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Cartel stability when costs are heterogeneous
International Journal of Industrial Organization 17 (1999) 717–734
Cartel stability when costs are heterogeneous R. Rothschild* Department of Economics, Lancaster University, Lancaster LA1 4 YX, UK Received 30 March 1997; received in revised form 30 April 1997; accepted 30 June 1997
Abstract This paper addresses a neglected question in the literature of cartel stability and the use of trigger strategies to maintain such stability. We employ a model in which market demand is linear, and involving n firms, each operating subject to one of possibly n different cost functions. We show that the stability of the cartel may depend crucially upon the relative efficiencies of the firms, and argue that in such models explicit attention must be given to this possibility. 1999 Elsevier Science B.V. All rights reserved. Keywords: Cartel stability; Trigger strategies; Heterogeneous costs JEL classification: D43
1. Introduction One of the best developed areas of research in the field of industrial organization analysis is into the existence and application of mechanisms for the maintenance of collusive agreements. Amongst the most familiar of the versions of such mechanisms is that proposed by Friedman (1971). The essence of Friedman’s approach, and that of the work of numerous subsequent contributors to the subject, is that firms can maintain collusive agreements by means of the threat of reversion to comprehensive noncooperation
* Fax: 144-1524-594244. E-mail address: [email protected] (R. Rothschild) 0167-7187 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 97 )00052-0
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in the event that deviation occurs. The punishment strategy which underpins this result is that firms weigh the short-term gains from deviation against the losses which subsequently arise from punishment. If the latter exceed the former, then deviation is deterred. Although the class of models which secure such outcomes has been widely studied, the underlying detail has occasionally received rather more sketchy attention, so that some potentially valuable insights may have been overlooked. The reason for this is, in part, due to the fact that when the cartel comprises all firms in the industry, either (a) the analysis has been conducted for general forms of the firms’ cost and revenue functions, or (b) where specific functional forms have been employed, the assumption has typically been that all firms are identical in terms of their costs, even if there may be a degree of differentiation amongst their products. The exception to this approach can be found in the literature of partial cartels made up of a ‘dominant’ firm facing a competitive ‘fringe’. In this type of model, as Donsimoni (1986) has shown, a degree of cost heterogeneity can be accommodated. But on the other hand, apart from Rothschild (1992), trigger strategies are typically not required to support collusion [for a discussion of these and related issues, see Martin (1993) and Shaffer (1995)]. The purpose of the present paper is to apply a Friedman strategy to an n-firm collusive oligopoly made up of firms which produce a homogeneous product, each subject to one of possibly n different marginal cost functions. We shall show that the introduction of the possibility of heterogeneity of such costs, quite apart from enhancing the realism of the model, helps to refine some issues which might have otherwise remained a matter of conjecture. The paper is organised as follows. In Section 2 we set out the model; Section 3 analyses the implications of cost heterogeneity in depth; Section 4 offers some concluding comments. The Appendices contain the more detailed mathematical workings.
2. The model We consider an industry comprising n firms, indexed i, i 5 1, 2, . . . ,n. We suppose that n > 3. Each i produces a quantity qi of a homogeneous product, subject to a cost function of the form c i (qi ) 5 bi qi 1 ai q 2i / 2, where bi > 0 and ai , ai . 0 ;i are not necessarily identical for all i. There are no fixed costs, so that marginal and average costs are bi 1 ai qi and bi 1 ai qi / 2 respectively. There are two features of the cost function which we would wish to emphasise at the outset. The first is that it implies that the cartel problem does not become trivial in the sense in which it would do so if all firms had different but constant marginal costs. In the latter case, it would clearly be practical for the firm with the lowest costs to produce the entire cartel output. In the model which we develop
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here, however, and as we show by means of an example,1 a switch of production exclusively to the most efficient firm would raise industry costs. The second feature of the cost function to emphasise is that fixed costs are taken to be zero. This simplifying assumption is common in the literature, partly because it reduces computational complexity but also because it can be made without any loss of generality. It will become apparent from our exposition of the important behavioural equilibria, that provided that fixed costs are not so high as to force firms out of business, the relative magnitudes of the payoffs to firms from different actions are unaffected by the omission of a fixed cost component. The demand curve in inverse form is p 5 1 2 Q, where
O q, n
Q5
i
i 51
and Q ,1. The essence of the ‘trigger strategy’ which is employed here is as follows. Firms adhere to the cartel agreement for as long as all fellow cartelists do so; in the event of deviation, the ‘loyal’ members of the coalition revert forever to the static (Cournot) noncooperative equilibrium. This forces the deviant to do the same. The threat of such retaliation is a deterrent to a prospective deviant if the latter’s (typically one-period) gain from cheating is no greater than the (discounted) per-period losses which arise from the subsequent punishment. More formally, consider an infinite replication of a single stage game involving all n firms in the industry, in which there exists for each firm i a pure strategy denoted by the vector z i . At any stage t, t [[0, `], z i (t ) indicates an action (a choice of output) s it , which is a function of the actions of all firms i in all stages to t 21, and which yields a payoff pi (sit , s 2it ), where s 2i are the actions of all firms m ch c other than firm i. For each i let s it , s it and s it denote the output of i corresponding respectively to collusion, defection from the collusive equilibrium, and Cournot m ch m noncooperative behaviour, respectively, and let pi (s m it , s 2it ), pi (s it , s 2it ) and c c pi (s it , s 2it ) be the associated payoffs. Then a trigger strategy for the game can be defined as follows: z i 1 5 s im z it 5
H
sm i s ci ,
if z 2i ( x ) 5 s m 2i , x 5 1, . . . , t 2 1; otherwise.
Let r, r.0, be the rate of interest. The essence of the trigger strategy is that any collusive equilibrium can be sustained if, for some common discount parameter m, m 51 /(11r), the following condition is satisfied:
1
See Appendix A for an illustrative example involving two firms.
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m m m pi (s ch it , s 2it ) 2 pi (s it , s 2it ) m > ]]]]]]]] ch m c c pi (s it , s 2it ) 2 pi (s it , s 2it )
(1)
3. The analysis We consider first the derivations of output and profits at the level of both the cartel and the individual firm. In order to keep the analysis as simple as possible, we shall suppose initially that bi 50, ;i, and reserve for later a discussion of the implications of relaxing this restriction. Given bi 50, ai can be thought of as an ‘inefficiency parameter’ for firm i. Since the marginal costs of all i are linearly increasing in qi , the industry marginal cost curve (that is, the ‘aggregate’ marginal costs of the member firms) is the horizontal sum of the costs of the individual firms. It is helpful to define
Oa n
S5
21 i
i 51
so that 1 /S is the slope of the marginal cost curve of the cartel. It is straightforward to show that the joint-profit maximising price and output of the cartel are 11S ]] 1 1 2S and S /(1 1 2S) respectively (see Appendix B). Cartel profits are then S ]]] 2(1 1 2S)
(2)
In the interests of economy of notation,2 we let A;11S and B;112S. Consider now the output and profit sharing rule which the cartel might adopt. If the cartel is to minimise the total costs of producing a given output level, it should allocate production amongst the firms in such a way that marginal costs are equalised.3 It is well known that if the marginal costs of one firm exceed those of another, then it will pay to switch production from the former to the latter until this
2 We adopt this procedure only in the body of the text. In order to present the derivations as clearly as possible we use the more detailed notation in the Appendix. 3 This rule, in the spirit of Patinkin (1947), is only one of several possible bases for allocating cartel output. It does, however, have the advantage of being both intuitively reasonable in that it derives from a cost minimisation process, and simple to implement.
