FS IV 95 - 5
C o u r n o t O l i g o p o l y a n d t h e T h e o r y o f S u p e r m o d u l a r G a m e s
Rabah Amir
March 1995
ISSN Nr. 0722 - 6748
F o r s c h u n g s s c h w e r p u n k t M a r k t p r o z e ß u n d U n t e r n e h m e n s e n t w i c k l u n g ( I I M V )
R e s e a r c h U n i t
M a r k e t P r o c e s s e s a n d
C o r p o r a t e D e v e l o p m e n t ( I I M )
Zitierweise/Citation:
Rabah Amir, Cournot Oligopoly and the Theory of Supermodular Games, Discussion Paper FS IV 95 - 5, Wissenschaftszentrum Berlin, 1995.
Wissenschaftszentrum Berlin für Sozialforschung gGmbH, Reichpietschufer 50, 10785 Berlin, Tel. (030) 2 54 91 - 0
♦
C o u r n o t O l i g o p o l y a n d t h e T h e o r y o f S u p e r m o d u l a r G a m e s *
We reconsider the Cournot oligopoly problem in light of the theory of supermodular games. Invoking the recent ordinal version of this theory proposed by Milgrom and Shannon, we generalize Novshek's existence result, derive the associated uniqueness result, give an extension of a classical existence result under symmetry, and provide conditions making a Cournot oligopoly into a log-supermodular game (with the natural order on the action sets). We also provide extensive and precise insight as to why decreasing best-responses are widely regarded as being "typical" for the Cournot model with production costs. Several illustrative examples are provided.
ZUSAMMENFASSUNG
D a s C o u r n o t O l i g o p o l u n d d i e T h e o r i e s u p e r m o d u l a r e r S p i e l e
In diesem Beitrag wird das Cournot Oligopol im Licht der Theorie supermodularer Spiele betrachtet. Ausgehend von der jüngsten ordinalen Fassung dieser Theorie, die von M ilgrom und Shannon vorgelegt wurde, wird Novsheks Existenzergebnis verallgemeinert und die Eindeutigkeit abgeleitet. Es wird eine Erweiterung des klassischen Existenzergebnisses unter Symmetrie vorgestellt und Bedingungen angegeben, unter denen ein Cournot Oligopol in ein log-supermodulares Spiel überführt werden kann (unter der Annahme der natürlichen Ordnung des "action sets" (evtl.
Handlungsmöglichkeiten)). Außerdem werden umfassende und präzise Einsichten erörtert, die verdeutlichen, warum fallende Reaktionsfunktionen weitgehend als typisch für das Coumot Modell mit Produktionskosten angesehen wird. Es werden mehrere illustrative Beispiele geliefert.
This work was sparked by conversations with Claude d'Aspremont while the author was a CORE fellow.
Special thanks go to him and to Isabel Grilo for continued interaction on this topic. The author is also grateful to Erwin Amann, Jean Gabszewicz and Jean-Francois Mertens for their helpful comments.
Research support from the DFG at the University of Dortmund is gratefully acknowledged, as is the wonderful hospitality at both CORE and Dortmund.
1 Preliminaries 1.1 Introduction
This paper brings together two independent lines o f research in industrial economics and game theory: Cournot equilibrium and the theory o f supermodular games. Cournot's (1838) work arguably represents the starting point o f both formalized economic theory and game theory. Yet to this day, this model remains one o f the most frequently used frameworks o f analysis in economic theory.
On the other hand, the theory o f supermodular optimization/games, introduced by Topkis (1978, 1979) and further developed by Vives (1990), Milgrom-Roberts (1990) and Milgrom- Shannon (1991), provides an elegant unifying approach, based on lattice-theoretic arguments, to characterize games with strategic complementarities.
Several links between these two strings o f literature have already been made, but typically not by the authors who wrote on Cournot equilibrium themselves. Thus, McManus (1962, 1964) proved in a self-contained manner that every symmetric Cournot oligopoly with a convex cost function possesses a symmetric Cournot equilibrium. Likewise, Roberts- Sonnenschein (1976) independently rediscovered this remarkable result (under the assumptions o f linear costs and bounded production capacities), still unaware o f any connection it might have with the work o f Tarski (1955) and o f Topkis (1968, 1978). It was years later that this connection was finally precisely established by Vives (1990), as follows. The best-response correspondence, mapping aggregate output o f all other firms into one firm's set o f best reply outputs, cannot have any slope (strictly) below -1: This corresponds to a straightforward application o f Topkis's (1978) monotonicity theorem. Furthermore, every set-valued function satisfying the above condition on slopes must intersect the 45° line: This is essentially a special case o f Tarski's (1955) intersection point theorem. Recently, Kukushkin (1992) extended McManus's existence result to accommodate different upper bounds on production across firms.
