of which is a best response to the other N − 1 strategies.

Let us suppose now that each %rm in a subset composed of S %rms (S6N ) isexogenously forced, during a given interval of time, to marginally reduce its produc-tion in a neighborhood of the prevailing equilibrium. Without loss of generality, wemay assume that this subset is composed of the %rst S %rms. Thus, for i = 1; : : : ; S,%rm i is forced to produce u∗i (t) + �ih(t)¿ 0 for all t ∈ [0; �], where the �i’s are arbi-trarily small positive parameters, h(t) is a negative piecewise continuous function and�∈ (0;mini Ti). As for the N − S %rms outside the constrained subset, each of them isaware of the perturbation that the %rst S %rms are facing and will adjust its decisionaccordingly. Notice that we could just as well have considered an exogenously imposedmarginal expansion of the production path by choosing h(t) to be positive.


Each of the constrained %rms considers its payo8 over the interval [0; �] as given.Its problem now reduces to determining its extraction rate for t¿ �, with stock xi(�)given by

xi(�) = x0 −∫ �

0(u∗i (t) + �ih(t)) dt: (3)

In the open-loop case, it does this by choosing at date t = 0 its extraction path {u(t)}for t¿ �, taking as given the full extraction paths of its rivals. In the closed-loop case,it chooses at t = 0 its extraction rule u(t; x(t)) for t¿ �, taking as given the extractionrules of its rivals. As for the unconstrained %rm j; j = S + 1; : : : ; N , it continues tochoose at t=0 its extraction path (rule) for t¿ 0 in the open-loop (closed-loop) gametaking as given that of its N − 1 rivals.

We will let v(t) = (v1(t); v2(t); : : : ; vN (t)) denote the vector of equilibrium extractionrates when the S %rms are constrained. Thus, vi(t)=u∗i (t)+�ih(t) for i=1; 2; : : : ; S andt ∈ [0; �] and v(t) = u∗(t) for �i = 0. Notice that the value function of %rm i’s program,which we will denote Vi, depends on the parameters �1; �2; : : : ; �S and � as well as onthe initial stocks. In particular, when evaluated at �i = 0 for i = 1; 2; : : : ; S, it yields thevalue of its program in the initial unconstrained equilibrium being considered.

3. The case of open-loop equilibria

Consider %rst the case of open-loop equilibria. In that case, an unconstrained equi-librium must satisfy, in addition to (1) and (2), the following set of conditions:

@i

@ui− �i = 0; (4)

�i = r�i; (5)

�i(Ti)¿ 0; �i(Ti)xi(Ti) = 0; (6)

i(u(Ti)) − �i(Ti)ui(Ti) = 0; (7)

where �i is the shadow value of %rm i’s resource stock. Notice that the transversalitycondition (6) implies x(Ti) = 0, since �i(Ti) = �i(0)e−rTi by (5) and �i(0) is positive,x0 being %nite.

Suppose each of the %rst S %rms is imposed a marginal reduction of its rate ofextraction over the interval [0; �], as de%ned in the previous section. Given �, thisexogenously imposed contraction of its production path will, at least in a neighborhoodof the unconstrained equilibrium, be bene%cial to %rm i, for i = 1; 2; : : : ; S, if and onlyif

dVi(�1 = · · · = �N = 0) =@Vi(�1 = · · · = �N = 0)

@�id�i

+S∑

j=1; j �=i

@Vi(�1 = · · · = �N = 0)@�j

d�j ¿ 0:


Clearly, since it is evaluated at a neighborhood of the unconstrained equilibrium(i.e., at �i = 0),

@Vi(�1 = · · · = �N = 0)@�i

= 0;

and we are left only with the second term, which will generally not vanish. 5

For simplicity, we assume hereafter that all of the identical S constrained %rms aretreated identically, so that �i = �, for i = 1; 2; : : : ; S. Then d�i=d�= 1, and we may write

dVi(� = 0)d�

=S∑

j=1; j �=i

@Vi(� = 0)@�j

; (8)

where it must be understood that letting � go to zero means letting each of the �i’s goto zero simultaneously.