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cost differential has been eliminated. As we show in Appendix C, such an allocation rule is equivalent to requiring that each cartel member’s share of the combined output and profit reflects its inefficiency parameter in a precise sense. Specifically, each i produces qi 5 Q /Sai 5 1 / [ai B] to obtain
p mi 5 1 / [2ai B]
(3)
It is easy to show that the derivative of the expression in (3) is decreasing in ai and increasing in a2i , 2i ±i. To do this, we set 1 n21 S 5 ] 1 ]] ai a2i Differentiation of Eq. (3) with respect to ai yields
a2i [a2i 1 2(n 2 1)] 2 ]]]]]]]]2 2[a2i (2 1 ai ) 1 ai (n 2 1)] Differentiation with respect to a2i yields
ai (n 2 1) ]]]]]]]] [a2i (2 1 ai ) 1 2ai (n 2 1)] 2 Thus, we have Proposition 1. A firm’ s share in the output of the cartel, and the profits which it obtains from membership, are smaller as the firm’ s inefficiency parameter is relatively larger, and larger as the inefficiency parameters of the other firms are relatively larger. In the analysis which follows we shall be concerned to analyse the effect of changes in the relative size of the inefficiency parameter on the incentives which a firm might have to deviate from the cartel agreement. In order to do so, we shall assume that the magnitude of S remains unchanged when ai changes. In practice, of course, ceteris paribus, changes in the size of ai will induce changes in S. However, since we are in effect examining the implications for an individual firm of being more or less efficient relative to the other members of the cartel, this is equivalent in the case of an increase in ai to ‘interchanging’ a less with a more
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efficient firm, all else unchanged.4 It is appropriate therefore to regard S as unaffected by this transposition. Suppose that firm i chooses to deviate from the allocation which it obtains at the joint profit maximum. We suppose here that deviation takes the form which is conventionally assumed in models of this type: the deviant takes the allocated cartel outputs of all other firms as given, and selects the output which maximises its own profits. The inverse demand curve facing the deviating firm i in this case is
Oq
n21
pi 5 1 2 qi 2
m 2i
,
2i51 m
where q 2i is the output of firm 2i at the cartel point. Setting
Oq
n21
m 2i
;Z
2i 51
yields pi 512qi 2Z. If firm i deviates, then this yields
5
ai A 1 1 qi 5 ]]]] ai B(2 1 ai ) The increase in output through deviation lowers market price to
a 2i A 1 ai (2 1 S) 1 1 ]]]]]] ai B(2 1 ai ) which yields deviation profits of
p
ch i
(ai A 1 1)2 5 ]]]] 2a 2i B 2 (2 1 ai )
(4)
We note here that deviation profits are always positive. Moreover, the derivative of the RHS of Eq. (4) with respect to ai is (ai A 1 1)(a i2 A 1 3ai 1 4) 2 ]]]]]]] 2a 3i B 2 (2 1 ai )2 which is always negative.
4 Since S is assumed to remain unchanged when ai rises, there is an obvious sense in which firm i can be thought to become less efficient relative to the industry. It is, however, instructive to compare this formulation of the problem, which highlights the differences between the profit outcomes to firms according to their inefficiency parameters, with that which assumes that changes in ai induce changes in S. Although we shall not pursue the latter formulation, we provide some examples in Appendix G which show how changes in ai and S, inter alia, might affect the incentive to deviate. 5 The basis for this result is set out in Appendix D .
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These observations yield the following: Proposition 2. Deviation profits are always positive for all values of the inefficiency parameter, and it is easily verified that these profits exceed the firm’ s profits at the cartel point.6 However, the gains from deviation fall as the deviant is less efficient. The intuition which underlies this result is straightforward. The greater the relative efficiency of the deviant the greater is the effect of deviation on total output. By implication, the lower is the relative efficiency of the deviant, the smaller is the gain from deviation since the output increase is relatively small. An interesting and important question which arises at this point concerns the impact of deviation upon the profits of the firms which remain loyal to the cartel agreement. The profits, p L , of any one of these firms – denoted j, j ±i – after deviation has occurred, are found by multiplying their original cartel output by the difference between the post deviation price and the average cost of such output. This yields
a 2i B 1 2ai A 1 2 p Lj 5 ]]]]] 2ai aj B 2 (2 1 ai )
(5)
which is always positive. It is clear that the post-deviation profits of any firm j are determined in part by both its own inefficiency parameter, aj , and that of the deviant, ai . We observe that the derivative of the RHS of Eq. (5) with respect to ai is
ai (ai S 2 2) 2 2 ]]]]] a 2i aj B 2 (2 1 ai )2 which is negative if ai is relatively small. By contrast, the derivative of the RHS of Eq. (5) with respect to aj is
a 2i B 1 2ai A 1 2 ]]]]] 2 2ai a 2j B 2 (2 1 ai ) which is always negative. These results suggest the following: Proposition 3. The profits to a loyalist post-deviation are always positive, but given the inefficiency parameter of the deviant the size of these profits is smaller as the inefficiency parameter of the loyalist is relatively larger. By contrast, given the inefficiency parameter of the loyal firm, the profits of that firm are decreasing
6
See Appendix E.
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in the parameter of the deviant, provided that this parameter is relatively small. Otherwise, however, the profits of the loyal firm are larger as the deviant is relatively more inefficient. The intuition here rests on the observation that when aj is relatively small, firm j’s equilibrium output is relatively large, so that a given fall in price due to deviation will have a proportionately larger impact upon j’s profits. An implication is that the greater the relative efficiency of the deviant, the larger is the price fall induced by deviation and the smaller is the profit to the loyalist. Conversely, the less efficient is the loyalist relative to the other firms, the smaller are the profits it obtains subsequent to any given deviation, since the decrease in price may be large but the increase in output will be relatively small. The profits to each firm i at the Cournot equilibrium 7 are (1 2 D)2 p ci 5 ]]] 2(2 1 ai )
(6)
where
Oq
n21
D;
c 2i
2i 51
c and q 2i is the Cournot output of firm 2i. It is clear that the profits to each firm at the Cournot equilibrium are lower as the individual firm’s inefficiency parameter is relatively larger, and the parameters of the other firms are relatively smaller. But we should note here that an increase in ai does have an effect on D of a kind which we argued earlier need not arise in the case of S. The reason for this is that whereas S is determined by all firms’ inefficiency parameters (so that ‘interchanging’ firms leaves S unaffected), D is determined by the outputs (and hence, indirectly, by the parameters) of all firms other than firm i. We shall return to the implications of this fact later. Having established the profits to individual firms i at the joint-profit maximum, deviation output and Cournot equilibrium, respectively, we consider now the conditions under which a firm with inefficiency parameter ai will be deterred from m ch c deviating. More specifically, we use the values of p i , p i and p i given in Eq. (3), Eq. (4) and Eq. (6) to obtain the necessary and sufficient condition to sustain the trigger strategy as an equilibrium:
ai S(2 2 ai S) 2 1 m > ]]]]]]]]]]]] 2 2 a i [DB (D 2 2) 1 S(2 1 3S)] 2 2ai A 2 1
7
See Appendix F.
(7)
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We show in the Appendix 8 that there exist plausible values of the parameters ai , S and D which ensure that, for some cases at least, the RHS of Eq. (7) is less than unity. If this were not the case, then clearly deterrence could not occur. The next step is to consider the effect of an increase in ai on the incentive to deviate. As it stands, the RHS of Eq. (7) is somewhat cumbersome, especially when derivatives are being taken, and cannot be interpreted unambiguously. However, it is possible to rewrite the RHS of the condition in Eq. (7) as equivalent to ch 12pm i /p i ]]]] 1 2 p ci /p ch i
(8)
and thereby to simplify the expression for the purposes of differentiation. Before proceeding to take the appropriate derivatives we note that an increase in ai , reflecting, as before, an ‘interchanging’ of a firm with a lower inefficiency parameter with a firm possessing a higher one, will also raise D. This occurs because a less efficient firm will produce a smaller output at the Cournot point than would one with a lower parameter. The effect of an increase in ai will therefore be made up of the direct effect of that increase, and the indirect effect of the increase in D. We note first that the derivative of the numerator of Eq. (8) with respect to ai is 2B(a S 2 1) ]]]] (ai A 1 1)3 which, since ai must be greater than 1 /S by definition, is always positive. Note also that D does not appear in the numerator. The derivative of the denominator of Eq. (8) with respect to ai is 2
2
2ai B (1 2 D) 2 ]]]] (ai A 1 1)3 which is always negative. Finally, the derivative of the denominator of Eq. (8) with respect to D is 2a 2i B 2 (1 2 D) ]]]] (ai A 1 1)2 which is always positive. We conclude that an increase in ai increases the numerator of Eq. (8) but has an ambiguous effect upon the denominator, depending upon whether the positive effect of the resulting increase in D is sufficient to offset the negative effect of the
8
See Appendix G.
