An analysis of the complex dynamic behaviour of nonlinear oligopoly models with time delays

Chaos, Sohtom & Fractals Vol. 7, No. 12, pp. 2049-2065, 1996 Copyright Q 1996 Elsevier Science Lfd

Printed in Great Britain. All rights reserved 096&0779/96 $15.00 + O.OC

PII: s0960-0779(%)ooo71-9

An Analysis of the Complex Dynamic Behaviour of Nonlinear Oligopoly Models with Time Delays

CARL CHIARELLA and ALEXANDER KHOMIN

School of Finance and Economics, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australia

Abstract-We consider the fate of output in the Cournot oligopoly model when the equilibrium is locally unstable. We discuss types of nonlinearities which may be present to bound the motion and introduce time lags in production and information which serve as bifurcation parameters. We apply the Hopf bifurcation theorem to determine conditions under which limit cycle motion is born, and use computer simulations to investigate the nature of the attractors generated by such models. Copyright 0 1996 Elsevier Science Ltd.

1. INTRODUCTION

The literature on the stability of the oligopoly model, in both discrete and continuous time, is one of the richest in economic dynamics. Okuguchi [l] gives a thorough discussion of many of the key results and generalisations to multi-product firms are contained in Okuguchi and Szidarovsky [2].

Typically stability conditions relate to a firm’s marginal costs and marginal revenues, time lags in production and information processing as well as industry structure. The focus of much research has been to enlarge as far as possible the regions of stability of the Cournot model, with regions of instability being considered as regions to avoid as it was assumed (or hoped) that no economy would operate there. The neglect of the regions of instability of the Cournot model was also due to a lack of appropriate technical tools. However, the advances in recent decades in the qualitative theory of nonlinear differential equations and the theory of nonlinear dynamical systems as expounded in Arnold [3], Guckenheiner and Holmes [4] and Jackson [5] have given the economic theorist both the inclination and the tools to analyse the behaviour of oligopoly models in the regions of local instability.

Once the model enters the region of local instability we would expect that nonlinearities and time lags in the governing economic relationships operate so that quantities and prices converge to some stable (possibly quite complicated) attractor. For the economic theorist wishing to analyse the Cournot model in its region of local instability the task is twofold. Firstly, to pinpoint the relevant nonlinearities and time lags. Secondly, to be able to say as much as possible about the nature of the attractor, particularly relating its salient features to the underlying economic relationships. The tools which the modem theory of nonlinear dynamical systems puts at our disposal for this task are (a) bifurcation theory, which enables us to identify regions where certain types of self-sustaining oscillating attractors are born, (b) centre-manifold theory, which allows us to reduce the dimensionality of the differential system under consideration to a manifold on which the attractor lies, and (c) numerical techniques which allow us to explore the types of attractors which may be generated as well as bifurcation diagrams with respect to key parameters of the model.

2049

?OS(l C‘. CHIARELLA and A. KWOMIN

One of the earliest studies of the complex nonlinear dynamics which is possible in the Cournot oligopoly model is that of Rand [6] who shows that simple nonlinear reaction functions can yield a very rich dynamics. Seade [7] analyses the instability of the Cournot oligopoly model and establishes conditions under which the Cournot solution to the oligopoly problem is either unstable or a saddle point with an unstable manifold of dimension at most I. Al-Nowaihi and Levine [S] give sufficient conditions for instability when the adjustment process is continuous and show that these may be significant when the number of firms is small. Furth [9] gives an extensive discussion of the stability and instability of the Coumot model. In contrast to Seade he gives examples of equilibria whose unstable manifold has dimension 2. He also considers the question of whether there is a unique unstable interior equilibrium and shows that whenever there is a unique unstable interior equilibrium there is at least one stable boundary equilibrium. Puu [lo] considers a discrete time duopoly model and shows how simple cost and demand functions can yield the type of nonlinear reaction functions considered by Rand, and he numerically simulates a number of complex attractors under various assumptions about expectations. This analysis has been extended in Puu ill].