M ore recently, in another remarkable development, Novshek (1985) showed that every n- firm oligopoly (with a homogeneous product) has a Cournot equilibrium if each firm's marginal revenue is a decreasing function o f the aggregate output o f all other firms. His insightful proof is also self-contained and made no explicit use o f lattice-theoretic arguments. See also Bamon- Fraysse (1985) for related work. Again, the link to the lattice-theoretic framework was made
by Vives (1990) for the duopoly case as follows: (i) The above condition on each firm's revenue is equivalent to the submodularity o f its payoff function in own output and aggregate output o f other firms, and (ii) the composite best-response map is nondecreasing and hence has a fixed point by Tarski's fixed-point theorem (see next subsection), which is easily seen to be a Cournot equilibrium. Put differently, if the natural order on one o f the two firms' action sets is reversed, then the duopoly is a supermodular game, as observed in Milgrom-Roberts (1990).
Clearly both these arguments break down when the number o f firms is three or more. In this case an ingenious aggregation procedure, crucially exploiting the fact that each firm's payoff depends on other firms' outputs only via their sum, is needed to establish the existence o f a Cournot equilibrium. Independently rediscovered by Novshek (1985), this argument is originally due to Selten (1970).
The literature covered so far in our review invokes, explicitly or implicitly, Topkis's cardinal notion o f supermodularity. In a recent development, Milgrom-Shannon (1991) propose an ordinal version o f Topkis's complementarity conditions which is definitely a weaker requirement, preserving the desired monotonicity conclusion for parametric optimization problems. The long list o f well-known economic models presented by Milgrom-Shannon (1991) makes a convincing case o f the value-added o f this ordinal extension for a variety o f applications. As a result, the compass o f the theory o f supermodular games is nicely enlarged.
The purpose o f the present paper is to investigate the extent to which the use o f ordinal complementarity may enrich the existing results on Cournot equilibrium. We derive three general results: (i) every Cournot duopoly with a log-concave decreasing demand function, and essentially arbitrary (increasing) cost functions, is an ordinally supermodular game (when one firm's output space is given the reverse order), (ii) every (symmetric) oligopoly with linear production costs, bounded production capacities and log-convex net-of-cost demand function is a log-supermodular game (with the original order on output spaces) and (iii) every symmetric oligopoly with linear costs and arbitrary (not necessarily monotone) demand function has a symmetric Cournot equilibrium.
A description o f the above results is provided next in more detail. The most important consequences o f (i) is that every such duopoly has a Cournot-Nash equilibrium (in pure strategies). Furthermore, with the additional assumption o f convex cost functions, the Cournot equilibrium is shown to be unique, implying in particular that the Cournot duopoly is then a dominance-solvable game. As an existence result, (i) is a significant generalization o f
Novshek's (1985) theorem, as is argued later via examples. Nevertheless, the hypothesis o f (i) is, in fact, far from necessary, due to a fundamental bias which the presence o f production costs introduces in favor o f decreasing best-response correspondences. In the context o f linear costs, we derive considerably weaker conditions preserving the conclusion o f (i), and provide a sharp example suggesting that these new assumptions cannot be further weakened. Finally, a precise and detailed discussion is given, justifying the widespread perception among users o f the Cournot framework, that decreasing best responses form the "typical" or even "normal"
case, at least in the presence o f costs.
While the property o f supermodularity does not hold for the general «-firm oligopoly (n>2) under the hypothesis o f (i), existence o f the Cournot equilibrium still follows in that case from the aggregation procedure alluded to earlier. On the other hand, (ii) may be viewed as a partial dual result o f (i). In the absence o f costs, this duality is complete (i.e., as far as the revenue function is concerned). However, as mentioned earlier, the mere presence o f (nontrivial) production costs shifts the balance towards submodular profit functions or decreasing best- responses. Examples o f demand functions satisfying the assumption o f (ii) are given.
While (iii) is not o f particular interest in industrial organization, it may be promising as a new approach for proving existence o f Cournot-Walras equilibria in limited classes o f strategic exchange economies with symmetric "oligopolists": see Codognato-Gabszewicz (1990), Roberts-Sonnenschein (1977)...
The rest o f this paper is organized as follows. Subsection 1.2 provides a brief summary o f all relevant lattice-theoretic notions in the context o f real decision and parameter spaces, which is sufficient for our results and much simpler to exposit. Section II, III and IV essentially develop results (i), (ii) and (iii) above, respectively, along with the associated discussions and interpretations. Finally, Section V is devoted to all o f the proofs.
1.2 Mathematical Background and Terminology
In an attempt to make this paper self-contained, we provide a summary o f all lattice- theoretical notions and results invoked, here, in the simplest framework for our needs. Thus every theorem presented here is only a special case o f the original one.