Since the %rms are assumed identical in all respects (see footnote 3) and since werestrict attention to equilibria that are symmetric at � = 0, then

@i(u∗)@vk

=@i(u∗)@v′k

for all k; k ′ �= i

and@Vi(x(�))

@xk=

@Vi(x(�))@x′k

for all k; k ′ �= i:

Furthermore, conditions (4)–(7) must hold when � = 0. Making use of those facts tocalculate the right-hand side of (8), we show in full detail in the appendix that

dVi(� = 0)d�

=∫ �

0

[@i(u∗)@vk

e−rt − @Vi(x∗(�))@xk

e−r�]

×[(N − S)

dvjdvi

+ (S − 1)]h(t) dt; (9)

where k denotes any %rm other than %rm i and j denotes any of the (symmetrical)%rms in the unconstrained subset {S + 1; S + 2; : : : ; N}. Thus, the expression inside thesecond set of square brackets is the derivative with respect to � (evaluated at �= 0) ofthe production of the N − 1 %rms other than %rm i, N − S of which are free to adjusttheir production, the S − 1 others being constrained just as %rm i.

We may think of the total e8ect on the equilibrium pro%ts of %rm i of a simultaneousexogenous contraction of the output paths of the S %rms as being composed of a directe8ect, due to the contraction of its own output path, and an indirect e8ect, due to thecontraction of the output paths of the S− 1 other constrained %rms and the adjustmentof their output paths by the N − S unconstrained %rms. However, as noted above,a marginal contraction of %rm i’s production has no e8ect on its equilibrium pro%tsin a neighborhood of an unconstrained equilibrium, by de%nition of this open-loopequilibrium. Hence the direct e8ect vanishes when evaluated in a neighborhood ofthe uncontrained open-loop equilibrium. It is useful to consider brieGy the economic

5 This is simply a form of the envelope theorem.


intuition behind this, since it will help understand why, as we will show in the nextsection, the direct e8ect does not vanish in the case of a closed-loop equilibrium. Aswe will see, this is because of strategic feedback e8ects which are present in the caseof a closed-loop equilibrium and not in that of an open-loop equilibrium. First, notethat a reduction of the %rm’s production at any date t ∈ [0; �] will necessarily result inan equivalent increase in the stock it has remaining at �. But along the equilibrium pathof an open-loop equilibrium, the shadow value of a unit of resource stock, �i(t), mustgrow at the rate of interest, by condition (5). This means that the present value of amarginal unit of the resource left unexploited at date t ∈ [0; �], given by �i(t)e−rt , mustbe equal to the present value of a marginal unit left unexploited at date �, given by�i(�)e−r�. Since the latter measures the present value of the contribution of a marginalunit of resource stock to the value of %rm i’s optimal program from date � on, thevalue of the reduction of the %rm’s production at any date t ∈ [0; �] will be exactlyo8set by the value of the increase in the stock it has remaining at �, thus cancelingout any direct e8ect. Hence a marginal contraction of %rm i’s open-loop equilibriumproduction path only has an indirect e8ect on its pro%ts, via the change it induces inthe extraction path of its rivals.

To analyze this indirect e8ect, let

Eo1(t) =

@i(u∗)@vk


e−r� (10)

and

Eo2(t) =

[(N − S)

dvjdvi

+ (S − 1)]h(t): (11)

The superscript o denotes the fact that each of those expressions is evaluated along anopen-loop equilibrium.

Consider %rst the expression for Eo2(t). This measures the e8ect of the exogenous

contraction on the combined production of %rm i’s N−1 rivals: Eo2(t)¡ 0(¿ 0) means

that the combined reaction of the (N−1) other %rms is to reduce (increase) production.Of the N − 1 rivals of %rm i, S − 1 are also constrained to reducing their production,while the other N − S can freely adjust their rate of extraction in reaction to thereduction of %rm i’s own rate of extraction. Because by assumption we are dealingwith strategic substitutes, dvj=dvi is negative and each of those N − S %rms reacts byincreasing its own production. Hence the sign of Eo

2(t) depends, at any t, on the sizeof the constrained and unconstrained subsets, and Eo

2(t) can change sign with respectto t.

