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A Continuous Approach to Oligopolistic MarketEquilibriumSjur D. Flåm, Adi Ben-Israel,
To cite this article:Sjur D. Flåm, Adi Ben-Israel, (1990) A Continuous Approach to Oligopolistic Market Equilibrium. Operations Research38(6):1045-1051. http://dx.doi.org/10.1287/opre.38.6.1045
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A CONTINUOUS APPROACH TO OLIGOPOLISTICMARKET EOUILIBRIUM
SJUR D. FLAMUniversity of Bergen, Bergen, Norway
ADI BEN-ISRAELRutgers University, New Brunswick, New Jersey
(Received July 1987; revisions received January, July 1989; accepted September 1989)
We provide an algorithm for computing Coumot-Nash equilibria in a market that involves finitely many producers.The algorithm amounts to following a certain dynamical system all the way to its steady state, which happens to be anoncooperative equilibrium. The dynamics arise quite naturally as follows. Let each producer continuously adjust theplanned production, if desired, as a response to the current aggregate supply. In doing so, the producer is completelyguided by myopic profit considerations. We show, under broad hypothesis, that this adjustment process is globally,asymptotically convergent to a Nash equilibrium.
The methods for computing economic equilibriahave been developed substantially since the
breakthrough of Scarf and Hansen (1973). In partic-ular, the approaches involving homotopy techniques(see Zangwill and Garcia 1981), or variational ine-qualities are currently very much in vogue (Dafermos1980, 1982, 1983, Dafermos and Nagumey 1984,Harker 1984, Nagurney and Aronson 1988, Pang andChan 1982).
The latter methods are iterative and may oftennaturally be interpreted as describing how individualplayers sequentially adjust their actions. Specifically,the disequilibrium behavior of any participant is toupdate an interim plan at discrete instants of time.This is done by accounting as best as possible for thelatest actions carried out by all other players. Moreprecisely, to revise a player's plan at any stage amountsto solving a private, decentralized optimization prob-lem depending on what the rivals are actually doing.Thus, algorithms inspired by the rich theory of varia-tional inequalities (Auslender 1976, Kinderlehrer andStampacchia 1980) have strong intuitive appeal: theyreflect endogenous processes that operate in marketsand may bring about an equilibrium. Alternatively,these algorithms may be interpreted as portraying afictitious preplay.
This paper is motivated in part by two objectives.For one thing, it seems just as natural and attractiveto change plans continuously; for another, one shouldavoid overreactions. Therefore, we model continuous
and marginal adaptations, and are thus led to considera differential system whose stability properties meritspecial attention. In fact, one purpose of this paper isto analyze, in the context of oligopolistic competition,the asymptotic system behavior when everyonechanges their strategy in accordance with profitsignals.
Another major objective of our analysis is to allowfor individual profit functions that are nondifferenti-able in the classical sense. Our motivation for thegreater generality is twofold. First, it saves us fromimposing the differentiability assumptions that domi-nate the literature. Second, it opens up the treatmentof important problem instances when, for example,individual cost functions are not stated explicitly butrather are derived from minimization. In particular,we can allow for discontinuously increasing marginalcosts, which arise naturally when there are severalproduction lines with different efficiencies and capac-ities; see the example in Section 4.
We show, under broad conditions, that myopic andprofit-oriented adjustments, adopted continuously byall noncooperative players, will take the entire com-munity on a ride to a Cournot-Nash equilibrium.Thus, we actually provide a very natural algorithmthat is completely in line with fairly simplistic reason-ing on how profit incentives would affect individualdecisions. One advantage of locating a market balancepoint in this way is evident. We can contend with onlymarginal adaptations of each player as we go; there is
Subject classifications: Games/group theory, noncooperative; equilibria in Cournot oligopolies. Programming, nonlinear: algorithm for computingequilibria with nonsmooth data.
no need to solve individual problems optimally underway.
Similar results have been obtained by Rosen (1965)in a pathbreaking paper. His dynamic system includesa unique velocity derived from Lagrangians involvingsmooth functions. By contrast, we accommodate fornondifferentiable functions that yield a possibly non-unique, perhaps even discontinuous velocity. Also,our objective is marginal profit, not a Lagrangian.