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increase in ai . If this is not (resp. is) the case, then the RHS of Eq. (7) can increase (resp. decrease) as ai rises, and we have: Proposition 4. If the effect of an increase in ai on the absolute magnitude of D is relatively small, then the incentive to deviate is increasing in the prospective deviant’ s inefficiency parameter. Otherwise, the converse is true. It is not possible to establish in general whether one or the other outcome will occur. But it is possible to offer an intuition for the general result. If the increase in D which occurs as a consequence of the ‘interchanging’ of the more with the less efficient firm is large, then Cournot profits may be relatively low. On the other hand, the one-period gain to the deviant, given that it is now relatively less efficient, may be quite small. In this case, the incentive to deviate diminishes. Conversely, if the increase in D is relatively small, then the attractiveness of any deviation is increased. There are two issues which arise at this point. The first concerns the possibility that, in the event of deviation, ‘loyal’ firms may also wish to adjust their output, but not necessarily to the Cournot level; the second, and related, question concerns the possible role of ‘cyclical’ factors in influencing cartel stability and, in particular, the initial choice of collusive output. We turn now to these topics. When costs are symmetrical, all firms have the same temptation (or disinclination) to deviate. When costs are heterogeneous, a given firm may not itself wish to deviate but might anticipate that another will do so. Given such a possibility, it is interesting to consider the optimal response by a ‘loyal’ member of the cartel to a deviation by another firm. Suppose that the response takes place in the period in which the deviation occurs. Let i be the deviating firm, and j the loyal member. Appendix D illustrates the basis for calculating the deviant’s output. The calculation which j makes involves subtracting i’s original output as well as its own from the original cartel output, and then adding back i’s deviation output. By using the technique employed to obtain Eq. (4), firm j can select an optimal output for itself. The profits for j are then: [aj (a 2i A 1 ai (2 1 S) 1 1) 1 ai (2 1 ai )] 2 ]]]]]]]]]]] 2a 2j a 2i (2 1 aj )(2 1 ai )2 B 2
(9)
It is straightforward to show that the derivative of Eq. (9) with respect to aj is negative, but that it is positive with respect to ai . The implication is that the more (less) efficient is the loyalist (deviant), the larger are the profits to the loyalist when responding optimally to a deviation. An implication of an optimal response by firm j to a deviation by i is suggested in an example in Appendix H. In this case, it must be true that j obtains larger profits than it would by not adjusting [the outcome in the latter case is given in Eq.
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(5)]. It is, perhaps, less obvious that, depending upon the parameter D, j obtains larger profits than it would in the Cournot equilibrium [see Eq. (6)]. The second question which we wish to address is the role of ‘cyclical’ factors in determining the stability of cartels. Rotemberg and Saloner (1986) consider a framework in which industry demand takes a form similar to pt 5 1 1 et 2 Q t where et is the realisation in period t of a random disturbance. In interpreting their formulation we assume that pt >0. The underlying assumption in the model is that these (observed) disturbances or shocks are independently and identically distributed with expected value zero, with the implication that future demand is unrelated to current demand.9 Rotemberg and Saloner argue that a prospective deviant will make its choice subject to the actual or realised value of et , but will anticipate the punishment which is associated with the expected value of et (i.e. zero). In such circumstances deviation can be profitable, so that cartels may become unstable in states of high demand. A further implication of this result is that cartels may choose to increase their output beyond the strictly collusive level in a given state in order to reduce the incentive for deviation, and that therefore periods of high demand may be characterised by relatively low prices. In order to consider some implications of the cyclical model, we modify the expressions in Eq. (3) and Eq. (4) by multiplying each by (11 et )2 . Then the gains from deviation in state et [the difference between Eq. (4) and Eq. (3)] are (1 1 et )2 (ai S(ai S 2 2) 1 1)2 ]]]]]]]] 2a 2i B 2 (2 1 ai ) which can easily be shown to be increasing in et . Given a value for the discount parameter m, it is comparatively simple to identify the output for each firm in state et which will just discourage deviation. This is done by substituting the modified expressions from Eq. (3) and Eq. (4), as well as that in Eq. (6), into Eq. (7), and setting the resulting ratio equal to the given value of m. The value of et which solves the equation is then the highest demand shock consistent with cartel stability. Substitution of this value of et into the expression which gives firm i’s optimal deviation output in state et enables us to identify the minimum output of each firm which is consistent with such stability. It can be seen from the numerical examples given in Appendix I that this implies a larger per-firm output, and a lower cartel price, than would be justified for a monopoly in that state. It is also possible to show that, as firms become
9
For a more general model which relaxes this relatively strong assumption, see Haltiwanger and Harrington (1991).
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absolutely less efficient, the sustainable cartel output (for given m ) is smaller, and the collusive price therefore higher. Unfortunately, however, as in the ‘noncyclical’ version of our model, the impact of a change in the relative efficiency of a given firm on the incentive to deviate is ambiguous.
3.1. Some implications for the model when the parameter bi may be positive The analysis so far has proceeded on the basis of the simplifying assumption that, for each i, the parameter bi in the cost function c i (qi )5 bi qi 1 ai q i2 / 2 is zero. At this point we wish to consider the implications of relaxing this restriction. The simplest way of modifying the model while retaining its capacity to yield interesting insights is to assume that ai is the same for all i, while the values of b may vary across the firms. Clearly, however, it is the case that if the industry price were so low as to lie below bi for some firm i, then the industry marginal cost curve would not be defined. We therefore impose the restriction that, given the cost functions of all n firms, the industry price p c at the Cournot equilibrium be greater than the b of the (possibly not unique) least efficient firm. This assumption ensures that all firms are always active in the industry. Let b¯ be the b of the least efficient firm. Then cartel marginal and average costs are b¯ 1Q /S and b¯ 1Q / 2S respectively. Cartel profits at the joint-profit maximising output Q m 5S(12 b¯ ) /B are therefore S(1 2 b¯ )2 ]]] 2B or n(1 2 b¯ )2 ]]] 2(a 1 2n) if S5n /a. Note that the last expression is increasing in cartel membership. Let mc*5 b¯ 1Q m /S be the cartel marginal cost at output Q m . The optimal output and profit allocation rule gives to each i the share mc* 2 bi ui ; ]]]]] n i 51 (mc* 2 bi )
O
For given i the cartel profits, deviation profits and Cournot equilibrium profits are, respectively,
ui S(1 2 b¯ )2 p mi 5 ]]]] 2B
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[ui S( b¯ 2 1) 2 S( b¯ 1 1) 1 bi B 2 1] 2 p ch i 5 ]]]]]]]]]] 2B 2 (2 1 a ) and (1 2 D 2 bi )2 p ci 5 ]]]] 2(2 1 a ) Substitution of the three foregoing expressions into Eq. (7) yields the condition for cartel stability. The derivatives of this condition with respect to the variables a, ui and D are highly ambiguous. However, it is possible to show that if n53 and all firms have identical b ’s Eq. (7) increases with bi . This is because the derivative of the equivalent expression Eq. (8) is positive if bi 1D ,1, a condition which must be satisfied if individual firms’ Cournot equilibrium outputs are, in turn, to be positive. The implication is that, at least for some values of the parameters, the incentive to deviate is increasing as the prospective deviant is more inefficient. In this respect the conclusion accords with that embodied in proposition 4 for the case in which the bi ’s are normalised to zero. The analysis set out in this paper shows the importance of giving due attention to the costs of firms in a cartel. This is, of course, not to suggest that the number of firms is immaterial. In fact, the size of the cartel is especially important in the context of partial cartels of the type considered by Shaffer (1995), or where trigger strategies are not employed in order to sustain collusion. Nevertheless, although the following proposition captures an obvious insight, it embodies a principle which is frequently suppressed: Proposition 5. The stability of a cartel depends not only on the number of members, but also on their individual and aggregate costs.
