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Market price and the stability of cartels
Economics Letters North-Holland
127
15 (1984) 127-137
MARKET PRICE AND THE STABILITY
OF CARTELS
R. ROTHSCHILD University of L.ancasier, Lancaster LA 1 4 YX, UK University of New South Wales, Kensington 2033, N.S. W. Australia Received
27 September
1984
We analyse a relationship between market price and the stability of cartels. yields insight into the size of the stable cartel relative to the economy.
The approach
The question of cartel stability has recently received attention from a number of writers. The framework of discussion typically involves a cartel and a competitive fringe. The cartel is assumed to be made up of firms which determine market price by equating marginal cost with marginal revenue, while the competitive fringe comprises firms which take the cartel price as given and determine output by equating this price with their marginal costs. As is well known, the larger output produced by the members of the competitive fringe yields to each such firm greater profits than those obtained by members of the cartel. The natural question is therefore under what circumstances the cartel will be ‘stable’. A principal result obtained by one group of researchers is that there exist stable cartels for all economies containing a finite number of firms [see, for example, Donsimoni et al. (1981) and d’Aspremont et al. (1983)]. For the purposes of this analysis, two types of stability have been identified: the cartel is said to be (i) externaZZy stable when entry into the cartel from the competitive fringe would raise market price insufficiently to increase the prospective member’s profits, and (ii) internally stable when the exit of a member would so depress price that he would gain nothing by joining the competitive fringe. 0165-1765/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
128
R. Rothschild / Market price and cartel stability
In this paper we consider a connection between cartel stability as defined here, and the prevailing market price. We shall show, inter alia, that a formulation in these terms yields valuable insight into the relationship between the size and the stability of cartels. We assume, with d’Aspremont et al., that the economy comprises N firms, producing a homogeneous output. We suppose that all firms have identical differentiable cost functions C(q), with marginal cost C’(q) which we take to be increasing in output q. The industry demand function is Q(p), where Q is the combined output of the cartel and the competitive fringe, and Q’(p) < 0. Let the cartel contain K firms, K 2 0, and the competitive fringe N - K. The output of each member of the fringe at price p is given by its supply function and denoted S(p). The demand curve facing the cartel is derived by subtracting the output of the fringe from the market demand at every price. Thus, the demand curve facing the cartel can be identified as
(1) The demand facing the individual firm in the cartel at given price is taken here to be Q”/K. We set Q”/K = qc and assume qc( p) > 0 and qC’(P) < 0. Suppose that, initially, the market is perfectly competitive, and that all firms sell the quantity q” at price p”_ Let p* be the market price which would be associated with the existence of a one-member cartel (i.e., K = 1, or the price leadership case), and qc the output of the price leader. It then follows that a firm will leave the fringe to become the price leader if, for that firm,
(2) Rewriting
(2) yields
( p” + Ap’)( q” - Aq’) -p”qo > AC,
ACSO.
(3)
Here, Ape and Aq” are to be interpreted as absolute values, and represent the changes in price and quantity necessary to secure maximum profits for the firm in its new status, given the quantity supplied by the fringe. The inequality in (3) implies a general condition on p which can be written in the following way: let the subscript i denote the i th prospective member of the cartel, p’ the price prevailing in the market, and A:: and
R. Rothschild / Market price and cartel stability
129
Aq: the change in price and output necessary to maximise i’s profits on entering the cartel. Then the cartel is externally stable at p’ if
p’ > b’,: (qi/Aq: -
1) - AC,/Aq:.
(4)
Now consider again the case where K = 1. The condition necessary to ensure internal stability here is found be rewriting (2) in a somewhat different way. Since we are considering the impact of exit from the cartel, we rewrite (2) to obtain
( p* - Ap*)( q= + Aq’) -p*q’
< AC,
(5)
where Ap* and Aq” have an interpretation analogous to that of Ape and Aq’. Manipulation as before yields a general condition for internal stability: let the subscript j denote the jth prospective member of the competitive fringe, p’ the price prevailing in the market, and A”p;and Aqs the change in price and output necessary to maximise j’s profits on joining the competitive fringe. Then the cartel is internally stable at p’ if p’ < A$ (q,/dq;
+ 1) + AC,/Aq;.
(6)
(4) and (6) together yield a necessary condition for the existence of both external and internal stability at any price p’. It is possible to make the number of firms (in both cartel and fringe) an element in the analysis. Given p’, let the sales of each member of the competitive fringe be LY/(N - K). Suppose now that one firm leaves the fringe and joins the cartel, and that the sales of this firm become /3/(K + 1). Then
and it follows, from (4) that
p’
[I
a(K+ 1) (Y(K+~)-/~(N-K)
(7)
The inequality in (6) can be approached in a similar fashion. Given p’, let the sales of each member of the cartel be y/K. Suppose that one firm
130
R. Rothschild/ Market price and cartelstnbdity
leaves the cartel and joins the fringe, become 6/(N - K + 1). Then,
Aq= ;-,_;,,
and
that
the sales of this firm
a
and it follows, from (6) that
P”APN-,+l
[1 y(N-
y(N-K+l) K+ 1)-s(K),
I+l]+(t$)N_,,,.
