The chaotic duopolists revisited

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Journal of Economic Behavior & Organization Vol. 33 (1998) 385-394

The chaotic duopolists revisited T. Puu Department of Economics, Umed University, 90187 Urned, Sweden Received 8 January

1996; accepted

15 May 1996

Abstract The adjustment process of three oligopolists is studied, under Coumot and Stackelberg action. It is demonstrated that with an iso-elastic demand function and constant marginal costs, the system can result in periodic or in chaotic behaviour. In particular, the case with two identical and one different oligopolists is focused, which turns into a virtual duopoly, though displaying a wider set of bifurcations than can occur in genuine duopoly. 0 1998 Elsevier Science B.V. JEL classification: D43; (C73?) Keywords:

1.

Duopoly;

Oligopoly;

Chaos; Bifurcation

Introduction

Duopoly and oligopoly, despite being first steps from monopoly towards perfect competition, are analytically not cases of intermediate complexity, but more complicated than any of the extremes. This is because oligopolists consider not only the behaviour of consumers but also of the competitors, and possible retaliations to their own actions. The formal theory of oligopoly goes as far back as back as to Cournot, Augustin, 1838, who treated the case with no retaliation at all, so that in every step each oligopolist acts as if the latest steps taken by the competitors will be repeated. The process was assumed to lead to a steady state, nowadays called Coumot equilibrium, though it is by no means certain that it is stable. Exactly a Century later, von Stackelberg, Heinrich, 1938 made some ingenious extensions of the Coumot model, by assuming that any one of the competitors might try to become a “leader” by taking the reactions of the competitors as governed by Coumot’s assumptions

into explicit

such a situation

consideration

when devising

their own actions.

would, of course, depend on whether the competitors

0167-2681/98/$19.00 (0 1998 Elsevier Science B.V. All rights reserved PIISO167-2681(97)00064-4

The tenability

of

were content with

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ofEconomic Behavior & Org. 33 (1998) 385-394

adhering to his Coumot-like behaviour, i.e., being “followers”. They might take up the challenge by trying to become leaders themselves. The outcome of such warfare would depend on long term production conditions for all the competitors, and on their current financial strengths. In the end, one of the oligopolists might ultimately force the others out of the market, thus becoming a monopolist. They might also agree on collusive behaviour, provided law admits this, or they might return to the Coumot equilibrium, or to a Stackelberg equilibrium, provided the oligopolists agree to accept leadership/ followership roles in a consistent way. It has been realised that the Coumot model may lead to cyclic behaviour, and Rand, David, 1978 conjectured that under suitable conditions the outcome would be chaotic. His purely mathematical treatment does, however, not include any substantial economic assumptions under which this becomes true. The present author, Puu, Tiinu, 1991, supplied such substantial assumptions in terms of an isoelastic demand curve and constant marginal costs. The case treated was duopoly, and it was shown that both simple and complex dynamics could arise. In the present article, the model is extended to the case of three oligopolists, which introduces new interesting complexities.

2. The cournot

model

Assume an isoelastic demand function, such that price, denoted by p, is reciprocal to the total demand. Provided demand equals supply, it is made up of the supplies of the competitors, denoted X, y and z. Thus p=

l

(1)

x+y+z

This demand function is not unproblematic, because it does not yield a reasonable solution to the collusive case, the reason being that when the possibility of making total supply zero is considered, price can go to infinity and total revenue remain constant. On the other hand, total costs would vanish, and the oligopolists could get the entire revenue without incurring any costs. Such a solution is purely formal and does not have any economic substance. This absurdity does not occur with any of the other solutions, as the presence of a positive supply by any competitors always keeps the price finite. We could also easily remedy this technical problem by adding any positive constant in the denominator of Eq. (1). As this would not change any substantial conclusions, but has the disadvantage of making all the formulas much more messy, we abstain from this, and just point out the little complication. Suppose, next that the duopolists produce with constant marginal costs, denoted a, b and c respectively. The profits of the firms become accordingly u=

x

-_ax

xfyfz v=

y

x+y+z

-

by

(2) (3)

Z Puu/J. of Economic Behavior & Org. 33 (1998) 385-394

387

z - cz

w=

(4)

x+y+z

The first firm would maximize U(x, y, z) with respect to x, the second V(x, y, z) with respect to y and the third W(x, y, z) with respect to z. Equating the partial derivatives to zero, we can solve for the reaction functions

Y=

Taking these as simultaneous equations, Cournot point. The solution is

we can solve for the quantities

produced in the

2(b + c - a)

x=

(8)