In this paper we wish to initiate a study of the evolution of output in continuous time oligopoly models when the equilibrium is locally unstable. To this end we discuss in Section 2 the types of nonlinearities which could come into play to bound the time paths of output. We also introduce time lags in firms’ expectations about other firms’ reponses. which act as bifurcation parameters in the birth of limit cycle motion of the time path of output.

In Section 3 we discuss the modelling of time lags which is inspired by the studies of Howroyd and Russell [l2] and Russell et ul. [ 131. Essentially they introduce three types of lag: a production lag in adjustment of firms’ output to its desired output; an information lag in the receipt of information about rival firms’ output; and an ‘own’ information lag about firms’ own output. In Section 4 we focus mostly on the first two of these within the framework of the Cournot oligopoly modei and, for the special case of identical firms, apply the Hopf bifurcation theorem to consider conditions under which limit cycle motion is born. This involves a relationship between the time lags and cost and demand functions in the model. In Section 5 we use the simple cost and demand functions considered by Puu ]lO, 111 and investigate numerically the types of attractors that emerge in the case of two and three firms. Section 6 draws some conclusions and indicates directions of future research.

2. 4 NONLINEAR OLIGOPOLY MODEI.

I\ natural way to obtain sustained output fluctuations in oligopoly models is to introduce time lags and nonlinearities into the standard formulation. Time lags can be introduced as rn Chapter 7 of [ij or aa in 112. 133. Typically firms adjust their output to desired output with a Lag (due to the production lag referred to in the previous section) and LW an adaptive expectations scheme to form their expectation of the other firms’ output (due to lags in information about the rivals’ output or about their own output). A range of nonlinearities is possible and include the following:

ial introducing noniineur cmt ,functmns UFIL~~OI. dernmd functions. These can have the effect of inducing nonmonotonic reaction functions, as for example is the case with the hyperbolic demand function considered by [ 10. 111. We shall in fact consider this latter demand function in our formulation below.

(h) Introducing constraints @FE output udjustment. Suppose for example that xf is the desired output of firm i and that the firm adjusts its actual output .Y, to the desired level

Nonlinear oligopoly models with time delays 2051

with a lag according to: ii = ki (XT - xi), ki > 0. However, it may be costly for the firm to make large output changes; indeed output changes above or below certain levels may be impossible. This effect could be captured by assuming a relationship between ii and xi as shown in Fig. 1. A specific functional form displaying these characteristics would be

K,(y) = pi tanh k’y ( 1 Pi

, Cki7 Pi ’ O)* (1)

We introduce such a nonlinearity in the simulations reported in Section 5. As stated earlier we assume a nonlinear demand function of the form

p = b I

Eli, (b ’ O), (2) i=l

where p is the price of output. Defining Qi = Cj+iXj as the output of rival firms from the point of view of firm i and

assuming constant marginal cost ci for firm i, firm i’s profit is given by

pi = bXi

x, + Q(e’ - cixi, I I

(3)

where Q I”’ is firm i’s expectation of rivals’ output Qi. The profit maximising condition for firm i is

b bxi

x I + QI” - (Xi + QIe’)* - Cj = 0. (4)

Assuming that each firm acts as a Cournot oligopolist, eqn (4) implies a reaction function relationship between xi and Qr’ which we write as

x. zz g.[Q!@] ,LlY (i = 1, . . ., n), (5) where the function gi is defined by

Fig. 1

205’ C. CHIARELLA and A. KHOMIN

This reaction i%rction has the backward bending shape displayed in Fig. 2a initially proposed in [6] and explored in the form of (6) in [lo, 111 in discrete time ohgopoly models. The system of equations

x, - gj(Qj) = 0. (i = 1. . .) n), (7)

defines the Nash-Cournot equilibrium. Tf we define the average marginal cost as

c, + c2 + . . . + c L; = --------2,

I?

then the (interior) equilibrium point is given by

(8)

(9)

In order to ensure that all equilibria are positive we require that the condition

c; ‘c (c; + c2 + L + 2c, -t . ’ i- cJ/n, ( 10)

be satisfied for all i. This condition states in essence that the marginal costs of the different firms should not be too different.

