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Competitive advantage and collusive optima
International Journal of Industrial Organization 5 (1987) 351-367 North-Holland
COMPETITIVE
ADVANTAGE
AND
COLLUSIVE
OPTIMA
Richard SCHMALENSEE*
Massachusetts Institute of Technology, Cambridge, MA 02139, USA Final version received April 1987 Cost differences may often be important in real oligopolies. If side payments are impossible the standard cartel objective, total industry profits, lacks plausibility when costs differ. This essay considers plausible alternatives. Four different technologies for effecting collusion, which are essentially identical when costs are equal, define sets of possible profits. Axiomatic bargaining models are used to select unconstrained optima from among these possibilities. Simple functional forms are employed. Low-cost firms with large shares in Cournot equilibrium may have little to gain from collusion. If collusion is effective, low cost firms will have relatively low estimated conjectural variations.
1. Introduction Once beyond existence theorems, most treatments of oligopoly theory focus o n t h e s y m m e t r i c , e q u a l - c o s t case. t B u t D e m s e t z (1973), P o r t e r (1985) a n d o t h e r s h a v e s t r e s s e d t h e e m p i r i c a l i m p o r t a n c e o f l o n g - l i v e d differences in efficiency, b r o a d l y d e f i n e d , a m o n g c o m p e t i n g sellers, 2 a n d it h a s b e c o m e s t a n d a r d to a l l o w for s u c h d i f f e r e n c e s in e c o n o m e t r i c s t u d i e s o f p a r t i c u l a r industries. T h e s y m m e t r i c case is t h u s o f d o u b t f u l e m p i r i c a l relevance. This e s s a y is c o n c e r n e d w i t h c o l l u s i o n w h e n c o s t s differ. It is often a s s u m e d w i t h o u t m u c h d i s c u s s i o n t h a t c o l l u d i n g firms a l w a y s a t t e m p t to *Much of the research reported here was performed while I was a visitor at CORE. I am indebted to CIM (Belgium) for financial support, to Jacques Thisse for arranging my visit, and to him and my other hosts at Louvain for making my stay pleasant and productive. I am also grateful to participants in seminars at CORE, LSE, and the Norwegian School of Economics and Business Administration for helpful discussions of earlier versions of this essay and to Ian Ayres, Paul Geroski, Tracy Lewis Garth Saloner, Margaret Slade, anonymous referees, and, especially, Rob Masson for useful comments and suggestions. The usual waiver of liability applies, of course. aSee, for instance, Friedman (•983). There are a few exceptions, of course. Osborne and Pitchik (1983) assume equal costs and use the Nash (1953) variable-threat bargaining model to analyze the effects of capacity differences on collusive outcomes. Demange and Ponssard (1985) examine the effects of differentiation on non-cooperative price-setting equilibria when costs are different. ZNote also that in the FTC Line-of-Business data for 1975, industry dummy variables and market share together explain only about 20~ of the sample variance of business unit profitability [Schmalensee (1985)]. Thus intra-industry differences in profitability are much more important than inter-industry differences, even when the effect of market share is controlled for.
0167-7187/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
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R. Schmalensee, Competitive advantage and collusive optima
maximize total industry profits. 3 But, as Bain (1948) argued almost 40 years ago in a comment on Patinkin's (1947) classic paper, this objective makes sense only when side payments are possible.'* When side payments are not possible, total industry profits may have to be reduced in order to attain an equitable division of the gains from collusion. Colluding firms must solve non-trivial bargaining problems. In what follows, plausible unconstrained solutions to these problems are examined in order to shed light on plausible cartel objectives when costs differ. I thus neglect three potentially important constraints on collusion. First, it is rarely trivial to reach a collusive agreement, and this task is likely to be harder when costs differ than in the symmetric case because firms' preferred prices differ. [See, for instance, Blair and Kaserman (1985, pp. 145146) or Scherer (1980, pp. 156-160).] Second, collusion requires not only that an agreement be reached, but that it be stable against cheating, s Finally, potential entrants may provide an effective check on collusion. The analysis proceeds as follows. Section 2 sets out basic assumptions and notation. Section 3 describes and compares four alternative collusion technologies, three of which are equivalent in the special symmetric case. Thes~ serve to define the feasible sets of profit vectors, and the corresponding profit-possibility frontiers, from which a cartel can choose. Section 4 then presents the solution concepts from axiomatic bargaining theory that are employed to characterize plausible choices and thus alternative unconstrained collusive optima. 6 Section 5 describes the resulting optima in terms of changes in profits and consumers' surplus, and section 6 examines the corresponding implicit conjectural derivatives. Section 7 provides a brief summary.
2. Assumptions and notation I consider a market for a homogeneous product in which one low-cost firm faces competition from N identical high-cost sellers. This permits me to vary the intensity of non-cooperative rivalry in a tractable fashion. In most of the analysis, the market demand function is taken to be linear. With appropriate choice of units for money and output, the inverse demand 3See, for instance, Osborne (1976) and Clarke and Davies (1982), '*See also Fellner (1949), Bishop (1960), and, especially, Osborne and Pitchik (1983). 5See, for instance, Porter (1983) and, for an interesting precursor, Orr and MacAvoy (1965). Note also that stability constraints will generally lower profits even when side payments are possible. aNon-cooperative bargaining theory [see Sutton (1986) for an overview] provides a more satisfactory approach in principle. But for the present study I need simple, general solutions that can be applied to a static model without adding restrictive dynamic assumptions, and noncooperative theory has yet to produce such solutions.
R. Schmalensee, Competitive advantage and collusive optima
353
function can thus be written as P = P(Q) = 1 -- Q,
(1)
where P is market price and Q is total output. The low-cost firm, which will be referred to as the leader or firm l in what follows, is assumed to have costs given by C(qO = (1 - O,)q~,
(2)
where q~ is the leader's output and 0a is a constant between zero and one. The assumption of constant unit costs and neglect of capacity constraints is consistent with a focus on long-lived differences in costs or products. Let 171 be the leader's profit. The N high-cost firms, which will be referred to as followers or firms of type 2, also have constant unit costs, C(qz)=(1-82)q2,
(3)
where q2 is the output of a single follower, and 8 z is a constant between zero and one. The two cost parameters must satisfy 82<8~ <282. The second of these inequalities ensures that the followers' costs are below the leader's monopoly price. Let N i l 2 be total followers' profit. Since the followers have identical and constant costs, profits can be shifted among them by shifting outputs even if side payments are impossible. We thus lose no generality by dealing throughout with average follower output and profit, q2 and FI e, respectively. These functional form assumptions produce a model that is algebraically simple, at least as compared to other asymmetric oligopoly models. This model also has the convenient property that as long as attention is limited to relative (i.e., percentage) changes in such quantities as profit and consumers' surplus, only the ratio R = 0 1 / 8 2 matters. (From the preceding paragraph, 1 < R < 2.) The market is thus effectively described by two parameters, R and N. On the other hand, the use of specific functional forms necessarily limits the generality of the results. The parameter R reflects both cost differences and the potential profitability of the market considered, which is determined by the levels of 81 and 02. To provide a closer link to observables, we can also describe the market by N and S, where S is the share of the leader in Cournot equilibrium and N remains the number of fringe firms. The relation among these parameters is as follows: R = N ( 1 + S ) / ( N + 1 -- S).
(4)
354
R~ Schmalensee, Competitive advantage and collusive optima
Increasing S or N, with the other parameter held constant, increases the leader's cost advantage. The larger is N, the more intense is competition, and the larger the leader's cost advantage must be to sustain any given market share. I assume that if collusion does not occur, the market is in Cournot equilibrium. The Cournot point serves in what follows as a benchmark for evaluating the effects of collusion as well as the status quo point in collusive bargaining. The Cournot assumption is familiar and tractable. It has the realistic implication, not shared by the natural Bertrand alternative, that high-cost sellers have positive market shares. And, following Kreps and Scheinkman (1983), it assigns central importance to capacity decisions and is thus consistent with the long-run focus of this inquiry. Note that under this assumption the leader/follower distinction refers only to market shares; there is no behavioral asymmetry.
3. Collusion technologies This section describes four general methods of affecting collusion, along with the corresponding profit-possibility frontiers and contract loci in output space.
Side payments (SP). If the colluding firms can make side payments (or merge), all production will be done by the leader. Total collusive output will thus equal the leader's monopoly output, 0~/2, and total profit will equal the leader's monopoly profit, (01)2/4. (Note that this neglects any short-run fixed costs associated with followers' capacity.) This simplest technology is represented by the straight line labeled SP in fig. 1 and by the similarly-labeled point in fig. 2. In both figures, the point labeled C corresponds to the Cournot equilibrium. [Figs. 1 and 2 are approximate representations of the case N = 2 and R=1.143 (or S=0.445), a case with relatively unimportant cost differences.] It is probably best to think of the SP technology as a standard of comparison, rather than as a realistic possibility in many situations. Market sharing (MS). If side payments are ruled out for legal or other reasons, each firm's earnings derive only from its own production and sales. In Bain's (1948) phrase, 'earnings follow output'. Profit possibilities are thus restricted as compared to the side payments technology, since production must be inefficient if both types of firms receive positive profits. The market sharing technology is most commonly assumed in such situations: firms are assumed to set and abide by output quotas. For constant, unequal marginal costs and any demand function satisfying the second-order conditions, the profit-possibilities frontier is strictly convex, as is the curve labeled MS in
R. Schmalensee, Competitive advantage and collusive optima
(80z __
T
355
$p
~0z) 2 4N
(002/4
HI
Fig. 1. Collusion technologies in profit space. q2 02 2N
~
c
0112 Fig, 2. Collusion technologies in output space.
ql
fig. 1. 7 R i s i n g m a r g i n a l c o s t is n e c e s s a r y ( b u t n o t sufficient) for t o t a l i n d u s t r y p r o f i t t o a t t a i n a l o c a l m a x i m u m w i t h b o t h q t a n d q2 p o s i t i v e w h e n c o s t s differ. W h e n d e m a n d a n d c o s t a r e g i v e n b y eqs. (1)-(3) a b o v e , it is s t r a i g h t f o r w a r d to use t h e f i r s t - o r d e r c o n d i t i o n s for m a x i m i z a t i o n of 172 s u b j e c t to a l o w e r b o u n d o n /11 t o s o l v e f o r p r o f i t s o f t h e t w o t y p e s of firms as f u n c t i o n s of t o t a l o u t p u t a l o n g t h e c o l l u s i v e f r o n t i e r , /7 t = ['(0~ - Q ) 2 ( 2 Q - o z ) ] / ( o t - o2),
(Sa)
N I l 2 = [(0z - Q)z(01 - 2 Q ) ] / ( o l - 02).