Our modeling of stabilizing economic forces has fewfeatures in common with the tatonnement of Walras(see Hahn 1982). The latter process, often refiected inexistence proofs for general equilibrium, as in Debreu(1982), invokes a central player who quotes andadjusts prices. By contrast, this paper deals with apartial equilibrium model of imperfect competition,when no auctioneer will appear. However, one featureis common with tatonnement: the Walras mechanismrelies on the monotonicity of excess demand; that is,the gap between demand and supply tends to decreasewith higher prices. Similarly, in our context, the mon-otonicity of marginal revenues will play a crucial rolethroughout the analysis.
We also emphasize that the gobal asymptotic con-vergence of our procedure corresponds exactly to thestability (and uniqueness) of the equilibrium. Stabilityproperties of the Cournot oligopoly solution havearoused considerable interest. Compared to the earlierstudies of Hahn (1962) and Seade (1980), which werecast in terms of goal-seeking behavior, our analysisemploys economic forces that seem more natural andeasier to compute.
The paper is organized as follows. Section 1 recallsthe classical Cournot model of oligopolistic competi-tion and states a theorem on optimality conditions.The algorithm for finding a noncooperative stableoutcome is presented in Section 2. There we proveglobal asymptotic convergence when marginal profitsare strictly monotone. Several sufficient conditions forstrict monotonicity are given in Section 3. Section 4provides a numerical example involving nonsmoothcost functions. The final section remarks on possibleextensions.
1. THE MODEL
Following Murphy, Sherali and Soyster (1982) wedefine the model of market conOict as follows. Thereare finitely many firms labeled by / E /. Each decidesits output level ^, > 0, of a homogenous good, therebyincurring a production cost c,(9,). Prices piQ) are
determined in the marketplace by the total supply
General game theory (see Owen 1982) documents thatunder economically plausible assumptions there willexist a Coumot-Nash noncooperative equilibriumg* = iqT ),G/ characterized by individual stability. Thatis, for each / G /, g* maximizes the profit
of this producer with respect to ^, > 0 when qj = qffor all j ¥" i. Various features of the oligopoly modelmay be exploited to establish specialized existencetheorems (consult Novshek 1985, Roberts andSonnenschein 1976, Shubik 1984, or Szidarovskyand Yakowitz 1977). For completeness, and to adver-tise the role subsequently played by monotonicity, westate one result in this vein.
Suppose that piQ) and the marginal profitd-Kiiq)/dqi of any firm / are both nonincreasing inqi > 0. Also suppose that the average monopoly profitis asymptotically negative, that is
Then there exists at least one Coumot-Nashequilibrium.
In fact, the profit 7r,(^) need not even be partiallydifferentiable in a classical sense. It suffices to considersuperdifferentials as defined in convex or nonsmoothanalysis (consult Rockafellar 1970 or Clarke 1983).To see this, recall that the, supposedly maximal,monotonicity of any firm's marginal profit ensuresthat its objective function is concave vAxh respect toits own production level (Rockafellar, Theorem 24.3).Therefore, we need, by standard existence theorems(see e.g., Aubin 1984, Theorem 12.2) only verify thateach qi can be restricted to some fixed compact convexset. But (1) tells that it is better to produce nothingiqt = 0) than to choose any production plan which isoutside a sufficiently large set.
Granted the existence of equilibria, we briefly char-acterize such points in variational terms. For simplic-ity, the marginal profit, possibly nonunique, of player/ will henceforth be denoted by
(2)
Theorem 1. (Optimality Conditions). Suppose thateach profit function ir, is partially superdifferentiablewith respect to qt at a feasible point q*. Then for
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g* > 0 to be a Cournot-Nash equilibrium it is neces-sary that:
VI. For all i G /, there exists a partial gradientf*Gfiiq*)ofiri(g'*) with respect to qi such that
{Qi-QT) • f*^^ whenever9,>0.
Equivalently, withf* = (JT)i^i we have
(q-q*) -f^O provided that q^O.