Note that in the case of identical firms (i.e. I’, = r? = . 1 = c, = c) eqn (9) reduces to

,u = L’ c n i 1

1 -- 2. 3 for all i. (11) n

Figure 2b illustrates the graphical determination of the equilibrium in the special case of duopol y .

3. MODELLMG THE LAG STRUCTURE

We assume that each firm adjusts to the desired level of output with a lag as discussed in Section 2. i.e.

i = k.[g.(Q!“) - x ] i III : * (12)

where ki (>O) (i = 1,. . ., n) are speed of adjustment parameters. This specification of the adjustment process may be viewed as a linearisation of the nonlinear function K, specified in Section 2, in which case ki = K:(O).

Russel et al. [ 131 introduce a time lag Ti that firm i experiences in obtaining information about rivals’ output as well as a lag Si in firm i obtaining or implementing information

i4 !b)

Pip. 2


about its own output. In terms of our current notation their adjustment process would be written as

pi = ki[gi(Ql”‘(t - Ti)) - Xi(t - A’,)]. (13)

Equation (13), being a differential-difference equation, presents the technical difficulty that its eigenvalue spectrum is infinite and hence application of the Hopf bifurcation theorem may become an analytically intractable problem. We must therefore address the problem of finding some way of modelling time lags which will allow applications of finite-dimensional bifurcation techniques. The simplest way seems to be to adopt continuously distributed lags. Thus firm i’s expectation of rivals’ output could be written as

Q’e)(t) = L I ‘W.(t - s)Qi(s)ds, I (14)

0

where Wi(t) is a weighing function that firm i applies to rival firms’ previous output. An appropriate class of weighting function would be

7 m 3 1,

m = 0, (15)

where Ti > 0 and m is an integer. This class has the following properties:

(a) for m = 0, weights are exponentially declining with the most weight being given to the most recent output;

(b) for m 2 1, zero weight is assigned to the most recent output, rising to maximum weight at a point Ti time units in the past and declining exponentially to zero thereafter;

(c) the area under the weighting function is unity for all Ti, for all m; (d) the function becomes more peaked around t = Ti as m increases. Indeed for m = 4

the function may for all practical purposes be regarded as very close to a Dirac delta function (i.e. a unit impulse function) centred at t = Ti, i.e.

I ‘W(t - S; Ti, m)Qi(s)ds z Qi(t - Ti),

0 (W

is not an unreasonable approximation for m 3 4. (e) as Ti + 0, the function tends to a Dirac delta function, i.e.

lim W(t - S; Tiy m) = d(t - s), T,+O

so that

liioltW(t - S; Tip m)Qi(s)ds = Q(t)* 0

Wb)

The weighting function W(t; Ti, m) is illustrated in Fig. 3 for various values of m. This weighting function has been used by Invernizzi and Medio [14] to model time lags in economics.

Adopting the same family of weighting functions for any firm’s own information lag we apply the approximation

‘W(t - S; Si, I)xi(s)ds ~ xi(t - Si),

2024 C. CHIARELLA and A. KHOMIN

Fia 3. The weighting function W

and allow for a different degree I of exponential weighting from that applied to the expectation formation of rival firm’s output

The dynamics of the economy are then governed by the integro-differential equation system

i; = k,[g,(Q!“‘f - X,], (17a)

Q;“(t) -7 j]‘W(i c: I’,. m;)Qj(S)ds, (17b) /

X,(t) = i “W(r -- .A’: S,, i,)x,(s)ds, (17c) li

for i = 1, 2, . . .: n. By differentiating (17b) m, -i- 1 times and (17~) 2, + 1 times we are able to reduce the system (17) to a set of N ordinary differential equations, where