A function/7 : -» /? is supermodular [submodular] if, for all Xj > x2,j^i > y 2,
F '(x I, y i) - F ( x 2, y l) > [< ]F {xj,y2) - F ( x 2, y 2). (1.1)
F has the single-crossing property or SCP [dual SCP] in (x,y) if
F '(xI, y 2) - F ( x 2, y 2) > [ < ] Q ^ F ( x 1, y l) - F { x 2, y l)>[<]0. (1.2) It is obvious that (1.1) implies (1.2), while the converse is generally not true. Note that (1.1) is a cardinal notion, while (1.2) is ordinal. Thus, the SCP is sometimes also referred to as ordinal supermodularity.
For x & R +, let A (x) = [a J(x),ar2(x ) ]c 7 ? +, with anda20 being real-valued functions. A(x) is ascending (in x) if a, and a2 are nondecreasing in x. The following results on monotone maximizers are central to our approach.
Theorem 1.1 (Topkis . (1978)) I f F is upper semi-continuous a n d supermodular [sub- modular], and ?l( ) is ascending [descending], then the maximum a n d minimum selections o f
y*(x) = arg m ax ^e^ x^ F ( x ,y ) are nondecreasing [nonincreasing] in x.
If F is strongly supermodular [submodular], defined by a strict inequality in (1.1), then the conclusion o f Theorem 1.1. holds for every selection o f j * Q .
Theorem 1.2 (Milgrom-Shannon (1991)) I f F is upper semi-continuous and has the SCP [DSCP], and is ascending, then the conclusion o f Theorem 1.1 holds.
If F has the strong SCP [strong DSCP], defined by (1.2) with a strict inequality on the RHS o f the => sign, then the conclusion o f Theorem 1.2 holds for every selection o f ,y*( ) .
The fixed-point theorems associated with this framework are due to Tarski (1955).
Theorem 1.3 L et C c : R + be a compact interval, and B .C —>C be a nondecreasing function. Then B has a fix e d point.
The following is really a specific case o f an intersection point theorem [Tarski(1995), p.250], o f two curves. Letting one o f the curves be the 45° line yields
Theorem 1.4 L et K a R+ be a compact interval, and B .K —>1K be such that
—— — > - l fo r all x, ^ x 2,y , e P (x ,) a n d y 2 e B ( x 2). Then B has a fix e d point.
x, - x2
Note that -1 could be replaced by -A, for any A > 0 , without invalidating the theorem. The crucial feature is that (some selection of) B should have no downward jumps: see also Vives (1990).
The following terminology will be used repeatedly in this paper. A function F:R+-+ R is log-concave [convex] if the function Log F is concave [convex]. Similarly, a function
G :R+x R +-> R is log-supermodular if Log G is supermodular.
A game with compact real action spaces is supermodular [ordinally supermodular] if each payoff function is supermodular [ordinally supermodular], under a specified order on each o f the action spaces, and u.s.c. in own actions. Supermodularity is typically interpreted as a complementarity property: The marginal returns to increasing a player's strategy are higher if the other player uses a higher strategy. Naturally, if the order on one player's action set is reversed, supermodularity then characterizes a substitutability property in two-player games.
2 Cournot Duopoly as a Supermodular Game
A Cournot duopoly game is fully described as follows: There are two firms facing an industry (inverse) demand function P ( ) , and producing a homogeneous good with cost functions C ,() and C2( ) . With x , y > 0 denoting the outputs o f Firms 1 and 2, and P ,C ,,C 2 mapping R + into itself, the profits o f Firm i (say 1) are given by
n ,( x ,j / ) = xP(x + ^ ) - C ,( x )
Our main result gives minimal independent conditions on the demand and cost functions that guarantee that the Cournot duopoly represents an ordinally supermodular game. A discussion o f the various assumptions and results is presented at the end o f this section.
Theorem 2.1 The Cournot duopoly is an ordinally supermodular game i f the follow ing assumptions hold:
(A 1) P ty is strictly decreasing, and log-concave.
(A 2) C, ( ) is strictly increasing and left-continuous, i = 1,2.
(A 3) There exists Q > 0 such that QP(Q) - C ^Q ) < 0 fo r all Q > Q ,i = 1,2.
The proofs are all given in the last section. We now state a number o f important related results.
The following immediate consequence o f Theorem 2.1 is the most fundamental one.
Corollary 2.2 Under the hypothesis o f Theorem 2.1, the Cournot duopoly game has a Nash equilibrium (in pure strategies).
Uniqueness o f Cournot-Nash equilibrium further requires decreasing returns in the production process.
Theorem 2.3 In addition to the hypothesis o f Theorem 2.1, assume that Ct is convex. Then the Cournot duopoly has a unique Nash equilibrium in pure strategies.
Then, the following important observation follows directly from the theory o f supermodular games, cf. Milgrom-Roberts (1990).
Corollary 2.4 Under the hypothesis o f Theorem 2.3, the unique Cournot equilibrium is also the unique mixed-strategy Nash equilibrium and the unique correlated equilibrium.
Moreover, the game is dominance solvable.
