Collusion, Competition, and Conjectures

Collusion, Competition, and ConjecturesAuthor(s): John McMillanSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 17, No. 4(Nov., 1984), pp. 788-805Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135074 .

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Collusion, competition, and conjectures J 0 H N M c M I L L A N / University of Western Ontario

Abstract. This paper investigates whether it is possible to narrow down the set of dynamic oligopoly equilibria by adding to the model an adjustment process in which firms' conjectures about rivals' reactions are based on observations of actual actions in the past. If a non-trivial adjustment takes place, the only equilibria of this system are the extreme outcomes of implicit collusion and competition; in particular, the Coumot point is not a stationary state. If the identical firms always act identically, implicit collusion evolves. Although convergence occurs from some particular initial conditions, in general the adjustment process modelled here is unstable.

Collusion, concurrence et conjectures. Ce m6moire cherche a decouvrir s'il est possible de r6duire l'ensemble des 6quilibres d'un oligopole dynamique en ajoutant au modele traditionnel un m6canisme d'ajustement par lequel les firmes construisent des conjectures quant aux r6actions de leurs rivales a partir d'observations de leurs d6cisions pass6es. Si un ajustement significatif a lieu, les seuls 6quilibres du systeme sont les points extremes engendr6s par la collusion implicite et la concurrence; en particulier, le point de Coumot n'est pas un 6tat stationnaire. Si les memes firmes agissent toujours de la meme facon, la collusion implicite s'impose. Meme s'il y a convergence a partir des conditions particuli6res de d6part, en g6n6ral, le processus d'ajustement propos6 par le modele est instable.

INTRODUCTION

This paper examines the formation of expectations about rivals' actions in a dynamic model of oligopoly. Consider an oligopoly game in which the Cournot-Nash equilibrium is unique. If the game is played only once, the firms need not be ignorant of their rivals' planned outputs. Each decision-maker can imagine himself in the position of any other decision-maker and can deduce what the other will do. The mutual consistency nature of the Cournot-Nash equilibrium (each firm is doing the best it can, given that the other firms are producing their Coumot-Nash outputs)

I am grateful to John Chilton, Ignatius Horstmann, Peter Howitt, Glenn MacDonald, Peter Morgan, Michael Parkin, Alan Slivinski, and Victoria Zinde-Walsh for comments and advice. The Social Sciences and Humanities Research Council of Canada provided research support. An earlier version was presented at the Canadian Economic Theory Conference, Edmonton, May 1982 and at the International Institute of Management, Berlin, August 1982.

Canadian Journal of Economics / Revue canadienne d'Economique, XVII, No. 4 November / novembre 1984. Printed in Canada / Imprime au Canada

0008-4085 / 84 / 788-805 $1.50 ? Canadian Economics Association



Collusion, competition, and conjectures / 789

means that each firm rationally plays its Cournot-Nash strategy. The expectation that others will play Cournot-Nash is a rational expectation (Johansen, 1982).

Dynamic models of oligopoly, in contrast, have a surfeit of equilibria. The repetition of plays allows the possibility of enforcing tacit collusion as an equilibrium; but there are many other equilibria as well (Friedman, 1977). The multiplicity of equilibria raises a question about the very meaning of equilibrium. At an equilibrium, each firm maximizes its profit given the others' actions. But if there are multiple equilibria, how can a firm know what its rivals are going to do? This paper suggests that one way of resolving this problem is to model an adjustment process in which firms learn by observation about the decision rules their rivals are using.

LEARNING ABOUT STRATEGIES

Consider a discretetime, infinite-horizon model of oligopoly. If firms are able to respond to the past actions of their rivals, there are many e quilibria. The firms' strategies may make current output depend upon others' past outputs according to a function that is either continuous (as in the reaction-function models of Friedman, 1977, ch. 5) or discontinuous (as in the trigger-strategy models of Friedman, 1977, ch. 8). Although there are infinitely many equilibria, the standard formulation presumes each firm knows, with certainty and correctly, the strategy each of its rivals is following. How can the firm know this? This paper offers the interpretation that the firm arrives at this knowledge as a result of observations of the rivals' actual outputs in the past.

If firms are to be modelled as behaving completely rationally during the learning process, two features would have to be modelled. First, in a fully dynamic model, a firm should consider the effects of its current actions on its future profits. Away from a stationary state in such a dynamic model there would be a trade-off between current profits and information; the firm might vary its current output levels, sacrificing some current profits, in order to obtain more accurate information about its rivals' reactions and therefore higher profits in the future. Second, the firm should not behave as though it were learning about nature; it should instead take account of the fact that it is learning about the behaviour of a game-playing rival. To make an optimal decision each firm would have to predict not only its rivals' outputs, but also its rivals' predictions of its output, its rivals' predictions of its prediction of their outputs, and so on in infinite regress.

This paper is an exploratory attempt to incorporate learning about reactions in a dynamic oligopoly model. The paper side-steps the difficulties of modelling rational learning behaviour by making two strong assumptions.

First, although the model is explicitly dynamic, firms are assumed myopically to maximize current profits only. (It seems to be a common feature of analyses of adjustment processes that agents are assumed to exhibit some such myopia. Compare, for example, with the models of convergence to rational-expectations equilibrium - Blume, Bray, and Easley, 1982.)

