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The kinked demand curve
International Journal of Industrial Organization 6 (1988) 373-384. North-Holland
THE KINKED DEMAND
CURVE*
A Game-Theoretic Approach V. B H A S K A R University College, London WCIE 6BT, UK
Final version received August 1987 In a simple model of duopoly, firms' price moves are modelled as an extensive form game where firms can respond to undercutting without delay. When firms are not too dissimilar, kinked demand strategies enforcing an arbitrary price may be Nash equilibria; however, these strategies are dominated and perfect equilibrium is unique at the minimum optimal common price. Rather than implying price rigidity, kinked demand strategies are a device for ensuring compliance with a collusive price leadership.
1. Introduction In its t r a d i t i o n a l f o r m u l a t i o n the k i n k e d d e m a n d curve [Sweezy (1939), Hall a n d H i t c h (1939)'1 has been a t h e o r y of price rigidity. It argues t h a t firms will desist from reducing price since t h e i r rivals m a t c h price cuts; however, since price increases are n o t m a t c h e d , n o firm will initiate a price increase. T h e resulting d i s c o n t i n u i t y in the m a r g i n a l revenue curve implies that the m a r k e t price will n o t r e s p o n d to small c h a n g e s in costs or d e m a n d . Due to the i n d e t e r m i n a c y of the kink a n d the a r b i t r a r i l y a s y m m e t r i c p a t t e r n of b e h a v i o u r p o s t u l a t e d , the t h e o r y is n o w largely d i s c r e d i t e d J T h e empirical evidence appears, on the face of it, c o n t r a d i c t o r y (see Reid's 1981 survey) the objective evidence denies t h a t prices are m o r e rigid in industries where the t h e o r y o u g h t to be applicable whereas a large n u m b e r of m a n a g e r s claim, w h e n interviewed, to face such a n a s y m m e t r y in c o m p e t i t o r s ' responses, implying t h a t their d e m a n d curves are indeed kinked, This p a p e r a t t e m p t s to *Presented at the 14th Annual Conference, European Association for Research in Industrial Economics, Many of the ideas in this paper are due to discussions with David Soskice who has been most generous in allowing me to use his unpublished notes. I am grateful to Giacomo Bonanno, Paul Klemperer, Meg Meyer and John Vickers for discussions and to Paul Geroski and two anonymous referees for their comments on an earlier version of this paper. The usual disclaimer applies. ~See Stigler's obituarial view (1978). In his textbook on oligopoly theory Friedman (1983, p. xiv) writes, 'I have unhesitatingly omitted certain developments that have long been staple fare but that I find to be without merit (such as) ... the celebrated kinky oligopoly demand curve due to Sweezy', 0167-7187/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
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provide a reformulation of the theory which not only meets these theoretical objections but is consistent with both types of evidence. Merton Peck's classic study of the aluminium industry (1961) proposed a modified version of the theory in which the equilibrium price was uniquely determined. Peck questioned the kinked demand theory's assertion that a firm's competitors will not match price increases; if all firms benefit from a higher common price, a price increase will be followed since the initiator will cancel it otherwise. This in turn makes it attractive for firms to initiate a price rise when all firms prefer a higher common price. The equilibrium must hence be at a price where at least one firm does not prefer a higher common price. Defining each firm's optimal common price, as the price which maximizes its profits given that its rivals are pricing in line, Peck argued that the unique equilibrium would be the lowest of these optimal common prices. At this price no firm would have an incentive to reduce price since the outcome would be a lower common price which hurts all. Nor would other firms be able to force the price upward since the firm with the lowest optimal common price has no incentive to match their price increases, Section 2 formalized these arguments by considering a duopoly setting price for a single trading period. I consider physically identical products while allowing for a finite cross elasticity of demand due to customer loyalties and switching costs [Okun (1981), Klemperer (1986)]. The key assumption is that firms can respond to their rival's undercutting without delay - this presumes that secret price reductions are not possible, an assumption which may be tenable in markets with a large number of buyers, as Stigler (1964, p. 47) suggests, since 'no one has yet invented a way to advertise price reductions which brings them to the attention of numerous customers but not to that of any rival'. Price moves are modelled as an extensive form game. Firms simultaneously announce initial prices; if prices differ, the firm pricing higher can reset its price. If it undercuts its competitor on the second move, the latter in turn is given the option of changing its price. The move sequence ends when one firm does not undercut so that its rival has no need to reply. These price moves are assumed to take place sufficiently rapidly that no trade takes place until the final prices are announced; transitory profits are assumed negligible as in the kinked demand theory. This game is equivalent (in the sense that equilibrium outcomes are identical) to a game where the firm pricing higher initially restricts itself on the second move to prices above or equal to its rival. F o r expositional purposes I therefore use the two-move game. The main result is that firms will match undercutting in the relevant range (provided that they are not too dissimilar) and perfect equilibrium is unique at the minimum optimal common price. Interestingly, kinked demand strategies enforcing prices below the minimum optimal common price may be Nash equilibria but these strategies are dominated for both players and
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will hence not be adopted. I also find that the minimum optimal common price is inapplicable if the firms are too dissimilar since the firm with the higher preferred price chooses not to match its rival's lower price on the second move - the unique equilibrium is of a Stackelberg type, the firm with the lowered preferred price acting as a leader, while the follower prices above it. These roles are not exogenously imposed but emerge naturally.2 Section 3 gives a simple example showing how the type of equilibrium can be related to differences in costs or demand elasticities. Section 4 interprets these results; rather than implying price rigidity, this theory provides a rigorous basis for a type of price leadership where the leader uses a kinked demand strategy to enforce the minimum optimal common price. The ability of firms to respond without delay to undercutting ensures that the unique noncooperative equilibrium is collusive; this approach to oligopolistic collusion avoids some of the problems which have arisen in the repeated games approach.