I, )i ,Y = vm .t- Xl, + 3rr. LA / I 1 1-I

For example, if I, = m, = 1, for all i, the system (17) reduces to

for i := 1 . 2. . . n. which is a system of 5n ordinary differential equations. If firms” own information lags are ignored (i.e. S, z 0) then we are dealing with the

integro-differential equation system


i. = k.[g.(Q!“‘) - x.1 I ,,I 1 ) (19a)

Q?‘(t) = lW(t - S; Ti, mi)Qi(S)ds. (19b)

Differentiating equation (19b) mi + 1 times we are able to reduce the system (19) to a set of M ordinary differential equations where

M = ~mi + 2n. i=l

For example, if m, = 1, for all i, the system (19) reduces to

i. = k.[g.(Q!“‘) - x.] I III I 7 (20il)

0s”’ = +[zl - Qi”], (20b) I

ii = $[Qi - Zi], (2Oc) I

for i = 1, . . ., IZ, which is a system of 3n equations. Most of the numerical simulations which we report below are based on this system.

In order to render the eigenvalue analysis of the next section tractable we shall there focus on the important special case in which the n firms are identical, so that

xi = X, Qi = (n - 1)x, Xi = X, for all i, (21a)

k, = k, Ti = T, Si = ST gi = g, for all i. @lb)

The system (16) then reduces (note we are in addition assuming that all firms have the same initial values) to

i = k[g((n - 1)x”‘) - X] 7

x@)(t) = I

‘W(l - s; T, m)x(s)ds 3 0

X(t) = I

‘VV(t - s; S, l)x(s)ds, 0

(2221)

(22b)

(22c)

where x@)(t) is one firm’s expectation at time t of the output of a typical rival firm. The equilibrium point (pi, eie’, Xi) of the integro-differential system (17) for the case of

it different firms is given by

gi(~le’) = gi, (2321)

p’ = Q. I I) (23b)

pi =Xi. (23~)

This last set of equations reduces to

gi CZj = fi, ( 1

(24) jti

which is familiar from the standard discussion of the Cournot oligopoly model as given, for example, in [l] and yields the equilibrium point we have already discussed in Section 2.

2m C’. CHIARELLA and A. KHOMIN

Another form of modelling continuously distributed lags, an alternative to eqn (14). would be

Q;“(t) = 1’ W,(r - s)Q;(s)ds. (S; B O), (2.5) J i - 0

with T, in eqn (15) appropriately chbsen so that the area under the weighting function W, is unity over the interval (t, t - $). This formulation involves taking a weighted average of rival firms’ previous output over the previous 6, time periods rather than over the entire past history of previous outputs. From an economic point of view this formulation would be preferable to the one we have adopted as it seems a little unreasonable to assume that firms will adjust to past outputs right back to initial time especialiy after the economy has been evolving for some time. However with lag structures of the type (25) it is not possible to reduce the dynamics of the oligopoly to a system of ordinary differential equations. Rather we would need to deal with a Volterra integro-differential system (which could be reduced to a delay-differential system). Hence our analysis in this paper focuses on the lag structure of eqn (14). We leave for future research analysis of the nonlinear oligopoly model with the economically more realistic lag structure of equation (25).

4. BIFURCATION ANALYSfS OF THE DYNAMICS OF THE COURNOT MODEL

4. i The general case

In order to analyze the local dynamic behaviour of the integro-differential equation system (17) around the equilibrium (23) we consider the linearised system. Using x(t), Q(f) to now denote deviations of these variables from their equilibrium levels, the linearisation of (23) may be written

i,(t) = k, yt fW( I

I - s: Ti. m,)Q;(s)ds - i&r - s; S,, i.)xi(s)ds]. ., 0

(26)

where yi = S:(&i). We obtain the characteristic equation of the set of linear Volterra integro-differential

equations (26) by employing the techniques expounded by Miller [15]. The essential idea of this approach is to seek solutions of the form

x,(t) = I’( en’,, ii = 1, . ., n). (27)

Substituting this last expression into (26) and allowing t -+ CC yields

mW(s; S,. l,)e-” ds

that is.