As for the expression Eo1(t), it is the sum of two terms. The %rst term is the present

value of the e8ect on the pro%t of %rm i of a change in the extraction rate of %rmk, k �= i, at date t ∈ [0; �]. This term is negative, by assumption: resources i and k,i �= k, are substitutes in demand. The second term is the present value of the e8ect ofa change in %rm k’s stock at � on the value of the optimal program of %rm i overthe interval [�; Ti]. This term is also negative. The expression Eo

1(t) can therefore beof either sign. Furthermore, it can change sign over the interval [0; �].

In a purely static Cournot oligopoly framework, the sign of Eo2(t) is all that matters

in order to determine the e8ect on the pro%t of the constrained %rm i: the exogenously


imposed contraction will be bene%cial to %rm i if and only if Eo2(t)¡ 0, so that the

combined reaction of the N − 1 other %rms is to reduce production (see Gaudet andSalant, 1991). This will be the case if the number (S) of %rms being constrainedis suFciently large relative to the number (N − S) of %rms left unconstrained. Anexplicit condition on the minimum number of constrained %rms for Eo

2(t)¡ 0 is derivedexplicitly in Gaudet and Salant (1991), where it is shown to depend on the curvatureof the demand and cost functions. Notice that this static Cournot oligopoly outcome isobtained in the limiting case where the stocks of the resource are in%nite. In that casean additional unit of reserves is worthless, so that the marginal e8ect on the optimalvalue of the remaining program of %rm i at any date t of resources left in the groundby any %rm, including itself, is zero (i.e., @Vi(x∗(�))=@xk =0 for k =1; 2; : : : ; N ). HenceEo

1(t) reduces to (@i(u∗)=@vk)e−rt , which is negative, since, by assumption, resourcesi and k are substitutes in demand. 6 In fact, when the resource stocks are in%nite, theproblem reduces to a in%nite sequence of static problems.

Since Eo1(t) can be of either sign when the stocks of the resource are %nite, knowing

the sign of Eo2(t) is no longer suFcient to establish whether the exogenous perturbation

is pro%table or not for each %rm in the constrained subset, as it was in the static Cournotcase. This is because we may have dVi(� = 0)=d�¿ 0 with Eo

2(t)¿ 0 as well as withEo

2(t)¡ 0. We may also have dVi(� = 0)=d�¡ 0 with Eo2(t) of either sign.

Consider for instance the case of a duopoly where one of the %rms is forced toreduce its production over the interval [0; �]. We, therefore, have N = 2 and S = 1and hence Eo

2(t) = (dvj=dvi)h(t)¿ 0. For the sake of argument, assume Eo1(t)¿ 0 for

all t ∈ [0; �]. 7 The fact that Eo1(t)¿ 0 for all t ∈ [0; �] means that the competition

which results from an extra unit of stock to the other %rm at � is more harmful tothe constrained %rm than is the competition from an extra unit of production by theother %rm at some date t ∈ [0; �]. If this were a static oligopoly, we would thereforeget the not surprising result that forcing one of the %rms to marginally reduce itsproduction cannot increase its equilibrium pro%ts. In the present context, however,since Eo

1(t)¿ 0; dVi(� = 0)=d�¿ 0 and the constrained %rm gains. Thus, we have theparadoxical situation where even a duopolist would gain from an exogenously imposedmarginal contraction of its production.

With more than two %rms in the industry, we may have Eo2(t)¡ 0 if the number of

%rms in the constrained subset is suFciently large. As pointed out above, in the staticCournot oligopoly, this is suFcient for each %rm in the subset to bene%t from the outputcontraction they are being imposed. The reason is that although each unconstrained %rmreacts by increasing its output, the combined reaction of a constrained %rm’s N − 1rivals is to reduce production, hence resulting in a fall in the total industry outputand an increase in price. 8 In the dynamic resource oligopoly, however, the change inoutput by each of the N − 1 rivals at any date t ∈ [0; �]—an increase for each of the

6 Recall that h(t) is negative by assumption.7 This is clearly possible. For example, it can be veri%ed that for the inverse demand system given by

Pi(u1; u2) = u−�i + �u−�

j ; i �= j; i; j = 1; 2, with �; �∈ [0; 1] and � some positive number, it occurs whenever�¡�.