4. Concluding comments This paper has explored some implications of the well-known trigger strategy framework for the maintenance of cartel stability. The focus has been upon the implications for the strategy of the existence of differences amongst firms’ marginal costs, or ‘inefficiency parameters’. An important insight to be gained from this study is that the effectiveness of the strategy as a deterrent to deviation depends crucially, and in a potentially quite complex way, upon the relative efficiencies of the deviant and the loyalists. More specifically, it becomes apparent that while some firms might be deterred from deviating, others will be willing to go ahead. Thus, if a trigger strategy is to be an effective deterrent, then the condition Eq. (1) (or, equivalently, Eq. (7)) must be
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satisfied for all of the potentially profitable deviants, and hence for a complex set of cost and output configurations. It is clearly the case that the results obtained here depend to a large extent on the particular form of the demand and cost functions which we employ. Whilst it would obviously be desirable to have more general forms of these functions yield comparably detailed insights, this is unfortunately not possible. At the same time, however, the approach adopted here does have the virtue of drawing attention to some of the possible implications of cost heterogeneity, and to this extent it constitutes both a contribution to an important area of analysis, and a basis for further work.
Acknowledgements I would like to thank the Editor, S. Martin, and two anonymous referees, for numerous very helpful comments on an earlier draft. Responsibility for any errors which remain is entirely my own.
Appendix A
Numerical example illustrating the increase in costs due to switching all production to the lowest cost firm We consider two examples involving the cost function
ai q 2i ]] c i (qi ) 5 bi qi 1 2 where, for the sake of simplicity, we suppose that bi 50. In the first example, we take the output of the cartel as given; in the second, we suppose that the most efficient firm produces its profit maximising output. Let n52. Suppose that Q525, a1 52, and a2 53. The cost minimisation rule for allocating shares in this case gives
a2 Q q1 5 ]]] 5 3Q / 5 5 15. a1 1 a 2 Similarly,
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a1 Q q2 5 ]]] 5 2Q / 5 5 10. a1 1 a 2 It is easily shown that in this case total cartel cost, or a1 q 12 / 21 a2 q 22 / 2, is 375. If production were undertaken exclusively by firm 1 (the lower cost producer), then cost would rise to 625 (525 2 ). Suppose, as an alternative, that output is not given. Let the demand function be p512Q. If a1 52 and a2 53, and both firms produce according to the share allocation rule identified above, then respective profits are 0.09375 and 0.0625, respectively. These sum to a cartel profit of 0.15625. But now suppose that the less efficient firm ceases production, and that the more efficient firm produces its profit maximising output. Then profits are 0.125,0.15625. We conclude that, in this case too, it is not profitable to shut down the less efficient firm.
Appendix B
Derivation of joint-profit maximising cartel output, price and profit If p512Q, then cartel marginal revenue is 122Q. Set marginal revenue equal to aggregate marginal cost, Q /S, to obtain Q5S /(112S). Substitute into the demand function to obtain p5(11S) /(112S). Cartel profits are found by appropriate substitution in the expression Q[ p2ac(Q)], where average costs for the cartel at output Q are given by 1 / 2(112S).
Appendix C
Derivation of individual firm’s output allocation and profit at the cartel price Refer again to Appendix A for the output allocation rule which emerges from the maximisation exercise. For the case involving more than two firms this rule, as it relates to the output of (say) firm 1, can be generalised to Q ]]]] n a1 i 51 a 21 i
O
This corresponds in an obvious way to the rule which we apply in the paper to any i, that is,
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qi 5 Q /Sai . Given this output allocation rule, the profits of firm i at price p5(11S) /(112S) are found by appropriate substitution in the expression qi [ p2ac(qi )] where, given qi and ai , the average costs of the firm are ai qi / 2.
Appendix D
The basis for the derivation of the deviant’s output and profit The value of Z in the text is the original cartel output, minus the amount originally produced by the deviant:
ai S 2 1 S 1 ]] 2 ]]] 5 ]]] 1 1 2S ai (1 1 2S) ai (1 1 2S)
Appendix E
Demonstration of the relationship between deviation profits and profits at the cartel point (ai (1 1 S) 1 1)2 ai S(ai S 2 2) 1 1 1 p ich 2 p im 5 ]]]]]] 2 2 2 ]]]] 5 ]]]]]] 2 a (1 1 2S) 2a i (2 1 ai )(1 1 2S) 2a i2 (2 1 ai )(1 1 2S)2 i which is positive if S ±1 /ai , which is true by definition.
Appendix F
Derivation of Cournot equilibrium profits Each i chooses qi to maximise pi , given D. This yields qi 5(12D) /(21 ai ). The resulting price is given by 12qi 2D, and equals [(12D)(11 ai )] /(21 ai ). Given average costs, profits are derived by appropriate substitution.
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Appendix G
Numerical examples of the deterrence condition in Eq. (7) We show first that there exist parameter values for which the RHS of Eq. (7) is smaller than unity, and therefore that deterrence is feasible if m is sufficiently large: Suppose n53. Set ai 56, ;i, so that S50.5. Since in this case D52qi , it follows that qi ;(12D) /(21 ai )51 /(41 ai )51 / 10. Hence D51 / 5. Substitution of the appropriate values into Eq. (7) yields a value of 0.510. Now suppose that the common inefficiency parameter for the three firms increases to ai 58. Then S53 / 8, and D51 / 6. Substitution into Eq. (7) yields a value of 0.507. Finally, suppose ai 5100. Then S53 / 100, and D51 / 52. Substitution yields as the deterrence condition m >0.5 As indicated earlier, it is instructive to consider the implications for the deterrence condition of allowing S to vary as ai becomes larger. The examples given here suggest that, in this case, as ai becomes larger, the Eq. (7) becomes smaller. We conclude that in this case a decrease in the inefficiency parameter for a given firm may encourage deviation.
Appendix H
An example to illustrate the potential gains from adjusting optimally to the deviant’s output Suppose n56. Set ai 50.75 and aj 50.5. Let n21 1 S ; ]] 1 ] ai aj If D .0.51, then the condition in Eq. (7) for a prospective deviant j is m .0.8; for a prospective deviant i the condition is m .1.9. It follows that if m is sufficiently close to unity, then i will deviate but j will not. If i deviates, then [from Eq. (9)] firm j obtains profits of 0.0482 by choosing its optimal response (all other firms’ outputs unchanged). These profits compare with the 0.039 obtainable by not reacting [see Eq. (5)], and the Cournot equilibrium profits [see Eq. (6)] of 0.0480. It is clear that these outcomes depend crucially upon the value of D. It is easy to find examples in which the incentives to deviate may be reversed and where
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Cournot equilibrium profits exceed those available even if the loyal firm responds optimally.
Appendix I
An illustration of the Rotemberg and Saloner result We employ an example from Appendix G. Suppose n53. Let ai 58, ;i, S53 / 8, and D51 / 6. Then, if m 50.13, a value of et 50.1 satisfies our deterrence condition with equality. Substitute this value of et into the optimal deviation output for firm i: (1 1 et )(ai A 1 1) ch q it 5 ]]]]] ai B(2 1 ai ) and multiply by n to obtain cartel output Q t 50.28 and corresponding pt 50.72. These values compare with 0.21 and 0.79, respectively, the output and price for a monopoly in state et . Now suppose that ai 5100, ;i, S53 / 100 and D51 / 52. Then the value of m which satisfies the deterrence condition with equality for et 50.1 is 0.002. It follows that if deviation is deterred in the former case (i.e. when ai 58), then it will also be deterred for this set of parameters. In other words, for these parameters at least, the output level which deters deviation when firms are absolutely efficient will also deter cheating when the firms are absolutely less efficient, and consequently given m the cartel output consistent with stability may be smaller (and price higher) the less efficient are the firms involved.