A;(A)v, + R,(J”)CVi = 0. (i z ITi

where for notational convenience we set

Ri(~) = i

-k,ri(ilT*!m, + l)-‘mlil’ -kiy;(aT, + I)-‘.

L (28)

Nonlinear oligopoly models with time delays

From (27) we obtain the characteristic equation

AI@) B,(A) B,(A) . . . BI (4

Bz(4 A2(4 B2(A) . . . B2(4

= 0.

Bn (4 Bn(4 B,(A) . . . A,(4

(30)

In the most general case, eqn (30) will be a polynomial in A. of degree N. In this paper we do not attempt an analysis of this most general case, but rather prefer

to focus on some important special cases in order to obtain a feel for the nature of the dynamic behaviour of the oligopoly model with nonlinearities and time delays.

4.2. The case of identical firms

As discussed in Section 3 the dynamics of the model in this case are governed by eqn (22), whose linearisation may be written as

k(t) = ky(n - l)L’W(t - s; T, m)Q(s)ds - ki’W(t - s; S, l)x(s) ds, (31)

where y = g’(( n - l)Z). Recalling the expression (11) for 2 in the case of identical firms we note that

y=;L -1. ( 1 n -1 We observe that

y = 0 for II = 2, y < 0 for n > 2.

The characteristic equation assumes the forms

A(AS + 1) - + 1 (“m’ )‘^+l+k(~+l~+l-ky(n-l)(*~+I)=O,

‘+‘(AT + 1) + k(AT + 1) - ky(n - 1) 1+1

= 0,

~(~+l)“l(~+l~+l+k(~+l)m+l-ky(n-l)(~+l~+l= 0,

(32~1)

Wb)

for 1 = 0, m > 0:

(3321)

for 1 > 0, m = 0,

Wb)

for 1 > 0, m > 0.

(33c)

Clearly a wide range of behaviour of the roots ,J is possible depending on m, I and S, 1’. We shall focus on the case when the firm’s own information lag S is much smaller than T, the information lag about rival firms. We thus set S = 0 in eqn (33~) which reduces to

A+k= ky(n - 1) (AT/m + l)m+l’

(34a)

for m > 0, and (from eqn (33b)) to

(‘. (‘H1AREL.I.A and A. KHOMIN

(34b)

for m = 0. When the equilibrium is locally unstable we seek to demonstrate the existence of limit

cycle motion by applying the Hopf bifurcation theorem. This approach requires determination of regions in the parameter space where the characteristic equation has pure complex roots.

Consider first the case m = 0 when the weighting function is exponentially declining. The eigenvalues are easily calculated as the roots of the quadratic equation

It is a straightforward matter to verify that the real parts of the roots of (35) are negative. given the signs of k. 1’ and 7‘.

For the case of a general value of m graphical analysis of (34a) (see Fig. 4) indicates that for m odd, there is only one real root: which is negative. For m even there are at most two real roots, both of which must be negative.

Analysis of the complex roots of the characteristic equation (34a) for general m seems difficult without resorting to numerical analysis, so we concentrate on the case m = 1. which reduces (34a) to

(2 + kj(A7‘ + lj’ - ky(t7 -- 1) = 0. (36)

To determine parameter values for which (36) may have pure complex roots we seek w such that I. = rtiu satisfies (36). Substituting into (36) and equating real and imaginary parts we find that (o must simultaneously satisfy

((; 2kT --. - + 1 - (,I’ q -- - 7 = .--- k(l .._._ - __-.__ (n - 1)~) _._.. 7‘: 7‘(2 + kT)

(37)

Equating the two expressions for 0’ we find the combination of parameter values for which pure complex roots are possible, namely.

kT I- 1 cz 0,

, m+l

(38j

Nonlinear oligopoly models with time delays 20%

If we regard y as a bifurcation parameter then (38) shows that the real part of the complex roots changes sign at the critical value y* given by

+kT+2, 1 which is graphed as a function of kT in Fig. 5. Comparing this value of y* with the value of y in eqn (32a) we find that n > 19 is required before bifurcations will occur in this case of identical firms. The numerical simulations of the next section indicate that in the case of unequal firms bifurcations occur at much lower values of n.