(‘I
The justification for bringing out the element of ‘elasticity’ hitherto left implicit in the two definitions of cartel stability, is clear. By doing so, we have made it possible to cast the analysis explicitly in terms of price, and to identify precisely conditions under which cartels will form and how, once formed, they will change in size as price itself changes. Although extended discussion of these conditions is not possible here, some general observations are in order. The conditions in (7) and (8) together imply that, ceteris paribus, (i) the price necessary to ensure both external and internal stability will be lower, the smaller is the effect on price of entry into or exit from the cartel; and (ii) the price necessary to ensure external (internal) stability will be lower, the larger is the ratio a//3 (y/S). Moreover, if CY/~ # (N - K)/( K + 1) then, (iii) when K-C (N + 1)/2, ceteris paribus, the price necessary to ensure both external and internal stability is lower, as K is smaller; but, (iv) when K > (N + 1)/2, ceteris paribus, the price necessary to ensure both external and internal stability is lower, as K is larger. Finally, (v), ceteris paribus, if LX/P = (N - K)/( K + l), then the price necessary to ensure external stability is infinite. The results given here provide some insight into the relative influences of cartel size (measured in terms of K) and cartel share (measured in terms of y). They give little support to the intuition that, given price, the potential for instability is greater, the larger is the cartel relative to the economy. Specifically, (ii) and (iii) show that, ceteris paribus, if K < (N + 1)/2, then the price necessary to ensure internal stability is lower, as y is larger and K is smaller (i.e., as the share of each member of the cartel is larger); (ii) and (iv) show that, ceteris paribus, if K > (N + 1)/2, then the price necessary to ensure external stability is lower, as y is smaller and K is larger (i.e., as the share of each member of the cartel is smaller). The result given in (v) shows a discontinuity in the relationship between price
R. Rothschild / Market price and cartel stability
131
and cartel stability. If the shares of a given firm are the same in both cartel and fringe, then there is no finite price which will ensure external stability of the cartel.
References d’Aspremont, C., A. Jacquemin, J.-J. Gabszewicz and J.A. Weymark, 1983, On the stability of collusive price leadership, Canadian Journal of Economics XVI, 17-25. Donsimoni, M.-P., N.S. Economides and H.M. Polemarchakis, 1981, Stable cartels, Document de travail no. 8112 (Institut National de la Statistique et des Etudes Economiques, Paris).
127
15 (1984) 127-137
MARKET PRICE AND THE STABILITY
OF CARTELS
R. ROTHSCHILD University of L.ancasier, Lancaster LA 1 4 YX, UK University of New South Wales, Kensington 2033, N.S. W. Australia Received
27 September
1984
We analyse a relationship between market price and the stability of cartels. yields insight into the size of the stable cartel relative to the economy.
The approach
The question of cartel stability has recently received attention from a number of writers. The framework of discussion typically involves a cartel and a competitive fringe. The cartel is assumed to be made up of firms which determine market price by equating marginal cost with marginal revenue, while the competitive fringe comprises firms which take the cartel price as given and determine output by equating this price with their marginal costs. As is well known, the larger output produced by the members of the competitive fringe yields to each such firm greater profits than those obtained by members of the cartel. The natural question is therefore under what circumstances the cartel will be ‘stable’. A principal result obtained by one group of researchers is that there exist stable cartels for all economies containing a finite number of firms [see, for example, Donsimoni et al. (1981) and d’Aspremont et al. (1983)]. For the purposes of this analysis, two types of stability have been identified: the cartel is said to be (i) externaZZy stable when entry into the cartel from the competitive fringe would raise market price insufficiently to increase the prospective member’s profits, and (ii) internally stable when the exit of a member would so depress price that he would gain nothing by joining the competitive fringe. 0165-1765/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
128
R. Rothschild / Market price and cartel stability
In this paper we consider a connection between cartel stability as defined here, and the prevailing market price. We shall show, inter alia, that a formulation in these terms yields valuable insight into the relationship between the size and the stability of cartels. We assume, with d’Aspremont et al., that the economy comprises N firms, producing a homogeneous output. We suppose that all firms have identical differentiable cost functions C(q), with marginal cost C’(q) which we take to be increasing in output q. The industry demand function is Q(p), where Q is the combined output of the cartel and the competitive fringe, and Q’(p) < 0. Let the cartel contain K firms, K 2 0, and the competitive fringe N - K. The output of each member of the fringe at price p is given by its supply function and denoted S(p). The demand curve facing the cartel is derived by subtracting the output of the fringe from the market demand at every price. Thus, the demand curve facing the cartel can be identified as
(1) The demand facing the individual firm in the cartel at given price is taken here to be Q”/K. We set Q”/K = qc and assume qc( p) > 0 and qC’(P) < 0. Suppose that, initially, the market is perfectly competitive, and that all firms sell the quantity q” at price p”_ Let p* be the market price which would be associated with the existence of a one-member cartel (i.e., K = 1, or the price leadership case), and qc the output of the price leader. It then follows that a firm will leave the fringe to become the price leader if, for that firm,
(2) Rewriting
(2) yields
( p” + Ap’)( q” - Aq’) -p”qo > AC,
ACSO.