Furthermore, we obtain
(3)
(4)
CP. qf •f* = 0 with qfequivalently
q* . f* = 0 with o* & 0
0 andff =s 0 for all i or
and /*«0. (5)
Conversely, when each x, is also pseudoconcave (seeBazaraa and Shetty 1979) with respect to qi at q*,then (4) or (5) are sufficient conditions for q* to be anequilibrium.
Remark. The proof of Theorem 1 will not be givenbecause VI is the standard variational inequality for-mulation of optimality conditions in convex program-ming (see Auslender). The equivalent statement (CP)in the form of a nonlinear complementarity problem(Karamardian 1971) results from the observation thatthe feasible domain is the nonnegative orthant. Weshall fmd (4) particularly useful in the subsequentanalysis.
Assumption 1. Hereafter we tacitly assume that atleast one Cournot-Nash equilibrium exists, and thatthe optimality conditions (4) or (5) are in force.
2. THE ALGORITHM
Suppose that at some instant t of continuous time, theplanned (intended) aggregate supply Q{t) is known.Producer / uses this information to explore whetherhe should change production in a direction thatbelongs to the set
(6)
If the partial derivative in (6) is unique, such a changewill bring about a maximal rate of increase in theprofit that accrues to player i. When (6) is multivalued,it is a closed interval, and a natural choice would beto select the direction that has a minimal norm. Thisis the direction of steepest ascent (see Shor 1985,Theorem 1.11). Thus, when 0 ef{q(,t)), a change inq, is not justified. Otherwise, the one-sided derivative
(left or right) that is closest to the origin, should beselected.
In all events, some care is needed to avoid introduc-ing negative quantities. Therefore, marginal profitsmust suitably be modified at boundary points of thefeasible region. Specifically, for an arbitrary initialpoint q{0) = (^,(0)),e/, we shall follow a dynamicsystem whose velocity is loosely defined by the differ-ential inclusion
E Fiq{t) for almost every (a.e.) t^O (7)
where F is defined by modifying marginal profits as
Fiiq):=f(q) when ^, > 0 (8)
otherwise. (9):= {<pf := max(O, V,)| ^, e
Note that the right-hand side of (7) may be discontin-uous or contain several elements (see Champsaur,Dreze and Henry 1977 or Henry 1972).
We now describe properties of a steady-state ^* ofsystem (7) that satisfies, by definition, the generalizedequation
0 G F{q*). (10)
Proposition 1. (Steady States are NoncooperativeEquilibria). Let individual quantities change accord-ing to the law (7-9). Then under the necessary andsufficient conditions of Theorem 1 any steady state isa Cournot-Nash equilibrium.
Proof, A steady-state q* that satisfies (10) must by(8-9) furnish generalized gradients fAq") E.(diTi/dqi)(q*) such that 'P^iq*) = 0 when q* > 0 or'PH<1*) = O otherwise. In any case, <Pi{q*) « 0 andqf'Piiq*) = 0. Thus, the nonlinear complementarityconditions (CP) of Theorem 1 are satisfied.
Next, we show that any solution to system (7)converges, irrespective of the starting point ^(0), to amarket equilibrium provided that marginal profitsf(q), as defined in (2), constitute a strictly monotonefunction
Here/is said to be monotone (strictly monotone) if
(q - q') • (<P - f) '^O (respectively, < 0) (11)
whenever q, q' •^O,qjtq' and -P Ef(q),<P' Ef(q').\fq'sfO'\% fixed in (11), then/is said to be monotone(strictly monotone) with respect Xo q'.
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Theorem 2. (Convergence to Coumot-Nash Equilib-ria). Suppose that marginal profits f{q) are strictlymonotone with respect to q* in the sense that {\ I) is astrict inequality whenever 0 « ^ # 9*. Then the equi-librium q* is unique, and every solution q{-) of{l),whatever the initial point q(0) > 0, will approach q*asymptotically, that is
lim q{t) = q*.
