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On the sustainability of collusion in differentiated duopolies
Economics Letters North-Holland
33
40 (1992) 33-37
On the sustainability of collusion in differentiated duopolies R. Rothschild Lancaster Received Accepted
Vniuersity, Lancaster,
UK
7 May 1992 15 July 1992
We consider a rule designed to maintain collusion in a duopoly made up of firms producing differentiated products. We analyse both price-setting and quantity-setting models, and show that product differentiation plays a crucial role in determining the effectiveness of this rule. Our results contrast with those obtained by earlier writers.
A great deal of attention has in recent years been given to the problem of maintaining cartel stability in the face of strong individual incentives to cheat. Amongst the best-known of the strategies which have been devised to this end is that of Friedman (1983). The essence of the approach is the threat that if deviation from the collusive agreement were to occur, then the ‘loyal’ members of the coalition would revert with a one-period lag to noncooperation. Where such punishment strategies take their most severe form, this reversion is of infinite duration, so that the prospective deviant must compare the short-term gain with the loss which will result from punishment in perpetuity. If the discount rate is sufficiently low, then such deviation will be deterred. There are several aspects of this literature which are of interest. The first concerns the credibility of the threat by the loyal members of the coalition to implement their punishment strategy. As Rothschild (1985) and al-Nowaihi and Levine (1992), amongst others, have argued, the implementation of the threat may leave the punishers with lower profits than they can obtain by allowing the deviant to go unpunished. Indeed, it is by now well-established that loyal firms might profitably ‘renegotiate’ [see Farrell and Maskin (1989)l once deviation has occurred, and thereby obtain higher profits than they would if they implemented their punishment. The second aspect concerns the focus of discussion on industries in which the good is homogeneous. Recent contributions by Deneckere (1983) and Ross (1992) have attempted to take explicit account of the effect of product differentiation on the incentive to deviate when collusion is maintained by threats of the type proposed by Friedman. Their results suggest that when firms compete in prices, the probability that the cartel will be stable is initially decreasing, but subsequently increasing, in the degree of product homogeneity, so that the cartel is most likely to be stable when the products are perfect substitutes. A third aspect involves the need to compare the effectiveness of threat strategies in differentiated oligopolies in which firms may use price as the principal strategic Correspondence
01651765/92/$05.00
to: R. Rothschild,
Lancaster
0 1992 - Elsevier
University,
Science
Lancaster
Publishers
LA1 4YX, UK.
B.V. All rights reserved
34
R. Rothschild / Sustainability of collusion in differentiated duopolk
variable, with those in which the firms adjust output. Such a comparison has been made for a duopoly by Deneckere (1983). He shows that when products are good substitutes, collusion is better supported in price setting games than in quantity games, while the reverse is the case when substitutability is low. However, as Majerus (1988) has demonstrated, this result is not confirmed as the number of firms in the industry increases. The purpose of this paper is to report some results which address these three concerns, and which contrast with those offered by previous authors. In order to do so, we consider a differentiated duopoly in which the firms (denoted 1 and 2) produce their output qi, i = 1,2, subject to identical, constant marginal costs. Without loss of generality we set these costs equal to zero. The measure of substitutability of the output of the two firms is d, d 2 0, where d = 0 implies that the markets are effectively independent, and d = cc,implies that the products are perfect substitutes. The demand curve facing the industry is of the form Q = 1 -p, where Q = q1 + q2. Given d and the choice of strategic variable, let X-“‘,arch and n-’ respectively represent for each duopolist the cartel profits, the profits from deviation, and the profits at the noncooperative equilibrium. If both firms have the same discount rate (Y, cy > 0, then deviation from the cartel agreement will be deterred as long as