2. The game in extensive form Firms /1 and B play an extensive form game with the following move order: they simultaneously announce initial prices a~,bl (a firm's price will be denoted by the corresponding lower case letter). If these differ, the firm pricing higher (say B) can announce b2; if it undercuts by choosing b2 b ~ for every history where a t > b 1. B's strategy is similarly defined. Let Z = Z ~ × Z b denote the strategy space. The 2After writing this paper I have come across a discussion paper by Kalai and Satterthwaite (1986) which assumes that firms always match the lowest price initiallyquoted, not allowingfor the possibility that firms either fail to match price or that they undercut further; this paper requires firms to choose optimally, thereby identifying the conditions under which price matching and the minimum optimal common price result and those situations where the equilibrium is Stackelberg. In a similar vein, Lutter and Logan (1987) show that a policy of guaranteeing lowest prices can facilitate collusion,
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V. Bhaskar, The kinked demand curve
payoffs R a and R b from a strategy pair depend only upon the prices last announced (the outcome), since price moves are assumed instantaneous. I require equilibrium strategies to be subgame perfect Nash equilibria, and undominated by any other strategy. Subgame perfection ensures that players choose their second move optimally and the Nash criterion ensures that strategies are best responses, cra dominates a '~ iff it does as well as ~r'~ against all of B's strategies and does strictly better against at least one of B's strategies, i,e., iff: R~(a~, a °) > Ra(a '", O"b) ~t o.b ~ z~b and R"(a", a b) > R~(cr~, a b) for some a b e 2 b, I call such an equilibrium perfect in short - it is indeed trembling-hand perfect if the appropriate generalization of the concept is made for this game with infinite pure strategies. 3 While I set out the arguments in the more general case when products are not completely homogeneous, due to horizontal differentiation and switching costs, the theory also applies with perfect homogeneity. The assumptions on ,firms' payoff functions are:
A1.
Payoffs are continuous twice differentiable functions of the two prices; products are substitutes so that aR"/Ob and c3Rb/aa>O; payoffs are strictly concave with respect to own price.
A2.
a2Ra/ab Oa and c~ZRb/c3ac3b> 0 so that the products are strategic complements and the two 'reaction functions '4 c~(b) and fl(a) are upward sloping [see Bulow, Geanakoplos and Klemperer (1985)]. The reaction functions are contractions, i.e., dc~/db < 1 and d~/da < 1, ensuring the existence of a unique Bertrand equilibrium (a**,b**). A3. Define A's c o m m o n price profits Ca(a) as its profits when both firms price at the same price a. C" is strictly concave in the c o m m o n price and is maximum at a*, A's optimal common price. Similarly, Cb(b) is concave with b* as B's optimal c o m m o n price. A4.
For expositional conciseness alone, let A uniformly have a lower price preference, a * < b * and a * * < b * * . N o t h i n g essential hinges on this assumption. It is useful to define prices ~ (A's threshold price) and /~ (B's threshold price) at which their rival's reaction function prescribes price matching behaviour. Since the reaction functions are contractions (A2) it follows that 3Defined as the limit, as e, goes to zero, of a sequence of Nash equilibria in completely mixed strategies, where the probability density corresponding to a price available at any information set must be at least ~. As is well known dominated strategies cannot be trembling-hand perfect. ~A slightlymisleadingterm since they describe a firm's optimal price as a function of its rival's price rather than actual reactions in this game. I defer to established usage.
V. Bhaskar, The kinked demandcurve fl(a)~a
as
a~fi,
a(b)~b
as
b~.
377
A firm which is undercut initially will try, on the second move, to get as close to the price prescribed by its reaction function without undercutting its rival. Hence A will choose a2 = b l if b 1 >/~ and az=a(bl) if b 1 4 and bz=fl(al) if a t < & as shown by the curve X Y Z in fig. 1. A's payoff from a price a~ > bl (in equilibrium A will price higher initially) can therefore be fully specified - if al >4, A gets the common price profit C"(al) since B matches a~ on the second move, On the other hand if al
Proof B's second move is optimal as we have seen. From fig. 2 it is seen that a* is the unique maximizer o f A's payoff; it is hence his unique best response (hence undominated) when B prices higher on the first move. While B's best response first move is not unique - any price which is not below a* would do as well given that he has to move down to a* on the second move, the choice of b* dominates all alternatives by doing strictly better when A deviates from a* in some way. b* does better than a price above it, should A deviate with a price above b*, since the outcome is b* rather than a price above it. b* does better than a price below it should A price higher than B initially, since the common price will then be closer to b*. It is easily checked that b* does no worse than these alternatives whatever A's deviation. It may be similarly shown that A's choice of a* dominates all others. This ensures not merely that the strategy pair a* is undominated, subgame perfect and Nash, but also that it is uniquely so, since a*" and a.b dominate all alternatives. Rational players have no reason to adopt any other strategy. 5Obviously,despite the kink in A's payoff function it may have only a single maximum. Once again, this maximum may be either at a* or & so that only these possibilities need be considered.
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V. Bhaskar, The kinked demand curve
0c /
/x
t;*
a**
gl Fig. 1
Ra
Ca ~
a*
Fig. 2
If firms are identical, equilibrium is at the (identical) optimal common price which is the monopoly price. It is simple to prove that these results hold in the more general game set out at the beginning of this section where a player may undercut on the second move. Since the moves end when one player does not undercut the other, it is clear that the player with the higher price must have moved last. Subgame perfection requires that this last move be optimal; hence any outcome to the right of (and below) X W V in fig. 1 may be ruled out, since it would imply that A has chosen an excessively high price on the last move if he chose a lower price so as to be on X W V , he would do better, Similarly, any outcome to the left of (and above) X Y Z may be ruled out since B would be choosing an exessively high price on the last move. Further if B undercuts below a*, this implies that both final prices must be below a*. Since (a*, a~') is better for B than any common price pair below it on X Y (due to the
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concavity of common price profits), and since point Y in turn is better than any other point on Y Z (since products are substitutes), and since any point below Y Z is worse than a point on it, it follows that B can only do worse by undercutting below a* on the second move. I now show that the traditional kinked demand stalemate is a Nash equilibrium where players adopt dominated strategies. Suppose both players choose some price p on the first move, where a * > p > & This is a Nash equilibrium which is Pareto-inferior to the minimum optimal common price; neither player benefits from pricing higher initially given that his rival chooses p since he would have to move down to p on the second move. We have already seen that this strategy pair is not rational since players are adopting strategies which are dominated by the alternative of choosing their respective optimal common prices initially. In the second possible case when A's payoff is globally maximum at ~, the unique perfect equilibrium has A announcing ~ on the first move while B announces b*. B then moves down on the second move to a price fl(~) which is greater than 4. A and B act as Stackelberg leader and follower respectively, these roles emerging naturally without being exogenously imposed.