References Donsimoni, M.-P., 1986. Stable heterogeneous cartels. International Journal of Industrial Organization 3, 451–467. Friedman, J., 1971. A noncooperative equilibrium for supergames. Review of Economic Studies 38, 1–12. Haltiwanger, J., Harrington Jr, J., 1991. The impact of cyclical demand movements on collusive behavior. Rand Journal of Economics 22, 89–106. Martin, S., 1993. Advanced Industrial Economics. Blackwell, Oxford. Patinkin, D., 1947. Multi-product firms, cartels and imperfect information. Quarterly Journal of Economics 62, 173–205. Rotemberg, J., Saloner, G., 1986. A supergame-theoretic model of price wars during booms. American Economic Review 76, 390–407. Rothschild, R., 1992. The stability of dominant-group cartels. In: Gee, J.M.A., Norman, G. (Eds.), Market Strategy and Structure. Harvester Wheatsheaf, New York. Shaffer, S., 1995. Stable cartels with a Cournot fringe. Southern Economic Journal 61, 744–754.
Cartel stability when costs are heterogeneous R. Rothschild* Department of Economics, Lancaster University, Lancaster LA1 4 YX, UK Received 30 March 1997; received in revised form 30 April 1997; accepted 30 June 1997
Abstract This paper addresses a neglected question in the literature of cartel stability and the use of trigger strategies to maintain such stability. We employ a model in which market demand is linear, and involving n firms, each operating subject to one of possibly n different cost functions. We show that the stability of the cartel may depend crucially upon the relative efficiencies of the firms, and argue that in such models explicit attention must be given to this possibility. 1999 Elsevier Science B.V. All rights reserved. Keywords: Cartel stability; Trigger strategies; Heterogeneous costs JEL classification: D43
1. Introduction One of the best developed areas of research in the field of industrial organization analysis is into the existence and application of mechanisms for the maintenance of collusive agreements. Amongst the most familiar of the versions of such mechanisms is that proposed by Friedman (1971). The essence of Friedman’s approach, and that of the work of numerous subsequent contributors to the subject, is that firms can maintain collusive agreements by means of the threat of reversion to comprehensive noncooperation
* Fax: 144-1524-594244. E-mail address: [email protected] (R. Rothschild) 0167-7187 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 97 )00052-0
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in the event that deviation occurs. The punishment strategy which underpins this result is that firms weigh the short-term gains from deviation against the losses which subsequently arise from punishment. If the latter exceed the former, then deviation is deterred. Although the class of models which secure such outcomes has been widely studied, the underlying detail has occasionally received rather more sketchy attention, so that some potentially valuable insights may have been overlooked. The reason for this is, in part, due to the fact that when the cartel comprises all firms in the industry, either (a) the analysis has been conducted for general forms of the firms’ cost and revenue functions, or (b) where specific functional forms have been employed, the assumption has typically been that all firms are identical in terms of their costs, even if there may be a degree of differentiation amongst their products. The exception to this approach can be found in the literature of partial cartels made up of a ‘dominant’ firm facing a competitive ‘fringe’. In this type of model, as Donsimoni (1986) has shown, a degree of cost heterogeneity can be accommodated. But on the other hand, apart from Rothschild (1992), trigger strategies are typically not required to support collusion [for a discussion of these and related issues, see Martin (1993) and Shaffer (1995)]. The purpose of the present paper is to apply a Friedman strategy to an n-firm collusive oligopoly made up of firms which produce a homogeneous product, each subject to one of possibly n different marginal cost functions. We shall show that the introduction of the possibility of heterogeneity of such costs, quite apart from enhancing the realism of the model, helps to refine some issues which might have otherwise remained a matter of conjecture. The paper is organised as follows. In Section 2 we set out the model; Section 3 analyses the implications of cost heterogeneity in depth; Section 4 offers some concluding comments. The Appendices contain the more detailed mathematical workings.
2. The model We consider an industry comprising n firms, indexed i, i 5 1, 2, . . . ,n. We suppose that n > 3. Each i produces a quantity qi of a homogeneous product, subject to a cost function of the form c i (qi ) 5 bi qi 1 ai q 2i / 2, where bi > 0 and ai , ai . 0 ;i are not necessarily identical for all i. There are no fixed costs, so that marginal and average costs are bi 1 ai qi and bi 1 ai qi / 2 respectively. There are two features of the cost function which we would wish to emphasise at the outset. The first is that it implies that the cartel problem does not become trivial in the sense in which it would do so if all firms had different but constant marginal costs. In the latter case, it would clearly be practical for the firm with the lowest costs to produce the entire cartel output. In the model which we develop
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here, however, and as we show by means of an example,1 a switch of production exclusively to the most efficient firm would raise industry costs. The second feature of the cost function to emphasise is that fixed costs are taken to be zero. This simplifying assumption is common in the literature, partly because it reduces computational complexity but also because it can be made without any loss of generality. It will become apparent from our exposition of the important behavioural equilibria, that provided that fixed costs are not so high as to force firms out of business, the relative magnitudes of the payoffs to firms from different actions are unaffected by the omission of a fixed cost component. The demand curve in inverse form is p 5 1 2 Q, where
O q, n
Q5
i
i 51
and Q ,1. The essence of the ‘trigger strategy’ which is employed here is as follows. Firms adhere to the cartel agreement for as long as all fellow cartelists do so; in the event of deviation, the ‘loyal’ members of the coalition revert forever to the static (Cournot) noncooperative equilibrium. This forces the deviant to do the same. The threat of such retaliation is a deterrent to a prospective deviant if the latter’s (typically one-period) gain from cheating is no greater than the (discounted) per-period losses which arise from the subsequent punishment. More formally, consider an infinite replication of a single stage game involving all n firms in the industry, in which there exists for each firm i a pure strategy denoted by the vector z i . At any stage t, t [[0, `], z i (t ) indicates an action (a choice of output) s it , which is a function of the actions of all firms i in all stages to t 21, and which yields a payoff pi (sit , s 2it ), where s 2i are the actions of all firms m ch c other than firm i. For each i let s it , s it and s it denote the output of i corresponding respectively to collusion, defection from the collusive equilibrium, and Cournot m ch m noncooperative behaviour, respectively, and let pi (s m it , s 2it ), pi (s it , s 2it ) and c c pi (s it , s 2it ) be the associated payoffs. Then a trigger strategy for the game can be defined as follows: z i 1 5 s im z it 5
H
sm i s ci ,
if z 2i ( x ) 5 s m 2i , x 5 1, . . . , t 2 1; otherwise.
Let r, r.0, be the rate of interest. The essence of the trigger strategy is that any collusive equilibrium can be sustained if, for some common discount parameter m, m 51 /(11r), the following condition is satisfied:
1
See Appendix A for an illustrative example involving two firms.
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m m m pi (s ch it , s 2it ) 2 pi (s it , s 2it ) m > ]]]]]]]] ch m c c pi (s it , s 2it ) 2 pi (s it , s 2it )
(1)
3. The analysis We consider first the derivations of output and profits at the level of both the cartel and the individual firm. In order to keep the analysis as simple as possible, we shall suppose initially that bi 50, ;i, and reserve for later a discussion of the implications of relaxing this restriction. Given bi 50, ai can be thought of as an ‘inefficiency parameter’ for firm i. Since the marginal costs of all i are linearly increasing in qi , the industry marginal cost curve (that is, the ‘aggregate’ marginal costs of the member firms) is the horizontal sum of the costs of the individual firms. It is helpful to define
Oa n
S5
21 i
i 51
so that 1 /S is the slope of the marginal cost curve of the cartel. It is straightforward to show that the joint-profit maximising price and output of the cartel are 11S ]] 1 1 2S and S /(1 1 2S) respectively (see Appendix B). Cartel profits are then S ]]] 2(1 1 2S)
(2)
In the interests of economy of notation,2 we let A;11S and B;112S. Consider now the output and profit sharing rule which the cartel might adopt. If the cartel is to minimise the total costs of producing a given output level, it should allocate production amongst the firms in such a way that marginal costs are equalised.3 It is well known that if the marginal costs of one firm exceed those of another, then it will pay to switch production from the former to the latter until this
2 We adopt this procedure only in the body of the text. In order to present the derivations as clearly as possible we use the more detailed notation in the Appendix. 3 This rule, in the spirit of Patinkin (1947), is only one of several possible bases for allocating cartel output. It does, however, have the advantage of being both intuitively reasonable in that it derives from a cost minimisation process, and simple to implement.