Theorem 2.1 is essentially a generalization o f Novshek's result. Nevertheless, monotonicity o f C ,Q is crucial for the validity o f Theorem 2.1 (as is easily observed from the proof), but not for Novshek's result. Naturally, this monotonicity assumption is not at all an imposition since it is dictated by economic considerations. N ote that start-up costs are still allowed, at zero and elsewhere. Also, observe that our Assumption (A 3 ) is clearly implied by Novshek's assumption that the inverse demand function crosses the horizontal axis.
Novshek's fundamental assumption is that each payoff is (differentiably) submodular in own output and others' aggregate output: The duopoly is then a cardinally supermodular game. By contrast, our assumptions make the duopoly an ordinally supermodular game (in the sense o f Milgrom-Shannon (1991), which may well fail the cardinal test, as argued below. Novshek's condition was stated in the following equivalent form
P '(Z ) + Z P "(Z ) < 0, VZ > 0. (2.1)
It is easily seen to be stronger than log-concavity o f P ( ) . Two examples o f valid and plausible demand functions, satisfying the conditions o f our result here, but not (2.1), are (with a being some positive constant):
E x a m p l e 2 . 5
P(Z) = l(Z - a ) 2 i f
‘.O i f
0 < Z < a Z > a
E x a m p l e 2 . 6
P(Z ) = e “flZ, Z > 0 .
The verifications being left to the reader, observe that both examples portray convex demand functions, the latter being a limiting case satisfying our conditions since Log P(-) is linear (more on Example 2.6 later).
Now we examine the extent to which the assumptions o f Theorem 2.1 could be weakened while preserving its conclusion. As stated, Assumptions (A.1) - (A.3) are relatively easy to verify for a given specification o f the functions P ( ),C i() and C2( ) . While this is certainly o f value in applications, it turns out that under some circumstances, an appreciable relaxation o f these assumptions is possible. In particular, under linear’ costs, the following generalization is noteworthy.
Theorem 2.7 Consider a Cournot duopoly -with a strictly decreasing inverse dem and function P(-) and constant average costs cl and c2 . L et q f b e the smallest optimal monopoly output fo r Firm i, and qt = P ~ ^ ( c \ i = 1,2. I f / ’ Q = P ( - ) - ct is log-concave on
= 1,2, then the Cournot duopoly is an ordinally supermodular game with a unique Nash equilibrium.
As stated, this result is very tight in the sense o f requiring minimal hypothesis, but has the drawback o f requiring the a priori knowledge o f q™,i = 1,2. Nevertheless, the improvement over Theorem 2.1 is notable even if we require log-concavity o f P over all o f [0,#f], z'=l, 2.
Furthermore, the conclusions o f Corollaries 2.2 - 2.4 are still valid under the hypothesis o f Theorem 2.7: This is easily seen from the proof o f the latter. Following is an example which sharply illustrates the scope o f Theorem 2.7 as well as its tightness.
Exam ple 2.8 Let
P(Z) = ^ . Z > O , a n d C lW = ^ .
Then the monopoly optimal output q m solves — 4} and is given by = 1. Also q solves ^ 4 = 4 , or q = 3. N ow A x) = ^ i _ 4 = 4 ^ 5 ) and as is easily
checked.
Hence, even though P is log-convex (everywhere) here, it turns out that the (net-of-cost) demand function P is log-concave on the interval <7", 7] (for this example) since
PP" —P '1 <0
on [1,3]. Note though that P is log-convex on [0,1]!The reaction function o f Firm 1 (with cost function 4 ) is
w { 0
which is easily seen to be decreasing in conformity with Theorem 2.7.
Thus, this example nicely captures the possibility that the presence o f (linear) costs may make a Cournot duopoly into a game with strategic substitutes when the costless version o f the same game exhibits stategic complementarities (in the original order on output spaces). An elaborate discussion related to this point is presented at the end o f Section 3. Finally, observe that the bounds q m and q contained in the statement o f Theorem 2.7 are essentially the tightest possible, as shown in this example.
3 The n-Firm Cournot Oligopoly
When the number o f firms n > 2 , the Cournot oligopoly is not a supermodular game under the hypothesis o f Theorem 2.1. Neither Vives's (1990) composite best-response approach, nor Milgrom-Roberts's (1990) action reordering trick will reveal strategic complementarities. On the other hand, the Cournot oligopoly is certainly an ordinally submodular game under the conditions o f Theorem 2.1.
Fortunately, a construction due to Selten (1970) and Novshek (1984, 1985), relying on the fact that the profits o f each firm only depend on its own output and on the aggregate output o f all other firms, would establish existence o f a Cournot equilibrium under the hypothesis o f Theorem 1.2. As discussed in Section 2, Novshek's existence result actually assumes stronger conditions than our Theorem 2.1, making the oligopoly into a cardinally submodular game.
This construction is not reviewed here. Nevertheless we give a statement o f the result under consideration while omitting its proof.