(Sb)
7With two firms or two groups of identical firms, the profit-possibilities frontier is obtained by maximizing H~ subject to the constraint 17l - k > 0 , for k between zero and firm l's monopoly profit. The constraint will be binding, and the corresponding multiplier, 4~, will thus be positive. By the envelope theorem, at an optimum q~ will equal - d H ~ / d k = - d / / j d H , where H~ is the constrained maximum of I/2. The profit-possibilities frontier, H 2 =H~'(H0, will thus be convex if and only if d~b/dk is negative. Totally differentiating the first,order conditions and the constraint and solving by Cramer's rule yields d~b/dk=-[(1-~b)2(P')2+XtKp(2P'+A)+X2(2(oP'+A) ~bXIX2]/D, where P'=dP(Q)/dQ, A=(d2P(Q)/dQ2)(~bqi+q2), Xi=d2Cu/dq~ for i = l , 2 , and D is the determinant of the bordered Hessian, which is positive by the second-order conditions. In the constant cost case, X1 = X 2 = 0 , and strict convexity follows unless q~= I. Solving the firstorder conditions, ~b= 1 implies equality of marginal costs, which was ruled out by assumption.
356
R. Schmalensee, Competitive advantage and collusive optima
As total industry output falls from the leader's monopoly output, 01/2, to the followers' monopoly output, 02/2, total profit falls, and the industry moves to the northwest along the MS locus in fig. 1. Unless only the leader produces, total profits are lower than along the SP locus. The same first-order conditions yield an explicit expression for the contract curve labeled MS in fig. 2,
qz={[201+O2]-[(201-Oz)Z-8ql(Ol-O2)]uz}/4N-q~/N.
(6)
This curve is strictly convex, as drawn. [Bishop (1960, p. 948) asserts the convexity of the contract curve in this case.]
Market division (MD). When the relevant frontier in a bargaining problem is non-concave, one normally thinks of using mixed strategies to convexify the feasible set. This seems unnatural here, however. If a coin were to be flipped once to decide which type of firm were permitted to monopolize the market, some enforcement mechanism would be required to compel the loser(s) to exit. Alternatively, if a coin were to be flipped many times, so that average profits per period equaled expected profits in the limit, firms would have to start up and shut down frequently. Neither scenario resembles any obvious example of actual cartel behavior. But an alternative convexification device does correspond to a frequentlyobserved pattern of cartel behavior: the firms can divide the market, s That is, each actual or potential customer can be assigned to a single firm. If all customers are identical, as I wilt assume for simplicity, and firm 1 is allocated a fraction W* of them, its inverse demand curve is given by P=P(ql/W*), and it will charge its monopoly price. Firms of type 2 will similarly charge their higher monopoly price to the customers they have been allocated. Since each type's output and profit are linear in W*, the M D frontier is linear in both fig. 1 and fig. 2. Because market division requires different prices for the same product when sellers' costs differ, it is clearly feasible only when some mechanism can be used to rule out arbitrage at moderate cost. Starting from any point on the MS contract curve, if each firm is given a share of customers equal to its market share at that point, all will be able to increase profits by changing price. Thus M D is a more efficient technology than MS, though MD is of course less efficient than SP, as fig. 1 shows. The differences among these technologies grow with cost differences. When costs are equal, the SP, MS, and M D loci coincide in both fig. 1 and fig. 2. In this symmetric case, and only in this case, the division of monopoly output among colluding sellers does not affect total industry profit. 8Blair and Kaserman (1985, ch, 7) discuss this device at some length. For brief treatments, see Stigler (1964) and Scherer (1980, pp. 168-175).
R. Schmalensee, Competitive advantage and collusive optima
357
P r o p o r t i o n a l r e d u c t i o n ( P R ) . In many situations in which side payments are impossible, arbitrage will prevent market division. Moreover, the complexity of the market sharing technology when costs differ may make long and complex negotiations necessary, especially when firms are imperfectly informed about their rivals' costs, and this may entail unacceptable antitrust risks. In such situations, colluding sellers may resort to simple procedures to set o u t p u t quotas. One procedure that immediately suggests itself is to maintain m a r k e t shares at their non-collusive (Cournot) values and reduce the output of all sellers proportionately. This procedure constrains the firms to move along the PR contract curve in fig. 2. By doing so they can, for instance, reach a point on the MS contract curve and enjoy the corresponding profits. 9 In the symmetric case, this point maximizes total industry profit and divides it equally among all sellers. When costs differ, however, this point has no special attraction. In general, movements in toward the origin along the PR curve in fig. 2 produce movements away from the Cournot point, C, along the PR profitpossibility locus in fig. 1. l° As fig. 1 indicates, the PR technology m a y be substantially less efficient than any of the others when costs differ.
4. Collusive solution concepts W e must now specify how colluding firms select a point on the relevant profit-possibility frontier. This section briefly presents the solution concepts from axiomatic bargaining theory that I employ for this purpose. It is not my intention to argue that reasonable bargainers must behave according to any one of these solution concepts. I use them merely to identify ranges of plausible cartel optima. [This section relies heavily on Roth's (1979) excellent development of axiomatic bargaining theory.] The solution concepts discussed below all produce collusive outcomes at which all parties are at least as well off as at the status quo point for all values of N and R. The implied cartel optima thus lie to the northeast of C in fig. 1, between the two dashed lines. In what follows an asterisk denotes a collusive outcome and a superscript 'c' denotes the Cournot status quo point. N a s h . In the classic Nash (1950) solution, the product of the parties' gains,
(~, -
~
H1)(IIz
,
-
c
H2)
N
,
is maximized along the relevant frontier. 9Even though fig. 2 suggests that the proportional reduction technology can reach a point on the MD contract curve, it should be clear that total profits at that point are less than the corresponding MD profits, since the market is shared by firms charging a single price. 1°The tangency of this latter locus and the MS contract curve always occurs above the Cournot point. It occurs to the left of that point, so that the leader is worse off than in Cournot equilibrium, if the leader's cost advantage is large enough. [The conditions are S(1 +S)> 1 and N > (1 - $2)/(S 2 + S - 1). These imply S > 0.781 when N = 1; this critical value declines to 0.681 in the limit as N increases.]
R. Schmalensee, Competitive advantage and collusive optima
358
Kalai-Smorodinsky (K-S). To compute the solution proposed by Kalai and Smorodinsky (1975), define H]" as the maximum profit the leader can get on the relevant frontier when all followers get at least their status quo profit, and define H~' as the maximum profit a single follower can receive when all other firms receive at least their status quo profit. (Under the PR technology /7]" is given by the point in fig. 1 where the PR locus is vertical, and, since all followers must receive the same profit under this technology, //~' is given by the highest point on that locus on or to the right of the dashed vertical line.) The K - S solution is then the point on the frontier at which all parties' actual gains are the same fraction of their maximum gains, c
(n?
m
c
*
c
- FI1)/(H 1 - Hi) = (H z - nz)/(n
m
c
2 - nz).
(7)
If the profit-possibility frontier is linear, as implied by the SP and M D technologies, it is easy to show (using the ability to allocate total follower profit arbitrarily among followers) that the Nash and K - S solutions coincide.
Equal gains (EG). Roth (1979, pp. 92-97) suggests the possible relevance of solutions in which the absolute gains to all parties are equal, *
c
(H1-H1)=(H2-H2),
*
C
(8)
Along the SP frontier, it is easy to show that the Nash, K-S, and EG solutions all coincide. Thus all firms gain an equal dollar amount from collusion in all three solutions. (In relative terms, of course, the followers gain more than the leader because their status quo profits are lower.) Along the flatter M D frontier in fig. 1, the EG solution lies to the left of the N a s h / K - S solution. The latter thus gives the low-cost firm a greater absolute gain than each high-cost firm. If the PR technology is employed, no solution satisfying (8) exists when N = 1. And when N = 2, the EG solution maximizes firm l's profit on the PR frontier.
W*=S. In the MD technology, a natural focal point [in the sense of Schelling (1960)] is provided by the allocation of customers to firms in the original Cournot equilibrium. Accordingly, for this technology only, I consider solutions in which W*, the fraction of customers assigned to the leader, equals S, the leader's initial market share. Each follower then receives a fraction ( 1 - S)/N of the market demand curve. 11 HAnother apparently plausible focal point under MD or MS is where S*, the leader's share of collusive output, is equal to his Cournot share, S. But under both technologies this 'solution' makes the leader worse off than at the status quo point when he has a large cost advantage.