By differentiating (36) implicitly with respect to y and evaluating the derivatives at A. = ?icc, (with w given by (37)) we find that

-k(2kT + 1)

= 2T2[(2kT + 1)2 + k*(l - (n - l)~)~] < 0. (40)

Thus by the Hopf bifurcation theorem (see e.g. Guckenheimer and Holmes [4]) we can assert the existence of a limit cycle for y in the neighbourhood of y = y*. Since R(A) decreases as y increases through y* the model would display the dynamic behaviour illustrated in Fig. 6, if the limit cycle were stable. To analyse the question of the stability of the limit cycle we resort to numerical methods in Section 5.

4.3. Allowing an own information lag

It is of some interest to see how the results of the previous section are perturbed if we allow the typical firm to have a lag concerning its own information. It seems difficult to obtain analytical results for general I and m. So initially we focus on the special case 1 = m = 0, in which case the characteristic equation reduces to the cubic

A(AS + l)(IT + 1) + k(ilT + 1) - ky(n - l)(AS + 1) = 0. (41)

This equation is best viewed as the intersection of the cubic A(AS + l)(AT + 1) with the straight line k[y( n - 1)s - T]A + k[y(n - 1) - 11, as illustrated in Fig. 7, which is drawn on the assumption that S << T is the most likely situation.

A Y* 1 KT

I >

I -8

(n-1,

Fig. 5

C. CHIARELLA and A. KHOMIN

Fig. 6

k[y(n-: j-i j

\\\\ /c[y(n-IL-T]l+ kly(n-1)-l]

\\

Fig. 7. Introduction of an own information lag S (I = 0, m = 0).

In order to establish the parameter conditions under which complex roots are possible we set II = +io in eqn (41). Equating real and imaginary parts we find that o.? must simultaneously satisfy

. (42)

These two expressions for trr? become equal when

We note that this condition can never be fulfilled when 1’ 5 0. Hence no complex roots are possible in this special case and the disposition of the roots must be as shown in Fig. 7, i.e. three negative real roots indicating local stability of the equilibrium.

A4nother special case which would extend the analysis of Section 4.2 would be 1 = 0, m == 1, in which case the characteristic equation becomes the quark

qAS -+- l)(U -I- I):! = ky(n - I)(hS i- 1) - k(AT + I)“. (44)

This may be viewed as the intersection of the quartic on the left-hand side and the quadratic on the right-hand side of eqn (44). Consideration of the maximum point of the


quadratic and its intersection with the vertical axis indicates that for n sufficiently large there should be at least one pair of complex roots. See Fig. 8.

The parametric conditions for the existence of a pair of complex roots is

TS[l + 2kT - ky(n - l)S12 = (T + 2S)(kT2 + S)(l + 2kT - ky(n - 1)s)

f k(1 - r(n - 1)). (45)

We do not attempt here to apply the Hopf bifurcation theorem to the critical value of y implied by eqn (45).

5. NUMERICAL SIMULATIONS

It is of interest to test the insights gained from the case of identical firms in the previ.ous section on the general case of differing firms. This can only be accomplished by means of numerical simulations. We consider a three-firm model and employ the following data set:

Cl = 0.03, c2 = 0.29, c3 = 0.3,

T1 = 1, T2 = 1, T3 = 2,

kl = 1, k2 = 1, k3 = 0.42,

and b = 0.5. These values have been chosen from numerical analysis of the maximum real part of the eigenvalues &,, R of the Jacobian matrix of the dynamical system (20) and the equilibrium viability conditions (10) in the case of three firms. Thus Fig. 9 displays kaXR as a function of c2 in the interval 0.27 < c2 < 0.33 for various values of T3, and Fig. 10 displays jlmaxR as a function of T3 for T1 = T2 = 1 and c2 = 0.29. By means of such analysis, values of cl, c2, c3, T,, T2, T3, kl, k2, k3 have been chosen close to values at which kaXR changes from negative to positive. It is our conjecture that these values are Hopf bifurcation values but confirmation awaits further analysis.