Theorem 3.1 Consider an n-firm Cournot oligopoly with the inverse dem and function and the cost function satisfying Assumptions (A I) - (A3) o f Theorem 2.1. Then each firm 's best response correspondence, mapping the aggregate output o f all the other (n-1) fir m s into the firm 's set o f best-responses, has only nondecreasing selections. Furthermore, a Cournot-Nash
equilibrium exists.
Finally, Theorem 3.2, below, gives conditions under which the Cournot oligopoly is a supermodular game (with the natural order on firms' outputs). Except for requiring either bounded production capacities or no variable costs, these conditions are in some respect dual to those o f Theorem 2.1 (for the revenue function). More on this point later.
We say that Firm z has (exogenously) fixed production capacity K t if its output cannot exceed K t(j = l,...,zz) independent o f the output levels o f the other firms. N ote that such capacities can be incorporated in the basic Cournot framework simply by assuming C, (x) to be large enough for all x > K t (e.g., infinite).
Theorem 3.2 Consider an n-firm Cournot oligopoly with fix e d production capacities K x, ...,K n, and linear cost functions C,.(x) = cIx, x e[0 ,X ,], i = l,2 ,...,n . Assume that P Q - cf is log-convex on [ 0 , ^ , + . . . + ^ ] . Then the oligopoly is a log-supermodular game in the natural order on output sets. Furthermore, i f the K t 's are all equal, the equilibrium set is a chain along the diagonal in [0, Ä-]” .
We now provide an overall discussion o f this result. The condition on demand is equivalent to log-convexity o f P Q - c , where c is the largest unit cost across firms, as is easily verified.
In particular, this assumption is stronger than log-convexity o f P ( ) : See the detailed discussion below.The conclusion o f Theorem 3.2 would generally be considered atypical, if not perverse, by many users o f the Cournot oligopoly framework. At first glance, it might even appear that log-convexity o f P ( ) , which is clearly stronger than convexity o f P ( ) , would imply quasi-convexity o f profits in own output, thus leading to extreme solutions only.
However, this is not true. Log-convexity o f P ( ) means P "P - P '2 > 0 whereas quasi- concavity o f I Ii can be shown to require P "P - 2 P ' < 0 : Clearly, the two inequalities are compatible. Here is a class o f examples:
Example 3.3
P(Z) = 1
(Z + l)“
Z > 0, a > 0
P ( ) is clearly log-convex, for all a > 0. The reaction curves and the unique (symmetric) Cournot equilibrium are (a verification o f the condition P " P - 2 P '2 < 0 for quasi-concavity is left to the reader here):
r(x ) = .-£±j. if a > 1 a-1
+oo if a < 1 and
— 1— i f a > 2
a - 2
+ o o i f a < 2
*
X
It is also left to the reader to verify that the introduction o f variable costs (even if linear) would destroy the (log-) supermodularity o f the game, as in Example 2.8.
We now take a closer look at the partial duality between Theorem 2.1 and 3.2. As far as the revenue term is concerned, there is obviously a complete duality between the two results. In this regard, it is o f interest to note that in Example 2.6, the inverse demand function P (Z ) = e~aZ satisfies the assumptions o f both theorems, and consequently, in the absence o f production costs, the best responses should be both nondecreasing and nonincreasing, hence constant! This is indeed the case, as the reader may easily verify. Thus each player has a dominant strategy here (equal to £ ).
However, the incorporation o f nondecreasing cost functions automatically destroys this kind o f duality, in favor o f nonincreasing best-responses. This is very well illustrated by Example 2.8, and can also be seen in the proof o f Theorem 2.1. In the context o f linear costs, we can further elaborate on this. Assuming for simplicity that c, = c2 = c , one can consider the effective strategy space to be A = {(x,y):x, y > 0 and x + y < P " '(c )J and reformulate the demand function as P ( ) = P ( ) - c on A , as is done in the proof o f Theorem 2.7. However, no matter how log-convex P ( ) is here, P cannot be log-convex on A since P p ,“1(c)] = 0 implies that (sa y )ylo g P (0 ) + J$ lo g
/ j p _;(c) = -00 <
l o g with 0 and P _;(c), here, being the test points in checking log-convexity. Therefore, Theorem 3.2 would not apply to P , as defined above. Furthermore, this discussion also shows that the crucial requirement, th a tP be > 0 in Theorem 3.2, is (implicitly) contained in the assumption that P () is log- convex.In addition, also observe that the sections A^ (say) o f A defined by A^ = {x > 0 :(x ,y ) c A | = > 0:x < P ] ( c ) - y j are not ascending in y , but in (-j). Taking as objective the log o f a firm's profits, within the framework o f Topkis's Theorem, the two difficulties just described completely capture - at least in the linear cost case - the basic asymmetry which the presence o f costs creates in favor o f nonincreasing best-responses.
Another interesting asymmetric feature lies in the fact that the profit function o f a Cournot firm cannot be cardinally supermodular in the outputs in the original order (even in the duopoly case). To see this, simply set the cross-partial o f profits > 0 and own output equal to 0.