Second, rather than learning about functions (as would be necessary if the model



790 / John McMillan

were as in Friedman, 1977, chs 5, 8), each firm is modelled as learning about a single parameter only. On the basis of past observations of its rivals' behaviour, each firm estimates an apparent correlation between changes in its own actions and changes in its rivals' actions and uses this apparent correlation in making its current output decision. These apparent correlations are the firms' conjectural variations. Unlike the standard conjectural-variations model' in which conjectures are given exogenously, in the model of this paper the conjectures are endogenous.2

The standard conjectural-variations model has firms believing something that is literally false. Each firm makes its profit-maximizing output decision under the conjecture that there is a simple functional relationship between its rivals' outputs and the output it chooses to produce; whereas in fact the rivals themselves are solving maximization problems. This paper proposes that the way to make sense of this conjectural-variations functional relationship is to interpret it not as a perceived causal relationship, but instead as an apparent correlation based on past experience. The model to be presented has each firm, on the basis of its observations of its rivals' outputs, modifying over time its conjectures about the relationship between its own output changes and its rivals' output changes. The nature of the equilibrium depends upon what the firms observe during the dynamic adjustment process.

It is a common criticism of the standard conjectural-variations model that it confuses statics and dynamics. The standard model is literally static; actions happen only once. Yet the conjectures describe firms' reactions to changes in the other firms' actions; so the model is implicitly dynamic. The present model is not subject to this criticism, because, although the firms behave myopically, actions occur in real time.3

THE OLIGOPOLIST'S DECISION

Consider an oligopolistic industry producing a homogeneous commodity. Firms make their output decisions simultaneously, once in each time period. The number of firms is fixed. The market inverse demand function is f:

Pt (.f ( I ) (1)

where Pt represents the price in period t, and xit 2 0, i = 1, . n, is the output of firm i in period t. Assumef ? 0, and f'(x) < 0 whenever f(x) > 0. All firms face the same decreasing-returns-to-scale cost function c: c' > 0, c"> 0. The cost function and the

1 The literature on conjectural variations was surveyed by Kamien and Schwartz (1983). 2 Compare with Irving Fisher's criticism of the Cournot equilibrium: 'But, as a matter of fact, no

businessman assumes either that his rival's output or price will remain constant any more than a chess player assumes that his opponent will not interfere with his effort to capture a knight. On the contrary, his whole thought is to forecast what move the rival will make in response to one of his own. He may lower his price to steal his rival's business temporarily or with the hope of driving him out of business entirely. He may take great care to preserve the modus vivendi so as not to break the market and provoke a rate war. He may raise his price, if ruinously low, in hopes that his rival, who is in the same difficulty, may welcome the change, and follow suit. The whole study is a "dynamic" one, and far more complex than Cournot makes it out to be' (1898, 126-7).

3 Kalai and Stanford (1983) showed that the equilibria of the standard conjectural-variations model are Nash equilibria in a fully specified repeated game; see also Marschak and Selten (1978).




demand function do not change over time. In order to concentrate on the firms' interactions with each other rather than with their environment, assume each firm has perfect knowledge about its own cost function, the market demand function, and the price that ruled at any time in the past. It is not necessary that each firm be able to observe its rivals' past outputs directly because each firm's profit depends on the sum of the rivals' outputs, not the individual outputs; this sum can be deduced from knowledge of the demand function, the market price, and the firm's own output. Nor is it necessary that the firm know its rivals' cost functions. On the basis of their knowledge about their rivals' past actions, the firms must somehow predict what their rivals will do in the current period. No binding agreements among the firms are possible, and the firms cannot communicate.

Define conjectural variations as follows. Denote the current period by t. Suppose that whenever firm i plans to change its output from one period to the next, by xit - xit- 1, it predicts that its rivals' outputs will change proportionately. Denote the sum of i's rivals' actual outputs in period t - 1 by xv 1, and i's belief about the sum of its rivals' outputs in period t by _i*. Then i's beliefs about the correlation between its own actions and its rivals' actions are represented by the conjectural-variations term bit, where

t_it = bit(xit- xit-1) + xi1 i - 1. n, (2)

that is, if xit # xit- 1

bit = (t-it -x_i'-1)1(xit -xit-1), i -1, .. n. (3)

The time superscript on the conjectural-variations term bit is necessary because this term will change over time during the transition to a conjectural equilibrium.4

Suppose that the oligopoly game is played once in each time period, but that firms seek in each period to maximize only current profits. At each point in time t during the adjustment process, the ith firm has a point estimate bit of the relationship between its own output changes and the sum of its rivals' output changes, as in (3).

In the demand function (1), substitute firm i's prediction of the other firms' total output, j_it, for their actual total output, x<it. This yields firm i's perceived demand function. Firm i's perceived demand function has as arguments only firm i's own current output, the parameters of the industry demand function, and the previous period's output levels; it does not depend on the (unknown) current output decisions of firm i's rivals. Given its beliefs about its rivals' behaviour as summarized in (2), firm i predicts that its profit in period t will be

7Tit = xitf([II + bit]xit -bixit- 1 + x-i- 1) - c(xit), i 1, ... , n. (4)

Maximizing this single-period profit with respect to xi (assuming an interior solution)

f[(I + bit)xit - bitxit-1 + x-it-l] + (1 + bit)xit'

f'[(l + bit)xit - btxit -1 + xt 1j - c' (xt) 0, i = 1, ... , n. (5)

4 The conjectural-variations models of Sato and Nagatani (1967) and Seade (1980, 24-5) have a dynamic structure similar to this model's, except that the conjectural-variations term is fixed exogenously.