3. An example In this section I use a simple example with linear demand and constant unit costs to illustrate the theory and demarcate the conditions under which the minimum optimal common price and the Stackelberg equilibrium arise. To consider the implications of cost differences, let both firms face symmetric demand curves, but let B have a higher unit cost. Without loss of generality A's cost can be taken to be zero since we can measure prices net of this level. Let q" and qb denote the respective demands, let c be B's cost level, and remember that prices are simply denoted by the respective lower case letters. 7>6>0,
qa=X-Ta+6b,
(1)
qb = X -- 7b + 6a,
(2)
R" = X a -
(3)
~a 2 + 6ab,
R b = (b - c) ( X -
7b + 6a).
(4)
The respective optimal common prices, which maximize R a and R b when a = b, are
a* = X / E 2 ( ~ - 6)'1,
J.I,O,-- D
(5)
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b*=[X+(7-6)c]/[2(y-6)],
(6)
so that a*
C"(a*)=X2/4(7-6).
(7)
A's profits as a Stackelberg leader can be derived by substituting B's reaction function fl(a)=[X+yc+6a]/2y into his profit function. In the Stackelberg equilibrium A's price c~ and profits are ~ = [2yX + 6X +yac]/[2(2y 2 - 62)], sa([l) =
[27X + 6X + yac] 2/[8y(2y2 -- 62)].
(8) (9)
The equilibrium will be at a*, the minimum optimal common price, or at a, the Stackelberg equilibrium, depending upon whether Ca(a*) is greater or less than S"(a). Examining the two expressions, it is clear that B's cost level only affects A's Stackelberg payoff, not affecting his common price payoff. Since Sa(a) is increasing in c, it is clear that there is a critical cost difference below which the mimimum optimal common price applies, and above which the equilibrium is Stackelberg. Further, it is the more efficient firm which plays a leadership role, whether in the Stackelberg equilibrium, or in choosing the optimal common price. The above example also has a demand interpretation. Suppose that both firms have identical costs but that B's demand curve is
qb=X+Tz-yb+fa,
z>0.
(10)
In other words B has a larger market share due to an intercept on the price axis which is z greater than A's and consequently also has a more inelastic demand. Such a difference in demand elasticity due to differing market shares can arise naturally in markets with consumer switching costs [see Klemperer (1986)], where an established firm with a large number of locked-in customers faces a more inelastic demand than an entrant; the specific linear model set out here can arise if in addition there is some underlying horizontal differentiation between products. Since in a linear model an upward shift in the demand curve is equivalent (in terms of equilibrium pricing) to an increase in cost, this case is analytically equivalent to the previous case of cost differences. We merely have to replace c by z in the expressions and it is easy to see that there is a critical level of z below which the minimum optimal common price holds, and above which the Stackelberg equilibrium results. 6 The economic interpretation is simply that 6The only differenceis that B's payoff will be higher in the case where he has a larger market share.
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the larger the number of locked-in customers, the more inelastic is B's demand so that it finally pays B to price above A in equilibrium. Notice further that, in contrast with the standard presumption, it is the firm with the smaller market share which plays the leadership role in setting price, whether in Stackelberg equilibrium, or in choosing the common price.
4. Implications of the theory Since the minimum optimal common price is easily generalized to more than two firms, it is clearly a theory of price determination under oligopoly. A striking implication of the theory is that if firms differ, there will be one firm such that any small changes in its cost or demand conditions will alter the minimal optimal common price and the actual price level, whereas changes in costs/demand conditions for the other firm will have no impact upon the price? While this contradicts the price rigidity argument, since a general increase in industry costs or demand will raise prices, it suggests instead that the minimum optimal common price firms plays a leadership role in setting the price. This firm selects the industry price and prevents competitors from deviating from this price by following a kinked demand strategy. Since all firms except the price leader will face kinked demand curves, this also reconciles the presumed conflict between the objective evidence (denying price rigidity) and the subjective evidence that a large number of managers claim to face demand curves which are kinked at the prevailing price. A comparison with more conventional theories of price leadership is in order. Ono's (1982) analysis of Stackelberg price leadership shows that a firm voluntarily chooses the leadership role only if it has an overwhelming cost advantage or very high quality products. When cost or demand differences are not very large, Ono shows that every firm prefers the follower's rote so that the Nash equilibrium is the only stable solution. Ono suggests that a farsighted firm might accept the undesirable leadership role since its profits would be higher than in the Nash equilibrium. This is unsatisfactory since the reluctant leader has every incentive to deviate from its role in a single shot game; further, if a low-cost firm assumes leadership, it may earn less profits than its high-cost rival. In our theory the Stackelberg equilibrium similarly emerges when one firm has an overwhelming cost advantage or very elastic demand. However, even the absence of such conditions, choosing the common price confers no disadvantage since the followers cannot choose 7A caveat - the equilibrium ca~ashift discontinuously from the minimum optimal common price to Stackelberg due to a marginal increase in the cost (or demand) of the firm with the higher preferred price. In fig. 2 an increase in B's costs can raise A's threshold price ~ and his Stackelberg payoff while leaving A's common price profits unaltered. In this case both firms' prices must fall discontinuously when B's costs rise. For other instances of such discontinuous shifts see Bonanno (1987).