In view o f the special nature o f the situation described by Theorem 3.2, an intuitive interpretation o f this result is certainly warranted. Log-convexity being a strong form o f convexity, the inverse demand function here decreases at a rapid rate as total output is increased. This leads to the surprising outcome that a given increase in own output generates more extra revenue for a firm (algebraically) when the rival firms are producing a higher joint output. Furthermore, it is not hard to see that the presence o f production costs would generally destroy this complementarity property. On the other hand, if P is actually increasing on part o f its domain, the latter property would be enhanced: whence the irrelevance o f a monotonicity assumption on P in Theorem 3.2.
Two final remarks on the limitations imposed in Theorem 3.2 on the production-cost side are in order. First, if the restrictions on capacities are lifted, i.e., if K { = +oo, z = 1 ,...,« , then the Theorem only holds if P (co) > c , which is realistic only when c = 0, as is clear from the above discussion. Second, nonlinear costs can be accommodated. The appropriate assumption on demand is then less crisp: P (x + T )- A f x ) must be log-supermodular in (x, T ), where At
C (x )
is Firm i ’s average cost curve, i.e., A f x ) = ■ ' ■ . Implicit here then is the fact, as discussed x
earlier, that P (x + T) - A,(x) > 0, zi = 1,..., n, for all x e [0, K t 1 and all Y e [0, Z K j ].
4 Symmetric Cournot Duopoly and Lattice-Theoretic Analysis
In this section, we concentrate on the relationship between zz-firm symmetric Cournot duopoly (characterized by the fact that all firms have the same cost function C () ) and the theory o f supermodular games/optimization. The first result, proved by McManus (1962, 1964) and independently rediscovered by Roberts-Sonnenschein (1976), provides surprisingly weak conditions under which a symmetric Nash equilibrium exists. While the use o f lattice-theoretic arguments makes the proof essentially trivial, it is to the above authors' credit that they w rote completely self-contained proofs, not being aware o f the work o f Tarski (1955) and Topkis (1968). This link was first made in the perceptive paper o f Vives (1990): we are only presenting an alternative proof here, together with a new extension o f this classical result.
Theorem 4.1 Every n-firm symmetric Cournot oligopoly with a nonincreasing left- continuous dem and function P(-) satisfying lim xP(x) = a (for some constant a) a n d a
nondecreasing convex cost function C ()h a s a symmetric Nash equilibrium (in pure strategies).
Since the proof o f Theorem 4.1 (see next section) makes use o f cardinal supermodularity (Topkis (1978)), one might ask whether anything more could be said by invoking the analogous ordinal notion (Milgrom-Shannon (1991)). The answer is that, with a linear cost function, the demand need not be monotone decreasing for the conclusion o f Theorem 4.1 to remain valid. This simple result could have important applications in Cournot-Walras models (see e.g., Codognato-Gabszewicz (1991)) where monotonicity o f the induced "demand functions" generally does not hold, due to general equilibrium effects.
Theorem 4.2 Every n-firm symmetric Cournot oligopoly with an upper semi-continuous demand function PC) satisfying lim xP(x) = a (for some constant a) a n d linear cost function
V 7 X->00 X 7
C(-) has a symmetric Cournot equilibrium.
A brief discussion o f the two results o f this section now follows. As described earlier, Theorem 4.1 goes back three decades. We are presenting it here for the sake o f completeness and as a vehicle for arguing that Tarski's intersection point theorem is generally not crucial in economic applications. As shown in our proof o f Theorem 4.1, Tarski's better-known fixed- point theorem, applied to a properly modified best-response map, is also adequate. A side advantage o f the latter result, not pursued in this paper, is that this would allow for a more direct application o f several recent results such as the comparative statics theorem for supermodular games, independently proved by Sobel (1988) and Milgrom-Roberts (1990).
As for Theorem 4,2, note that log-supermodularity may alternatively be used for its proof.
Potential applications may be in special classes o f Cournot-Walras models, due to the hypothesis o f linear costs. It does not seem possible to extend this result to more general, say convex, cost functions.
5 Proofs
All proofs for this paper are given below. In addition to the fundamental results o f Section 1, our arguments will frequently make use o f the following easily proved result (for a proof o f Part (ii), see e.g., Amir (1991)).
Lem m a 5.1 L et f , g :R + -» R , f be a [strictly] concave function, a n d g a [strictly] convex function. Then the real-valuedfunction
i) (x, y ) - > f ( x + y ) is [strictly] submodular on R + x R + ; ii) ( x ,y ) - > f ( x - y ) is [strictly] supermodular on the lattice
(p = { (x ,y ) :y > 0 and x > y } ;
iii) (x, y ) -» g (x + y ) is [strictly] supermodular on R + x R +.
We now proceed to the proofs o f our main results.