792 / John McMillan

Equation (5) implicitly defines the ith firm's profit-maximizing output as a function of its own and its rivals' outputs in the previous period and of its conjecture (represented by bit) about its rivals' actions in the current period.

The second-order condition is

2(1 + bit)f'( ) + (1 + bit)2xi/f"(Q) - c" (xit) < 0, i =1, ... , n, (6)

for all xi' ' 0. Each firm's output decision is based on its point estimate bit of the conjectural-

variations term. This point estimate is based on past observations of actual actions. Each firm makes its output decision as in (5). At the end of period t each firm can compare its rivals' actual total output, x_i', with what it had predicted, 4_it. Thus at the end of period t, the firm receives new information about this apparent relationship; the firm observes the actual ratio of its opponents' output change to its own output change,

P3i t=(X-it -X-it-1)1(xit -xit-1), il.,n. (7)

(Contrast with equation (3).) With this new information, the firm revises for period t + I its point estimate bit' I of the conjectural-variations term; suppose there is an updating operator Bit which uses the new information 3it to generate a new point-estimate of the conjectural-variations parameter bit+':

bit+ I = Bit(bit, P3it). (8)

The function Bit will be assumed to be continuously differentiable, with 3Bitf/it > 0, so that observations do affect conjectures and the updating process is not trivial. Note that if xit = xit-' but x_it 4 x_it- 1, then it is undefined. Assume that Bit is such that in this case bit+ 1 is large and possibly infinite. (Further properties of Bit are stated in (9).) An example of an updating process which has these properties is an adaptive process: bit+l- oibit + (1 - cti)3it, 0 < (xi < 1.

In order to behave as described here, the firms need minimal information. They need to know their own cost functions, the market demand function, and past prices. They need not know their rivals' costs, how many rivals there are (though they must know there exists at least one rival), or the identity of the rivals. (The model assumes for the sake of simplicity that the firms face identical costs, but the firms themselves need not know this.) Thus the firms may have much less information than they would need in order to compute their Coumot-Nash (rational-expectations) strategies; behaving as in this model may be the best they can do.

EQUILIBRIUM

Once the adjustment process has ended, no firm will change its output. Thus it, being a ratio of output changes (equation (7)), is undefined. It seems reasonable, since no firm receives any new information in this case, to postulate that the updating operator Bit has the property that the conjecture bit is left unchanged when the observation Pit is a ratio of zeros. (If xit = xit- I andx<it = x<it-1, then from (2), (_it - x_it; that is, the output firm i predicted its rivals would produce is the same as the output they actually produce, given their beliefs bj t, j # i. Thus bit, upon which firm




i based its prediction, is confirmed; the updated value, bit"l, should be unchanged.) Assume, moreover, that an observation Pit, which is defined and different from the ex ante belief bit, results in a changed ex post belief bit' 1 f bit. These properties of the updating operator are summarized in the following assumption: for all firms i = 1,

, n,

bit+ I bit if xit = xit-' and x_it = xit- (9a)

bit+' bit only if either bit Pit or xit x xit-- and x-it _ X-i_- (9b)

(9b) says that ex ante beliefs are left unchanged ex post only if they are confirmed by observation. Note that (9a) leaves open the possibility that, if outputs do change so that Pit is not a ratio of zeros, then a confirmation of beliefs does not necessarily mean that the beliefs remain unchanged (that is, it may happen that bit+ 1 f bit even though pit = bit). Clearly these requirements would be satisfied if beliefs were updated adaptively.

Define an equilibrium of this trial-and-error learning process to be a vector of outputs and conjectures (xI*, ..., xn*; bI*, ..., bn*) such that each firm's output xi maximizes its profit, given its beliefs bi* about its rivals' reactions and given that the equilibrium vector of outputs (xl *, .. ., xn*) was also produced in the previous period. From (9a), this definition of equilibrium implies that conjectures bi* do not change once equilibrium has been reached. (Throughout, asterisks will be used to denote equilibrium values of variables.)

It follows from this definition that equilibrium points can be found simply by putting xit- 1 = xit for all i = 1, .. ., n in equation (5), for given values of bit, i = 1, ..., n, and solving for xit. In the standard conjectural-variations model (with the conjectures exogenously fixed) there are infinitely many equilibria. A similar phenomenon occurs in this model, in the sense that, for a wide range of initial beliefs bil, i = 1, .. ., n, there is a corresponding set of initial output levels xiP, i = 1 ... , n, such that no firm has any incentive to change its output. For example, suppose firstly that b1l = 0, i = 1, ..., n. Then, from (5), x1 = xio if xi0 is the Coumot level of output. Thus, from (9a), no revision of beliefs will occur; bi2 = bil = 0. In other words, if all firms start with Coumot conjectures and producing Coumot outputs, they will stay at the Coumot equilibrium forever. Secondly, suppose that, each firm's initial conjecture is bil = -1. Then xi =xi if f(xi? + x-0) = c'(xi?); that is, price equals marginal cost. Thus if initial output levels sum to the perfectly competitive output level, and if each firm's initial conjecture is bil = -1, the system will stay at perfect competition. Thirdly, suppose all firms initially have the conjecture bi = n - 1. If the initial outputs are Xmln, where Xm represents the joint-profit maximizing level of industry output, then (5) says industry marginal revenue equals marginal cost; total industry profit is maximized. No firm has any incentive to revise its beliefs, and the system stays at the joint-profit maximizing equilibrium.