Proof. Let 9(-) be an arbitrary solution of system (7).Then by definition q{-) is absolutely continuous.Define a Lyapunov function F(see Arnold 1974) tomonitor the progress of ^ ( 0 in terms of the distanceto q* as
We assert that there exists 5 > 0 such that q{t)GKimplies
Along the trajectory ^(•) we have
V(t) = liQiit) - qn'P^t) (a.e. (12)
for some selection "^,(0 G Fi{q(t)), i e /. Then"^,(0 Gfi(Q(t)) a.e. except when some ^,(0 = 0 andall elements of fi{q{t)) are nonpositive. In this case,the corresponding term / in (12) equals zero, and ismajorized by
0.
Thus, (12) implies
a.e. (13)
for some selection <P,(0 Bfi(q(t)), i E. I. Now select,for each / G /, some / * G fAq*) that satisfies thecomplementarity condition (CP) in Theorem 1. Themonotonicity assumption (11) and the variationalinequality (VI) entail that the right-hand side of (13)is majorized by
Thus, (4) tells us that K(0 « 0 a.e., which by the factthat V{-) is absolutely continuous, implies that it isnonincreasing. Assume that
V{t) —> some V > 0 as ? —» 00.
Then the entire trajectory q{-) stays outside the openball Br(q*) with center q* and radius
r := (2uy^\
Also, the solution ^ (0 will eventually, i.e., for allt ^ some to, be trapped in the compact set
K := clBr.d(
F{q{t)), t > to} < -5.
If not, we may find sequences q' G K, V G F{q')such that
O^iq" - q*) • 'P" -^0 as 1/^00.
Here the inequality follows from monotonicity andthe fact that OG F{q*). Since F is an upper semicon-tinuous correspondence with compact values, F(K) isitself compact (Aubin and Cellina 1984, Proposition1.1.3). Therefore, without loss of generality, supposethat q' -* q G K and <P' ^<P e. F{q). However, bystrict monotonicity we arrive at the contradiction
0 = lim {q' - q*) = (q - q*) • (<P - 0) < 0.
Thus, V(t) « —5 for almost all sufficiently large t.This contradicts that lim,_oo V{t) > 0. Consequently,V{t)—*O as t ^ -1-00, and this immediately yields thedesired conclusion.
Remark. We have not discussed the existence of solu-tions to system (7). This issue is treated by Aubin andCellina. In particular, their Corollary 6.5.2 could bedirectly applied to give the preceding Theorem 2provided that (11) is strengthened to apply for allq* > 0. Kondor (1981) has studied the local stabilityof (7) when the velocity is unique and continuous.
3. ON STRICT MONOTONICITY
In the Introduction we mentioned the well known factthat for existence it is common that individual profitdecreases at the margin as production is expandedpartially. The algorithm proposed in Section 2, imi-tates a sort of preplay communication and computa-tion (a Vorspiel), and requires joint monotonicity ofthe entire marginal profit structure as defined in (11).This section identifies some conditions that lead tothe satisfaction of (11).
To this end, we shall separate cost, which is likelyto be marginally monotone, from the more trouble-some revenues. More precisely, assume henceforththat all individual cost functions c, are convex. Thenclearly, marginal revenues will come to occupy ourattention. Below we supply a list of conditions thatcan often be used as an a priori check for the mono-tonicity of marginal revenues.
Proposition 2. (On the monotonicity of marginalrevenues).
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Oligopolistic Market Equilibrium Approach / 1049
/. Suppose that the matrix 3 (,3
is well defined, negative definite and integrable alongstraight lines. Then marginal revenues are strictlymonotone.
ii. Suppose that the inverse demand curve is decreas-ing at the equilibrium q*, that is
( G - (14)
for all Q ^ 0. Then marginal revenues are strictlymonotone decreasing under each of the followingconditions:
a. S (9. - QT){P'{Q)qi - P'{Q*)Qr) ^ 0
for all q » 0 different from q* with at least one of theinequalities (14) or a being strict. In particular, Itsuffices that p is linear and strictly decreasing {negativedefinite).
b. The price p{Q) is strictly concave as a functionofQ.
c. The price p{Q) is strictly convex as a function ofQ,and
p"iQ)Q<o. (15)
Proof. For i see Rosen. For ii.a recall that p{Q) +P'(Q)Qi is the marginal revenue of producer /. Thusmonotonicity at Q* is due to
• (P(Q) + P'{Q)q, - P(Q*) - p'iQ*)q*)
= Q-Q*) • (PiQ) - P(Q*))
+ S (?- - QT) • (p'iQ)q> - p'iQ*)qr)i
where both terms are nonpositive by supposition.The proof of ii.b and c follows the lines of Harker
(1986), Corollaries 1 and 2. For ii.b see also Shubik,Theorem 5.2.