P ro o f of Theorem 2.1 We first show that the profit function Eli (say) satisfies the dual strong single-crossing property on R + x R +. To this end, first observe that since log P(-) is concave (by Assumption A 1), log P (x + y ) is submodular in (x ,y ) e P + x R +, by Lemma 5.1 (a), i.e., for any x1 > x 2, y { > y 2, we have
log P(x, + <y1) —lo g P (x 2 + y ,) < lo g P (x , + y 2) - l o g P ( x 2 + y 2)
This reduces to
p (xi + P i) P (*i + ^ ) + y , ) ~ P (x 2 + y 2)
Since we want to prove the bracketed version o f (1.2), assume that x,P(x, + y 2) — C ,(x,) < x 2P (x 2 + y f ) — C,(x2) .
Substituting (5.1) into the RHS o f (5.2) yields -P(*i+-y2)
P ^ + j / i ) P (x 2 + C,(^2)
or, multiplying across by f (*.+?■) P (x1+?2) ’
(5.1)
(5.2)
+y,)~
P(x, + y 2)C ,(xx) < x 2P (x2 + y i ) - ^ ( * i + / i )
P ( x , + j 2)^ ( * 2 ) (5-3) Since P(x, + J , ) < P(x, + .y2) and C (x ,)> C (x 2) from the hypothesis, it follows from (5.3) that
x,P(x, + y ,) - C, (x,) < x2P (x 2 + J , ) - C, (x2) (5.4) Since (5.2) implies (5.4), every selection out o f Player l's best-response correspondence x*(j>) = argm axx>0|x P (x + y ) - C 1(x)j is nonincreasing by Theorem 1.2 (note that the use o f max is justified here since I I 1 is u.s.c. in x and the effective strategy (i.e., output) set o f a firm is obviously contained in [o, ß ] , since Q is larger than the (largest) optimal monopoly output by Assumption (A 3), which itself is larger than any point on x*(y), by the above monotonicity conclusion).
By taking the reverse order in one firm's strategy, the Cournot duopoly is then an ordinally supermodular game (see Vives (1990) for an alternative version o f this argument). □ P ro o f o f C orollary 2.2: This follows immediately from Theorem 2.1, via the results on
ordinally supermodular games. See Milgrom-Shannon (1991). □
Proof of T heorem 2.3: We first show that all the slopes o f every selection o f x*() are larger than or equal to -1. Here, instead o f considering Firm l's best-response problem as max^>0{xP(x + y ) ~ Q ( x ) J , it is more convenient to think o f Firm 1 as choosing cumulative output z = x + y , given output y by Firm 2; then we have
m a x { ( z - y ) P ( z ) - C l ( z - y ) } (5.5)
From Lemma 5.1, it is easily seen that the maximand in (5.5) is strictly supermodular in (z,y) on the lattice and (p = \{z, y ) : y > 0 and z > y } since the term - y P ( z ) is strictly supermodular and the terms zP (z) and -C \(z - y ) are supermodular on (p. Moreover, the feasible set [y,+°°) is ascending in y. Hence, by Topkis's Theorem, every selection out o f the
arg m ax z* (y )in (5.5) is nondecreasing. This is equivalent to saying that all slopes o f every selection out o f x ’(y ) = z * ( y ) - y are larger th a n -1.
Furthermore, we know from Theorem 2.1 that every selection o f x*(y) is nonincreasing or, in other words, has no slopes above 0. Hence, every selection ofx*(y) has no slope outside the interval [-1,0], which implies that x*(y) is a (Liptschitz) continuous function; in fact, a nonexpansive function. Uniqueness o f Nash equilibrium then follows from a fairly standard argument: see Lemma 2.3 in Amir (1991) for a detailed proof (o f exactly the same point, but in a different context),or Friedman (1983). □
P ro o f of C orollary 2.4 This follows directly from the main result o f Milgrom-Roberts (1990), as extended to ordinally supermodular games by Milgrom-Shannon (1991). □ P ro o f of Theorem 2.7 The effective strategy space for Firm 1, say, given that Firm 2's output is y is clearly the set |x :0 < x < y } : it will turn out that viewing the Cournot duopoly as a generalized game1 is convenient here.
Since costs are linear (hence convex), we know from the proof o f Theorem 2.3, that every selection o f the cumulative best-response correspondence x*(y) + y is nondecreasing in y, so that in particular x*(y) + y>q™ = min {x*(o)j, for all y > 0 . Since x*(y) P _1( c ,) - y , it follows x*(_y) e [ < - y ,P ~ x(c,) - .y] = for ally. L etZ = {(x,y):0 < y < P ~ }(c,) ,x e Z y } . We can thus consider Firm l's best-response problem as m ax^logx[p(x + y ) - c j : x e Z ^ J . Considering the reverse order for y on [o .-P -fe )] , it follows that Z is a sublattice o f
R + x R +, and that the maximand above is supermodular in (x,-y) since l o g [ P ( ) - c ,J is concave on by hypothesis. Since Z^ is ascending in (-y) or descending in y , we conclude from Topkis's Theorem, that the max and min selections o f x ‘(y) are nonincreasing (with the natural order on y). Hence the duopoly game is supermodular, as in Theorem 2.1.