Thus there are infinitely many conjectural equilibria. By varying the initial conjectures bil, any outcome between perfect competition and the joint-profit maximum can be maintained as an equilibrium for some particular allocation of initial



794 / John McMillan

outputs. It was assumed, however, in discussing these equilibria that the system started at one of the equilibria with the firms' initial beliefs mutually consistent. Having exactly such a set of initial conditions is a measure-zero accident; a slightly different initial value of any variable would mean that adjustments necessarily occur. It will now be shown that if output adjustments do occur, the set of possible equilibria is much smaller.

COLLUSION AND COMPETITION

Suppose that the system does not start with outputs and conjectures such that no firm wants to change its output; suppose instead that some transition does take place. Outputs change over time, so that Pit is defined. Note that from (7),

I+ pit xi t xi-) xit - Xii ) (10)

Thus the following relationship must hold among the observations [it:

either Pitf3 -1 for all i =1, ... ,n (la)

n

or 3 [1/(1 + it)]= 1. (1 Ib)

(Note that (1 a) occurs when total industry output does not change from period t - 1 to period t; (1 lb) occurs when it does change.)

If a non-trivial transition does take place, so that the beliefs bit are altered in the light of the observations [it, then assumption (9b) ensures that the property (11) of the observations is inherited by any steady-state beliefs bi*:

either bi* -1 for all i = 1, ... , n (12a) n

or 3 [I/(I + bi*) -1. (12b)

This is because, if Pit is not a ratio of zeros and if [it # bit for some t, bit cannot be a steady-state set of conjectures because (9b) would imply some revision of conjectures so that bit" ft bit.

Condition (12) limits the outcomes that can be reached as endpoints of this learning process. In particular, (12) is not satisfied by the Cournot conjectures (bi* = 0, i = 1, ... , n); it is satisfied, however, by perfectly competitive conjectures (bi* = -1, i = 1, ... , n) and by joint-profit maximizing conjectures (bi* n - 1, i = 1, ... , n).

More generally, as is shown in the appendix, at the equilibria associated with the conjectures (12), it is as if the market maximizes the weighted sum of profits J.= l1 OiTi for some al, . ., an (with some -i non-zero but not necessarily positive).

If J,= not is maximized for some oxi > 0, i = 1, ... , n, then the equilibrium lies on the profit-possibility frontier. (Such outcomes will be referred to in this paper as implicitly collusive.) These equilibria occur when (12b) holds and bi* > 0 for all i 1, ..., n. To prove this, rewrite (5) as

[f/(I + bi*)] + xi*f' - [ci'(xi*)/(I + bi*)] = 0. (13)




Sum over all firms, using Ji= In 1/(1 + bi*) -

n

f +Xf' [ci'(xi*)/(1 + bi*)], (14)

where X denotes total industry output. With bi* ', 0 0 I /((I + bi*) ' 1. Thus (14) says that industry marginal revenue equals a weighted average of each firm's marginal cost. Clearly this is a necessary condition forXf(X) - ZI=In c[X/(l + bi*)] to be maximized; that is, for total industry profit to be maximized subject to the constraint that the ith firm has a market share of 1/(1 + bi*). Varying the bi*'s (with bi* 0 O) traces out the profit-possibility frontier. Note one extreme case which satisfies (12b) with bi* - 0, and therefore is on the profit-possibility frontier: suppose for one i, bi* = 0, while bj* = oc, j-1, . . ., i - 1, i + 1, ..., n. In an equilibrium with such conjectures, the firms j, for j f i, produce nothing, while firm i produces as a monopolist.

The alternative possibility, also consistent with (12), is that the equilibrium maximizes Ej= IXI jaj with (x; < 0 for at least one j = 1, ... , n. This occurs when bi* < 0 for at least one i = 1, ... , n. (To see this, note from the appendix that (A2) shows that bi* < 0 if and only if rfii < 0, using lTik to denote 8kTi/Xk. Since 1-ik < 0 for i # k, (A 12) then holds if and only if some vj s are strictly positive and some sjs are strictly negative.) Thus, in the weighted sum of profits being maximized at equilibrium, some firms have negative weight. At equilibrium, given the conjectures, no firm will want to change its output; however, with rii < 0, the direct effect of a decrease in firm i's output would be an increase in all firms' profits (thus this outcome is clearly not on the profit-possibility frontier). From (1 2b), if there is a bi* < 0, then there is a bk* c 1 (because for bi* < 0 either bi* '-1 or O b*> -1; in the latter case (12b) can hold only if there is a bk* < -- 1). If bk* - 1, then (5) implies that for firm k, price is less than or equal to marginal cost at equilibrium. For these reasons the outcome in which one or more of the bi* 's (or equivalently one or more of the oxj s) is negative will be referred to as cutthroat competition. A particular example of this is perfect competition (b.* -1 for all i 1, .. , n).

To illustrate these equilibria, consider duopoly. Condition (12) reduces to the simpler statement b I * = 1/b2*. Either both b I* and b2* are positive, in which case the outcome is implicitly collusive; or both bI* and b2* are negative, in which case the outcome is one of cutthroat competition. In the former case total industry profit is maximized subject to market shares being 1/(1 + bl*) and b1*/(I + bl*). In the latter case, (A2) implies IlI < 0 and 'X22 < 0. Thus, from (A12), one of the Uxs is positive and the other negative: at the equilibrium, a weighted difference of the firms' profits is maximized. Since both bi*s are negative, the firms produce more in total than they would at the Coumot equilibrium.5

Thus, because of the existence of the algebraic relationship (11) among the

5 These two kinds of outcomes are reminiscent of the 'collusion' and 'warfare' solutions of Bishop (1960). They are different, however, because Bishop made the unorthodox assumption that each firm seeks to maximize a weighted sum of its own and its rivals' profits. In this model each firm maximizes its own profit, given its conjectures; the effect, however, is that at equilibrium it is as if the market maximizes a weighted sum of profits.