(1) The questions which we wish to consider are the following. First, what is the effect of product differentiation on the incentive to deviate from the cartel? Second, is the incentive to punish affected by the existence of product differentiation? Third, are the results different according to whether the duopolists compete in price or in output? Fourth, are differentiated-product cartels which compete in price likely to be more or less stable than those that compete in output? We consider the two cases in turn. In the first, the duopolists compete in prices; in the second, competition takes the form of adjustment in quantities. We show that the stability of the cartel depends crucially upon both the degree of product substitutability and the choice of strategic variable. Bertrand behaviour (price competition) If price is the strategic variable, then the demand for the output of firms 1 and 2 is given by
and
(3) The condition for a Bertrand noncooperative
equilibrium is
(4)
R. Rothschild / Sustainability of collusion in differentiated duopolies
for i = 1,2; j = 2,1. Solving yields pi = l/(2 + d/2), qi = (2 + d)/P(4 q? =
1 +d/2
3.5
+ &I and
(5)
2( 2 + d/2)2
for i = 1,2. If both firms sell at the joint-profit maximising price and share the collusive output, then each obtains rTTm = l/8. By contrast, if one firm adheres to the collusive agreement while the other undertakes a profit-maximising price-deviation, then [from (411,the deviant sets a price (4 + d)/(4(2 + d)). Substitution in (3) to establish the deviant’s output, and some manipulation, yields
ch_
?r
-
(4+dj2 64(2 +d)
(6)
’
Substitution of these values of r”‘, rch and rc in (1) yields the condition on (Ywhich must be satisfied if deviation is to be deterred. This condition,
cul
8(2 + d) (4+d)2
(7)
’
depends exclusively on the substitutability of the two products. Since rch > P iff d > 0, deviation is initially profitable as long as the degree of substitutability is positive. More generally, the likelihood of deviation from the cartel - and hence the probability of cartel instability - is monotonically increasing in d. This result contrasts with those of Deneckere (19831, Majerus (1988) and Ross (1992). Let 7rL, &=
d2-4d-8 32(2+d)
(8)
’
be the profits to the loyal member of the cartel once the cheater has undertaken its profit-maximising deviation. lf the threat of reversion to noncooperative behaviour is to be credible, then we require that for the loyal firm, rrc 2 r ‘. Some substitution establishes that this requires d2(8 + d2 + 4d)
32(4 + d)2(2 + d)
2 0,
which is always true. In summary: Deviation from the collusive agreement in a price-setting duopoly is profitable only if the degree of substitutability between the products is greater than zero. Moreover, the greater the degree of substitutability, the greater is the incentive to deviate and therefore the more unstable the cartel will be. By contrast, the threat to punish such deviation is always credible and independent of the substitutability of the products.
36
R. Rothschild / Sustainability of collusion in differentiated duopolies
Cournot behaviour (quantity competition)
If quantity is the strategic variable, then the noncooperative
equilibrium condition is
= 0
for i = 1,2; j = 2, I. Solving yields qi = (1 + d)/(4 =_
r-
(1+4(2+4 (4+3d)2
(9) + 3d), pi = (1 + d/2)/(2
+ 3d/2)
and
(10)
.
As before, if both firms sell at the joint-profit maximising price and share the collusive output, then each obtains T”’ = l/8. If one firm adheres to the collusive agreement and continues to produce the output $, while the other undertakes a profit maximising quantity deviation, then [from (911 ch
_
Tr
-
(4 + WC8 + 34 64(2 +d)2
(11)
’
As in the previous case, rch - TP > 0 iff d > 0. Substitution of mm, rCh and rrc in (1) then yields the condition on (Ywhich must be satisfied if such deviation is to be deterred: 8d( 2 + d)2 (y < (4 + 3d)2(4
(12)
+ d) .
Once again, the condition on LYis determined exclusively by the degree of substitutability of the two products. However, since the derivative of the right-hand side of (12) with respect to d is 64(2 + d)(d2
+ 3d + 4)
(4 + 3d)3(4
+ d)2
> 0,
the quantity-setting cartel contrasts in an obvious way with one in which price is the strategic variable. In the latter case, the cartel becomes less stable as the products become more homogeneous; in the former, the cartel becomes more stable with increasing homogeneity. As before, let rrL be the profits to the loyal member of the cartel once the cheater has undertaken its profit-maximising deviation. Since the output of the loyalist is i, rL=
8 + 3d
(13)
32(2 + d) .
Subtracting (10) from (13) we obtain -
d(5d2 + 16d + 16) 32(4 + 3d)2(2
+ d)
< 0.