Remark. With linear demand and quadratic costfunctions, the differential system (7) and the general-ized equation (10) become linear. The tractable fea-tures of such problem instances (tantamount to linearcomplementarity) has also been stressed by Marcotte(1984).
s• \
. . .
ss
VI
•-1
,
V\
si
- -
1
- - -
1
_
...
- •
. . .
5.00 10.00Time
Figure 1. Symmetric oligopoly with discontinuousmarginal revenues.
4. AN EXAMPLE
For illustration, consider an oligopoly facing (inverse)convex, nonsmooth demand
p(Q) = max(O, 9 - Q)
which is choked off at the price 9, and exceeds thequantity 9 only if the good is offered for free. Notethat individual revenues are nonsmooth atQ = 9.
Figure 1 illustrates the symmetric case when eachout of three oligopolists has constant marginal costCiiQi) = 1- The initial point ^(0) is (4, 6, 8) and theequilibrium q* = (2, 2, 2). As long as Q(0 > 9,no firm obtains revenues. In fact, until t = 3, whenQ(3) = 9, all reduce their production simply to avoidcost. From time 3 onward revenues flow in, and akink is then observed in all production profiles.Since Q* < 9, one may claim that nonsmoothnesswas really not essential here.
Therefore, Figure 2 repeats the same story, butwith unequal and discontinuous marginal cost. Infact, c,'(9,) = 0.5(/ + 1) except that c',(qi) = 3 whenqi > 2.25. We start at ^(0) = (3, 6, 8) and the equilib-rium q* = (2.25, 1.92, 1.42) is reached somewhat
-̂
\-
\i
-s\- -
2 q^
-
-
i q
= •
i
- -
10.00Time
15.00 20.00
Figure 2. Asymmetric oligopoly with discontinuousmarginal costs and revenues.
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before t = 12 using stepsize At = 0.01. Kinks in ^,(0are observed fairly early as the production of firm 1 istemporarily reduced below the threshold 2.25.
For integration, we have invariably employedEuler's method, and relied on the user friendly simu-lation package STELLA developed by Richmond,Peterson and Vescuso (1987).
5. CONCLUDING REMARKS
The algorithm in Section 2 has been stated in contin-uous time. It has an entirely parallel discrete timeversion. When actually computing (approximate)solutions one must discretize time, and, possibly, con-front numerical problems associated with the methodof integration. Such problems are beyond the scope ofthis paper. However, one issue should be noted:namely, that we are not seeking the solution of aninitial value problem (7). Rather, the asymptotics of(7) is the only interesting issue. Thus, we can affordto integrate (7) inaccurately and restart from newlyobtained estimate equilibria. Clearly, as we approachthe true equilibrium it is appropriate to switch locallyto a Newton-type method. In fact, (7) is a so-calledgradient system which is likely to converge slowly inthe vicinity of a stationary point.
We emphasize that discrete time versions of thealgorithm in Section 2 difTer basically from thesequential (Gauss-Seidel or Jacobi type) methodsusing complete optimization at each intermediatestage, as done by Harker (1984). Instead, (7) contendswith making only gradient steps.
We have presented results in such a way that gen-eralizations to multicommodity markets should befairly straightforward. The spatial structure of marketsand the role played by transportation have not beenmentioned. Harker (1986) discusses such extensions.These, as well as the traffic equilibrium models ofDafermos (1980, 1982), are likely to be amenable toalgorithms of the preceding type. As in homotopytechniques (Zangwill and Garcia), we invariably endup with a differential system. However, this system islikely to be governed by nonsmooth, even nonuniqueforces.
ACKNOWLEDGMENT
This research has been sponsored in part by a grantfrom the Royal Norwegian Council of Industrial andScientific Research.
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