1 A generalized game is one in which the joint action space is not the Cartesian product of the individual player's action spaces. In other words, for such games, a player's action set depends on the action vector of the other players.
Together with the earlier monotonicity property o f the cumulative best-response, we conclude, as in the proof o f Theorem 2.3 that the slopes o f x ‘( ) are all in the interval [-1,0].
A similar argument applies to Firm 2. Note here that we are implicitly using the definition o f supermodular generalized games, as in Topkis (1979). Uniqueness o f Cournot equilibrium
follows now as in Theorem 2.3. □
P ro o f o f Theorem 3.2 Let f be the aggregate output o f Firms 2, 3, . . ., n, and consider F irm l. Since l o g n ,( x ,y ) = logx[P (x + y ) - c 1] = logx + log[P(x + y ) - c 1], and since log[P(.) - c j is convex on [0,X’1+...+X'n] (by assumption), we. conclude from Lemma 5.1 that l o g l l ! is supermodular in (x,T). Furthermore, the output space here is [0 ,2 fJ for each player, and this is obviously a complete lattice. Hence, the Cournot oligopoly here is a log- supermodular game, and a Cournot equilibrium exists, possibly with each firm producing .
Next, we show that, whenever the 's are equal, asymmetric equilibria are not possible.
As this is a general fact about symmetric supermodular games, we give a general proof, by contradiction. Let (x ,,x 2,...,x n) be an asymmetric equilibrium (thus with at least two x / s being distinct). Assume then, w.l.o.g. that x, = max.{x(} and x2 = m i n j x j , so that x, > x 2.
Since the game is symmetric, every permutation o f (x ,,x 2,...,x n) is also an equilibrium.
Consider, in particular, (x ,,x 2,x 3...,x n) and (x2,x ,,x 3...,x n). These being equilibria says that Firm 1 strictly decreases its output from x, to x2 as the other firms' output vector increases from (x2,x 3...,x n) to (x ,,x 3,...,x n) , which contradicts the fact that Firm l's best-response is nondecreasing (since the game is log-supermodular). This completes the proof o f Theorem 4.3.
□
P ro o f of Theorem 4.1 For extra clarity, aggregate outputs are always denoted by capitalized letters while single firm outputs are not.
Here, the strategy space o f each firm is easily seen to be compact, say [o, ö ] , in view o f the assumptions on P ( ) and C ( ) . Each payoff is easily seen to be u.s.c. in own output, C ( ) being necessarily continuous. Consider (say) Firm 1 with output x, with Y representing aggregate
output o f all other firms. Motivated by symmetry, define a cumulative best-response map B from [o,(« - l ) ß ] to the set o f all its subsets by
(x '+ y ). (5.6)
n
where x ' is a best-response by Firm 1 to total output Y, i.e.,
x 'e a r g m a x |x /’(x + y ) - C ( x ) J (5.7)
It is easily verified that B indeed has the power set o f p ),(« -l)(? j|a s its range. From an argument similar to that following (5.5), we conclude that B and B , defined respectively as the max and min selections out o f B, are nondecreasing.2 Hence, by Tarski's fixed-point- theorem, B (say) has a fixed-point Yo. If x'q is the (largest) best-response in the sense o f (5.7) to Yo, then
Yo = !-(x’0+ r 0) or x'o =
n n - \
which says that each o f n firms producing x'o is a symmetric Cournot equilibrium. This
completes the proof o f Theorem 4.1. □
P ro o f of Theorem 4.2 Here C(x) = ex (by assumption). Once more, the use o f max is justified in the best-response problem for similar reasons as in the previous proof. The proof
now proceeds along the lines o f the previous one, after we show that the maximand in
<5-8) has the SSC property on <p = |(K ,Z ).Y > Q,Z > r j . To this end, let Z, > Z 2,Y1 > Y2 and assume that
( z , - y 4 / > ( z , ) - c] > ( z 2 - r 2)[p (z 2) - c ] . (5.9)
This conclusion does not necessarily hold for every selection of B since the hypothesis of Theorem 4.1 only yields (nonstrict) supermodularity of (5.5).
2
Since log(Z - Y) is strictly supermodular on (p by Lemma 5.1, it follows that
(z 2 - i ’2)(z 1 - r 1) > ( z 2 - r 1)(z 1- r 2) ( 5 i o )
Substituting (5.10) into the RHS o f (5.9) yields
(Z, - ^ ( Z l ) - c ] > (?2 C1 > or
(z, - ri)[/>(Zi) - d > (Z2 - r,)[p (z2) - c] (5.1 i)
Since (5.9) implies (5.11), the maximand in (5.8) has the strong SCP, and the proof may be
completed as in Theorem 4.1. □
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