796 / John McMillan

different firms' observations, and because this relationship must be inherited by any equilibrium conjectures bi* so that (12) holds, the only possible outcomes that can be reached as the endpoints of an adjustment process are implicit collusion (of which joint-profit maximization is a special case) and cutthroat competition (of which perfect competition is a special case). The unsatisfactory property of the standard conjectural-variations model, the fact that almost any outcome can be an equilibrium, is evidently ameliorated when we make conjectures endogenous.

CONVERGENCE

To illustrate the implications for conjectural equilibrium of observations of apparent correlations between output changes, suppose the system is initially in one of the conjectural equilibria described above. Now suppose that at time t an exogenous change in each firm's costs or in market demand disturbs the system. Suppose each firm has perfect information about this exogenous disturbance. Then each firm will change its output according to (5); xit x xit-1, i = 1, ... , n. Each firm will observe an apparent correlation 3it, and the relationship (10) among these 3its will hold. Thus if the original conjectures bit do not satisfy (12) (that is, the original equilibrium was neither one of cutthroat competition nor on the profit-possibility frontier), some or all of the firms will receive disconfirming information about their conjectures: bit t P3it By (9b), therefore, revision of conjectures is called for, so that bit" ft bit, and the original equilibrium is upset by the exogenous cost or demand change. Suppose, on the other hand, that the predisturbance conjectures do satisfy (12); in particular, suppose initially bit =- 1, so that the system is initially at the perfectly competitive equilibrium. By symmetry, the initial outputs are the same for each firm; xit = xjt, i, j = 1, ..., n. Moreover, from (5), each firm will react to the disturbance symmetrically, so that xit+1 xj t+1, i = 1, n. From (7), sincexit = xjt, i,j 1,

I n, the ith firm's observation it = (n - 1) ft -1; the original conjectures bit =

-1 are refuted. Suppose instead that initially the system is at the joint-profit maximizing equilibrium, with bit = n - 1, i = 1, ... , n. Again, by symmetry, xit+ 1 = x it+ 1, i, m = 1, ...,I n. Thus, since xit = xj t, i, j = I1 ...,I n, Pit = n -1; therefore pit = bit, i = 1, ..., n and the original conjectures are confirmed. The joint-profit maximizing conjectural equilibrium is not disturbed by the observations of reactions following an exogenous change. 6

Thus, although this model has a continuum of equilibria, the conjectures upon which most of these equilibria are based are destroyed by the new information about rivals' reactions that results from exogenous changes. (Note, however, that only the first step in the adjustment was examined; the possibility that conjectures might eventually return to their original values has not been ruled out here.)

The state of the system at time t is described by the vector of outputs and beliefs yt (x1t, ... I Xnt, b1t+ , ... , bnt+ 1). The transition to yt+ 1 is defined by (5), (7), and (8).

6 It is easy to find examples of non-symmetric (that is, non-joint-profit-maximizing) equilibria on the profit-possibility frontier, which are upset by new information of this sort. This does not rule out the possibility that some non-symmetric profit-possibility-frontier equilibria (or non-symmetric cutthroat-competition equilibria) may not be disturbed by such exogenous changes.




Equations (5), (7), and (8) implicitly define a system of non-linear first-order difference equations in yt. Is this difference-equation system stable? We now demonstrate convergence from some particular initial outputs and conjectures; in the next section we consider the question of convergence from arbitrary initial conditions.

Consider first the special case in which the initial outputs sum to the perfectly competitive level of industry output, x_*? + xio - Xc; and the initial conjectures are bi -1, i = 1, ..., n. Then from (5), in period I each firm will produce xi' = XCIn. Thus each firm will observe 13i = [(n - 1)XC/n - xio]l(XC/n - x10) -1. Thus bi = -1, and the system reaches the perfectly competitive equilibrium after one adjustment.

Suppose instead that the initial variables are symmetric: x,? = xio and bi1 = bj1 for all i, j = 1, ... , n, but that the initial outputs are not profit maximizing given the initial beliefs, so that the outputs will change (xi? ft xi, i 1, ... , n) and the firms will receive observations i', i = 1, .. i , n. Then, from the symmetry, at each stage of the process each firm will observe Pi = (n - 1), i = 1, ... , n, t = 1, 2, 3, .... Since each firm repeatedly observes i' -- (n - 1), if the updating operators are the same for each firm and are contractions, then eventually the beliefs bit must converge upon (n - 1). Once this happens, no further convergence of the bits will occur; however, the outputs xit will still change. A result due to Sato and Nagatani (1967) now becomes relevant. Sato and Nagatani investigated the stability of a conjectural-variations model in which conjectures are fixed and showed that a sufficient condition for convergence of outputs with fixed conjectures is, in the notation of this paper,

- o <bi ai- 1<0 (15a)

n

Ibiail + Z lajI < 1, i = 1, ... , n, (l5b) J=1 j#i

where

ai [f' + (1 + bi)xitf']/[2(1 + bi)f' + (1 + bi)2xitf" - c"(x1)]. (16)