R. Schmalensee, Competitive advantage and collusive optima
359
5. Collusive optima Under the SP and M D technologies, the equilibria corresponding to the Nash/K-S, EG, and (for M D only) W * = S solutions can be obtained explicitly as functions of N and S. Under the PR technology, the K - S and EG equilibria can be also obtained analytically. Even though the model used here is about the simplest imaginable, however, all the formulae involved are too complex to be informative. Moreover, under the MS and PR technologies, numerical solution of equations is necessary to obtain Nash equilibria, and the K - S and E G equilibria must also be treated numerically under the MS technology. Accordingly, [ adopt a purely numerical approach in this section and the next. Collusive optima are compared for some illustrative parameter values, and general relations revealed by computation of many equilibria are discussed, 12 Consistent with the remarks at the start of section 4, I emphasize patterns that hold across solution concepts. Changes in R. Table 1 describes alternative collusive optima when N = 1 for three values of R. When costs are equal, the natural symmetric duopoly collusive equilibrium (which is obtained by applying any of section 4's solution concepts to any of section 3's collusion technologies) raises both firms' profits, lowers consumers' (Marshallian) surplus, and reduces total surplus compared to the Cournot benchmark. When costs are different, production is inefficient in the initial Cournot equilibrium. The possibility of rationalizing the production gives rise to potential gains from collusion that are not present in the symmetric case. SP collusion takes full advantage of this possibility and even produces an overall net gain in (Marshallian) social welfare if R > ( 6 N + 1 6 ) / ( 5 N + 1 2 ) . But all collusive optima computed under the other three technologies, which necessarily fail to rationalize production completely, involve a net social loss. In all the asymmetric optima I have examined, the leader's percentage gain from collusion is below the followers', and the difference rises with increases in R. The greater the leader's cost advantage, all else equal, the less he has to gain from collusion. As R approaches 2, the initial Cournot equilibrium approaches a monopoly for the leader, so that his maximum possible gain from collusion goes to 0. If the SP technology is ruled out, all parties, including buyers, are generally made noticeably worse off at collusive optima, Price is higher because it must reflect, in part, the preferences of the high-cost followers, And
12All the equilibria discussed in the text were computed for N= 1,2,..., 10, for R ranging over the interval (1.0,2.0). Programs were written in IBM/MicrosoftBASIC, Release 2.0, and executed on an IBM PC. The author will be happy to supply copies of all programs employed to anyone sending a suitable formatted diskette.
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360
Table 1 Collusive optima withN=l. Percentage surplus gain Value of R
Technology/solution
Leader
Followers
Consumers
(S=0.50)
Symmetric
Alt a
12.50
12.50
-43.75
1.143 (S = 0.60)
SP
All
16.67
37.50
MD
Nash/K-S EG W* = S
9.86 8.55 6.67
16.99 19.29 22.50
MS
Nash K-S EG
9.50 9.50 8.22
PR
Nash K-S EG b
6.67 6.40 .
Net
~o profit sacrifice
1.000 -31.25
0
- 36.00
-5,88
0
-41,73 -41.84 -42.00
- 14.31 - 14.47 - 14,71
8,95 9.13 9,37
18.69 18,69 22.49
--41.95 -41.95 - 42.20
- 14.57 - 14.57 - 14.92
9,33 9.33 9.50
20.14 21.60
- 36,46 -42.24 .
- 12,36 - 15.06
9,97 9.75
.
.
1.400 (S=0.75)
1.727 (S = O.9O)
SP
All
12.50
112.50
-23.44
MD
Nash/K-S EG W* = S
7.17 4.84 2,08
32.91 43.58 56.25
-31.41 - 32.05 - 32.81
-8.55 -9.40 - 10,42
2.08
10.42 11.25 12.24
MS
Nash K-S EG
5.81 5.81 3.78
25.18 25.18 34.00
-38.43 -38.43 - 39.14
-12.78 -12.78 - 13.62
12,05 12.05 12.82
PR
Nash K S EG
2.08 1.70 -
31.73 40,82 -
- 23.86 - 32.53 -
- 7.80 - 11.34
14.25 13.79
SP
All
5.09
4t2.50
-9.75
MD
Nash/K-S EG W* = S
3.87 1.94 0,28
105.02 157,31 202.50
- 13.82 - 14.85 - 15,75
-2.06 -3.24 -4.26
4.51 5.66 6.56
MS
Nash K-S EG
2.51 2.52 0.96
48,01 47.95 78.08
-27.24 - 27.24 - 29.01
-8.41 - 8.41 - 9,8 t
6.35 6.35 7.41
PR
Nash K-S EG
0.28 0,21
42,93 61.27 -
- 9.85 - 14.29
- 3,24 -4.82 -
8,42 8.27
" N a s h , K - S , and E G solutions coincide. bThe P R / E G solution does n o t exist for N = 1.
2.56
0
0
R. Schmalensee, Competitive advantage and collusive optima
361
total industry profits are not maximized because production is not fully rationalized. The last column in table 1 gives the percentage by which total industry profits fall short of their maximum value. For all values of N and all non-SP equilibria, this profit sacrifice is maximized for moderate values o f R. As R approaches i, the sacrifice goes to 0 because the symmetric case is approached. At the other extreme, as R approaches 2, the sacrifice goes to 0 along with the potential gains from SP collusion. The difference between the MD and MS solutions are relatively small, as are those between the MS/Nash and MS/K-S equilibria. These patterns reflect the near-linearity of the MS profit-possibility frontier for most values of N and R. [This was noted by Bishop (1960, p. 948).] The PR solution is close to the other two in some cases, though it is generally clearly inferior from the leader's viewpoint. Because it precludes rationalizing production by increasing the leader's share, the extent to which total output can be restricted is limited, and consumers are often better off under PR collusion than if the MD or MS technologies had been employed. In the examples of MD collusion described in table 1, the W * = S solution lies to the left of the Nash/K-S and EG solutions along the MD frontier. This implies that the Nash, K-S, and EG solutions involve W*> S, so that the leader's share of customers (and, afortiori, of output) is increased by collusion. Similarly, the leader's market share exceeds S at all MS collusive optima in table 1. This reflects the potential gains from rationalizing production. But when R is small and N is large, Nash, K-S, and, especially, EG collusion may involve reducing the leader's share. Potential rationalization gains are small when R is near 1, and the followers' bargaining power rises as N increases. Changes in N. Table 2 shows the effect of changes in N, with R held constant at a moderate level, In the symmetric case, increases in N simply increase the intensity of competition at the status quo point and thus increase the sellers' gains and buyers' losses produced by collusion. When costs differ, however, increases in N also increase the status quo share and aggregate bargaining power of the high-cost followers. The first effect increases the potential gains from rationalizing production. But the second increases the profit sacrifice made in the interests of equity when side payments are impossible, as the last column in table 2 shows. Followers' percentage gains tend to increase with N, reflecting both their lower Cournot profits and their increased bargaining power. For small values of R and N, the leader's relative gains tend to rise with N, reflecting the increasing potential gains from rationalizing production. Otherwise, the bargaining power effect dominates, and increases in N reduce the leader's relative (and, afortiori, absolute) gains. (The borderline values of N and R depend on the technology and solution concept, as table 2 reveals.)
R, Schmalensee, Competitive advantage and collusive optima
362
Table 2 Collusive o p t i m a with R = 1,40. Percentage surplus gain Value of N 1 (S=0.75)
3 (S=0.59)
5 (s=o,53)
Technology/solution
Leader
Followers
Consumers
Net
~o profit sacrifice
Symmetric (R=I)
All a
12.50
12.50
-43.75
-31.25
sP
All
12.50
112,50
-23.44
2.08
MD
Nash/K-S EG W* = S
7.17 4.84 2.08
39,21 43.58 56.25
- 31.41 - 32.05 - 32.81
- 8.55 - 9.40 - 10.42
10.42 11.25 12.24
MS
Nash K-S EG
5.81 5,81 3.78
25.18 25.18 34.00
-38,43 -38.43 -39.14
-12.78 -12.78 -13.62
12.05 12.05 12,82
PR
Nash K-S EG b
2.08 1.70 .
31.73 40.82 . .
- 23.86 - 32.53 .
- 7,80 - 11.34
14.25 13.79
0 0
Symmetric (R = I)
All
56.25
56.25
-60.93
-21.88
0
SP
All
16.31
306.25
-36.72
4.88
0
MD
Nash/K-S EG
W*=S
12,47 7.25 7.08
119.52 t36.19 136.74
-48.48 -49.37 -49,40
- 14.61 -16,09 - 16.13
18,58 19.99 20.04
MS
Nash K-S EG
l 1.32 11.47 6.52
107.28 106.80 122.48
- 54.70 - 54.68 -55.48
- 19,24 - 19.20 -20.58
20.30 20,26 21.60
PR
Nash K-S EG
7.02 6.19 6.29
104.99 119.I9 118.18
-39,74 -47.75 -47.11
-12.78 - 16.64 -16.31
22,87 22.08 22.11
Symmetric (R = i)
All
104.17
104.17
-65.97
-23,44
sP
All
15.63
493,06
-41.38
6.43
MD
Nash/K-S EG
12.86 7.15 10.34
210.74 229.48 219.01
-54.49 -55.28 - 54,84
-17.37 -18,81 -18.01
22.36 23,71 22.96
W*=S
0 0
MS
Nash K-S EG
11.84 12,14 6.62
195,24 194.25 212.51
-60.02 -59.98 -60,63
-21,89 -21.82 -23.13
24,02 23.95 25.24
PR
Nash K-S EG
10.05 9.29 6.62
184.56 196.84 212,50
-47.21 -52.24 -60.94
-15.32 - 17.97 -23.32
25.68 25.13 25.24
"Nash, K - S , and E G solutions coincide, bThe P R / E G solution does not exist for N = 1.
R, Schmalensee, Competitive advantage and collusive optima
363
Tables 1 and 2 indicate that in the presence of substantial competitive advantage, corresponding to a large R, the leader's gains from collusion are likely to be small relative to status quo profits when side payments are impossible. While the leader's potential relative gains do not approach 0 especially rapidly as R approaches 2, they seem small relative to likely yearto-year fluctuations for values of R and N that might be encountered in practice. (Note also that the figures shown deal only with e c o n o m i c profits; the corresponding percentage increases in the a c c o u n t i n g profits reported to shareholders are likely to be much smaller.) It thus seems likely that the absence of gains commensurate with the costs and (legal and other) risks involved tends to make collusion less frequent than otherwise in the presence of substantial competitive advantage. 13 IBM has been accused of m a n y things, but not of colluding with its rivals.