(a + b + c)* 2(c + a - b)

(9)

’ = (a + b + c)* 2(a + b - c)

(10)

’ = (a + b + c)~ Substituting point

back

into

Eqs. (5)-(7)

we

find

the

f_, = (b + c - a)* (a + b + c)* v = (c + a - b)’ (a + b + c)* w

=

(a+ b -

cJ2

(a + b + c)*

profits

in

the

Cournot

(111

(12)

(13)

Not surprisingly, the firm with the lowest marginal cost will make the largest profit. It will, of course make an even larger profit if it becomes a Stackelberg leader, but so will the other firms as well. We will see more about this case in the sequel, but let us for the moment look closer at the dynamic Cournot adjustment process. First of all, we have to find out something about the stability of the Coumot point as defined by Eqs. (8)-(10). Unless the Coumot point is unstable, we cannot have either fluctuations or chaos in the long run. The Jacobian matrix at the Coumot point is composed of the partial derivatives of Eqs. (5)~(7), where we substitute for the variables

Z Pm/J.

388

of

Economic Behavior & Org. 33 (1998) 385-394

Fig. I. Level curves for Jacobian

from Eqs. (8)~(IO):

b+;;3a

b+z3u

c+a-3b

0

c to-36 46

rrf;;3c

atb-3c

0 M=

determinant.

46

(14)

0

4c

The determinant

of this matrix is Det(M) = (b + c - 3a)(c + a - 3b)(u + b - 3~) 32abc

(15)

The Coumot point loses stability when the absolute value of Eq. (15) equals unity. To find out more about the possible regions of instability, we note that Eq. (15) depends only on the ratio of marginal costs. With three marginal costs, there are of course two ratios. Putting b=ha and c=ka, everything except the two ratio factors cancel out, and we get: Det(M) = (h + k - 3)(h + 1 - 3k)(k + 1 - 2h) 32hk

(16)

In Fig. 1, we can see the contour lines for the value of the determinant in the plane of cost ratios. It has to be noted that loss of stability occurs both when the determinant equals 1

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of Economic Behavior & Org. 33 (1998) 385-394

and - 1, so there are five different regions where we can expect interesting phenomena when iterating Bqs. ( 15)-( 17).

389

dynamical

3. The Stackelberg case Before continuing with the dynamical analysis, let us just state the Stackelberg conditions. We should realise that now there are various hierarchies of leadership. It is possible not only to be a Stackelberg leader or else follow the Coumot reaction curve. As each firm now has two competitors, it is also possible to accept the leadership of one competitor, but treat the other competitor as a follower. The mathematical complexity of such cases, with three different levels of behaviour, however, grows beyond measure. We therefore in the sequel assume only two kinds of behaviour: being a leader, and following the reaction curve. Supposing the first firm is a leader and the other two followers, we can substitute from the reaction functions Eqs. (6) and (7) into the profit function Eq. (2), which then becomes (17) This leadership profit function depends on the output of the firm itself only, and we can equate its first derivative to zero. The resulting equation in n is readily solved

(18) The outputs of the other firms then are obtained

from Eqs. (6) and (7):

(u + b + c)(ac + (u - b)(b + c)) Y’

4a2(b + c)2

z = (u + b + c)(ub + (u - c)(b + c)) 4u2(b + c)~ Substituting profit

back from Eqs. (18)-(20)

into Eq. (2), we obtain the Stackelberg

(19)

(20) leadership

(21) It is easy to Eq. (1 l), so The profits substitution,

prove that U according to Eq. (21) is always at any oligopolist may again at any moment try to of the two followers in the Coumot point but they are of little particular interest for the

least as large as according to become a Stackelberg leader. are as easily obtained by moment.

4. The dynamic system Let us now write down the equations of iteration for the Cournot case by lagging Eqs. (5)-(7). We note that, we unlike the duopoly case treated in Butt, Tonu, 1991 do not

390

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deal with essentially independent mappings. As always, a system of difference equations can be manipulated to single equations, but they are of higher order, involving several previous values of the single variable, not just a lengthening of the basic period as in duopoly. In relation to this, it is no longer unessential whether all oligopolists adjust simultaneously, or in a certain order. All different assumptions in this respect lead to different dynamical systems. Assuming for the sake of symmetry that adjustment is simultaneous, we have

&+I = Yt+1

=

&+I

=

J+ J t’ Yt

~ a

Zr

Zr +

xt

~

b

xt + Y, ~

C

- y, -

Zt

(22)

-

Xl

(23)