In the special case currently under consideration, bi = n - 1, i ,. , n, and xit =

xj1 , i, j = 1, ... , n. Moreover, from the second-order condition (6), the denominator of ai is strictly negative. Thus the first inequality in (15a) is satisfied. The second inequality in (1Sa) follows from (15b). Given the symmetry, (15b) reduces to

2(n - 1) lail<K 1. (17)

Elementary manipulations, using (6), show that (17) is satisfied in the problem under consideration if

(2f' - c")/n(n - 2)xit <f" < [-2(2n - l)f' + c"]/(3n - 2)nxit. (18)

The term on the left of (18) is strictly negative and the term on the right is strictly positive; thus (18) says that the demand function is neither too concave nor too convex. Thus, if all firms start with identical non-equilibrium initial outputs and conjectures, all update their conjectures in the same way, and the updating operators



798 / John McMillan

are contractions, then the system converges on the joint-profit maximizing outcome provided (18) is satisfied.7

I N S T A B I L I T Y

The special examples in the previous section showed that there exist initial conditions from which the system converges upon the joint-profit maximum or perfect competition. Convergence is not, however, typical. It will now be shown that, if this learning process starts from arbitrary initial outputs and conjectures, it is unstable. The reason for this instability is straightforward. The firms are observing a ratio of changes i' = (xit - x7i- i)/(xit - xi- 1); if the system were to approach a steady state, both denominator and numerator would become small. Thus, small changes in one of the outputs would generate arbitrarily erratic changes in the observation Pi'. This difficulty necessarily arises in a model in which conjectures are affected by observations; it is inherent in the definition of conjectural variations as the ratio of output changes.

An equilibrium is stable in Liapunov's definition if nearby solutions stay nearby for all future time. Conversely, and more precisely, an equilibrium y* = (x1*, .... xn*; b1 *, ... , bn*) is unstable if there is a neighbourhood U of y* such that for every neighbourhood U1 of y* in U, there is at least one path yt, starting at y? E U1, which does not lie entirely in U (Hirsch and Smale, 1974, 185-7). Suppose the initial state is Y' = (XI* + E1 --.. Xn* + en; bI*, ..., bn*) for some non-zeroEl, ...E,en. (That is, initially conjectures are at their equilibrium values while outputs are near, but not at, their equilibrium values.) Denote the subsequent state by yl = (x11, .. , Xnl; b11, ... bnl). Taking a Taylor-series expansion around y*, using the first-order condition (5) and the implicit-function theorem (denoting IFi Ej by E-i),

Xi- {[bi*f' + bi*(l + bi*)xi*f"]Ei - [f' + (1 + bi*)Xi*f"]E_}

/[2(1 + bi*)f' + (1 ? b,*)2xi*f"- C"]. (19) Assume that

f' + (1 + bi*)xi*f" ft 0, for some i = 1, ... , (20)

the alternative case will be considered below. From (19), the change in the ith firm's output from period 0 to period 1 is

Xi- (xi* + Ei) {[-(2 + bi*)f' - (1 + bi*)xi*f" + c"]Ei - [f' + (1 + bi*)xi*f"]Ei}/[2(1 + bi*)f' + (1 + bi*)2xi*f" -c"]. (21)

The denominator of the right-hand side of (19) and (21) is non-zero because of the second-order condition (6). Since some output changes will occur, it can be presumed that the bi*s satisfy (12). By (12) it cannot be the case that all the bi*s are infinite: assume b1* < oo and that (20) holds for i = 1. The initial outputs are arbitrary: choose E1, E-1 so that (for -(2 + b1*)f' - (1 + bl*)xl*f" +c" ft 0):

7 On the other hand, it is easy to show that the sufficient condition for instability of this output- adjustment process due to Seade (1980, 24) is never satisfied in this special case, because it contradicts the second-order condition (6).




El {[=f + (1 + b1*)x1*%f]Ei}/[-(2 + bl*)f' - (1 + bl*)xl*f" + c"]. (22)

Then, from (21), the change in firm l's output from period 1 to period 2 is approximately zero:

X- (x* + E1) 0. (23)

Now choose E2, , . . e En such that their sum satisfies (22) and such that, using (2 1), the total change in the other firms' output is non-zero:

n

I Xj - (Xj* + E) ft 0. (24) j=2

(Clearly there are enough degrees of freedom so that this can be done: the point yo was not an equilibrium by construction, so that at least one firm j, j = 2, ... , n, will change its output.) Then from the definition of 3 (7), the first firm observes an arbitrarily large PI 1. Thus the updating rule (8) prescribes an arbitrarily large change in the conjectural-variations parameter bi2. Thus for every neighbourhood of an equilibrium point there is a path that leaves the neighbourhood, provided (20) holds. is a path that leaves the neighbourhood, provided (20) holds.