8 See Gaudet and Salant (1991) for further details on the static oligopoly case.


N − S unconstrained %rm and a decrease for the other S − 1 constrained %rm—resultsin an equivalent change in the opposite direction in its stock at date �. It may bethat the net outcome is Eo

1(t)¿ 0 for some subinterval of [0; �]—possibly the wholeinterval—and that the overall net e8ect over the interval [0; �] is detrimental to eachconstrained %rm, resulting in a reduction of the present value of its pro%ts even ifEo

2(t)¡ 0 for all t.This may happen even in the case where each of the N %rms is simultaneously

being forced to a marginal contraction of its output path over the interval [0; �]. In thatcase, S =N and Eo

2(t) = (N −1)h(t)¡ 0. Since the extraction paths of each %rms overthe interval [0; �] is exogenously given, this is equivalent to a game starting at � withx(�) as the vector of initial stocks. Since each %rm’s resource stock at � is larger thanit would have been along the unconstrained equilibrium path, each %rm’s discountedequilibrium rent from � on will be smaller. 9 If the gains from jointly producing closerto the monopoly output over the interval [0; �] are insuFcient to compensate for this,then Eo

1(t)¿ 0 and each %rm loses. Paradoxically, this means that if the N %rms wereto form a cartel to last some exogenously set interval of time [0; �], �¡T ∗

i , they mightchoose to expand their production path over that interval.

Although these results provide some important insights into the possible e8ect onthe present value of natural resource oligopolists of some exogenous constraint on theirproduction path, they su8er from the drawback that it is has been assumed the %rms cancommit at time zero to entire production paths. It is well known that unless the onlyinformation accessible to each %rm as the game evolves is that which it has availableat the initial date, such open-loop equilibria will not be subgame perfect. We nowturn to consideration of the e8ect of the same type of constraints in Markov perfectequilibria, where %rms commit to an extraction rule which is a function of time andof the vector of current resource stocks, rather to an extraction path.

4. The case of closed-loop equilibria

We will now assume that the %rms know, at each instant, the vector of remainingresource stocks and that they are able to adjust their extraction rates to this information.A production strategy speci%es for each date t a rate of extraction which depends on thevector of stocks remaining at that date. 10 We will put the emphasis, in this section,on deriving the additional e8ect that comes from considering an equilibrium where%rms simultaneously choose extraction rules rather than extraction paths and giving itan economic interpretation.

One important di8erence with the open-loop game considered in the previous sectionis that each %rm now imputes a shadow value to the resource stocks of its competitors,because its own decision rule depends on them. Thus, the current value Hamiltonian

9 Even though the previous production paths for t¿ � are still feasible, they cannot constitute an equilib-rium anymore, for this would mean leaving some of the stock unexploited.

10 We assume the strategies to be continuous with respect to t and continuously di8erentiable with respectto x.


corresponding to %rm i’s optimization problem in the closed-loop game is

Hi(u; x; �i; �ik) = i(u(t; x)) − �i(t)ui(t; x) −∑k �=i

�ik(t)uk(t; x);

where, as before, �i(t) is the shadow value associated to %rm i’s own stock, and now,in addition, �ik(t) represents the shadow value associated by %rm i to %rm k’s stock.