6. Implicit conjectural derivatives In this section I follow the approach introduced by Iwata (1974) and use sellers' implicit conjectural derivatives to describe collusive optima, This approach only makes sense under the MS or PR technology, when firms of both types are operating and sharing the market. Consider a market with T firms, all of which may have different cost functions, and let q_~ be the output of all firms except i. Then at any set of outputs, firm i's implicit conjectural derivative, 2~, is the value of dq_~/dq~ that would make its actual output profit.maximizing. Re-arranging the firstorder condition for maximization of firm i's profits, the implicit conjectural derivative consistent with equilibrium at any q~, q_~ pair is given by 2~ = -- ( P - CI + q~P')/ q~P',
(9)
where C~ is firm i's marginal cost, It is easy to show that 2~=--1 implies competitive (P = C') behavior, while 2~= 0 for all i at the Cournot benchmark. More generally, using the second-order condition for profit maximization, one can show that for fixed q_~, an increase in 2~ lowers q~. If 2~= 1, for instance, firm i behaves as i f it fears that increases in its output would be matched, unit for unit, by its rivals, and it accordingly produces less than it would if it felt rivals would not react at all (2~=0). Alternatively, it behaves as i f its output restrictions will be matched, and so restricts output more than in the Cournot case, It is hard to believe that actual oligopoly behavior - particularly collusive behavior - involves this sort of non-strategic maximization. Estimated values a3If cost differences generally served to enhance the stability of collusive arrangements, perhaps by facilitating the choice of a price leader, this conclusion would have to be qualified.
R. Schmalensee, Competitive advantage and collusive optima
364
of the 21 are best understood as providing a convenient s u m m a r y description of market behavior, which m a y in fact involve a complex mix of cooperative and non-cooperative elements. The 2~ are in a sense dual to the qt; their merit is comparability across markets. In this framework, collusion occurs when firms behave as if they feared that increases in their outputs would induce their rivals to produce more. Iwata (1974) has shown that the following c o n d i t i o n is satisfied at all points on the MS profit-possibilities frontier: T
~ = ~ z i = 1,
(10)
i=1
where the first equality defines ~, a n d z i = l / ( 1 +2i) for all i. [Eq. (10) is equivalent to Iwata's (1974, p. 961) eq. (6.12).] It is easy to show (using the relevant first-order conditions) that the 2i must all be positive on the MS frontier. It is only slightly harder to show that 7j > 1 above the MS contract curve in the output space of fig. 2.14 T h u s m o v e m e n t toward collusion involves increases in the 2~. Since 7~ increases w i t h o u t b o u n d as behavior approaches perfect competition (i.e., all 2~ a p p r o a c h - 1 ) , 1/tP provides a measure of the extent to which industry behavior is non-competitive. This quantity is generally confined to the unit interval, t h o u g h 7~ < 1 if the PR technology is used to reach a point strictly inside the MS contract curve. In the special case on which this paper focuses, where one leader faces N identical followers, MS collusion implies
N +(N- 1)21-2122=0.
(11)
Here 2t is the leader's conjectural derivative with respect to aggregate followers' output, and 22 is each follower's conjectural derivative with respect to the o u t p u t of the leader and the other N - 1 followers. This expression is positive under P R collusion at points outside the MS contract curve in fig. 2. It is negative at points inside that curve. Table 3 presents the implicit conjectural derivatives consistent with a variety of MS and P R collusive equilibria. In these a n d all other equilibria examined, 41 is smaller t h a n 22. As table 3 indicates, the difference between these parameters can be quite dramatic. If estimates of this sort were obtained econometrically in some market, one might be tempted to characterize the followers as more concerned a b o u t competitive reactions to o u t p u t 1*Assume that 2~> - 1. so that P>C'~ for all i. Then ~ is a continuous function of the vector of outputs. Since ~ is equal to 1 only on the MS contract curve, (W-I) must have the same sign at all points above this curve. Consider a set of small output changes, Aq, i = 1..... T,, of the same sign. Differentiating and using (9) to substitute for P', the corresponding changes in profits are found to be AHl=(P-C'i)(c~-ri)AQ, i= 1..... T, where al= dql/dQ>O for all i. Since the ai sum to unity, there exists a set of output decreases (increases) that raise the profits of all sellers if and only if 7~> 1 (~< t). Thus 7~> 1 above (~< 1 below) the MS contract curve.
R. Schmalensee, Competitive advantage and collusive optima
365
Table 3 Implicit conjectural derivatives at collusive optima. N=I
Value
Technology/
of R
solution
1.143
MS
Nash K-S EG
PR
1,400
1.727
21
N=3
N=5
),2
),l
),z
21
22
0.81 0,81 0.83
1.23 1.23 1.20
2,03 2,03 2.17
3.48 3.48 3.38
3.06 3.05 3.30
5.64 5.64 5.52
Nash K-S EG a
0.83 0.93
0,95 1,06 -
1,78 1.94 2.57
2.86 3,12 4.12
2.65 2.70 5.44
5.01 5,12 10.32
MS
Nash K-S EG
0.57 0.57 0.61
1.77 1.77 1.64
1.17 1,16 1.29
4.57 458 4.32
1.56 1.55 1.71
7.21 7.22 6.92
PR
Nash K-S EG
0.34 0.51 -
0.73 1.09 -
0.78 1.03 1.01
2.40 3.t9 3.12
1.08 1.29 1.73
4.39 5.22 7.00
MS
Nash K-S EG
0.32 0,32 0.37
3.12 3.12 2.72
0.57 0.56 0.63
7.30 7.34 6,75
0.69 0.69 0.75
11.21 11.28 10,63
PR
Nash K-S EG
0.11 0.17 -
0.59 0,88 -
0.22 0.32 0.29
1.85 2.67 2.39
0.29 0.39 0.44
3.25 4.47 4.97
~The PR/EG solution does not exist when N = 1.
increases than the leader. 1~ While this may be the right interpretation in many cases, such a pattern at least suggests the desirability of testing the collusive null hypothesis tp= 1. It is possible that the followers are restricting output substantially as part of a collusive bargain, rather than because they believe that total industry output is highly responsive to their own production decisions. When either the MS or the PR technology is employed, both 2 t and 22 tend to increase with N, as condition (11) suggests. Holding N constant, increases in R tend to lower 21 and to raise 22 in MS collusion. Under the PR technology, increases in R most often lower both parameters, but several exceptions are visible in table 3. As cost differences are increased, it becomes harder to make large proportional reductions in output without making the leader worse off. Consistent with this, all of the PR equilibria shown in table 3 correspond to 15Spiller and Favaro (1984) consider a market with 'dominant' and 'fringe' firms. They find that 2's for fringe firms were consistent with Cournot behavior. Among the dominant firms, however, smaller sellers had larger 2's. Table 3 implies that this pattern is consistent with collusion (and cost differences) among the dominant firms.
366
R. Schmalensee, Competitive advantage and collusive optima
points outside the MS contract curve, except for the three EG equilibria in the upper right-hand corner of the table.
7. Conclusions Symmetric oligopoly models may lack empirical relevance; they are certainly very special theoretically, Even with functional forms selected for tractability, relaxation of the assumption of symmetry gives rise to considerable algebraic complexity. Four technologies for effecting collusion that are essentially equivalent in the symmetric case are quite distinct when sellers' costs differ, and plausible bargaining outcomes can only be analyzed numerically. Numerical analysis of collusive optima implied by axiomatic bargaining theory reveals a variety of distinctions and effects that are neither present in the symmetric case nor sensitive to the axiomatic solution concept employed. In the absence of side payments, these flow mainly from changes in the balance between two forces. The first is the potential gain from rationalizing production by increasing the share of the most efficient producer. This force is strongest for moderate cost differences. The other force is the bargaining power of high-cost firms. This force increases with the number of inefficient producers. Two specific findings seem especially important for applied work. First, if a leading firm's cost advantage is substantial, its potential gains from collusion are relatively small. This suggests that collusion is unlikely to be observed in the presence of substantial competitive advantage, though the observed price may be close to the (leader's) monopoly level. Second, conjectural derivatives consistent with perfect collusion may vary substantially from firm to firm, depending on the nature of collusion, the importance of cost differences, and the number of sellers. Differences in econometrically estimated conjectural derivatives may thus reflect bargaining outcomes, not differences in rivalrous behavior.
References Bain, J.S., 1948, Output quotas in imperfect cartels, Quarterly Journal of Economics 62, 617-622. Bishop, R.L., 1960, Duopoly: Collusion or warfare?, American Economic Review 50, 933-961. Blair, R. and D.L. Kaserman, 1985, Antitrust economics (Irwin, Homewood, IL). Clarke, R. and S,W. Davies, 1982, Market structure and price-cost margins, Economica 49, 277-287. Demange, G. and J.-P. Ponssard, 1985, Asymmetries in cost structures and incentives toward price competition,International Journal of Industrial Organization 3, 85-100. Demsetz, H., 1973, Industry structure, market rivalry, and public policy, Journal of Law and Economics 16, 1-10. Fellner, W., 1949, Competition among the few (Knopf, New York). Friedman, J., 1983, Oligopoly theory (Cambridge University Press, Cambridge). Iwata, G. 1974, Measurement of conjectural variations in oligopoly, Econometrica 42, 947-966.
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Kalai, E. and M. Smorodinsky, 1975, Other solutions to Nash's bargaining problem, Econometrica 43, 513-518. Kreps, D.M. and J.A. Scheinkman, 1983, Quantity precommitment and Bertrand competition yield Cournot outcomes, Bell Journal of Economics 14, 326-337. Nash, J,F., 1950, The bargaining problem, Econometrica 28, 513-518. Nash, J.F., 1953, Two-person cooperative games, Econometrica 31, 128-140. Orr, D. and P.W. MacAvoy, 1965, Price strategies to promote cartel stability, Economica 32, 186-197. Osborne, D.K., 1976, Cartel problems, American Economic Review 66, 835-844. Osborne, MJ. and C. Pitchik, 1983, Profit-sharing in a collusive oligopoly, European Economic Review 22, 59-74. Patinkin, D., 1947, Multi-plant firms, cartels, and imperfect competition, Quarterly Journal of Economics 61, 173-205. Porter, M.E., 1985, Competitive advantage (Free Press, New York). Porter, R.H., 1983, Optimal cartel trigger price strategies, Journal of Economic Theory 29, 313-338. Roth, A.E., 1979, Axiomatic models of bargaining (Springer, New York). Schelling, T.C., 1960, The strategy of conflict (Harvard University Press, Cambridge, MA). Scherer, F.M., 1980, Industrial market structure and economic performance, 2nd ed. (RandMcNally, Chicago, IL). Schmalensee, R., 1985, Do markets differ much?, American Economic Review 75, 341-351. Spiller, P.T. and E. Favaro, 1984, The effects of entry regulation on oligopolistic interaction: The Uruguayan banking sector, Rand Journal of Economics 15, 244-254. Stigler, G.J., 1964, A theory of oligopoly, Journal of Political Economy 72, 44-61. Sutton, J., 1986, Non-cooperative bargaining theory: An introduction, Review of Economic Studies 53, 709-724.