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cost differential has been eliminated. As we show in Appendix C, such an allocation rule is equivalent to requiring that each cartel member’s share of the combined output and profit reflects its inefficiency parameter in a precise sense. Specifically, each i produces qi 5 Q /Sai 5 1 / [ai B] to obtain
p mi 5 1 / [2ai B]
(3)
It is easy to show that the derivative of the expression in (3) is decreasing in ai and increasing in a2i , 2i ±i. To do this, we set 1 n21 S 5 ] 1 ]] ai a2i Differentiation of Eq. (3) with respect to ai yields
a2i [a2i 1 2(n 2 1)] 2 ]]]]]]]]2 2[a2i (2 1 ai ) 1 ai (n 2 1)] Differentiation with respect to a2i yields
ai (n 2 1) ]]]]]]]] [a2i (2 1 ai ) 1 2ai (n 2 1)] 2 Thus, we have Proposition 1. A firm’ s share in the output of the cartel, and the profits which it obtains from membership, are smaller as the firm’ s inefficiency parameter is relatively larger, and larger as the inefficiency parameters of the other firms are relatively larger. In the analysis which follows we shall be concerned to analyse the effect of changes in the relative size of the inefficiency parameter on the incentives which a firm might have to deviate from the cartel agreement. In order to do so, we shall assume that the magnitude of S remains unchanged when ai changes. In practice, of course, ceteris paribus, changes in the size of ai will induce changes in S. However, since we are in effect examining the implications for an individual firm of being more or less efficient relative to the other members of the cartel, this is equivalent in the case of an increase in ai to ‘interchanging’ a less with a more
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efficient firm, all else unchanged.4 It is appropriate therefore to regard S as unaffected by this transposition. Suppose that firm i chooses to deviate from the allocation which it obtains at the joint profit maximum. We suppose here that deviation takes the form which is conventionally assumed in models of this type: the deviant takes the allocated cartel outputs of all other firms as given, and selects the output which maximises its own profits. The inverse demand curve facing the deviating firm i in this case is
Oq
n21
pi 5 1 2 qi 2
m 2i
,
2i51 m
where q 2i is the output of firm 2i at the cartel point. Setting
Oq
n21
m 2i
;Z
2i 51
yields pi 512qi 2Z. If firm i deviates, then this yields
5
ai A 1 1 qi 5 ]]]] ai B(2 1 ai ) The increase in output through deviation lowers market price to
a 2i A 1 ai (2 1 S) 1 1 ]]]]]] ai B(2 1 ai ) which yields deviation profits of
p
ch i
(ai A 1 1)2 5 ]]]] 2a 2i B 2 (2 1 ai )
(4)
We note here that deviation profits are always positive. Moreover, the derivative of the RHS of Eq. (4) with respect to ai is (ai A 1 1)(a i2 A 1 3ai 1 4) 2 ]]]]]]] 2a 3i B 2 (2 1 ai )2 which is always negative.
4 Since S is assumed to remain unchanged when ai rises, there is an obvious sense in which firm i can be thought to become less efficient relative to the industry. It is, however, instructive to compare this formulation of the problem, which highlights the differences between the profit outcomes to firms according to their inefficiency parameters, with that which assumes that changes in ai induce changes in S. Although we shall not pursue the latter formulation, we provide some examples in Appendix G which show how changes in ai and S, inter alia, might affect the incentive to deviate. 5 The basis for this result is set out in Appendix D .
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These observations yield the following: Proposition 2. Deviation profits are always positive for all values of the inefficiency parameter, and it is easily verified that these profits exceed the firm’ s profits at the cartel point.6 However, the gains from deviation fall as the deviant is less efficient. The intuition which underlies this result is straightforward. The greater the relative efficiency of the deviant the greater is the effect of deviation on total output. By implication, the lower is the relative efficiency of the deviant, the smaller is the gain from deviation since the output increase is relatively small. An interesting and important question which arises at this point concerns the impact of deviation upon the profits of the firms which remain loyal to the cartel agreement. The profits, p L , of any one of these firms – denoted j, j ±i – after deviation has occurred, are found by multiplying their original cartel output by the difference between the post deviation price and the average cost of such output. This yields
a 2i B 1 2ai A 1 2 p Lj 5 ]]]]] 2ai aj B 2 (2 1 ai )
(5)
which is always positive. It is clear that the post-deviation profits of any firm j are determined in part by both its own inefficiency parameter, aj , and that of the deviant, ai . We observe that the derivative of the RHS of Eq. (5) with respect to ai is
ai (ai S 2 2) 2 2 ]]]]] a 2i aj B 2 (2 1 ai )2 which is negative if ai is relatively small. By contrast, the derivative of the RHS of Eq. (5) with respect to aj is
a 2i B 1 2ai A 1 2 ]]]]] 2 2ai a 2j B 2 (2 1 ai ) which is always negative. These results suggest the following: Proposition 3. The profits to a loyalist post-deviation are always positive, but given the inefficiency parameter of the deviant the size of these profits is smaller as the inefficiency parameter of the loyalist is relatively larger. By contrast, given the inefficiency parameter of the loyal firm, the profits of that firm are decreasing
6
See Appendix E.
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in the parameter of the deviant, provided that this parameter is relatively small. Otherwise, however, the profits of the loyal firm are larger as the deviant is relatively more inefficient. The intuition here rests on the observation that when aj is relatively small, firm j’s equilibrium output is relatively large, so that a given fall in price due to deviation will have a proportionately larger impact upon j’s profits. An implication is that the greater the relative efficiency of the deviant, the larger is the price fall induced by deviation and the smaller is the profit to the loyalist. Conversely, the less efficient is the loyalist relative to the other firms, the smaller are the profits it obtains subsequent to any given deviation, since the decrease in price may be large but the increase in output will be relatively small. The profits to each firm i at the Cournot equilibrium 7 are (1 2 D)2 p ci 5 ]]] 2(2 1 ai )
(6)
where
Oq
n21
D;
c 2i
2i 51
c and q 2i is the Cournot output of firm 2i. It is clear that the profits to each firm at the Cournot equilibrium are lower as the individual firm’s inefficiency parameter is relatively larger, and the parameters of the other firms are relatively smaller. But we should note here that an increase in ai does have an effect on D of a kind which we argued earlier need not arise in the case of S. The reason for this is that whereas S is determined by all firms’ inefficiency parameters (so that ‘interchanging’ firms leaves S unaffected), D is determined by the outputs (and hence, indirectly, by the parameters) of all firms other than firm i. We shall return to the implications of this fact later. Having established the profits to individual firms i at the joint-profit maximum, deviation output and Cournot equilibrium, respectively, we consider now the conditions under which a firm with inefficiency parameter ai will be deterred from m ch c deviating. More specifically, we use the values of p i , p i and p i given in Eq. (3), Eq. (4) and Eq. (6) to obtain the necessary and sufficient condition to sustain the trigger strategy as an equilibrium:
ai S(2 2 ai S) 2 1 m > ]]]]]]]]]]]] 2 2 a i [DB (D 2 2) 1 S(2 1 3S)] 2 2ai A 2 1
7
See Appendix F.
(7)
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We show in the Appendix 8 that there exist plausible values of the parameters ai , S and D which ensure that, for some cases at least, the RHS of Eq. (7) is less than unity. If this were not the case, then clearly deterrence could not occur. The next step is to consider the effect of an increase in ai on the incentive to deviate. As it stands, the RHS of Eq. (7) is somewhat cumbersome, especially when derivatives are being taken, and cannot be interpreted unambiguously. However, it is possible to rewrite the RHS of the condition in Eq. (7) as equivalent to ch 12pm i /p i ]]]] 1 2 p ci /p ch i
(8)
and thereby to simplify the expression for the purposes of differentiation. Before proceeding to take the appropriate derivatives we note that an increase in ai , reflecting, as before, an ‘interchanging’ of a firm with a lower inefficiency parameter with a firm possessing a higher one, will also raise D. This occurs because a less efficient firm will produce a smaller output at the Cournot point than would one with a lower parameter. The effect of an increase in ai will therefore be made up of the direct effect of that increase, and the indirect effect of the increase in D. We note first that the derivative of the numerator of Eq. (8) with respect to ai is 2B(a S 2 1) ]]]] (ai A 1 1)3 which, since ai must be greater than 1 /S by definition, is always positive. Note also that D does not appear in the numerator. The derivative of the denominator of Eq. (8) with respect to ai is 2
2
2ai B (1 2 D) 2 ]]]] (ai A 1 1)3 which is always negative. Finally, the derivative of the denominator of Eq. (8) with respect to D is 2a 2i B 2 (1 2 D) ]]]] (ai A 1 1)2 which is always positive. We conclude that an increase in ai increases the numerator of Eq. (8) but has an ambiguous effect upon the denominator, depending upon whether the positive effect of the resulting increase in D is sufficient to offset the negative effect of the
8
See Appendix G.