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E Bhaskar, The kinked demand curve
their price unrestrictedly; if they deviate by undercutting, the leader matches their price. This contrasts with the usual analysis of Stackelberg leadership where the leader commits to a price and allows the followers to choose unrestrictedly. The theory also explains the p h e n o m e n o n of a single firm initiating changes in industry price which has often been noted [see Scherer (1980, pp. 176-184)] if we allow that the followers will be uncertain about the leader's optimal c o m m o n price and will face small costs of changing price. The followers will then be content to match the leader's price changes while the leader is willing to initiate these changes, knowing that its rivals will follow it. These factors also enable one to argue for a limited role for the original price rigidity argument in industries where firms are very similar, so that there is no clear leader. In some circumstances the price may be sticky even if every firm experiences a rise in its optimal c o m m o n price - while each firm is sure that its optimal c o m m o n price has risen, it is unsure if this is the case for all its rivals, and hence refrains from raising price for fear that it may have to be cancelled. Such a stalemate is more likely when industry demand increases rather than when nominal input costs rise, since the latter are likely to be c o m m o n knowledge while changes in rivals' marginal costs and demand elasticities as a result of demand shifts are m o r e difficult to predict, This may be a partial explanation for the puzzling lack of cyclical variability in industrial prices. 8 A significant aspect of the minimal optimal c o m m o n price is that it is a 'collusive' equilibrium which Pareto-dominates the N a s h - B e r t r a n d outcome. 9 The key assumption which ensures this result is that a firm can respond to its rival's price change quickly, so that undercutting is unprofitable. 1° This may, in some ways, be more satisfactory than the repeated game a p p r o a c h to non-cooperative collusion. While collusive outcomes are perfect equilibria in an infinitely repeated game, the F o l k theorem also states that the set of equilibria include every individually rational outcome; hence outcomes far worse than the Nash C o u r n o t or Bertrand outcomes are as possible as collusive ones. Similarly, while Radner (1980) shows that a collusive outcome can result from near-rational behaviour in a finitely repeated game, once aOkun, (1981, p. 165) summarizes the evidence, 'Econometrically, demand is found to have little, if any, influence (on prices) outside the auction market for raw materials'. In a more recent study of prices in the U.K., Bhaskar (1988a) finds the strong influence of costs and international competition with demand variables being insignificant. 9However the outcome, is not perfectly collusive, in the sense of being Pareto-optimal, unless firms are identical or products are completely homogeneous; in the absence of binding agreements, the ability to respond quickly to deviations ensures superior but not necessarily the best outcomes. 1°This result is in contrast to Marschak and Selten's (1974, 1978, 1979) who first tried to analyze such a quick response game in terms of 'restabilizing response functions' and found multiple equilibria, including sub-optimal ones such as the Bertrand outcome. Bhaskar (1988b) demonstrates that such response functions are not rational and sets up a general extensive form game which yields collusive'outcomes only.
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again the C o u r n o t or Bertrand outcomes remain perfect equilibria. Even if we restrict ourselves to collusive outcomes both these approaches still burden us with a multiplicity of such equilibria, with no obvious way of choosing a single outcome when firms are not identical, ix The alternative procedure used here dispenses with infinite repetition by allowing firms to match lower prices instantly so that the non-cooperative o u t c o m e is collusive; furthermore, equilibrium is unique. The assumption that transitory profits are negligible may be empirically valid for m a n y industries; alternatively, it m a y also be defended as an instance of b o u n d e d l y rational behaviour by firms. The m i n i m u m optimal c o m m o n price is applicable in industries with a small number of firms producing h o m o g e n e o u s or horizontally differentiated products, lz where rivals' actions can be m o n i t o r e d and secret price cutting is difficult. While Peck's careful study of the a l u m i n u m industry is the most detailed evidence in its favour, some of the case studies of price leadership are also consistent with it. It is a simple collusive equilibrium which can be easily used even when firms are non-identical; see Carlin and Soskice (1985) for an application to o p e n - e c o n o m y macroeconomics. By formalizing the kinked d e m a n d theory as an extensive form game I reinterpret it as a theory of price determination when firms can respond without delay to undercutting. While the kinked demand curve has been around for fifty years, it is largely discredited today. Perhaps this reinterpretation may enable it to retain some relevance. l~Friedman (1983) singles out the 'balanced temptation' point where each firm has the same ratio of the single period gain (from defecting from this equilibrium) to the subsequent losses (when firms revert to the Cournot outcome after defection). Such a rule requires each firm to have complete information on its rival's and own payoff functions; further, since such punishment strategies need not be unique [Abreu (1986)] the 'balanced temptation' point will not be unique. 121t may also be valid when products are vertically differentiated provided that the price differential required to compensate for the difference in quality is obvious and is accepted by both firms - the outcome will be the minimum optimal quality-adjusted common price. Of course the natural price differential may not be obvious in many industries and firms may have inconsistent views on it. If products are essentially identical (even if not completely homogeneous) this problem does not arise.
References Abreu, D., 1986, External equilibria of oligopolistic supergames, Journal of Economic Theory 39, 191-225. Bhaskar, V., 1988a, Pricing employment in the U.K.: An open economy model, Discussion paper 51 (Institute of Economics and Statistics, University of Oxford). Bhaskar, V., 1988b, Rational play in a quick response duopoly, Discussion paper 29 (Nuffield College, Oxford). Bonanno, G., 1987, Monopoly equilibria and catastrophe theory, Australian Economic Papers, forthcoming. Bulow, J.L., J. Geanakoplos and P. Klemperer, 1985, Multimarket oligopoly: Strategic substitutes and complements, Journal of Political Economy 93, 488-511.
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Carlin, W. and D. Soskice, 1985, Real wages, unemployment, international competitiveness and inflation; A framework for analysing closed and open economies, Mimeo. (University College, Oxford). Friedman, J.W., 1983, Oligopoly theory (Cambridge University Press, Cambridge). Hall, R.L. and C.J. Hitch, i939, Price theory and business behaviour, Oxford Economic Papers 2, 1245. Kalai, E. and M.A. Satterthwaite, 1986, The kinked demand curve, facilitating practices, and oligopolistic competition, Discussion paper no. 677 (The Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, IL). Klemperer, P., 1986, Entry deterrence in markets with consumer switching costs, Economic Journal Conference Papers 97, 99-117. Lutter, R. and J. Logan, 1987, Guaranteed lowest prices: Do they facilitate collusion?, Paper presented at the 14th Annual Conference, European Association for Research in Industrial Economics, Madrid. Marschak, T. and R. Selten, 1974, General equilibrium with price-making firms (Springer-Verlag, Berlin-Heidelberg. Marschak, T. and R. Selten, 1977, Oligopolistic economies as games of limited information, Zeitschrift fur die Gesamte Staatswissenschaft 133, 385-410. Marschak, T. and R. Selten, 1978, Restabilizing responses, inertia supergames and oligopolistic equilibria, Quarterly Journal of Economics 92, 71-93. Okun, A.M., 1981, Prices and quantities: A macroeconomic analysis (Blackwell, Oxford). Ono, Y., 1982, Price leadership: A theoretical analysis, Economica 49, 11-20. Peck, M.J., 1961, Competition in the aluminum industry 1945-58 (Harvard University Press, Cambridge, MA). Radner, R., 1980, Collusive behaviour in noncooperative epsilon-equilibria of oligopolies with long but finite lives, Journal of Economic Theory 22, 136-154. Reid, G.C., 1981, The kinked demand curve analysis of oligopoly (Edinburgh University Press, Edinburgh). Scherer, F,M., 1980, Industrial market structure and economic performance) Rand MeNally, Chicago, IL). Soskice. D., 1983, Notes on the minimum optimal common price, Mimeo. (University College, Oxford). Stigler, G,J., 1964, A theory of oligopoly, Journal of Political Economy 72, 44-61, Stigler, G.J., 1978, The literature of economics: The case of the kinked oligopoly demand curve, Economic Inquiry 16, 185-204. Sweezy, P.M., 1939, Demand under conditions of oligopoly, Journal of Political Economy 47, 568-573.