COMPETITIVE
ADVANTAGE
AND
COLLUSIVE
OPTIMA
Richard SCHMALENSEE*
Massachusetts Institute of Technology, Cambridge, MA 02139, USA Final version received April 1987 Cost differences may often be important in real oligopolies. If side payments are impossible the standard cartel objective, total industry profits, lacks plausibility when costs differ. This essay considers plausible alternatives. Four different technologies for effecting collusion, which are essentially identical when costs are equal, define sets of possible profits. Axiomatic bargaining models are used to select unconstrained optima from among these possibilities. Simple functional forms are employed. Low-cost firms with large shares in Cournot equilibrium may have little to gain from collusion. If collusion is effective, low cost firms will have relatively low estimated conjectural variations.
1. Introduction Once beyond existence theorems, most treatments of oligopoly theory focus o n t h e s y m m e t r i c , e q u a l - c o s t case. t B u t D e m s e t z (1973), P o r t e r (1985) a n d o t h e r s h a v e s t r e s s e d t h e e m p i r i c a l i m p o r t a n c e o f l o n g - l i v e d differences in efficiency, b r o a d l y d e f i n e d , a m o n g c o m p e t i n g sellers, 2 a n d it h a s b e c o m e s t a n d a r d to a l l o w for s u c h d i f f e r e n c e s in e c o n o m e t r i c s t u d i e s o f p a r t i c u l a r industries. T h e s y m m e t r i c case is t h u s o f d o u b t f u l e m p i r i c a l relevance. This e s s a y is c o n c e r n e d w i t h c o l l u s i o n w h e n c o s t s differ. It is often a s s u m e d w i t h o u t m u c h d i s c u s s i o n t h a t c o l l u d i n g firms a l w a y s a t t e m p t to *Much of the research reported here was performed while I was a visitor at CORE. I am indebted to CIM (Belgium) for financial support, to Jacques Thisse for arranging my visit, and to him and my other hosts at Louvain for making my stay pleasant and productive. I am also grateful to participants in seminars at CORE, LSE, and the Norwegian School of Economics and Business Administration for helpful discussions of earlier versions of this essay and to Ian Ayres, Paul Geroski, Tracy Lewis Garth Saloner, Margaret Slade, anonymous referees, and, especially, Rob Masson for useful comments and suggestions. The usual waiver of liability applies, of course. aSee, for instance, Friedman (•983). There are a few exceptions, of course. Osborne and Pitchik (1983) assume equal costs and use the Nash (1953) variable-threat bargaining model to analyze the effects of capacity differences on collusive outcomes. Demange and Ponssard (1985) examine the effects of differentiation on non-cooperative price-setting equilibria when costs are different. ZNote also that in the FTC Line-of-Business data for 1975, industry dummy variables and market share together explain only about 20~ of the sample variance of business unit profitability [Schmalensee (1985)]. Thus intra-industry differences in profitability are much more important than inter-industry differences, even when the effect of market share is controlled for.
0167-7187/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
352
R. Schmalensee, Competitive advantage and collusive optima
maximize total industry profits. 3 But, as Bain (1948) argued almost 40 years ago in a comment on Patinkin's (1947) classic paper, this objective makes sense only when side payments are possible.'* When side payments are not possible, total industry profits may have to be reduced in order to attain an equitable division of the gains from collusion. Colluding firms must solve non-trivial bargaining problems. In what follows, plausible unconstrained solutions to these problems are examined in order to shed light on plausible cartel objectives when costs differ. I thus neglect three potentially important constraints on collusion. First, it is rarely trivial to reach a collusive agreement, and this task is likely to be harder when costs differ than in the symmetric case because firms' preferred prices differ. [See, for instance, Blair and Kaserman (1985, pp. 145146) or Scherer (1980, pp. 156-160).] Second, collusion requires not only that an agreement be reached, but that it be stable against cheating, s Finally, potential entrants may provide an effective check on collusion. The analysis proceeds as follows. Section 2 sets out basic assumptions and notation. Section 3 describes and compares four alternative collusion technologies, three of which are equivalent in the special symmetric case. Thes~ serve to define the feasible sets of profit vectors, and the corresponding profit-possibility frontiers, from which a cartel can choose. Section 4 then presents the solution concepts from axiomatic bargaining theory that are employed to characterize plausible choices and thus alternative unconstrained collusive optima. 6 Section 5 describes the resulting optima in terms of changes in profits and consumers' surplus, and section 6 examines the corresponding implicit conjectural derivatives. Section 7 provides a brief summary.
2. Assumptions and notation I consider a market for a homogeneous product in which one low-cost firm faces competition from N identical high-cost sellers. This permits me to vary the intensity of non-cooperative rivalry in a tractable fashion. In most of the analysis, the market demand function is taken to be linear. With appropriate choice of units for money and output, the inverse demand 3See, for instance, Osborne (1976) and Clarke and Davies (1982), '*See also Fellner (1949), Bishop (1960), and, especially, Osborne and Pitchik (1983). 5See, for instance, Porter (1983) and, for an interesting precursor, Orr and MacAvoy (1965). Note also that stability constraints will generally lower profits even when side payments are possible. aNon-cooperative bargaining theory [see Sutton (1986) for an overview] provides a more satisfactory approach in principle. But for the present study I need simple, general solutions that can be applied to a static model without adding restrictive dynamic assumptions, and noncooperative theory has yet to produce such solutions.
R. Schmalensee, Competitive advantage and collusive optima
353
function can thus be written as P = P(Q) = 1 -- Q,
(1)
where P is market price and Q is total output. The low-cost firm, which will be referred to as the leader or firm l in what follows, is assumed to have costs given by C(qO = (1 - O,)q~,
(2)
where q~ is the leader's output and 0a is a constant between zero and one. The assumption of constant unit costs and neglect of capacity constraints is consistent with a focus on long-lived differences in costs or products. Let 171 be the leader's profit. The N high-cost firms, which will be referred to as followers or firms of type 2, also have constant unit costs, C(qz)=(1-82)q2,
(3)
where q2 is the output of a single follower, and 8 z is a constant between zero and one. The two cost parameters must satisfy 82<8~ <282. The second of these inequalities ensures that the followers' costs are below the leader's monopoly price. Let N i l 2 be total followers' profit. Since the followers have identical and constant costs, profits can be shifted among them by shifting outputs even if side payments are impossible. We thus lose no generality by dealing throughout with average follower output and profit, q2 and FI e, respectively. These functional form assumptions produce a model that is algebraically simple, at least as compared to other asymmetric oligopoly models. This model also has the convenient property that as long as attention is limited to relative (i.e., percentage) changes in such quantities as profit and consumers' surplus, only the ratio R = 0 1 / 8 2 matters. (From the preceding paragraph, 1 < R < 2.) The market is thus effectively described by two parameters, R and N. On the other hand, the use of specific functional forms necessarily limits the generality of the results. The parameter R reflects both cost differences and the potential profitability of the market considered, which is determined by the levels of 81 and 02. To provide a closer link to observables, we can also describe the market by N and S, where S is the share of the leader in Cournot equilibrium and N remains the number of fringe firms. The relation among these parameters is as follows: R = N ( 1 + S ) / ( N + 1 -- S).
(4)
354
R~ Schmalensee, Competitive advantage and collusive optima
Increasing S or N, with the other parameter held constant, increases the leader's cost advantage. The larger is N, the more intense is competition, and the larger the leader's cost advantage must be to sustain any given market share. I assume that if collusion does not occur, the market is in Cournot equilibrium. The Cournot point serves in what follows as a benchmark for evaluating the effects of collusion as well as the status quo point in collusive bargaining. The Cournot assumption is familiar and tractable. It has the realistic implication, not shared by the natural Bertrand alternative, that high-cost sellers have positive market shares. And, following Kreps and Scheinkman (1983), it assigns central importance to capacity decisions and is thus consistent with the long-run focus of this inquiry. Note that under this assumption the leader/follower distinction refers only to market shares; there is no behavioral asymmetry.
3. Collusion technologies This section describes four general methods of affecting collusion, along with the corresponding profit-possibility frontiers and contract loci in output space.
Side payments (SP). If the colluding firms can make side payments (or merge), all production will be done by the leader. Total collusive output will thus equal the leader's monopoly output, 0~/2, and total profit will equal the leader's monopoly profit, (01)2/4. (Note that this neglects any short-run fixed costs associated with followers' capacity.) This simplest technology is represented by the straight line labeled SP in fig. 1 and by the similarly-labeled point in fig. 2. In both figures, the point labeled C corresponds to the Cournot equilibrium. [Figs. 1 and 2 are approximate representations of the case N = 2 and R=1.143 (or S=0.445), a case with relatively unimportant cost differences.] It is probably best to think of the SP technology as a standard of comparison, rather than as a realistic possibility in many situations. Market sharing (MS). If side payments are ruled out for legal or other reasons, each firm's earnings derive only from its own production and sales. In Bain's (1948) phrase, 'earnings follow output'. Profit possibilities are thus restricted as compared to the side payments technology, since production must be inefficient if both types of firms receive positive profits. The market sharing technology is most commonly assumed in such situations: firms are assumed to set and abide by output quotas. For constant, unequal marginal costs and any demand function satisfying the second-order conditions, the profit-possibilities frontier is strictly convex, as is the curve labeled MS in
R. Schmalensee, Competitive advantage and collusive optima
(80z __
T
355
$p
~0z) 2 4N
(002/4
HI
Fig. 1. Collusion technologies in profit space. q2 02 2N
~
c
0112 Fig, 2. Collusion technologies in output space.
ql
fig. 1. 7 R i s i n g m a r g i n a l c o s t is n e c e s s a r y ( b u t n o t sufficient) for t o t a l i n d u s t r y p r o f i t t o a t t a i n a l o c a l m a x i m u m w i t h b o t h q t a n d q2 p o s i t i v e w h e n c o s t s differ. W h e n d e m a n d a n d c o s t a r e g i v e n b y eqs. (1)-(3) a b o v e , it is s t r a i g h t f o r w a r d to use t h e f i r s t - o r d e r c o n d i t i o n s for m a x i m i z a t i o n of 172 s u b j e c t to a l o w e r b o u n d o n /11 t o s o l v e f o r p r o f i t s o f t h e t w o t y p e s of firms as f u n c t i o n s of t o t a l o u t p u t a l o n g t h e c o l l u s i v e f r o n t i e r , /7 t = ['(0~ - Q ) 2 ( 2 Q - o z ) ] / ( o t - o2),
(Sa)
N I l 2 = [(0z - Q)z(01 - 2 Q ) ] / ( o l - 02).
