Cournot oligopoly and output adjustment costs

113

Economics Letters 7 (1981) 113-117 North-Holland Publishing Company

COURNOT OLIGOPOLY AND OUTPUT ADJUSTMENT COSTS Terry D. HOWROYD University

of New Brunswick,

Fredericton,

N. B., Cunudu E3B 5A3

John A. RICKARD Griffith University, Received

4 April

Nathan,

Queensland 41 II, Austruliu

1981

The stability of the Coumot oligopoly model is re-examined, taking into account possible costs which may be incurred when firms change their production output from one business period to the next. It is found that such output adjustment costs do not affect equilibrium outputs and prices, but do tend to have a stabilizing effect on the market.

1. Introduction Since the early work of Theocharis (1960), Fisher (1961) and others, the dynamics of the Cournot oligopoly model has been analysed extensively under varying assumptions. An excellent review of recent work can be found in Okuguchi (1976). To date, in the works cited above and elsewhere, no specific attention has been given to the adjustment costs which are frequently incurred by firms as a consequence of changing production output from one business period to the next. In practice, certain set-up, shut-down or other costs are incurred when a firm changes its production output. That is, in addition to the costs associated with producing a given output, there are frequently costs incurred in adjusting manpower, equipment, raw materials and other requirements from one level to another. The cost of producing at a new level, whether higher or lower than previously, is greater on the first occasion than on subsequent occasions. The main purpose of this paper is to examine the effect of output adjustment costs on the Cournot equilibrium output and the stability of 01651765/81/0000-0000/$02.50

0 1981 North-Holland

T. D. Howroyd, J.A. Rickurd / Cournot oligopo(v und output adjustment

114

costs

this output. The paper also compares and contrasts the stability of a model incorporating such costs with familiar results obtained when such costs are absent. The results of our analysis may be summarized as follows. Firstly, the presence of these adjustment costs appears not to change the equilibrium outputs and profits of the firms; it does, however, have an effect on the dynamic approach to equilibrium. Secondly, adjustment costs appear to have a stabilizing effect; it is possible for unstable situations in the absence of adjustment costs to become stable when adjustment costs are present. The latter agrees with our intuition; the presence of adjustment costs inhibits production variation by firms and thus enhances stability.

2. The model We suppose that there are n firms F,, . . . , F,, and demand function pi and cost function C, given by p, =

a, - b&x,,

c, =gi +cixi

that

firm E; has

(1)

+fd,xf

++,(xj

-x;-

)‘.

(2)

Here x, and x, are the outputs of firm 4 at time t and time t - 1 respectively. We shall make the obvious restrictions on the parameters b; >O,

c, >O,

We shall also require ai >CI,

g, 20,

e, 20. that b,ci + a,di > 0;

thus we ensure that each firm output; that individual profit ment of production; and that mum permissible output. As a b, +d,

>O,

(3)

(4)

may have overhead costs independent of positions start to improve at commencemarginal costs are still positive at maxiconsequence of these inequalities we have

2b, + d, + e, > 0,

(5)

so that in the absence of output adjustment costs individual marginal costs cannot decline as fast as price; and profits are subject to the law of diminishing returns.

T. D. Howrqvd, J.A. Rickard / Cournor ohgopo!lJ und output udjustment

costs

115

The last term in the expression for C, measures an output adjustment cost which is positive no matter whether the output decreases or increases from one period to the next, so that it could occur that a decrease in production is accompanied initially by an increase in total costs. In general, we expect the total cost of production to rise with increase in output and fall with decrease in output. However, in changing from one level of production to a higher or lower level of production, most firms incur adjustment costs. The costs are normally incurred when production is increased or decreased to some new level. However, if production is then maintained at the new level for a further business period then such costs will be absent. In summary, we expect the cost of producing at a particular level to be greater on the first occasion than on an immediately subsequent occasion. The profit function for firm F; is 57,

=p,x,

-

c,.

Under the Cournot assumption firm E; sets its production for the next period, x: say, by assuming its competitors do not change their outputs from the previous period, and then by finding the value of x, which maximizes 4; this is

x,*=h,’

i

a, -ci

-bj

2 x,- +

,#f

e,xip

where h, = 2 b, + d, + e,. Equilibrium x*=x I

1 =& 1’

i=l

i

(6)

,

is achieved

when

,..., n.

It follows that (2b, + d;)6, = 0, - c, - b, z

a,,

Jfi

and so the equilibrium outputs 6,, . . . ,S, are independent of e,, . . . ,e,. We see that equilibrium outputs, prices and profits are independent of the output adjustment costs. To describe the dynamics of the Cournot oligopoly model, Fisher used the difference equations x * -x- I

= k,(x:

-x,-

).

(8)

116

T.D. Howroyd, J.A. Rickard / Cournvt oligopo!v and output mijustment

costs

The constants k, are positive, and may be regarded as speeds of adjustment; they represent, in a sense, the degree of confidence that the firms have in their calculation of the optimal outputs XT. From (6) and (8) we obtain the linear system of difference equations x=a+(1-M)x-,

(9)

where x and x - are the column matrices with ith components x, and x,respectively, 1 is the identity matrix, M is the matrix KH( BA + B + D) in which A is the square matrix with all entries unity, and K, H, B,D are diagonal matrices with i th diagonal elements k;, h, ‘, b,, d, respectively. The difference eq. (9) is stable if and only if the eigenvalues X of the matrix I - M lie in the disk 1X 1< 1. But M is similar to the symmetric matrix N = ( KHB)-‘j2M( KHB)‘12, and consequently the eigenvalues of N, M and I - M are real. N is also positive definite since the quadratic form

xTj’!J~= (zk;/2h,-‘/2b’/2X.j2 + Zk;h;‘(b; I

+ d,)x,2,

is positive definite, and so the eigenvalues of N, M, are positive. We see from Gersgorin’s theorem (rows) [see, e.g., Woods (1978)] that the eigenvalues of M lie in the interval IhI dmaxk,h,-‘((n+ I

l)b, +d,),

and so the inequalities k,(( n + l)b, + d;) < 4b, + 2d, + 2e,, suffice to ensure stability of the difference of the Cournot equilibrium. We note that the eq. (7) may be written

(10) eq. (9), and thus the stability in matrix

form

The same argument as we used for the matrix N shows that the matrix BA + B + D is positive definite and hence non-singular. This proves the existence and uniqueness of the equilibrium S. In the special case when the firms are identical, dropping the subscripts on the parameters we find that the eigenvalues of M are k(b _t

T. D. Howroyd, J.A. Rickurd / Courno~ oligopoly and output adjustment costs

d)/(2b + d + e) (with multiplicity e), so that we have stability if k((n+

l)b+d)<4b+2d+2e,

11I

n - 1) and k((n + 1)b + d)/(2b + d +

(11)

finite oscillations about equilibrium output if the inequality in (11) is replaced by equality, and instability if the inequality in (11) is reversed. In the special case when e = 0, (11) gives the result obtained by Fisher in his Theorem 3.

References Fisher, F.M., 196 I, The stability of the Coumot oligopoly solution: The effects of speeds of adjustment and increasing marginal costs, Review of Economic Studies 28, 125- 135. Okuguchi, K., 1976, Expectations and stability in oligopoly models. Lecture Notes in Economics and Mathematical Systems 138 (Springer, New York). Theocharis, R.D., 1960, On the stability of the Cournot solution of the oligopoly problem, Review of Economic Studies 27, 133- 134. Woods, J.E., 1978, Mathematical economics (Longman. London).