THE KINKED DEMAND
CURVE*
A Game-Theoretic Approach V. B H A S K A R University College, London WCIE 6BT, UK
Final version received August 1987 In a simple model of duopoly, firms' price moves are modelled as an extensive form game where firms can respond to undercutting without delay. When firms are not too dissimilar, kinked demand strategies enforcing an arbitrary price may be Nash equilibria; however, these strategies are dominated and perfect equilibrium is unique at the minimum optimal common price. Rather than implying price rigidity, kinked demand strategies are a device for ensuring compliance with a collusive price leadership.
1. Introduction In its t r a d i t i o n a l f o r m u l a t i o n the k i n k e d d e m a n d curve [Sweezy (1939), Hall a n d H i t c h (1939)'1 has been a t h e o r y of price rigidity. It argues t h a t firms will desist from reducing price since t h e i r rivals m a t c h price cuts; however, since price increases are n o t m a t c h e d , n o firm will initiate a price increase. T h e resulting d i s c o n t i n u i t y in the m a r g i n a l revenue curve implies that the m a r k e t price will n o t r e s p o n d to small c h a n g e s in costs or d e m a n d . Due to the i n d e t e r m i n a c y of the kink a n d the a r b i t r a r i l y a s y m m e t r i c p a t t e r n of b e h a v i o u r p o s t u l a t e d , the t h e o r y is n o w largely d i s c r e d i t e d J T h e empirical evidence appears, on the face of it, c o n t r a d i c t o r y (see Reid's 1981 survey) the objective evidence denies t h a t prices are m o r e rigid in industries where the t h e o r y o u g h t to be applicable whereas a large n u m b e r of m a n a g e r s claim, w h e n interviewed, to face such a n a s y m m e t r y in c o m p e t i t o r s ' responses, implying t h a t their d e m a n d curves are indeed kinked, This p a p e r a t t e m p t s to *Presented at the 14th Annual Conference, European Association for Research in Industrial Economics, Many of the ideas in this paper are due to discussions with David Soskice who has been most generous in allowing me to use his unpublished notes. I am grateful to Giacomo Bonanno, Paul Klemperer, Meg Meyer and John Vickers for discussions and to Paul Geroski and two anonymous referees for their comments on an earlier version of this paper. The usual disclaimer applies. ~See Stigler's obituarial view (1978). In his textbook on oligopoly theory Friedman (1983, p. xiv) writes, 'I have unhesitatingly omitted certain developments that have long been staple fare but that I find to be without merit (such as) ... the celebrated kinky oligopoly demand curve due to Sweezy', 0167-7187/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
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provide a reformulation of the theory which not only meets these theoretical objections but is consistent with both types of evidence. Merton Peck's classic study of the aluminium industry (1961) proposed a modified version of the theory in which the equilibrium price was uniquely determined. Peck questioned the kinked demand theory's assertion that a firm's competitors will not match price increases; if all firms benefit from a higher common price, a price increase will be followed since the initiator will cancel it otherwise. This in turn makes it attractive for firms to initiate a price rise when all firms prefer a higher common price. The equilibrium must hence be at a price where at least one firm does not prefer a higher common price. Defining each firm's optimal common price, as the price which maximizes its profits given that its rivals are pricing in line, Peck argued that the unique equilibrium would be the lowest of these optimal common prices. At this price no firm would have an incentive to reduce price since the outcome would be a lower common price which hurts all. Nor would other firms be able to force the price upward since the firm with the lowest optimal common price has no incentive to match their price increases, Section 2 formalized these arguments by considering a duopoly setting price for a single trading period. I consider physically identical products while allowing for a finite cross elasticity of demand due to customer loyalties and switching costs [Okun (1981), Klemperer (1986)]. The key assumption is that firms can respond to their rival's undercutting without delay - this presumes that secret price reductions are not possible, an assumption which may be tenable in markets with a large number of buyers, as Stigler (1964, p. 47) suggests, since 'no one has yet invented a way to advertise price reductions which brings them to the attention of numerous customers but not to that of any rival'. Price moves are modelled as an extensive form game. Firms simultaneously announce initial prices; if prices differ, the firm pricing higher can reset its price. If it undercuts its competitor on the second move, the latter in turn is given the option of changing its price. The move sequence ends when one firm does not undercut so that its rival has no need to reply. These price moves are assumed to take place sufficiently rapidly that no trade takes place until the final prices are announced; transitory profits are assumed negligible as in the kinked demand theory. This game is equivalent (in the sense that equilibrium outcomes are identical) to a game where the firm pricing higher initially restricts itself on the second move to prices above or equal to its rival. F o r expositional purposes I therefore use the two-move game. The main result is that firms will match undercutting in the relevant range (provided that they are not too dissimilar) and perfect equilibrium is unique at the minimum optimal common price. Interestingly, kinked demand strategies enforcing prices below the minimum optimal common price may be Nash equilibria but these strategies are dominated for both players and
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will hence not be adopted. I also find that the minimum optimal common price is inapplicable if the firms are too dissimilar since the firm with the higher preferred price chooses not to match its rival's lower price on the second move - the unique equilibrium is of a Stackelberg type, the firm with the lowered preferred price acting as a leader, while the follower prices above it. These roles are not exogenously imposed but emerge naturally.2 Section 3 gives a simple example showing how the type of equilibrium can be related to differences in costs or demand elasticities. Section 4 interprets these results; rather than implying price rigidity, this theory provides a rigorous basis for a type of price leadership where the leader uses a kinked demand strategy to enforce the minimum optimal common price. The ability of firms to respond without delay to undercutting ensures that the unique noncooperative equilibrium is collusive; this approach to oligopolistic collusion avoids some of the problems which have arisen in the repeated games approach.