(14)
R. Rothschild / Sustainability of collusion in differentiated duopolies
37
In summary: Deviation from the collusive agreement in a quantity-setting duopoly is profitable only if the degree of substitutability between the products is greater than zero. In this case, stability of the cartel is monotonically increasing in d. The threat to punish such deviation is always credible and independent of the substitutability of the products. The foregoing results make the comparison of the two cases straightforward. As a final step it is useful to consider whether stability of the price-setting cartel implies stability of the quantity-setting cartel, or vice-versa. In order to do this, we need only establish whether the right-hand side of (7) is larger or smaller than the right-hand side of (12). Some simple manipulation shows that (7) is larger than (12) if
d2(d- 3) < 16 l+d
or d I 6. If this is so, then stability of a quantity-setting cartel implies that the price-setting cartel will also be stable. Conversely, if d > 6, then stability of the price-setting cartel implies that the quantity-setting cartel will also be stable. The results obtained in this paper contrast with those of Deneckere (1983) and Ross (1992). It would seem natural, as a next step, to investigate within the context of this model the relationship between product differentiation and the stability of n-firm cartels, and thereby to offer a possible contrast to the results obtained by Majerus (1988) for an n-firm extension of Deneckere’s analysis. It would also be interesting to consider the implications for the present case of applying a more severe punishment rule of the type proposed by Abreu (1986).
References Abreu, D., 1986, Extremal equilibria of oligopolistic supergames, Journal of Economic Theory 39, 191;225. Al-Nowaihi, A. and P. Levine, 1992, Credibility and the degree of collusion among oligopolists, Mimeo. (University of Leicester, Leicester). Deneckere, R., 1983, Duopoly supergames with product differentiation, Economics Letters 11, 37-42. Farrell, J. and E. Maskin, 1989, Renegotiation in repeated games, Games and Economic Behaviour 1, 327-360. Friedman, J., 1983, Oligopoly theory (Cambridge University Press, New York). Majerus, D., 1988, Price vs quantity competition in oligopoly supergames, Economics Letters 27, 293-297. Rothschild, R., 1985, Noncooperative behaviour as a credible threat, Bulletin of Economic Research 37, 245-248. Ross, T.W., 1992, Cartel stability and product differentiation, International Journal of Industrial Organization 10, 1-13.
33
40 (1992) 33-37
On the sustainability of collusion in differentiated duopolies R. Rothschild Lancaster Received Accepted
Vniuersity, Lancaster,
UK
7 May 1992 15 July 1992
We consider a rule designed to maintain collusion in a duopoly made up of firms producing differentiated products. We analyse both price-setting and quantity-setting models, and show that product differentiation plays a crucial role in determining the effectiveness of this rule. Our results contrast with those obtained by earlier writers.
A great deal of attention has in recent years been given to the problem of maintaining cartel stability in the face of strong individual incentives to cheat. Amongst the best-known of the strategies which have been devised to this end is that of Friedman (1983). The essence of the approach is the threat that if deviation from the collusive agreement were to occur, then the ‘loyal’ members of the coalition would revert with a one-period lag to noncooperation. Where such punishment strategies take their most severe form, this reversion is of infinite duration, so that the prospective deviant must compare the short-term gain with the loss which will result from punishment in perpetuity. If the discount rate is sufficiently low, then such deviation will be deterred. There are several aspects of this literature which are of interest. The first concerns the credibility of the threat by the loyal members of the coalition to implement their punishment strategy. As Rothschild (1985) and al-Nowaihi and Levine (1992), amongst others, have argued, the implementation of the threat may leave the punishers with lower profits than they can obtain by allowing the deviant to go unpunished. Indeed, it is by now well-established that loyal firms might profitably ‘renegotiate’ [see Farrell and Maskin (1989)l once deviation has occurred, and thereby obtain higher profits than they would if they implemented their punishment. The second aspect concerns the focus of discussion on industries in which the good is homogeneous. Recent contributions by Deneckere (1983) and Ross (1992) have attempted to take explicit account of the effect of product differentiation on the incentive to deviate when collusion is maintained by threats of the type proposed by Friedman. Their results suggest that when firms compete in prices, the probability that the cartel will be stable is initially decreasing, but subsequently increasing, in the degree of product homogeneity, so that the cartel is most likely to be stable when the products are perfect substitutes. A third aspect involves the need to compare the effectiveness of threat strategies in differentiated oligopolies in which firms may use price as the principal strategic Correspondence
01651765/92/$05.00
to: R. Rothschild,
Lancaster
0 1992 - Elsevier
University,
Science
Lancaster
Publishers
LA1 4YX, UK.