Any equilibrium must satisfy 11

@i

@ui− �i = 0; (12)

�i = r�i −∑k �=i

(@i

@uk− �ik

)@uk@xi

; (13)

�ik = r�ik −∑k �=i

(@i

@uk− �ik

)@uk@xk

; (14)

�i(Ti)¿ 0; �i(Ti)xi(Ti) = 0; (15)

�ik(Ti)xk(Ti) = 0; (16)

i(u(Ti; x(Ti))) − �i(Ti)ui(Ti) − �ik(Ti)uk(Ti) = 0: (17)

Other than the fact that each %rm now imputes a shadow value to the resourcestocks held by its competitors, there is another important di8erence that characterizes aclosed-loop equilibrium. It comes from the fact that each %rm now considers as giventhe other %rms’ extraction strategies—which are functions of the vector of resourcestocks—and not the other %rms’ extraction paths, as in an open-loop equilibrium. Thus,each %rm is aware of the inGuence of its own resource stock on the production decisionsof the other %rms. For this reason, there is now a strategic element to the value of the%rm’s own stock, which explains the second term of condition (13).

As is clear from condition (13), an immediate consequence of this strategic con-sideration is that the discounted shadow value to the %rm of a unit of resource stock(�i(t)e−rt) is not constant, as it was along an open-loop equilibrium. Therefore, alongthe equilibrium path of a closed-loop equilibrium, the present value of a marginal unitof the resource left unexploited at date t ∈ [0; �) is not equal to the present value of amarginal unit left unexploited at date � and, hence, not equal to the present value ofthe contribution of a marginal unit of resource stock to the value of the %rm’s optimalprogram from date � on. As a consequence, and unlike in the open-loop equilibrium,the %rm is not indi8erent, along the equilibrium production path of a closed-loop equi-librium, between a marginal decrease of its rate of extraction at some date t and acompensating marginal increase of its available stock at �¿ t. This is because each%rm i is aware of the fact that its rivals base their production decisions at any datet directly on the vector of remaining resource stocks of all n %rms, including that of%rm i.

11 See Basar and Olsder (1982) or Fudenberg and Tirole (1991, Chap. 13, pp. 521–523).


An exogenous contraction of %rm i’s extraction path over the interval [0; �] willtherefore now have a direct e8ect on its present value, unlike in an open-loop equi-librium. Indeed, suppose we constrain a subset of S %rms in the same way as we didin the open-loop case, with the contraction h(t) being still a function of time, so thatthe extraction rate of the constrained %rm is now vi(x(t); t) = u∗i (x(t); t) + �ih(t). Asin the open-loop case, we assume for simplicity that �i = � for i = 1; 2; : : : ; S. We then%nd (see the appendix, where the details of the derivation are spelled out) that alonga symmetric closed-loop equilibrium, if it exists,

dVi(� = 0)d�

=∫ �

0

[@i(u∗)@vk


e−r�] [

(N − S)dvjdvi

+ (S − 1)]

×h(t) dt +∫ �

0

[@i(u∗)

@vie−rt − @Vi(x∗(�))

@xie−r�

]h(t) dt; (18)

where, as in (9), k denotes any %rm other than %rm i and j denotes any %rm in theunconstrained subset {S + 1; S + 2; : : : ; N}. Note the additional term, when comparedto the open-loop case. This additional term captures the direct e8ect of the exogenouscontraction of %rm i’s production path on its present value.

Since optimality condition (12) requires

@i(u∗i (t))@ui

= �i(t);

and since at any date t (including t = �), we have

�ik(t) =@Vi(x∗(t))

@xkand �i(t) =

@Vi(x∗(t))@xi

;

if we de%ne

Ec1(t) =

@i(u∗)@vk

e−rt − �ik(t)e−r�; (19)

Ec2(t) =

[(N − S)

dvjdvi

+ (S − 1)]h(t): (20)

and

Ec3(t) = �i(t)e−rt − �i(�)e−r�; (21)

we may rewrite (18) as

dVi(� = 0)d�

=∫ �

0Ec

1(t)Ec2(t) dt +

∫ �

0Ec

3(t)h(t) dt: (22)

In a closed-loop equilibrium, Ec3(t) is not everywhere zero as it was in the open-loop

case, since condition (13) does not require that �(t) grow at the rate of interest r, ascondition (5) did in the open-loop case. From (21), it can be seen that the quantitativeimportance of Ec

3(t) depends on the di8erence between the present value of a marginalunit of the resource left unexploited at date t ∈ [0; �) and the present value of a marginalunit left unexploited at date �, which itself depends, amongst other things, on the rateof discount r.