Return now to consider the case in which (20) does not hold:

f' + (1 + bi*)xi*f" - 0 for all i 1, ... , n. (25)

Note incidentally that this implies n n

A [f'/(l + bi*) + xi*f' - 0, (26)

or, using X to denote total industry output,

f ' + Xf" 0 (27)

because of (12b). (It cannot be the case that both (25) and (12a) hold for some i.) Equation (27) is satisfied by the particular demand function

f(X) - k,-k2 ln X, (28)

where k, and k2 are strictly positive constants. If (25) holds, then it can be seen from (19) that, if the system starts with equilibrium conjectures and close-to-equilibrium outputs, it moves immediately to within a first-order approximation of the equilibrium point. However, suppose (25) holds and the system starts at an arbitrary point in the neighbourhood of equilibrium yo (xl * + El, * * *, Xn* + en; b1* + 81, ... , bn* + )n), with Ei ft 0, bi= 0, i = 1, ..., n. Taking a Taylor-series expansion around the equilibrium point y*,

Xi -xi* 1 {[bi*f' + bi*(l + bi*)xi*f"]Ei - ' + (1 + bi*)Xi*f"]E-i - xi*f' i}

/[2(1 + bi*)f' + (1 + bi*)2xi*f" -c"]. (29)

Using (25), this reduces to

xi-Xi* x -xi*f' bi/[2(l + bi*)f' + (1 + bi*)2xif" - c"]. (29)



800 / John McMillan

Thus the change in the ith firm's output from period 0 to period 1 is

X'- (X-* + EJ) x{[-xbi - 2Ei (1 + bi*)]f' - Ei(1 + bi*)2xi*f"f + EiC"}

/[2(1 + bi*)f' + (1 + bi*)2xf'f - c"]. (30)

If 68 and E, are chosen, so that

al = [-2EI(I + b *)f' - EI(I + bl*)2xI *f" + EIc'"]/xI*, (31)

then the change in the first firm's output is approximately zero. If 6j and Ej, for some j = 2, ... , n, are chosen so that the equivalent to (31) does not hold for j, then once

again the first firm will observe an arbitrarily large P3I . Thus, a path starting from a point satisfying (31) will leave any neighbourhood of y*.

Hence the system of first-order non-linear difference equations in yt = (xt, bt+ 1)

defined implicitly by equations (5), (7), and (8) is locally unstable. Since it is locally unstable, it is clearly globally unstable.

CONCLUDING COMMENTS

To model firms as conjecturing that there is a relationship between their own outputs and their rivals' outputs is a short-cut way to model oligopolistic interdependencies. The conjectures are literally false; each firm conjectures that its rivals' output decisions simply follow a rule making their outputs functions of its own output, when in fact the rivals are solving maximization problems. Presumably the firm believes that such a relationship exists because of its past experiences. This paper has modelled the development of such beliefs.8

In order to behave as in this model, the firms need very little information. They must know their own characteristics, the market demand function, and past market prices. They need not know anything about the other firms beyond the fact that there exists at least one rival firm.

In the model of this paper, if no adjustments take place there is a continuum of equilibria. Most of these equilibria, however, are eliminated if the firms observe actual output changes in their rivals; the new information firms obtain causes them to change their conjectures. The only possible endpoints of an adjustment process are implicit collusion and cutthroat competition. If the firms are initially at a symmetric conjectural equilibrium and there is an exogenous change in costs or demand, the

8 This model is complementary with another approach to making conjectures endogenous, that of 'rational' or 'consistent' conjectures. (See, for example, Bresnahan, 1981; Kamien and Schwartz, 1983; and Laitner, 1980.) The rational-conjectures models answer the question: Which conjectural- variations equilibria are not upset if firms investigate the correctness of their conjectures by experimenting with small output changes? There are a number of differences in time structure between the two approaches. First, the timing of firms' actions is explicit in the present model and only implicit in the rational-conjectures models. Second, this model examines the process which takes place before equilibrium is reached; the rational-conjectures models examine what happens if, after equilibrium has been reached, firms experiment. Because of this second difference, there is a third difference. It is crucial in this model that, within each period, firms make their decisions simultaneously (they cannot respond to the other firms' simultaneous actions). The rational- conjectures models have a Stackelberg-like implicit time structure: each firm is concerned with its rivals' reactions to its own deviation from equilibrium, regarding itself as a Stackelberg leader during this experiment.




subsequent adjustment will cause the firms to alter their conjectures unless the initial equilibrium was joint-profit maximizing. There exist initial outputs and conjectures from which the system converges to the joint-profit maximizing solution or the perfectly competitive solution. However, if the system starts from arbitrary initial conjectures and outputs, then the process of revision of conjectures is unstable, in the sense that there are points near any conjectural equilibrium such that, if the system reaches such a point, it then moves away from the equilibrium. This instability necessarily arises in a conjectural-variations model when conjectures are generated by observations; it is inherent in the standard definition of conjectural variations as the ratio of output changes.9

The results of this exercise in making conjectures endogenous are mixed. On the one hand, this model has succeeded in narrowing down the set of conjectural equilibria. On the other hand, the instability means that there is no guarantee that even those outcomes that remain as possible equilibria will ever be reached. The situations in which convergence does occur, however, are not without interest. Suppose the system starts at any symmetrical initial situation. For example, a symmetrical Coumot equilibrium might be disturbed by an exogenous change in costs or demand that is observed by all firms. Then the system will converge on the joint-profit- maximizing outcome. If the identical firms always act identically, implicit collusion evolves. The model thus provides some limited justification for the rejection of Coumot's equilibrium concept by Bertrand (1883), Chamberlin (1933), and others, on the grounds that rational oligopolists will recognize their mutual dependence and somehow find their way to a tacitly collusive solution.

APPENDIX

In this appendix it is proved that an equilibrium with conjectures satisfying (12) maximizes the weighted sum of profits >2= 1ncvjrU, with cxi ft 0 for some i.