2. The game in extensive form Firms /1 and B play an extensive form game with the following move order: they simultaneously announce initial prices a~,bl (a firm's price will be denoted by the corresponding lower case letter). If these differ, the firm pricing higher (say B) can announce b2; if it undercuts by choosing b2
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V. Bhaskar, The kinked demand curve
payoffs R a and R b from a strategy pair depend only upon the prices last announced (the outcome), since price moves are assumed instantaneous. I require equilibrium strategies to be subgame perfect Nash equilibria, and undominated by any other strategy. Subgame perfection ensures that players choose their second move optimally and the Nash criterion ensures that strategies are best responses, cra dominates a '~ iff it does as well as ~r'~ against all of B's strategies and does strictly better against at least one of B's strategies, i,e., iff: R~(a~, a °) > Ra(a '", O"b) ~t o.b ~ z~b and R"(a", a b) > R~(cr~, a b) for some a b e 2 b, I call such an equilibrium perfect in short - it is indeed trembling-hand perfect if the appropriate generalization of the concept is made for this game with infinite pure strategies. 3 While I set out the arguments in the more general case when products are not completely homogeneous, due to horizontal differentiation and switching costs, the theory also applies with perfect homogeneity. The assumptions on ,firms' payoff functions are:
A1.
Payoffs are continuous twice differentiable functions of the two prices; products are substitutes so that aR"/Ob and c3Rb/aa>O; payoffs are strictly concave with respect to own price.
A2.
a2Ra/ab Oa and c~ZRb/c3ac3b> 0 so that the products are strategic complements and the two 'reaction functions '4 c~(b) and fl(a) are upward sloping [see Bulow, Geanakoplos and Klemperer (1985)]. The reaction functions are contractions, i.e., dc~/db < 1 and d~/da < 1, ensuring the existence of a unique Bertrand equilibrium (a**,b**). A3. Define A's c o m m o n price profits Ca(a) as its profits when both firms price at the same price a. C" is strictly concave in the c o m m o n price and is maximum at a*, A's optimal common price. Similarly, Cb(b) is concave with b* as B's optimal c o m m o n price. A4.
For expositional conciseness alone, let A uniformly have a lower price preference, a * < b * and a * * < b * * . N o t h i n g essential hinges on this assumption. It is useful to define prices ~ (A's threshold price) and /~ (B's threshold price) at which their rival's reaction function prescribes price matching behaviour. Since the reaction functions are contractions (A2) it follows that 3Defined as the limit, as e, goes to zero, of a sequence of Nash equilibria in completely mixed strategies, where the probability density corresponding to a price available at any information set must be at least ~. As is well known dominated strategies cannot be trembling-hand perfect. ~A slightlymisleadingterm since they describe a firm's optimal price as a function of its rival's price rather than actual reactions in this game. I defer to established usage.
V. Bhaskar, The kinked demandcurve fl(a)~a
as
a~fi,
a(b)~b
as
b~.
377
A firm which is undercut initially will try, on the second move, to get as close to the price prescribed by its reaction function without undercutting its rival. Hence A will choose a2 = b l if b 1 >/~ and az=a(bl) if b 1 4 and bz=fl(al) if a t < & as shown by the curve X Y Z in fig. 1. A's payoff from a price a~ > bl (in equilibrium A will price higher initially) can therefore be fully specified - if al >4, A gets the common price profit C"(al) since B matches a~ on the second move, On the other hand if al
Proof B's second move is optimal as we have seen. From fig. 2 it is seen that a* is the unique maximizer o f A's payoff; it is hence his unique best response (hence undominated) when B prices higher on the first move. While B's best response first move is not unique - any price which is not below a* would do as well given that he has to move down to a* on the second move, the choice of b* dominates all alternatives by doing strictly better when A deviates from a* in some way. b* does better than a price above it, should A deviate with a price above b*, since the outcome is b* rather than a price above it. b* does better than a price below it should A price higher than B initially, since the common price will then be closer to b*. It is easily checked that b* does no worse than these alternatives whatever A's deviation. It may be similarly shown that A's choice of a* dominates all others. This ensures not merely that the strategy pair a* is undominated, subgame perfect and Nash, but also that it is uniquely so, since a*" and a.b dominate all alternatives. Rational players have no reason to adopt any other strategy. 5Obviously,despite the kink in A's payoff function it may have only a single maximum. Once again, this maximum may be either at a* or & so that only these possibilities need be considered.
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V. Bhaskar, The kinked demand curve
0c /
/x
t;*
a**
gl Fig. 1
Ra
Ca ~
a*
Fig. 2
If firms are identical, equilibrium is at the (identical) optimal common price which is the monopoly price. It is simple to prove that these results hold in the more general game set out at the beginning of this section where a player may undercut on the second move. Since the moves end when one player does not undercut the other, it is clear that the player with the higher price must have moved last. Subgame perfection requires that this last move be optimal; hence any outcome to the right of (and below) X W V in fig. 1 may be ruled out, since it would imply that A has chosen an excessively high price on the last move if he chose a lower price so as to be on X W V , he would do better, Similarly, any outcome to the left of (and above) X Y Z may be ruled out since B would be choosing an exessively high price on the last move. Further if B undercuts below a*, this implies that both final prices must be below a*. Since (a*, a~') is better for B than any common price pair below it on X Y (due to the
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concavity of common price profits), and since point Y in turn is better than any other point on Y Z (since products are substitutes), and since any point below Y Z is worse than a point on it, it follows that B can only do worse by undercutting below a* on the second move. I now show that the traditional kinked demand stalemate is a Nash equilibrium where players adopt dominated strategies. Suppose both players choose some price p on the first move, where a * > p > & This is a Nash equilibrium which is Pareto-inferior to the minimum optimal common price; neither player benefits from pricing higher initially given that his rival chooses p since he would have to move down to p on the second move. We have already seen that this strategy pair is not rational since players are adopting strategies which are dominated by the alternative of choosing their respective optimal common prices initially. In the second possible case when A's payoff is globally maximum at ~, the unique perfect equilibrium has A announcing ~ on the first move while B announces b*. B then moves down on the second move to a price fl(~) which is greater than 4. A and B act as Stackelberg leader and follower respectively, these roles emerging naturally without being exogenously imposed.
3. An example In this section I use a simple example with linear demand and constant unit costs to illustrate the theory and demarcate the conditions under which the minimum optimal common price and the Stackelberg equilibrium arise. To consider the implications of cost differences, let both firms face symmetric demand curves, but let B have a higher unit cost. Without loss of generality A's cost can be taken to be zero since we can measure prices net of this level. Let q" and qb denote the respective demands, let c be B's cost level, and remember that prices are simply denoted by the respective lower case letters. 7>6>0,
qa=X-Ta+6b,
(1)
qb = X -- 7b + 6a,
(2)
R" = X a -
(3)
~a 2 + 6ab,
R b = (b - c) ( X -
7b + 6a).