(Sb)
7With two firms or two groups of identical firms, the profit-possibilities frontier is obtained by maximizing H~ subject to the constraint 17l - k > 0 , for k between zero and firm l's monopoly profit. The constraint will be binding, and the corresponding multiplier, 4~, will thus be positive. By the envelope theorem, at an optimum q~ will equal - d H ~ / d k = - d / / j d H , where H~ is the constrained maximum of I/2. The profit-possibilities frontier, H 2 =H~'(H0, will thus be convex if and only if d~b/dk is negative. Totally differentiating the first,order conditions and the constraint and solving by Cramer's rule yields d~b/dk=-[(1-~b)2(P')2+XtKp(2P'+A)+X2(2(oP'+A) ~bXIX2]/D, where P'=dP(Q)/dQ, A=(d2P(Q)/dQ2)(~bqi+q2), Xi=d2Cu/dq~ for i = l , 2 , and D is the determinant of the bordered Hessian, which is positive by the second-order conditions. In the constant cost case, X1 = X 2 = 0 , and strict convexity follows unless q~= I. Solving the firstorder conditions, ~b= 1 implies equality of marginal costs, which was ruled out by assumption.
356
R. Schmalensee, Competitive advantage and collusive optima
As total industry output falls from the leader's monopoly output, 01/2, to the followers' monopoly output, 02/2, total profit falls, and the industry moves to the northwest along the MS locus in fig. 1. Unless only the leader produces, total profits are lower than along the SP locus. The same first-order conditions yield an explicit expression for the contract curve labeled MS in fig. 2,
qz={[201+O2]-[(201-Oz)Z-8ql(Ol-O2)]uz}/4N-q~/N.
(6)
This curve is strictly convex, as drawn. [Bishop (1960, p. 948) asserts the convexity of the contract curve in this case.]
Market division (MD). When the relevant frontier in a bargaining problem is non-concave, one normally thinks of using mixed strategies to convexify the feasible set. This seems unnatural here, however. If a coin were to be flipped once to decide which type of firm were permitted to monopolize the market, some enforcement mechanism would be required to compel the loser(s) to exit. Alternatively, if a coin were to be flipped many times, so that average profits per period equaled expected profits in the limit, firms would have to start up and shut down frequently. Neither scenario resembles any obvious example of actual cartel behavior. But an alternative convexification device does correspond to a frequentlyobserved pattern of cartel behavior: the firms can divide the market, s That is, each actual or potential customer can be assigned to a single firm. If all customers are identical, as I wilt assume for simplicity, and firm 1 is allocated a fraction W* of them, its inverse demand curve is given by P=P(ql/W*), and it will charge its monopoly price. Firms of type 2 will similarly charge their higher monopoly price to the customers they have been allocated. Since each type's output and profit are linear in W*, the M D frontier is linear in both fig. 1 and fig. 2. Because market division requires different prices for the same product when sellers' costs differ, it is clearly feasible only when some mechanism can be used to rule out arbitrage at moderate cost. Starting from any point on the MS contract curve, if each firm is given a share of customers equal to its market share at that point, all will be able to increase profits by changing price. Thus M D is a more efficient technology than MS, though MD is of course less efficient than SP, as fig. 1 shows. The differences among these technologies grow with cost differences. When costs are equal, the SP, MS, and M D loci coincide in both fig. 1 and fig. 2. In this symmetric case, and only in this case, the division of monopoly output among colluding sellers does not affect total industry profit. 8Blair and Kaserman (1985, ch, 7) discuss this device at some length. For brief treatments, see Stigler (1964) and Scherer (1980, pp. 168-175).
R. Schmalensee, Competitive advantage and collusive optima
357
P r o p o r t i o n a l r e d u c t i o n ( P R ) . In many situations in which side payments are impossible, arbitrage will prevent market division. Moreover, the complexity of the market sharing technology when costs differ may make long and complex negotiations necessary, especially when firms are imperfectly informed about their rivals' costs, and this may entail unacceptable antitrust risks. In such situations, colluding sellers may resort to simple procedures to set o u t p u t quotas. One procedure that immediately suggests itself is to maintain m a r k e t shares at their non-collusive (Cournot) values and reduce the output of all sellers proportionately. This procedure constrains the firms to move along the PR contract curve in fig. 2. By doing so they can, for instance, reach a point on the MS contract curve and enjoy the corresponding profits. 9 In the symmetric case, this point maximizes total industry profit and divides it equally among all sellers. When costs differ, however, this point has no special attraction. In general, movements in toward the origin along the PR curve in fig. 2 produce movements away from the Cournot point, C, along the PR profitpossibility locus in fig. 1. l° As fig. 1 indicates, the PR technology m a y be substantially less efficient than any of the others when costs differ.
4. Collusive solution concepts W e must now specify how colluding firms select a point on the relevant profit-possibility frontier. This section briefly presents the solution concepts from axiomatic bargaining theory that I employ for this purpose. It is not my intention to argue that reasonable bargainers must behave according to any one of these solution concepts. I use them merely to identify ranges of plausible cartel optima. [This section relies heavily on Roth's (1979) excellent development of axiomatic bargaining theory.] The solution concepts discussed below all produce collusive outcomes at which all parties are at least as well off as at the status quo point for all values of N and R. The implied cartel optima thus lie to the northeast of C in fig. 1, between the two dashed lines. In what follows an asterisk denotes a collusive outcome and a superscript 'c' denotes the Cournot status quo point. N a s h . In the classic Nash (1950) solution, the product of the parties' gains,
(~, -
~
H1)(IIz
,
-
c
H2)
N
,
is maximized along the relevant frontier. 9Even though fig. 2 suggests that the proportional reduction technology can reach a point on the MD contract curve, it should be clear that total profits at that point are less than the corresponding MD profits, since the market is shared by firms charging a single price. 1°The tangency of this latter locus and the MS contract curve always occurs above the Cournot point. It occurs to the left of that point, so that the leader is worse off than in Cournot equilibrium, if the leader's cost advantage is large enough. [The conditions are S(1 +S)> 1 and N > (1 - $2)/(S 2 + S - 1). These imply S > 0.781 when N = 1; this critical value declines to 0.681 in the limit as N increases.]
R. Schmalensee, Competitive advantage and collusive optima
358
Kalai-Smorodinsky (K-S). To compute the solution proposed by Kalai and Smorodinsky (1975), define H]" as the maximum profit the leader can get on the relevant frontier when all followers get at least their status quo profit, and define H~' as the maximum profit a single follower can receive when all other firms receive at least their status quo profit. (Under the PR technology /7]" is given by the point in fig. 1 where the PR locus is vertical, and, since all followers must receive the same profit under this technology, //~' is given by the highest point on that locus on or to the right of the dashed vertical line.) The K - S solution is then the point on the frontier at which all parties' actual gains are the same fraction of their maximum gains, c
(n?
m
c
*
c
- FI1)/(H 1 - Hi) = (H z - nz)/(n
m
c
2 - nz).
(7)
If the profit-possibility frontier is linear, as implied by the SP and M D technologies, it is easy to show (using the ability to allocate total follower profit arbitrarily among followers) that the Nash and K - S solutions coincide.
Equal gains (EG). Roth (1979, pp. 92-97) suggests the possible relevance of solutions in which the absolute gains to all parties are equal, *
c
(H1-H1)=(H2-H2),
*
C
(8)
Along the SP frontier, it is easy to show that the Nash, K-S, and EG solutions all coincide. Thus all firms gain an equal dollar amount from collusion in all three solutions. (In relative terms, of course, the followers gain more than the leader because their status quo profits are lower.) Along the flatter M D frontier in fig. 1, the EG solution lies to the left of the N a s h / K - S solution. The latter thus gives the low-cost firm a greater absolute gain than each high-cost firm. If the PR technology is employed, no solution satisfying (8) exists when N = 1. And when N = 2, the EG solution maximizes firm l's profit on the PR frontier.
W*=S. In the MD technology, a natural focal point [in the sense of Schelling (1960)] is provided by the allocation of customers to firms in the original Cournot equilibrium. Accordingly, for this technology only, I consider solutions in which W*, the fraction of customers assigned to the leader, equals S, the leader's initial market share. Each follower then receives a fraction ( 1 - S)/N of the market demand curve. 11 HAnother apparently plausible focal point under MD or MS is where S*, the leader's share of collusive output, is equal to his Cournot share, S. But under both technologies this 'solution' makes the leader worse off than at the status quo point when he has a large cost advantage.