B.V. All rights reserved
34
R. Rothschild / Sustainability of collusion in differentiated duopolk
variable, with those in which the firms adjust output. Such a comparison has been made for a duopoly by Deneckere (1983). He shows that when products are good substitutes, collusion is better supported in price setting games than in quantity games, while the reverse is the case when substitutability is low. However, as Majerus (1988) has demonstrated, this result is not confirmed as the number of firms in the industry increases. The purpose of this paper is to report some results which address these three concerns, and which contrast with those offered by previous authors. In order to do so, we consider a differentiated duopoly in which the firms (denoted 1 and 2) produce their output qi, i = 1,2, subject to identical, constant marginal costs. Without loss of generality we set these costs equal to zero. The measure of substitutability of the output of the two firms is d, d 2 0, where d = 0 implies that the markets are effectively independent, and d = cc,implies that the products are perfect substitutes. The demand curve facing the industry is of the form Q = 1 -p, where Q = q1 + q2. Given d and the choice of strategic variable, let X-“‘,arch and n-’ respectively represent for each duopolist the cartel profits, the profits from deviation, and the profits at the noncooperative equilibrium. If both firms have the same discount rate (Y, cy > 0, then deviation from the cartel agreement will be deterred as long as
(1) The questions which we wish to consider are the following. First, what is the effect of product differentiation on the incentive to deviate from the cartel? Second, is the incentive to punish affected by the existence of product differentiation? Third, are the results different according to whether the duopolists compete in price or in output? Fourth, are differentiated-product cartels which compete in price likely to be more or less stable than those that compete in output? We consider the two cases in turn. In the first, the duopolists compete in prices; in the second, competition takes the form of adjustment in quantities. We show that the stability of the cartel depends crucially upon both the degree of product substitutability and the choice of strategic variable. Bertrand behaviour (price competition) If price is the strategic variable, then the demand for the output of firms 1 and 2 is given by
and
(3) The condition for a Bertrand noncooperative
equilibrium is
(4)
R. Rothschild / Sustainability of collusion in differentiated duopolies
for i = 1,2; j = 2,1. Solving yields pi = l/(2 + d/2), qi = (2 + d)/P(4 q? =
1 +d/2
3.5
+ &I and
(5)
2( 2 + d/2)2
for i = 1,2. If both firms sell at the joint-profit maximising price and share the collusive output, then each obtains rTTm = l/8. By contrast, if one firm adheres to the collusive agreement while the other undertakes a profit-maximising price-deviation, then [from (411,the deviant sets a price (4 + d)/(4(2 + d)). Substitution in (3) to establish the deviant’s output, and some manipulation, yields
ch_
?r
-
(4+dj2 64(2 +d)
(6)
’
Substitution of these values of r”‘, rch and rc in (1) yields the condition on (Ywhich must be satisfied if deviation is to be deterred. This condition,
cul
8(2 + d) (4+d)2
(7)
’
depends exclusively on the substitutability of the two products. Since rch > P iff d > 0, deviation is initially profitable as long as the degree of substitutability is positive. More generally, the likelihood of deviation from the cartel - and hence the probability of cartel instability - is monotonically increasing in d. This result contrasts with those of Deneckere (19831, Majerus (1988) and Ross (1992). Let 7rL, &=
d2-4d-8 32(2+d)
(8)
’
be the profits to the loyal member of the cartel once the cheater has undertaken its profit-maximising deviation. lf the threat of reversion to noncooperative behaviour is to be credible, then we require that for the loyal firm, rrc 2 r ‘. Some substitution establishes that this requires d2(8 + d2 + 4d)
32(4 + d)2(2 + d)
2 0,
which is always true. In summary: Deviation from the collusive agreement in a price-setting duopoly is profitable only if the degree of substitutability between the products is greater than zero. Moreover, the greater the degree of substitutability, the greater is the incentive to deviate and therefore the more unstable the cartel will be. By contrast, the threat to punish such deviation is always credible and independent of the substitutability of the products.