(4)
The respective optimal common prices, which maximize R a and R b when a = b, are
a* = X / E 2 ( ~ - 6)'1,
J.I,O,-- D
(5)
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380
b*=[X+(7-6)c]/[2(y-6)],
(6)
so that a*
C"(a*)=X2/4(7-6).
(7)
A's profits as a Stackelberg leader can be derived by substituting B's reaction function fl(a)=[X+yc+6a]/2y into his profit function. In the Stackelberg equilibrium A's price c~ and profits are ~ = [2yX + 6X +yac]/[2(2y 2 - 62)], sa([l) =
[27X + 6X + yac] 2/[8y(2y2 -- 62)].
(8) (9)
The equilibrium will be at a*, the minimum optimal common price, or at a, the Stackelberg equilibrium, depending upon whether Ca(a*) is greater or less than S"(a). Examining the two expressions, it is clear that B's cost level only affects A's Stackelberg payoff, not affecting his common price payoff. Since Sa(a) is increasing in c, it is clear that there is a critical cost difference below which the mimimum optimal common price applies, and above which the equilibrium is Stackelberg. Further, it is the more efficient firm which plays a leadership role, whether in the Stackelberg equilibrium, or in choosing the optimal common price. The above example also has a demand interpretation. Suppose that both firms have identical costs but that B's demand curve is
qb=X+Tz-yb+fa,
z>0.
(10)
In other words B has a larger market share due to an intercept on the price axis which is z greater than A's and consequently also has a more inelastic demand. Such a difference in demand elasticity due to differing market shares can arise naturally in markets with consumer switching costs [see Klemperer (1986)], where an established firm with a large number of locked-in customers faces a more inelastic demand than an entrant; the specific linear model set out here can arise if in addition there is some underlying horizontal differentiation between products. Since in a linear model an upward shift in the demand curve is equivalent (in terms of equilibrium pricing) to an increase in cost, this case is analytically equivalent to the previous case of cost differences. We merely have to replace c by z in the expressions and it is easy to see that there is a critical level of z below which the minimum optimal common price holds, and above which the Stackelberg equilibrium results. 6 The economic interpretation is simply that 6The only differenceis that B's payoff will be higher in the case where he has a larger market share.
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the larger the number of locked-in customers, the more inelastic is B's demand so that it finally pays B to price above A in equilibrium. Notice further that, in contrast with the standard presumption, it is the firm with the smaller market share which plays the leadership role in setting price, whether in Stackelberg equilibrium, or in choosing the common price.
4. Implications of the theory Since the minimum optimal common price is easily generalized to more than two firms, it is clearly a theory of price determination under oligopoly. A striking implication of the theory is that if firms differ, there will be one firm such that any small changes in its cost or demand conditions will alter the minimal optimal common price and the actual price level, whereas changes in costs/demand conditions for the other firm will have no impact upon the price? While this contradicts the price rigidity argument, since a general increase in industry costs or demand will raise prices, it suggests instead that the minimum optimal common price firms plays a leadership role in setting the price. This firm selects the industry price and prevents competitors from deviating from this price by following a kinked demand strategy. Since all firms except the price leader will face kinked demand curves, this also reconciles the presumed conflict between the objective evidence (denying price rigidity) and the subjective evidence that a large number of managers claim to face demand curves which are kinked at the prevailing price. A comparison with more conventional theories of price leadership is in order. Ono's (1982) analysis of Stackelberg price leadership shows that a firm voluntarily chooses the leadership role only if it has an overwhelming cost advantage or very high quality products. When cost or demand differences are not very large, Ono shows that every firm prefers the follower's rote so that the Nash equilibrium is the only stable solution. Ono suggests that a farsighted firm might accept the undesirable leadership role since its profits would be higher than in the Nash equilibrium. This is unsatisfactory since the reluctant leader has every incentive to deviate from its role in a single shot game; further, if a low-cost firm assumes leadership, it may earn less profits than its high-cost rival. In our theory the Stackelberg equilibrium similarly emerges when one firm has an overwhelming cost advantage or very elastic demand. However, even the absence of such conditions, choosing the common price confers no disadvantage since the followers cannot choose 7A caveat - the equilibrium ca~ashift discontinuously from the minimum optimal common price to Stackelberg due to a marginal increase in the cost (or demand) of the firm with the higher preferred price. In fig. 2 an increase in B's costs can raise A's threshold price ~ and his Stackelberg payoff while leaving A's common price profits unaltered. In this case both firms' prices must fall discontinuously when B's costs rise. For other instances of such discontinuous shifts see Bonanno (1987).
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E Bhaskar, The kinked demand curve
their price unrestrictedly; if they deviate by undercutting, the leader matches their price. This contrasts with the usual analysis of Stackelberg leadership where the leader commits to a price and allows the followers to choose unrestrictedly. The theory also explains the p h e n o m e n o n of a single firm initiating changes in industry price which has often been noted [see Scherer (1980, pp. 176-184)] if we allow that the followers will be uncertain about the leader's optimal c o m m o n price and will face small costs of changing price. The followers will then be content to match the leader's price changes while the leader is willing to initiate these changes, knowing that its rivals will follow it. These factors also enable one to argue for a limited role for the original price rigidity argument in industries where firms are very similar, so that there is no clear leader. In some circumstances the price may be sticky even if every firm experiences a rise in its optimal c o m m o n price - while each firm is sure that its optimal c o m m o n price has risen, it is unsure if this is the case for all its rivals, and hence refrains from raising price for fear that it may have to be cancelled. Such a stalemate is more likely when industry demand increases rather than when nominal input costs rise, since the latter are likely to be c o m m o n knowledge while changes in rivals' marginal costs and demand elasticities as a result of demand shifts are m o r e difficult to predict, This may be a partial explanation for the puzzling lack of cyclical variability in industrial prices. 8 A significant aspect of the minimal optimal c o m m o n price is that it is a 'collusive' equilibrium which Pareto-dominates the N a s h - B e r t r a n d outcome. 9 The key assumption which ensures this result is that a firm can respond to its rival's price change quickly, so that undercutting is unprofitable. 1° This may, in some ways, be more satisfactory than the repeated game a p p r o a c h to non-cooperative collusion. While collusive outcomes are perfect equilibria in an infinitely repeated game, the F o l k theorem also states that the set of equilibria include every individually rational outcome; hence outcomes far worse than the Nash C o u r n o t or Bertrand outcomes are as possible as collusive ones. Similarly, while Radner (1980) shows that a collusive outcome can result from near-rational behaviour in a finitely repeated game, once aOkun, (1981, p. 165) summarizes the evidence, 'Econometrically, demand is found to have little, if any, influence (on prices) outside the auction market for raw materials'. In a more recent study of prices in the U.K., Bhaskar (1988a) finds the strong influence of costs and international competition with demand variables being insignificant. 9However the outcome, is not perfectly collusive, in the sense of being Pareto-optimal, unless firms are identical or products are completely homogeneous; in the absence of binding agreements, the ability to respond quickly to deviations ensures superior but not necessarily the best outcomes. 1°This result is in contrast to Marschak and Selten's (1974, 1978, 1979) who first tried to analyze such a quick response game in terms of 'restabilizing response functions' and found multiple equilibria, including sub-optimal ones such as the Bertrand outcome. Bhaskar (1988b) demonstrates that such response functions are not rational and sets up a general extensive form game which yields collusive'outcomes only.