Ec1(t) and Ec

2(t) have the same interpretation as Eo1(t) and Eo

2(t) in the previoussection. They di8er, however, by the fact that, being evaluated along a closed-loopequilibrium, not only are the equilibrium paths not the same, but the derivatives ofVi with respect to xk , k �= i and the derivative of vj with respect to � must takeinto account the fact that the extraction strategy of each %rm is now a functionof the vector of remaining resource stocks and that these remaining stocks will bea8ected by a contraction of the production path of the S %rms in the constrainedsubset.

The shadow value to %rm i of a unit of %rm k’s stock, �ik(t), is negative if a marginalincrease of %rm k’s stock at date t along the equilibrium path decreases the presentvalue, at t, of the Gow of pro%ts of %rm i. Assume this is the case along the equilibriumpath being considered. This is certainly plausible, given the structure of the problemand especially the fact that the goods are substitutes in demand. In that case, the signof Ec

1(t) is indeterminate. As for the sign of Ec2(t), it depends, as in the open-loop case,

on the relative size of the constrained and unconstrained subsets. The sign of Ec3(t)

will be negative, and hence Ec3(t)h(t) positive, if the discounted shadow value of %rm

i’s own stock, �i(t)e−rt , is growing over time along the equilibrium path. Hence, as inthe dynamic open-loop equilibrium and contrary to the static equilibrium, knowing thesign of Ec

2(t) is not suFcient to determine the sign of dVi(� = 0)=d�. In particular, itcannot be ruled out, once again, that forcing a single %rm in a nonrenewable resourceoligopoly to contract its production path over a given interval of time will improve itspro%ts.

5. Conclusion

Static—or steady-state, if it exists—Cournot oligopoly analysis predicts that if asingle %rm in the oligopoly was forced to contract its production marginally, its pro%tswould fall. On the other hand, if all the %rms were simultaneously forced to reducetheir production, thus moving the industry towards monopoly output, each %rm’s pro%twould rise. We have shown that these results may not hold in a dynamic oligopoly,such as a nonrenewable resource oligopoly, where the exogenous constraint would takethe form of a contraction of the %rm’s output path over some %xed interval of time:there are situations where a %rm will gain from being the lone %rm constrained inthis way and cases where each %rm will lose if all the %rms in the industry are soconstrained. As in the static oligopoly, such constraints have the e8ect of inducingall the unconstrained %rms, if any, to adjust their output at each date at which theconstraint holds. But in a dynamic oligopoly they also modify the initial conditionsof the remaining game that follows the lifting of the constraints. These e8ects mayhave opposite consequences for pro%ts and the net result will depend on the actualparameters. Clearly, the very intuitive results obtained from a static analysis of thee8ects of exogenous phenomena, such as strikes, taxes or quotas, which cause %rms tocontract their production path, can be misleading when the context is one of dynamicoligopoly. Their e8ects on pro%ts may be exactly reversed.


Acknowledgements

We wish to thank the Social Sciences and Humanities Research Council of Canadafor %nancial support.

Appendix

We show in this appendix that dVi(�=0)=d� is given by (19) in the case of open-loopequilibria and by (18) in the case of closed-loop equilibria.

Recall that Vi(x(t)) denotes the equilibrium value of the program of %rm i beginningat date t with resource stock x(t), as measured by the Gow of future pro%ts, discountedto date t, along the equilibrium path beginning at t. In particular, the value of con-strained %rm i’s equilibrium program at time t = 0 in a constrained equilibrium is afunction of �1; : : : ; �S and �, in addition to x(0). To denote this fact, we might writeVi(x(0); �1; : : : ; �S ; �)(=J (v(x(0)))). For any constrained %rm i∈{1; 2; : : : ; S}, we have

Vi(x(0); �1; : : : ; �S ; �) =∫ �

0e−rti(v) dt + e−r�Vi(x(�)):

Since v(x(0)) = u∗(x(0)) when �1 = · · ·= �S = 0, we will, by a slight abuse of notation,write Vi(�1 = · · · = �S = 0) to denote the equilibrium present value of %rm i evaluatedat �1 = · · · = �S = 0, that is at the initial unconstrained equilibrium.