R. Schmalensee, Competitive advantage and collusive optima
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5. Collusive optima Under the SP and M D technologies, the equilibria corresponding to the Nash/K-S, EG, and (for M D only) W * = S solutions can be obtained explicitly as functions of N and S. Under the PR technology, the K - S and EG equilibria can be also obtained analytically. Even though the model used here is about the simplest imaginable, however, all the formulae involved are too complex to be informative. Moreover, under the MS and PR technologies, numerical solution of equations is necessary to obtain Nash equilibria, and the K - S and E G equilibria must also be treated numerically under the MS technology. Accordingly, [ adopt a purely numerical approach in this section and the next. Collusive optima are compared for some illustrative parameter values, and general relations revealed by computation of many equilibria are discussed, 12 Consistent with the remarks at the start of section 4, I emphasize patterns that hold across solution concepts. Changes in R. Table 1 describes alternative collusive optima when N = 1 for three values of R. When costs are equal, the natural symmetric duopoly collusive equilibrium (which is obtained by applying any of section 4's solution concepts to any of section 3's collusion technologies) raises both firms' profits, lowers consumers' (Marshallian) surplus, and reduces total surplus compared to the Cournot benchmark. When costs are different, production is inefficient in the initial Cournot equilibrium. The possibility of rationalizing the production gives rise to potential gains from collusion that are not present in the symmetric case. SP collusion takes full advantage of this possibility and even produces an overall net gain in (Marshallian) social welfare if R > ( 6 N + 1 6 ) / ( 5 N + 1 2 ) . But all collusive optima computed under the other three technologies, which necessarily fail to rationalize production completely, involve a net social loss. In all the asymmetric optima I have examined, the leader's percentage gain from collusion is below the followers', and the difference rises with increases in R. The greater the leader's cost advantage, all else equal, the less he has to gain from collusion. As R approaches 2, the initial Cournot equilibrium approaches a monopoly for the leader, so that his maximum possible gain from collusion goes to 0. If the SP technology is ruled out, all parties, including buyers, are generally made noticeably worse off at collusive optima, Price is higher because it must reflect, in part, the preferences of the high-cost followers, And
12All the equilibria discussed in the text were computed for N= 1,2,..., 10, for R ranging over the interval (1.0,2.0). Programs were written in IBM/MicrosoftBASIC, Release 2.0, and executed on an IBM PC. The author will be happy to supply copies of all programs employed to anyone sending a suitable formatted diskette.
R. Schmalensee, Competitive advantage and collusive optima
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Table 1 Collusive optima withN=l. Percentage surplus gain Value of R
Technology/solution
Leader
Followers
Consumers
(S=0.50)
Symmetric
Alt a
12.50
12.50
-43.75
1.143 (S = 0.60)
SP
All
16.67
37.50
MD
Nash/K-S EG W* = S
9.86 8.55 6.67
16.99 19.29 22.50
MS
Nash K-S EG
9.50 9.50 8.22
PR
Nash K-S EG b
6.67 6.40 .
Net
~o profit sacrifice
1.000 -31.25
0
- 36.00
-5,88
0
-41,73 -41.84 -42.00
- 14.31 - 14.47 - 14,71
8,95 9.13 9,37
18.69 18,69 22.49
--41.95 -41.95 - 42.20
- 14.57 - 14.57 - 14.92
9,33 9.33 9.50
20.14 21.60
- 36,46 -42.24 .
- 12,36 - 15.06
9,97 9.75
.
.
1.400 (S=0.75)
1.727 (S = O.9O)
SP
All
12.50
112.50
-23.44
MD
Nash/K-S EG W* = S
7.17 4.84 2,08
32.91 43.58 56.25
-31.41 - 32.05 - 32.81
-8.55 -9.40 - 10,42
2.08
10.42 11.25 12.24
MS
Nash K-S EG
5.81 5.81 3.78
25.18 25.18 34.00
-38.43 -38.43 - 39.14
-12.78 -12.78 - 13.62
12,05 12.05 12.82
PR
Nash K S EG
2.08 1.70 -
31.73 40,82 -
- 23.86 - 32.53 -
- 7.80 - 11.34
14.25 13.79
SP
All
5.09
4t2.50
-9.75
MD
Nash/K-S EG W* = S
3.87 1.94 0,28
105.02 157,31 202.50
- 13.82 - 14.85 - 15,75
-2.06 -3.24 -4.26
4.51 5.66 6.56
MS
Nash K-S EG
2.51 2.52 0.96
48,01 47.95 78.08
-27.24 - 27.24 - 29.01
-8.41 - 8.41 - 9,8 t
6.35 6.35 7.41
PR
Nash K-S EG
0.28 0,21
42,93 61.27 -
- 9.85 - 14.29
- 3,24 -4.82 -
8,42 8.27
" N a s h , K - S , and E G solutions coincide. bThe P R / E G solution does n o t exist for N = 1.
2.56
0
0
R. Schmalensee, Competitive advantage and collusive optima
361
total industry profits are not maximized because production is not fully rationalized. The last column in table 1 gives the percentage by which total industry profits fall short of their maximum value. For all values of N and all non-SP equilibria, this profit sacrifice is maximized for moderate values o f R. As R approaches i, the sacrifice goes to 0 because the symmetric case is approached. At the other extreme, as R approaches 2, the sacrifice goes to 0 along with the potential gains from SP collusion. The difference between the MD and MS solutions are relatively small, as are those between the MS/Nash and MS/K-S equilibria. These patterns reflect the near-linearity of the MS profit-possibility frontier for most values of N and R. [This was noted by Bishop (1960, p. 948).] The PR solution is close to the other two in some cases, though it is generally clearly inferior from the leader's viewpoint. Because it precludes rationalizing production by increasing the leader's share, the extent to which total output can be restricted is limited, and consumers are often better off under PR collusion than if the MD or MS technologies had been employed. In the examples of MD collusion described in table 1, the W * = S solution lies to the left of the Nash/K-S and EG solutions along the MD frontier. This implies that the Nash, K-S, and EG solutions involve W*> S, so that the leader's share of customers (and, afortiori, of output) is increased by collusion. Similarly, the leader's market share exceeds S at all MS collusive optima in table 1. This reflects the potential gains from rationalizing production. But when R is small and N is large, Nash, K-S, and, especially, EG collusion may involve reducing the leader's share. Potential rationalization gains are small when R is near 1, and the followers' bargaining power rises as N increases. Changes in N. Table 2 shows the effect of changes in N, with R held constant at a moderate level, In the symmetric case, increases in N simply increase the intensity of competition at the status quo point and thus increase the sellers' gains and buyers' losses produced by collusion. When costs differ, however, increases in N also increase the status quo share and aggregate bargaining power of the high-cost followers. The first effect increases the potential gains from rationalizing production. But the second increases the profit sacrifice made in the interests of equity when side payments are impossible, as the last column in table 2 shows. Followers' percentage gains tend to increase with N, reflecting both their lower Cournot profits and their increased bargaining power. For small values of R and N, the leader's relative gains tend to rise with N, reflecting the increasing potential gains from rationalizing production. Otherwise, the bargaining power effect dominates, and increases in N reduce the leader's relative (and, afortiori, absolute) gains. (The borderline values of N and R depend on the technology and solution concept, as table 2 reveals.)
R, Schmalensee, Competitive advantage and collusive optima
362
Table 2 Collusive o p t i m a with R = 1,40. Percentage surplus gain Value of N 1 (S=0.75)
3 (S=0.59)
5 (s=o,53)
Technology/solution
Leader
Followers
Consumers
Net
~o profit sacrifice
Symmetric (R=I)
All a
12.50
12.50
-43.75
-31.25
sP
All
12.50
112,50
-23.44
2.08
MD
Nash/K-S EG W* = S
7.17 4.84 2.08
39,21 43.58 56.25
- 31.41 - 32.05 - 32.81
- 8.55 - 9.40 - 10.42
10.42 11.25 12.24
MS
Nash K-S EG
5.81 5,81 3.78
25.18 25.18 34.00
-38,43 -38.43 -39.14
-12.78 -12.78 -13.62
12.05 12.05 12,82
PR
Nash K-S EG b
2.08 1.70 .
31.73 40.82 . .
- 23.86 - 32.53 .
- 7,80 - 11.34
14.25 13.79
0 0
Symmetric (R = I)
All
56.25
56.25
-60.93
-21.88
0
SP
All
16.31
306.25
-36.72
4.88
0
MD
Nash/K-S EG
W*=S
12,47 7.25 7.08
119.52 t36.19 136.74
-48.48 -49.37 -49,40
- 14.61 -16,09 - 16.13
18,58 19.99 20.04
MS
Nash K-S EG
l 1.32 11.47 6.52
107.28 106.80 122.48
- 54.70 - 54.68 -55.48
- 19,24 - 19.20 -20.58
20.30 20,26 21.60
PR
Nash K-S EG
7.02 6.19 6.29
104.99 119.I9 118.18
-39,74 -47.75 -47.11
-12.78 - 16.64 -16.31
22,87 22.08 22.11
Symmetric (R = i)
All
104.17
104.17
-65.97
-23,44
sP
All
15.63
493,06
-41.38
6.43
MD
Nash/K-S EG
12.86 7.15 10.34
210.74 229.48 219.01
-54.49 -55.28 - 54,84
-17.37 -18,81 -18.01
22.36 23,71 22.96
W*=S
0 0
MS
Nash K-S EG
11.84 12,14 6.62
195,24 194.25 212.51
-60.02 -59.98 -60,63
-21,89 -21.82 -23.13
24,02 23.95 25.24
PR
Nash K-S EG
10.05 9.29 6.62
184.56 196.84 212,50
-47.21 -52.24 -60.94
-15.32 - 17.97 -23.32
25.68 25.13 25.24
"Nash, K - S , and E G solutions coincide, bThe P R / E G solution does not exist for N = 1.
R, Schmalensee, Competitive advantage and collusive optima
363
Tables 1 and 2 indicate that in the presence of substantial competitive advantage, corresponding to a large R, the leader's gains from collusion are likely to be small relative to status quo profits when side payments are impossible. While the leader's potential relative gains do not approach 0 especially rapidly as R approaches 2, they seem small relative to likely yearto-year fluctuations for values of R and N that might be encountered in practice. (Note also that the figures shown deal only with e c o n o m i c profits; the corresponding percentage increases in the a c c o u n t i n g profits reported to shareholders are likely to be much smaller.) It thus seems likely that the absence of gains commensurate with the costs and (legal and other) risks involved tends to make collusion less frequent than otherwise in the presence of substantial competitive advantage. 13 IBM has been accused of m a n y things, but not of colluding with its rivals.