Zr -

- Xl - Yl

5. Back to “duopoly” Experiments with iterating Eqs. (22)-(24), show that, provided two marginal costs are equal, an attractor is located in a plane embedded in the three-dimensional phase space of this system. In plain words, the actions of the firms with equal marginal costs tend to become equal over time. It is therefore a good starting point to study this essentially twodimensional process, again a kind of “duopoly”. The attractivity of the attractor may be sensitive to initial conditions, and this is again different from the case of the previous systems. To explore this, suppose b=c and assume that this is always true. Simulating the threedimensional process, as formulated by Eqs. (22)-(24), this will indeed become true, given suitable initial conditions, which by no means require initial equality between the two variables. The system then becomes

J

% - 2y*

x,+1 =

Ytfl

=

xt + Yr

J ~

b

- Xl -

(25) yr

There is a twin equation to Eq. (26), where z replaces y, but there is no need to reproduce it. The present system is different from the two-dimensional systems discussed in Puu, Tiinu, 1991 as it cannot be decomposed into independent maps of the first order. For Eqs. (25) and (26) we can compute the Coumot point: 2a ’ = (a + 2b)2 .X=

2(2b -a) (u + 26)’

(27)

(28)

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of Economic Behavior & Org. 33 (1998) 385-394

It could also have been found find out about the stability of equal marginal costs in Eq. (14), and (26) right away and calculate as

391

by substitution of b=c into Eqs. (8)-(10). To the Cournot point, we can substitute for the or else we can differentiate the system Eqs. (25) the determinant of its Jacobian matrix from 57, 58

Det = (a - 2b)(3a - 2b) 8ab This expression

can never become

1, but it assumes the value -1 for: b

m

a

2

-=2f--

(30)

Unlike the case discussed in Puu, T&u, 1991, the roots are not in reciprocity, so the marginal costs enter asymmetrically into the critical ratio. Fig. 2(a)-(c) display a sequence of events at a marginal cost ratio near the higher critical root according to Eq. (30). We first see a very slow approach to the stable Cournot equilibrium at a cost ratio close to the critical value. Next, we see the situation just after the Coumot point has lost stability. What we see is no longer the start of a period doubling cascade, but a Hopf bifurcation from fixed point to cycle. Finally, we see a picture of fully developed chaos. The shape of the chaotic attractor changes markedly when we add the possibility that the firms try to establish Stackelberg leadership according to Eq. (18) and its companions for the other two firms. We can again do this on a probabilistic basis in the simulations, as it is always better for each competitor to be a Stackelberg leader than to be in Coumot equilibrium. At this stage, we should also add a little technicality in the context of simulations. All the reaction functions, as specified in Eqs. (25) and (26), taken literally result in negative values for sufficiently large arguments. Such solutions to the optimisation problem of the firms are purely mathematical, and, of course, have no significance. In terms of economic substance, those cases happen when the competitors supply so much that in view of its own marginal costs, the firm in question cannot make any profits at all at any positive quantity produced. The consequence is then that the firm produces zero. Thus, all the reaction functions should be stated as the maximum of zero and the respective expression stated. For simplicity, we did not state this explicitly, but at simulations this fact has to be taken into consideration. This fact becomes particularly important when Stackelberg leadership is taken into consideration, because in some cases it is impossible to find parameter values that keep the Stackelberg solutions within the bounds where the mathematical expressions for the reaction functions stay positive. In Fig. 3, we display the chaotic case illustrated in Fig. 2(c), but with Stackelberg action added. Though two of the marginal costs are still equal, the attractor in three dimensional phase space is no longer embedded in a plane, but rather seems to live on some surface of a complex shape.

392

7: Puu/J. of Economic Behavior & Org. 33 (1998) 385-394

2b Fig. 2. (a) Slow approach to stable Coumot point just before bifurcation. (c) Fully developed chaos.

(b) Just after Hopf bifurcation

to cycle.

7: Puu/J.

of Economic Behavior & Org. 33 (1998) 385-394

Fig. 2. (Continued)

Fig. 3. Chaotic attractor

with Stackelberg

action added.

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References Coumot, Augustin, 1838. Rechereches sur les principes mammatiques de la theotie de la richesse, Hachette, Paris. von Stackelberg, Heintich, 1938. Probleme der unvollkommenen Konkurrenz, Weltwittschaftliches Archiv 48 (95). Rand, David, 1978. Exotic phenomena in games and duopoly models, Journal of Mathematical (173). Puu, Tow, 1991. Chaos in duopoly pricing, Chaos, Solitons, and Fractals I, 573.

Economics

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