Di8erentiating, we get

dVi =∫ �

0

N∑j=S+1

e−rt @i(v)@vj

dvj +S∑

‘ �=i;‘=1

e−rt @i(v)@v‘

h(t) d�l

dt

+ e−r�N∑

j=S+1

@Vi(x(�))@xj

dxj + e−r�S∑

‘ �=i;‘=1

@Vi(x(�))@x‘

dx‘

+∫ �

0e−rt @i(v)

@vih(t) d�i dt + e−r� @Vi(x(�))

@xidxi:

Since, for all #∈{1; 2; : : : ; N},

x#(�) = x0 −∫ �

0v#(t) dt;

we have

dx# = −∫ �

0dv# dt:

Therefore,

dxj = −∫ �

0dvj dt;

dxl = −(∫ �

0h(t) dt

)d�l


and

dxi = −(∫ �

0h(t) dt

)d�i;

and hence, if we assume for simplicity that �i = �, so that d�i = d� for i = 1; 2; : : : ; S,we can write

dVi

d�=∫ �

0

N∑j=S+1

e−rt @i(v)@vj

dvjd�

+S∑

‘ �=i;‘=1

e−rt @i(v)@v‘

h(t)

dt

− e−r�∫ �

0

N∑j=S+1

@Vi(x(�))@xj

dvjd�

dt +S∑

‘ �=i;‘=1

@Vi(x(�))@x‘

h(t)

dt

+∫ �

0

[e−rt @i(v)

@vi− e−r� @Vi(x(�))

@xi

]h(t) dt:

In the case of open-loop equilibria, we know by (4) that at � = 0,@i(v(t))

@vi= �i(t)

and by (5) that

�i(�) = er(�−t)�i(t):

Therefore,

e−rt @i(v)@vi

− e−r� @Vi(x(�))@xi

= 0;

since at any date t along the optimal path, we have

�i(�) =@Vi(x(�))

@xi:

Furthermore, the %rms being by assumption identical in all respects and the equilib-rium at � = 0 being symmetric, we have

@i(u∗)@vk

=@i(u∗)@vk′


and@Vi(x(�))

@xk=

@Vi(x(�))@xk′


and hence, as stated in (9),

dVi(� = 0)d�

=∫ �

0

[@i(u∗)@vk

e−rt−@Vi(x(�))@xk

e−r�] [

(N−S)dvjdvi

+(S−1)]h(t) dt;

where k denotes any %rm other than %rm i and j denotes any %rm in the unconstrainedsubset {S + 1; S + 2; : : : ; N}. It should also be noted that, in deriving the above, wehave made use of the fact that dvj=d� = (dvj=dvi)(dvi=d�) = (dvj=dvi)h(t).


In the case of closed-loop equilibria, at � = 0, we know by (13) that

�i(�) �= er(�−t)�i(t)

and, therefore,

e−rt @i(v)@vi

− e−r� @Vi(x(�))@xi

�= 0:

Hence, as stated in (18),dVi(� = 0)

d�=∫ �

0

[@i(u∗)@vk


e−r�] [

(N − S)dvjdvi

+ (S − 1)]

×h(t) dt +∫ �

0

[e−rt @i(u∗)

@vi− e−r� @Vi(x∗(�))

@xi

]h(t) dt;

where again k denotes any %rm other than %rm i and j denotes any %rm in theunconstrained subset {S + 1; S + 2; : : : ; N}.

References

Basar, T., Olsder, G.J., 1982. Dynamic Noncooperative Game Theory. Academic Press, New York.Bulow, J.I., Geanakoplos, J.D., Klemperer, P.D., 1985. Multimarket oligopoly: strategic substitutes and

complements. Journal of Political Economy 93, 488–511.Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA.Gaudet, G., Salant, S.W., 1991. Increasing the pro%ts of a subset of %rms in oligopoly models with strategic

substitutes. American Economic Review 81, 658–665.

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On the profitability of production perturbations in a dynamic natural resource oligopoly