v. Bhaskar, The kinked demand curve
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again the C o u r n o t or Bertrand outcomes remain perfect equilibria. Even if we restrict ourselves to collusive outcomes both these approaches still burden us with a multiplicity of such equilibria, with no obvious way of choosing a single outcome when firms are not identical, ix The alternative procedure used here dispenses with infinite repetition by allowing firms to match lower prices instantly so that the non-cooperative o u t c o m e is collusive; furthermore, equilibrium is unique. The assumption that transitory profits are negligible may be empirically valid for m a n y industries; alternatively, it m a y also be defended as an instance of b o u n d e d l y rational behaviour by firms. The m i n i m u m optimal c o m m o n price is applicable in industries with a small number of firms producing h o m o g e n e o u s or horizontally differentiated products, lz where rivals' actions can be m o n i t o r e d and secret price cutting is difficult. While Peck's careful study of the a l u m i n u m industry is the most detailed evidence in its favour, some of the case studies of price leadership are also consistent with it. It is a simple collusive equilibrium which can be easily used even when firms are non-identical; see Carlin and Soskice (1985) for an application to o p e n - e c o n o m y macroeconomics. By formalizing the kinked d e m a n d theory as an extensive form game I reinterpret it as a theory of price determination when firms can respond without delay to undercutting. While the kinked demand curve has been around for fifty years, it is largely discredited today. Perhaps this reinterpretation may enable it to retain some relevance. l~Friedman (1983) singles out the 'balanced temptation' point where each firm has the same ratio of the single period gain (from defecting from this equilibrium) to the subsequent losses (when firms revert to the Cournot outcome after defection). Such a rule requires each firm to have complete information on its rival's and own payoff functions; further, since such punishment strategies need not be unique [Abreu (1986)] the 'balanced temptation' point will not be unique. 121t may also be valid when products are vertically differentiated provided that the price differential required to compensate for the difference in quality is obvious and is accepted by both firms - the outcome will be the minimum optimal quality-adjusted common price. Of course the natural price differential may not be obvious in many industries and firms may have inconsistent views on it. If products are essentially identical (even if not completely homogeneous) this problem does not arise.
References Abreu, D., 1986, External equilibria of oligopolistic supergames, Journal of Economic Theory 39, 191-225. Bhaskar, V., 1988a, Pricing employment in the U.K.: An open economy model, Discussion paper 51 (Institute of Economics and Statistics, University of Oxford). Bhaskar, V., 1988b, Rational play in a quick response duopoly, Discussion paper 29 (Nuffield College, Oxford). Bonanno, G., 1987, Monopoly equilibria and catastrophe theory, Australian Economic Papers, forthcoming. Bulow, J.L., J. Geanakoplos and P. Klemperer, 1985, Multimarket oligopoly: Strategic substitutes and complements, Journal of Political Economy 93, 488-511.
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Carlin, W. and D. Soskice, 1985, Real wages, unemployment, international competitiveness and inflation; A framework for analysing closed and open economies, Mimeo. (University College, Oxford). Friedman, J.W., 1983, Oligopoly theory (Cambridge University Press, Cambridge). Hall, R.L. and C.J. Hitch, i939, Price theory and business behaviour, Oxford Economic Papers 2, 1245. Kalai, E. and M.A. Satterthwaite, 1986, The kinked demand curve, facilitating practices, and oligopolistic competition, Discussion paper no. 677 (The Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, IL). Klemperer, P., 1986, Entry deterrence in markets with consumer switching costs, Economic Journal Conference Papers 97, 99-117. Lutter, R. and J. Logan, 1987, Guaranteed lowest prices: Do they facilitate collusion?, Paper presented at the 14th Annual Conference, European Association for Research in Industrial Economics, Madrid. Marschak, T. and R. Selten, 1974, General equilibrium with price-making firms (Springer-Verlag, Berlin-Heidelberg. Marschak, T. and R. Selten, 1977, Oligopolistic economies as games of limited information, Zeitschrift fur die Gesamte Staatswissenschaft 133, 385-410. Marschak, T. and R. Selten, 1978, Restabilizing responses, inertia supergames and oligopolistic equilibria, Quarterly Journal of Economics 92, 71-93. Okun, A.M., 1981, Prices and quantities: A macroeconomic analysis (Blackwell, Oxford). Ono, Y., 1982, Price leadership: A theoretical analysis, Economica 49, 11-20. Peck, M.J., 1961, Competition in the aluminum industry 1945-58 (Harvard University Press, Cambridge, MA). Radner, R., 1980, Collusive behaviour in noncooperative epsilon-equilibria of oligopolies with long but finite lives, Journal of Economic Theory 22, 136-154. Reid, G.C., 1981, The kinked demand curve analysis of oligopoly (Edinburgh University Press, Edinburgh). Scherer, F,M., 1980, Industrial market structure and economic performance) Rand MeNally, Chicago, IL). Soskice. D., 1983, Notes on the minimum optimal common price, Mimeo. (University College, Oxford). Stigler, G,J., 1964, A theory of oligopoly, Journal of Political Economy 72, 44-61, Stigler, G.J., 1978, The literature of economics: The case of the kinked oligopoly demand curve, Economic Inquiry 16, 185-204. Sweezy, P.M., 1939, Demand under conditions of oligopoly, Journal of Political Economy 47, 568-573.