Figures lla and b display the x1, x2 and x2, x3 phase diagrams which are 2-cycles. An unsatisfactory aspect of these cycles is that at various points some outputs become negative.

The model will operate in an economically feasible region if firms adopt the variable speed of adjustment of eqn (1). We have chosen pi = 0, = /3, = 0.1 and obtained the economically viable cycles displayed in Fig. 12a and b.

Fig. 8. Introduction of an own information lag S (I = 0, m = 1).

C. CHIARELLA and A. KHOMIN

Maximum of real parts (0.331000,0.061776) 618 ) , , , , 1, , , , , ,I, II,, ,111 I,‘, 1111

Ts2

322

254

ReMax -123

--271

-419

0.270 0,278 0.285 0.293 0.301 0.308 0.316 0.323 0.331 C0.270000.--0.0566881)

Maximum of real parts (10.oooooO.0.06003 1) 600

471

LOB

ReMax -149

-261

0.10 1.34 2.58 3.81 5.05 6.29 l.S2 8.76 10.00 (0.1omoo,-0.038406) r3

For the value of t: chosen we experimented with a wide range of c,: 7‘,, ki around the above values but only ever obtained 2-cycles. Analysis of Hopf bifurcation diagrams is still to be carried out and may reveal period-doubling sequences but at this stage on the basis of numerical experiments performed we remain sceptical of this possibility in the triopoly model. We have experimented numerically with the l@firms case and have obtained


(a) (2.586902.0.478182) 0.478

0.269

0.164

0.255 ET

-938 0.241 0.576 0.91 I 1.25 1.58 1.92 2.25 2.59 (-0.093786,-0.359197) 5

@I (0.274263.0.478206) 0.478 , , , , , , ,

- x3vx2

I"" (,I, ,,I, ,I,, ,I,, ,,,I

: table 1 0.373 -

0.269 b

~lIIIIII~tlIIL’ll”“l”“l”“I”“l”‘~l -737 -302 133 568 0.100 0.144 0.187 0.231 0.274 (-0.073704,-0.35946l) *3

Fig. 11. (a) and (b)

qualitative behaviour similar to that of the 3-firms model. Technical programming constraints have prevented us from experimenting with the values of mi. Such further numerical simulations are left for future research.

6. CONCLUSIONS

We have formulated a continuous time oligopoly model with a nonlinear market demand function and allowed for nonlinear speeds of adjustment of firms’ output to desired output.

2064 C. CHIARELLA and A. KHCMIN

(a) (I .796480.0.224890) 0.225

0.164

426

122

I .os 1.17 t .26 1.35 1.44 ! .s3 1.62 I .71 1.80 (1.076758.-0.018 193) xi

@I (0.108492.0.224890) 0.225 , , , ,I, ,111,1,11,1111,1111,11,, ,,I,

(0.003696,-0.018193) T3

‘Turre lags are modelled by way of continuously distributed lags so as to allow reduction of the dynamics to a set of ordinary differential equations. We have obtained analytically Hopf bifurcations results for the special case of’ identical firms. ‘These results indicate that a large number of firms ( J> 19) is required before the conditions of a Hopf bifurcation can be satisfied. Numerical simulations of the case of unequal firms reveal that Hopf bifurcations occur even in the three-firm model. However the resulting limit cycles only become economically viable after introduction of a nonlinear speed of adjustment function. In spite


of a wide range of numerical experiments it has not seemed possible to obtain dynamic behaviour more complex than limit cycles. Certainly there is no evidence of the chaotic behaviour of the discrete time model of Puu [lo, 111 which contains most of the elements of our continuous time model. It is our conjecture that the continuously distributed lag over a fixed time interval or more elaborate expectation mechanisms of the type proposed by Brock and Hommes [16] will yield such complex behaviour. However we leave this analysis for future research.

REFERENCES

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An analysis of the complex dynamic behaviour of nonlinear oligopoly models with time delays