36
R. Rothschild / Sustainability of collusion in differentiated duopolies
Cournot behaviour (quantity competition)
If quantity is the strategic variable, then the noncooperative
equilibrium condition is
= 0
for i = 1,2; j = 2, I. Solving yields qi = (1 + d)/(4 =_
r-
(1+4(2+4 (4+3d)2
(9) + 3d), pi = (1 + d/2)/(2
+ 3d/2)
and
(10)
.
As before, if both firms sell at the joint-profit maximising price and share the collusive output, then each obtains T”’ = l/8. If one firm adheres to the collusive agreement and continues to produce the output $, while the other undertakes a profit maximising quantity deviation, then [from (911 ch
_
Tr
-
(4 + WC8 + 34 64(2 +d)2
(11)
’
As in the previous case, rch - TP > 0 iff d > 0. Substitution of mm, rCh and rrc in (1) then yields the condition on (Ywhich must be satisfied if such deviation is to be deterred: 8d( 2 + d)2 (y < (4 + 3d)2(4
(12)
+ d) .
Once again, the condition on LYis determined exclusively by the degree of substitutability of the two products. However, since the derivative of the right-hand side of (12) with respect to d is 64(2 + d)(d2
+ 3d + 4)
(4 + 3d)3(4
+ d)2
> 0,
the quantity-setting cartel contrasts in an obvious way with one in which price is the strategic variable. In the latter case, the cartel becomes less stable as the products become more homogeneous; in the former, the cartel becomes more stable with increasing homogeneity. As before, let rrL be the profits to the loyal member of the cartel once the cheater has undertaken its profit-maximising deviation. Since the output of the loyalist is i, rL=
8 + 3d
(13)
32(2 + d) .
Subtracting (10) from (13) we obtain -
d(5d2 + 16d + 16) 32(4 + 3d)2(2
+ d)
< 0.
(14)
R. Rothschild / Sustainability of collusion in differentiated duopolies
37
In summary: Deviation from the collusive agreement in a quantity-setting duopoly is profitable only if the degree of substitutability between the products is greater than zero. In this case, stability of the cartel is monotonically increasing in d. The threat to punish such deviation is always credible and independent of the substitutability of the products. The foregoing results make the comparison of the two cases straightforward. As a final step it is useful to consider whether stability of the price-setting cartel implies stability of the quantity-setting cartel, or vice-versa. In order to do this, we need only establish whether the right-hand side of (7) is larger or smaller than the right-hand side of (12). Some simple manipulation shows that (7) is larger than (12) if
d2(d- 3) < 16 l+d
or d I 6. If this is so, then stability of a quantity-setting cartel implies that the price-setting cartel will also be stable. Conversely, if d > 6, then stability of the price-setting cartel implies that the quantity-setting cartel will also be stable. The results obtained in this paper contrast with those of Deneckere (1983) and Ross (1992). It would seem natural, as a next step, to investigate within the context of this model the relationship between product differentiation and the stability of n-firm cartels, and thereby to offer a possible contrast to the results obtained by Majerus (1988) for an n-firm extension of Deneckere’s analysis. It would also be interesting to consider the implications for the present case of applying a more severe punishment rule of the type proposed by Abreu (1986).
References Abreu, D., 1986, Extremal equilibria of oligopolistic supergames, Journal of Economic Theory 39, 191;225. Al-Nowaihi, A. and P. Levine, 1992, Credibility and the degree of collusion among oligopolists, Mimeo. (University of Leicester, Leicester). Deneckere, R., 1983, Duopoly supergames with product differentiation, Economics Letters 11, 37-42. Farrell, J. and E. Maskin, 1989, Renegotiation in repeated games, Games and Economic Behaviour 1, 327-360. Friedman, J., 1983, Oligopoly theory (Cambridge University Press, New York). Majerus, D., 1988, Price vs quantity competition in oligopoly supergames, Economics Letters 27, 293-297. Rothschild, R., 1985, Noncooperative behaviour as a credible threat, Bulletin of Economic Research 37, 245-248. Ross, T.W., 1992, Cartel stability and product differentiation, International Journal of Industrial Organization 10, 1-13.