6. Implicit conjectural derivatives In this section I follow the approach introduced by Iwata (1974) and use sellers' implicit conjectural derivatives to describe collusive optima, This approach only makes sense under the MS or PR technology, when firms of both types are operating and sharing the market. Consider a market with T firms, all of which may have different cost functions, and let q_~ be the output of all firms except i. Then at any set of outputs, firm i's implicit conjectural derivative, 2~, is the value of dq_~/dq~ that would make its actual output profit.maximizing. Re-arranging the firstorder condition for maximization of firm i's profits, the implicit conjectural derivative consistent with equilibrium at any q~, q_~ pair is given by 2~ = -- ( P - CI + q~P')/ q~P',
(9)
where C~ is firm i's marginal cost, It is easy to show that 2~=--1 implies competitive (P = C') behavior, while 2~= 0 for all i at the Cournot benchmark. More generally, using the second-order condition for profit maximization, one can show that for fixed q_~, an increase in 2~ lowers q~. If 2~= 1, for instance, firm i behaves as i f it fears that increases in its output would be matched, unit for unit, by its rivals, and it accordingly produces less than it would if it felt rivals would not react at all (2~=0). Alternatively, it behaves as i f its output restrictions will be matched, and so restricts output more than in the Cournot case, It is hard to believe that actual oligopoly behavior - particularly collusive behavior - involves this sort of non-strategic maximization. Estimated values a3If cost differences generally served to enhance the stability of collusive arrangements, perhaps by facilitating the choice of a price leader, this conclusion would have to be qualified.
R. Schmalensee, Competitive advantage and collusive optima
364
of the 21 are best understood as providing a convenient s u m m a r y description of market behavior, which m a y in fact involve a complex mix of cooperative and non-cooperative elements. The 2~ are in a sense dual to the qt; their merit is comparability across markets. In this framework, collusion occurs when firms behave as if they feared that increases in their outputs would induce their rivals to produce more. Iwata (1974) has shown that the following c o n d i t i o n is satisfied at all points on the MS profit-possibilities frontier: T
~ = ~ z i = 1,
(10)
i=1
where the first equality defines ~, a n d z i = l / ( 1 +2i) for all i. [Eq. (10) is equivalent to Iwata's (1974, p. 961) eq. (6.12).] It is easy to show (using the relevant first-order conditions) that the 2i must all be positive on the MS frontier. It is only slightly harder to show that 7j > 1 above the MS contract curve in the output space of fig. 2.14 T h u s m o v e m e n t toward collusion involves increases in the 2~. Since 7~ increases w i t h o u t b o u n d as behavior approaches perfect competition (i.e., all 2~ a p p r o a c h - 1 ) , 1/tP provides a measure of the extent to which industry behavior is non-competitive. This quantity is generally confined to the unit interval, t h o u g h 7~ < 1 if the PR technology is used to reach a point strictly inside the MS contract curve. In the special case on which this paper focuses, where one leader faces N identical followers, MS collusion implies
N +(N- 1)21-2122=0.
(11)
Here 2t is the leader's conjectural derivative with respect to aggregate followers' output, and 22 is each follower's conjectural derivative with respect to the o u t p u t of the leader and the other N - 1 followers. This expression is positive under P R collusion at points outside the MS contract curve in fig. 2. It is negative at points inside that curve. Table 3 presents the implicit conjectural derivatives consistent with a variety of MS and P R collusive equilibria. In these a n d all other equilibria examined, 41 is smaller t h a n 22. As table 3 indicates, the difference between these parameters can be quite dramatic. If estimates of this sort were obtained econometrically in some market, one might be tempted to characterize the followers as more concerned a b o u t competitive reactions to o u t p u t 1*Assume that 2~> - 1. so that P>C'~ for all i. Then ~ is a continuous function of the vector of outputs. Since ~ is equal to 1 only on the MS contract curve, (W-I) must have the same sign at all points above this curve. Consider a set of small output changes, Aq, i = 1..... T,, of the same sign. Differentiating and using (9) to substitute for P', the corresponding changes in profits are found to be AHl=(P-C'i)(c~-ri)AQ, i= 1..... T, where al= dql/dQ>O for all i. Since the ai sum to unity, there exists a set of output decreases (increases) that raise the profits of all sellers if and only if 7~> 1 (~< t). Thus 7~> 1 above (~< 1 below) the MS contract curve.
R. Schmalensee, Competitive advantage and collusive optima
365
Table 3 Implicit conjectural derivatives at collusive optima. N=I
Value
Technology/
of R
solution
1.143
MS
Nash K-S EG
PR
1,400
1.727
21
N=3
N=5
),2
),l
),z
21
22
0.81 0,81 0.83
1.23 1.23 1.20
2,03 2,03 2.17
3.48 3.48 3.38
3.06 3.05 3.30
5.64 5.64 5.52
Nash K-S EG a
0.83 0.93
0,95 1,06 -
1,78 1.94 2.57
2.86 3,12 4.12
2.65 2.70 5.44
5.01 5,12 10.32
MS
Nash K-S EG
0.57 0.57 0.61
1.77 1.77 1.64
1.17 1,16 1.29
4.57 458 4.32
1.56 1.55 1.71
7.21 7.22 6.92
PR
Nash K-S EG
0.34 0.51 -
0.73 1.09 -
0.78 1.03 1.01
2.40 3.t9 3.12
1.08 1.29 1.73
4.39 5.22 7.00
MS
Nash K-S EG
0.32 0,32 0.37
3.12 3.12 2.72
0.57 0.56 0.63
7.30 7.34 6,75
0.69 0.69 0.75
11.21 11.28 10,63
PR
Nash K-S EG
0.11 0.17 -
0.59 0,88 -
0.22 0.32 0.29
1.85 2.67 2.39
0.29 0.39 0.44
3.25 4.47 4.97
~The PR/EG solution does not exist when N = 1.
increases than the leader. 1~ While this may be the right interpretation in many cases, such a pattern at least suggests the desirability of testing the collusive null hypothesis tp= 1. It is possible that the followers are restricting output substantially as part of a collusive bargain, rather than because they believe that total industry output is highly responsive to their own production decisions. When either the MS or the PR technology is employed, both 2 t and 22 tend to increase with N, as condition (11) suggests. Holding N constant, increases in R tend to lower 21 and to raise 22 in MS collusion. Under the PR technology, increases in R most often lower both parameters, but several exceptions are visible in table 3. As cost differences are increased, it becomes harder to make large proportional reductions in output without making the leader worse off. Consistent with this, all of the PR equilibria shown in table 3 correspond to 15Spiller and Favaro (1984) consider a market with 'dominant' and 'fringe' firms. They find that 2's for fringe firms were consistent with Cournot behavior. Among the dominant firms, however, smaller sellers had larger 2's. Table 3 implies that this pattern is consistent with collusion (and cost differences) among the dominant firms.
366
R. Schmalensee, Competitive advantage and collusive optima
points outside the MS contract curve, except for the three EG equilibria in the upper right-hand corner of the table.
7. Conclusions Symmetric oligopoly models may lack empirical relevance; they are certainly very special theoretically, Even with functional forms selected for tractability, relaxation of the assumption of symmetry gives rise to considerable algebraic complexity. Four technologies for effecting collusion that are essentially equivalent in the symmetric case are quite distinct when sellers' costs differ, and plausible bargaining outcomes can only be analyzed numerically. Numerical analysis of collusive optima implied by axiomatic bargaining theory reveals a variety of distinctions and effects that are neither present in the symmetric case nor sensitive to the axiomatic solution concept employed. In the absence of side payments, these flow mainly from changes in the balance between two forces. The first is the potential gain from rationalizing production by increasing the share of the most efficient producer. This force is strongest for moderate cost differences. The other force is the bargaining power of high-cost firms. This force increases with the number of inefficient producers. Two specific findings seem especially important for applied work. First, if a leading firm's cost advantage is substantial, its potential gains from collusion are relatively small. This suggests that collusion is unlikely to be observed in the presence of substantial competitive advantage, though the observed price may be close to the (leader's) monopoly level. Second, conjectural derivatives consistent with perfect collusion may vary substantially from firm to firm, depending on the nature of collusion, the importance of cost differences, and the number of sellers. Differences in econometrically estimated conjectural derivatives may thus reflect bargaining outcomes, not differences in rivalrous behavior.
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Kalai, E. and M. Smorodinsky, 1975, Other solutions to Nash's bargaining problem, Econometrica 43, 513-518. Kreps, D.M. and J.A. Scheinkman, 1983, Quantity precommitment and Bertrand competition yield Cournot outcomes, Bell Journal of Economics 14, 326-337. Nash, J,F., 1950, The bargaining problem, Econometrica 28, 513-518. Nash, J.F., 1953, Two-person cooperative games, Econometrica 31, 128-140. Orr, D. and P.W. MacAvoy, 1965, Price strategies to promote cartel stability, Economica 32, 186-197. Osborne, D.K., 1976, Cartel problems, American Economic Review 66, 835-844. Osborne, MJ. and C. Pitchik, 1983, Profit-sharing in a collusive oligopoly, European Economic Review 22, 59-74. Patinkin, D., 1947, Multi-plant firms, cartels, and imperfect competition, Quarterly Journal of Economics 61, 173-205. Porter, M.E., 1985, Competitive advantage (Free Press, New York). Porter, R.H., 1983, Optimal cartel trigger price strategies, Journal of Economic Theory 29, 313-338. Roth, A.E., 1979, Axiomatic models of bargaining (Springer, New York). Schelling, T.C., 1960, The strategy of conflict (Harvard University Press, Cambridge, MA). Scherer, F.M., 1980, Industrial market structure and economic performance, 2nd ed. (RandMcNally, Chicago, IL). Schmalensee, R., 1985, Do markets differ much?, American Economic Review 75, 341-351. Spiller, P.T. and E. Favaro, 1984, The effects of entry regulation on oligopolistic interaction: The Uruguayan banking sector, Rand Journal of Economics 15, 244-254. Stigler, G.J., 1964, A theory of oligopoly, Journal of Political Economy 72, 44-61. Sutton, J., 1986, Non-cooperative bargaining theory: An introduction, Review of Economic Studies 53, 709-724.