(3)
Here, Ape and Aq” are to be interpreted as absolute values, and represent the changes in price and quantity necessary to secure maximum profits for the firm in its new status, given the quantity supplied by the fringe. The inequality in (3) implies a general condition on p which can be written in the following way: let the subscript i denote the i th prospective member of the cartel, p’ the price prevailing in the market, and A:: and
R. Rothschild / Market price and cartel stability
129
Aq: the change in price and output necessary to maximise i’s profits on entering the cartel. Then the cartel is externally stable at p’ if
p’ > b’,: (qi/Aq: -
1) - AC,/Aq:.
(4)
Now consider again the case where K = 1. The condition necessary to ensure internal stability here is found be rewriting (2) in a somewhat different way. Since we are considering the impact of exit from the cartel, we rewrite (2) to obtain
( p* - Ap*)( q= + Aq’) -p*q’
< AC,
(5)
where Ap* and Aq” have an interpretation analogous to that of Ape and Aq’. Manipulation as before yields a general condition for internal stability: let the subscript j denote the jth prospective member of the competitive fringe, p’ the price prevailing in the market, and A”p;and Aqs the change in price and output necessary to maximise j’s profits on joining the competitive fringe. Then the cartel is internally stable at p’ if p’ < A$ (q,/dq;
+ 1) + AC,/Aq;.
(6)
(4) and (6) together yield a necessary condition for the existence of both external and internal stability at any price p’. It is possible to make the number of firms (in both cartel and fringe) an element in the analysis. Given p’, let the sales of each member of the competitive fringe be LY/(N - K). Suppose now that one firm leaves the fringe and joins the cartel, and that the sales of this firm become /3/(K + 1). Then
and it follows, from (4) that
p’
[I
a(K+ 1) (Y(K+~)-/~(N-K)
(7)
The inequality in (6) can be approached in a similar fashion. Given p’, let the sales of each member of the cartel be y/K. Suppose that one firm
130
R. Rothschild/ Market price and cartelstnbdity
leaves the cartel and joins the fringe, become 6/(N - K + 1). Then,
Aq= ;-,_;,,
and
that
the sales of this firm
a
and it follows, from (6) that
P”APN-,+l
[1 y(N-
y(N-K+l) K+ 1)-s(K),
I+l]+(t$)N_,,,.
(‘I
The justification for bringing out the element of ‘elasticity’ hitherto left implicit in the two definitions of cartel stability, is clear. By doing so, we have made it possible to cast the analysis explicitly in terms of price, and to identify precisely conditions under which cartels will form and how, once formed, they will change in size as price itself changes. Although extended discussion of these conditions is not possible here, some general observations are in order. The conditions in (7) and (8) together imply that, ceteris paribus, (i) the price necessary to ensure both external and internal stability will be lower, the smaller is the effect on price of entry into or exit from the cartel; and (ii) the price necessary to ensure external (internal) stability will be lower, the larger is the ratio a//3 (y/S). Moreover, if CY/~ # (N - K)/( K + 1) then, (iii) when K-C (N + 1)/2, ceteris paribus, the price necessary to ensure both external and internal stability is lower, as K is smaller; but, (iv) when K > (N + 1)/2, ceteris paribus, the price necessary to ensure both external and internal stability is lower, as K is larger. Finally, (v), ceteris paribus, if LX/P = (N - K)/( K + l), then the price necessary to ensure external stability is infinite. The results given here provide some insight into the relative influences of cartel size (measured in terms of K) and cartel share (measured in terms of y). They give little support to the intuition that, given price, the potential for instability is greater, the larger is the cartel relative to the economy. Specifically, (ii) and (iii) show that, ceteris paribus, if K < (N + 1)/2, then the price necessary to ensure internal stability is lower, as y is larger and K is smaller (i.e., as the share of each member of the cartel is larger); (ii) and (iv) show that, ceteris paribus, if K > (N + 1)/2, then the price necessary to ensure external stability is lower, as y is smaller and K is larger (i.e., as the share of each member of the cartel is smaller). The result given in (v) shows a discontinuity in the relationship between price
R. Rothschild / Market price and cartel stability
131
and cartel stability. If the shares of a given firm are the same in both cartel and fringe, then there is no finite price which will ensure external stability of the cartel.
References d’Aspremont, C., A. Jacquemin, J.-J. Gabszewicz and J.A. Weymark, 1983, On the stability of collusive price leadership, Canadian Journal of Economics XVI, 17-25. Donsimoni, M.-P., N.S. Economides and H.M. Polemarchakis, 1981, Stable cartels, Document de travail no. 8112 (Institut National de la Statistique et des Etudes Economiques, Paris).