Note first that in equilibrium xi' xi'- l, so that the first-order condition (5) can be written as (denoting drti/dxj by aij)

Tii + bi*7rij 0 O, i 1,..., n (A1)

where aij = 'rik for all j, k i. Since lij < O,

sign (Xrii) = sign (bi). (A2)

Rewrite (12) as n n n

(1 + bi*)- I fl (1 + bi*)=0. (A3) i#k

9 The reason for the instability is, as pointed out above, that Pi' is discontinuous at an equilibrium. There is a direction of generalization, which may help remove the model's instability. The firm's beliefs about its rivals' reactions are completely described in this model by a single number, bi'. It might be more satisfactory to represent the firm's conjectures by a subjective probability distribution over possible reactions of its rivals. After each new observation the firm would update its prior probability distribution. These probability distributions over time would follow a Markov process, the convergence properties of which could be studied. In such a model updated conjectures might be less sensitive to output changes in the last period than would be true in the present model.



802 / John McMillan

Equation (A3) holds if and only if

-bl* 1 . . . 1

1 -b2*1

D-- -0. (A4)

1 1 . . . -bn

To prove this, rearrange this determinant as

-bl* 1 1 . .1.11

1 + bl* -(1 + b2*) 0 0

D- 0 1+b2* -(? +b3*) 0

0 0 0 1? bnI* -(1 + bn*)

-bl* 1 1 . . 1

1 1 b2*

1 + bl*

n-I o 1?+b2* I1?b3* 0 171 (1 + bi*) I1 +b* 1 + b2* (A5)

i=l1

0 0 0 1+ bn* 1 +bn-I*

Adoptthe following abbreviation: Yi b1*, Y2 (1 + b2*)/(1+ bl*), Yn, (1 +

bn*)/'(1 + bn_1*). Then

-Yi 1 1 . . . 1

1 -Y2 0 0

0 1 -Y3

0 0 0 . . Yn




1 1 1 . . . 1

1 -Y3 0 0

0 1 Y40

0 0 0 .Yn

1 1 1

1 -t 0

I -)n-y l ^2w 1) n-

Y3-Y4 .. 'Yn +

I~~ O2 . Yn

0 0 . Yn

- (-l )47 1 Y2 * -Yn + (-1) 1Y3^Y4 ... * ?n (-Y)n 3Y4 - *Yn + n

+(-1l)(l)nk- H tyi + + (-1I)ny Ia + (-1)n'+. (A6) I k

Thus

n

D = H (1 + bi*) [(I)nY1lY2 .Y. y, + (-1)n 1(Y3Y4 ..Y n" + Y4 .., Yn + i =1

n n n +ywn +?1)] ( 1)n[bl* H (I+ bi*)- Z H (1 + bi*)-

i#k n n n

=)n l (I bi*) I H (1 + bi*) (A7) i-I t1~~~~~ ~ ~~~ t-1 k i#k

From (A7), (A4) holds if and only if (A3) holds. From (Al), bi = -maiil-rik for all k ft i. Substituting this in (A4) (using the k in -rak

to denote any k ft i):

IT11 1 . . . 1

1 T22

'lQT2k - 0. (A8)

1 1 . .I~ ITnn



804 / John McMillan

Notice that

-T 1 . . . 1 171k

T22 1 1 172k

Tnk

TIk 0 0

0 l2k 0

0 0 . Tnk

IT11 1T2k ,, nk

71k 722 1nk (A9)

71k lT2k *Tnn

From (A9) since lTik =rrj for all k, j ft i, since the determinant of a product is the product of the determinants, and since the second matrix on the left-hand side of (A9) is non-singular, (A8) holds if and only if

7T1 I 721 *7T I n

T12 722 7Tn2

. 0. (AIO)

171n 7T2n *7*nn

Condition (A 10) holds if and only if there exist xt ., OCn, with o # 0 for some j, such that




n

, OTjri = O, i-l, ... , n. (Al1) J-=1

That is, (A 10) is a necessary condition for j 1= c notj (for some coI, ..., o( not all zero but not necessarily positive) to be maximized.

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Bemheim, R. Douglas (1984) 'Rationalizable strategic behavior.' Econometrica 52, 1007-28 Bertrand, Joseph (1883) Review of Cournot's Recherches sur les Principles Mathematiques

de la TheWorie des Richesses. Journal des Savants, 499-508 Bishop, Robert L. (1960) 'Duopoly: collusion or warfare?' American Economic Review 50,

933-61 Blume, L.E., Bray, M.M. and D. Easley (1982) 'Introduction to the stability of rational

expectations equilibrium.' Journal of Economic Theory 26, 313-7 Bresnahan, T.F. (1981) 'Duopoly models with consistent conjectures.' American Economic

Review 71, 934-45 Chamberlin, Edward (1933) The Theory of Monopolistic Competition (Cambridge, MA:

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Economics 12, 119-38 Friedman, James W. (1977) Oligopoly and the Theory of Games (Amsterdam: North-

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and Linear Algebra (New York: Academic Press) Johansen, Leif (1982) 'On the status of the Nash type of non-cooperative equilibrium in

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Cournot games with fast reactions.' Discussion Paper No. 575, Center for Mathematical Studies in Economics and Management Science, Northwestern University

Kamien, Morton I. and Nancy Schwartz (1983) 'Conjectural Variations.' This JOURNAL 16, 191-211

Laitner, John (1980), "'Rational" Duopoly Equilibria,' Quarterly Journal of Economics, 94, 641-662.

Marschak, T. and R. Selten (1978) 'Restabilizing responses, inertia supergames, and oligopolistic equilibria.' Quarterly Journal of Economics 92, 71-93

Pearce, David G. (1984) 'Rationalizable strategic behavior and the problem of perfection.' Econometrica 52, 1029-50

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Collusion, Competition, and Conjectures