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increase in ai . If this is not (resp. is) the case, then the RHS of Eq. (7) can increase (resp. decrease) as ai rises, and we have: Proposition 4. If the effect of an increase in ai on the absolute magnitude of D is relatively small, then the incentive to deviate is increasing in the prospective deviant’ s inefficiency parameter. Otherwise, the converse is true. It is not possible to establish in general whether one or the other outcome will occur. But it is possible to offer an intuition for the general result. If the increase in D which occurs as a consequence of the ‘interchanging’ of the more with the less efficient firm is large, then Cournot profits may be relatively low. On the other hand, the one-period gain to the deviant, given that it is now relatively less efficient, may be quite small. In this case, the incentive to deviate diminishes. Conversely, if the increase in D is relatively small, then the attractiveness of any deviation is increased. There are two issues which arise at this point. The first concerns the possibility that, in the event of deviation, ‘loyal’ firms may also wish to adjust their output, but not necessarily to the Cournot level; the second, and related, question concerns the possible role of ‘cyclical’ factors in influencing cartel stability and, in particular, the initial choice of collusive output. We turn now to these topics. When costs are symmetrical, all firms have the same temptation (or disinclination) to deviate. When costs are heterogeneous, a given firm may not itself wish to deviate but might anticipate that another will do so. Given such a possibility, it is interesting to consider the optimal response by a ‘loyal’ member of the cartel to a deviation by another firm. Suppose that the response takes place in the period in which the deviation occurs. Let i be the deviating firm, and j the loyal member. Appendix D illustrates the basis for calculating the deviant’s output. The calculation which j makes involves subtracting i’s original output as well as its own from the original cartel output, and then adding back i’s deviation output. By using the technique employed to obtain Eq. (4), firm j can select an optimal output for itself. The profits for j are then: [aj (a 2i A 1 ai (2 1 S) 1 1) 1 ai (2 1 ai )] 2 ]]]]]]]]]]] 2a 2j a 2i (2 1 aj )(2 1 ai )2 B 2
(9)
It is straightforward to show that the derivative of Eq. (9) with respect to aj is negative, but that it is positive with respect to ai . The implication is that the more (less) efficient is the loyalist (deviant), the larger are the profits to the loyalist when responding optimally to a deviation. An implication of an optimal response by firm j to a deviation by i is suggested in an example in Appendix H. In this case, it must be true that j obtains larger profits than it would by not adjusting [the outcome in the latter case is given in Eq.
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(5)]. It is, perhaps, less obvious that, depending upon the parameter D, j obtains larger profits than it would in the Cournot equilibrium [see Eq. (6)]. The second question which we wish to address is the role of ‘cyclical’ factors in determining the stability of cartels. Rotemberg and Saloner (1986) consider a framework in which industry demand takes a form similar to pt 5 1 1 et 2 Q t where et is the realisation in period t of a random disturbance. In interpreting their formulation we assume that pt >0. The underlying assumption in the model is that these (observed) disturbances or shocks are independently and identically distributed with expected value zero, with the implication that future demand is unrelated to current demand.9 Rotemberg and Saloner argue that a prospective deviant will make its choice subject to the actual or realised value of et , but will anticipate the punishment which is associated with the expected value of et (i.e. zero). In such circumstances deviation can be profitable, so that cartels may become unstable in states of high demand. A further implication of this result is that cartels may choose to increase their output beyond the strictly collusive level in a given state in order to reduce the incentive for deviation, and that therefore periods of high demand may be characterised by relatively low prices. In order to consider some implications of the cyclical model, we modify the expressions in Eq. (3) and Eq. (4) by multiplying each by (11 et )2 . Then the gains from deviation in state et [the difference between Eq. (4) and Eq. (3)] are (1 1 et )2 (ai S(ai S 2 2) 1 1)2 ]]]]]]]] 2a 2i B 2 (2 1 ai ) which can easily be shown to be increasing in et . Given a value for the discount parameter m, it is comparatively simple to identify the output for each firm in state et which will just discourage deviation. This is done by substituting the modified expressions from Eq. (3) and Eq. (4), as well as that in Eq. (6), into Eq. (7), and setting the resulting ratio equal to the given value of m. The value of et which solves the equation is then the highest demand shock consistent with cartel stability. Substitution of this value of et into the expression which gives firm i’s optimal deviation output in state et enables us to identify the minimum output of each firm which is consistent with such stability. It can be seen from the numerical examples given in Appendix I that this implies a larger per-firm output, and a lower cartel price, than would be justified for a monopoly in that state. It is also possible to show that, as firms become
9
For a more general model which relaxes this relatively strong assumption, see Haltiwanger and Harrington (1991).
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absolutely less efficient, the sustainable cartel output (for given m ) is smaller, and the collusive price therefore higher. Unfortunately, however, as in the ‘noncyclical’ version of our model, the impact of a change in the relative efficiency of a given firm on the incentive to deviate is ambiguous.
3.1. Some implications for the model when the parameter bi may be positive The analysis so far has proceeded on the basis of the simplifying assumption that, for each i, the parameter bi in the cost function c i (qi )5 bi qi 1 ai q i2 / 2 is zero. At this point we wish to consider the implications of relaxing this restriction. The simplest way of modifying the model while retaining its capacity to yield interesting insights is to assume that ai is the same for all i, while the values of b may vary across the firms. Clearly, however, it is the case that if the industry price were so low as to lie below bi for some firm i, then the industry marginal cost curve would not be defined. We therefore impose the restriction that, given the cost functions of all n firms, the industry price p c at the Cournot equilibrium be greater than the b of the (possibly not unique) least efficient firm. This assumption ensures that all firms are always active in the industry. Let b¯ be the b of the least efficient firm. Then cartel marginal and average costs are b¯ 1Q /S and b¯ 1Q / 2S respectively. Cartel profits at the joint-profit maximising output Q m 5S(12 b¯ ) /B are therefore S(1 2 b¯ )2 ]]] 2B or n(1 2 b¯ )2 ]]] 2(a 1 2n) if S5n /a. Note that the last expression is increasing in cartel membership. Let mc*5 b¯ 1Q m /S be the cartel marginal cost at output Q m . The optimal output and profit allocation rule gives to each i the share mc* 2 bi ui ; ]]]]] n i 51 (mc* 2 bi )
O
For given i the cartel profits, deviation profits and Cournot equilibrium profits are, respectively,
ui S(1 2 b¯ )2 p mi 5 ]]]] 2B
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[ui S( b¯ 2 1) 2 S( b¯ 1 1) 1 bi B 2 1] 2 p ch i 5 ]]]]]]]]]] 2B 2 (2 1 a ) and (1 2 D 2 bi )2 p ci 5 ]]]] 2(2 1 a ) Substitution of the three foregoing expressions into Eq. (7) yields the condition for cartel stability. The derivatives of this condition with respect to the variables a, ui and D are highly ambiguous. However, it is possible to show that if n53 and all firms have identical b ’s Eq. (7) increases with bi . This is because the derivative of the equivalent expression Eq. (8) is positive if bi 1D ,1, a condition which must be satisfied if individual firms’ Cournot equilibrium outputs are, in turn, to be positive. The implication is that, at least for some values of the parameters, the incentive to deviate is increasing as the prospective deviant is more inefficient. In this respect the conclusion accords with that embodied in proposition 4 for the case in which the bi ’s are normalised to zero. The analysis set out in this paper shows the importance of giving due attention to the costs of firms in a cartel. This is, of course, not to suggest that the number of firms is immaterial. In fact, the size of the cartel is especially important in the context of partial cartels of the type considered by Shaffer (1995), or where trigger strategies are not employed in order to sustain collusion. Nevertheless, although the following proposition captures an obvious insight, it embodies a principle which is frequently suppressed: Proposition 5. The stability of a cartel depends not only on the number of members, but also on their individual and aggregate costs.
4. Concluding comments This paper has explored some implications of the well-known trigger strategy framework for the maintenance of cartel stability. The focus has been upon the implications for the strategy of the existence of differences amongst firms’ marginal costs, or ‘inefficiency parameters’. An important insight to be gained from this study is that the effectiveness of the strategy as a deterrent to deviation depends crucially, and in a potentially quite complex way, upon the relative efficiencies of the deviant and the loyalists. More specifically, it becomes apparent that while some firms might be deterred from deviating, others will be willing to go ahead. Thus, if a trigger strategy is to be an effective deterrent, then the condition Eq. (1) (or, equivalently, Eq. (7)) must be
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satisfied for all of the potentially profitable deviants, and hence for a complex set of cost and output configurations. It is clearly the case that the results obtained here depend to a large extent on the particular form of the demand and cost functions which we employ. Whilst it would obviously be desirable to have more general forms of these functions yield comparably detailed insights, this is unfortunately not possible. At the same time, however, the approach adopted here does have the virtue of drawing attention to some of the possible implications of cost heterogeneity, and to this extent it constitutes both a contribution to an important area of analysis, and a basis for further work.
Acknowledgements I would like to thank the Editor, S. Martin, and two anonymous referees, for numerous very helpful comments on an earlier draft. Responsibility for any errors which remain is entirely my own.
Appendix A
Numerical example illustrating the increase in costs due to switching all production to the lowest cost firm We consider two examples involving the cost function
ai q 2i ]] c i (qi ) 5 bi qi 1 2 where, for the sake of simplicity, we suppose that bi 50. In the first example, we take the output of the cartel as given; in the second, we suppose that the most efficient firm produces its profit maximising output. Let n52. Suppose that Q525, a1 52, and a2 53. The cost minimisation rule for allocating shares in this case gives
a2 Q q1 5 ]]] 5 3Q / 5 5 15. a1 1 a 2 Similarly,
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a1 Q q2 5 ]]] 5 2Q / 5 5 10. a1 1 a 2 It is easily shown that in this case total cartel cost, or a1 q 12 / 21 a2 q 22 / 2, is 375. If production were undertaken exclusively by firm 1 (the lower cost producer), then cost would rise to 625 (525 2 ). Suppose, as an alternative, that output is not given. Let the demand function be p512Q. If a1 52 and a2 53, and both firms produce according to the share allocation rule identified above, then respective profits are 0.09375 and 0.0625, respectively. These sum to a cartel profit of 0.15625. But now suppose that the less efficient firm ceases production, and that the more efficient firm produces its profit maximising output. Then profits are 0.125,0.15625. We conclude that, in this case too, it is not profitable to shut down the less efficient firm.
Appendix B
Derivation of joint-profit maximising cartel output, price and profit If p512Q, then cartel marginal revenue is 122Q. Set marginal revenue equal to aggregate marginal cost, Q /S, to obtain Q5S /(112S). Substitute into the demand function to obtain p5(11S) /(112S). Cartel profits are found by appropriate substitution in the expression Q[ p2ac(Q)], where average costs for the cartel at output Q are given by 1 / 2(112S).
Appendix C
Derivation of individual firm’s output allocation and profit at the cartel price Refer again to Appendix A for the output allocation rule which emerges from the maximisation exercise. For the case involving more than two firms this rule, as it relates to the output of (say) firm 1, can be generalised to Q ]]]] n a1 i 51 a 21 i
O
This corresponds in an obvious way to the rule which we apply in the paper to any i, that is,
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qi 5 Q /Sai . Given this output allocation rule, the profits of firm i at price p5(11S) /(112S) are found by appropriate substitution in the expression qi [ p2ac(qi )] where, given qi and ai , the average costs of the firm are ai qi / 2.
Appendix D
The basis for the derivation of the deviant’s output and profit The value of Z in the text is the original cartel output, minus the amount originally produced by the deviant:
ai S 2 1 S 1 ]] 2 ]]] 5 ]]] 1 1 2S ai (1 1 2S) ai (1 1 2S)
Appendix E
Demonstration of the relationship between deviation profits and profits at the cartel point (ai (1 1 S) 1 1)2 ai S(ai S 2 2) 1 1 1 p ich 2 p im 5 ]]]]]] 2 2 2 ]]]] 5 ]]]]]] 2 a (1 1 2S) 2a i (2 1 ai )(1 1 2S) 2a i2 (2 1 ai )(1 1 2S)2 i which is positive if S ±1 /ai , which is true by definition.
Appendix F
Derivation of Cournot equilibrium profits Each i chooses qi to maximise pi , given D. This yields qi 5(12D) /(21 ai ). The resulting price is given by 12qi 2D, and equals [(12D)(11 ai )] /(21 ai ). Given average costs, profits are derived by appropriate substitution.
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Appendix G
Numerical examples of the deterrence condition in Eq. (7) We show first that there exist parameter values for which the RHS of Eq. (7) is smaller than unity, and therefore that deterrence is feasible if m is sufficiently large: Suppose n53. Set ai 56, ;i, so that S50.5. Since in this case D52qi , it follows that qi ;(12D) /(21 ai )51 /(41 ai )51 / 10. Hence D51 / 5. Substitution of the appropriate values into Eq. (7) yields a value of 0.510. Now suppose that the common inefficiency parameter for the three firms increases to ai 58. Then S53 / 8, and D51 / 6. Substitution into Eq. (7) yields a value of 0.507. Finally, suppose ai 5100. Then S53 / 100, and D51 / 52. Substitution yields as the deterrence condition m >0.5 As indicated earlier, it is instructive to consider the implications for the deterrence condition of allowing S to vary as ai becomes larger. The examples given here suggest that, in this case, as ai becomes larger, the Eq. (7) becomes smaller. We conclude that in this case a decrease in the inefficiency parameter for a given firm may encourage deviation.
Appendix H
An example to illustrate the potential gains from adjusting optimally to the deviant’s output Suppose n56. Set ai 50.75 and aj 50.5. Let n21 1 S ; ]] 1 ] ai aj If D .0.51, then the condition in Eq. (7) for a prospective deviant j is m .0.8; for a prospective deviant i the condition is m .1.9. It follows that if m is sufficiently close to unity, then i will deviate but j will not. If i deviates, then [from Eq. (9)] firm j obtains profits of 0.0482 by choosing its optimal response (all other firms’ outputs unchanged). These profits compare with the 0.039 obtainable by not reacting [see Eq. (5)], and the Cournot equilibrium profits [see Eq. (6)] of 0.0480. It is clear that these outcomes depend crucially upon the value of D. It is easy to find examples in which the incentives to deviate may be reversed and where
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Cournot equilibrium profits exceed those available even if the loyal firm responds optimally.
Appendix I
An illustration of the Rotemberg and Saloner result We employ an example from Appendix G. Suppose n53. Let ai 58, ;i, S53 / 8, and D51 / 6. Then, if m 50.13, a value of et 50.1 satisfies our deterrence condition with equality. Substitute this value of et into the optimal deviation output for firm i: (1 1 et )(ai A 1 1) ch q it 5 ]]]]] ai B(2 1 ai ) and multiply by n to obtain cartel output Q t 50.28 and corresponding pt 50.72. These values compare with 0.21 and 0.79, respectively, the output and price for a monopoly in state et . Now suppose that ai 5100, ;i, S53 / 100 and D51 / 52. Then the value of m which satisfies the deterrence condition with equality for et 50.1 is 0.002. It follows that if deviation is deterred in the former case (i.e. when ai 58), then it will also be deterred for this set of parameters. In other words, for these parameters at least, the output level which deters deviation when firms are absolutely efficient will also deter cheating when the firms are absolutely less efficient, and consequently given m the cartel output consistent with stability may be smaller (and price higher) the less efficient are the firms involved.
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