Cartel Stability and the Curvature of Market Demand

Lambertini, Luca Working Paper Cartel Stability and the Curvature of Market Demand Quaderni - Working Paper DSE, No. 211 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (1994) : Cartel Stability and the Curvature of Market Demand, Quaderni - Working Paper DSE, No. 211, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5115 This Version is available at: https://hdl.handle.net/10419/159054 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ CARTEL STABILITY AND THE CURVATURE OF MARKET DEMAND Luca Lambertini* Dipartimento di Scienze Economiche# Università degli Studi di Bologna Strada Maggiore 45 40125 Bologna ITALY tel 39-51-6402600 fax 39-51-6402600 e-mail [email protected] and Linacre College Oxford OX1 3JA UNITED KINGDOM e-mail [email protected] December 1994 JEL Classification: L 13 Keywords: curvature of market demand, collusion, stability Abstract The stability of collusion is analysed for a family of demand functions whose curvature is determined by a parameter varying between zero and infinity. If demand is sufficiently convex, firms may prefer to act as quantity setters in order to increase cartel stability. Otherwise, price-setting behaviour enhances their ability to collude. *I wish to thank Guido Candela, Claudia Scarani and Dario Sermasi for insightful discussion and suggestions. The usual disclaimer applies. #Please send correspondence to the Italian address. 1. Introduction The issueof cartelstability has been widelyand deeplyanalysed under manyperspectives (see D’Aspremont and Gabszewicz (1986) for an exhaustive survey). Particularly noteworthy efforts have been devoted to the investigation of the bearings of heterogeneity among firms on cartelstability(D’Aspremontetal.(1983),Donsimoni(1985),Donsimonietal.(1986)).Another relevant stream of literature deals with the relationship between product differentiation and the stability of collusion (see, inter alia, Deneckere (1983), Chang (1991), Ross (1992) Rothschild (1992) and Häckner (1994)). The role of imperfect information in affecting firms’ ability to collude has also received attention (Green and Porter (1984), Rees (1985), Rotemberg and Saloner (1986)). Here I want to investigate what kind of linkages there exist between the curvature, i.e., the convexity/concavity of the market demand function, and the stability of a cartel made up by symmetric firms supplying a homogeneous good, under both quantity-setting and price-setting behaviour. It turns out that under Cournot behaviour the critical discount factor is increasing andconcaveintheparameterdescribingdemandcurvature,sothatcartelstabilityisadecreasing functionofthesameparameter,whileunderBertrandbehaviourthecriticalvalueofthediscount factor,andconsequentlycartelstability,areindipendentofthecurvatureofthedemandfunction. Theseresults revise and encompassthose already available in the literature (seeMajerus, 1988) as special cases. 2. The model I adopt the simple model described by Anderson and Engers (1992, p.129), assuming, in ordertosimplifycalculations,thatonlytwofirmsoperateinthemarket.Theysellahomogeneous product. The market demand function is defined by this demand function is always downward sloping, and can be either convex (when ) or concave (when ). Fixed costs can be assumed away without loss of generality, while it must be assumed that marginal costs are nil in order to obtain explicit solutions. Thus, firm i’s profit function coincides with her revenue. The collusive setting is analysed under the standard hypothesisthat afterdeviation agentsrevert to theNash equilibriumstrategies forever. Thenext subsectionisdevotedtotheanalysisofCournotbehaviour.Subsection2.2thenbrieflydescribes Bertrand behaviour. Q=1−pα,α>0; (1 α≤1 α≥1 1 2.1. Quantity-setting duopoly Consider first what happens if firms noncooperatively maximize single-period profits by simultaneously choosing quantitites. Firm i’s profit is given by The first oder condition for firm iis then yielding as the Nash equilibrium quantity, and as the noncooperative equilibrium profit (see Anderson and Engers, 1992, p.131). Ifinsteadfirmscollude,theycooperativelysetquantitiesinordertomaximizejointprofits: the generic first order condition is: yielding as a solution πi=qi(1−qi−qj) 1 α,i,j=1,2,i≠j.(2 ∂πi ∂qi=(1−qi−qj) 1 α  1−qi α(1−qi−qj)   =0,(3 qiN=α 1+2α(4 πi N=α (2α+1) α+1 α (5 ΠC=π 1+π 2;(6 ∂ΠC ∂qi=(1−qi−qj) 1 α(α − qi−qj−αqi−αqj) α(1−qi−qj)=0,(7 2 while each firm’s collusive profit amounts to A quick comparison between (4-5) and (8-9) shows that the cartel resticts both individual and total output, consequently enhancing profits. Let’s now consider unilateral deviation from the cartel agreement. Provided firm jsticks to her collusive output the deviation output by firm iis obtained from condition (4): yielding as the single-period deviation payoff. Deviation by firm ientails the following profit for firm j: Since, as it is easily checked, for all the punishment strategy is always credible. Thus, for collusion to be sustainable each firm’s discount factor must sutisfy the following condition: qiC=α 2(α + 1),(8 πi C=α 2(α + 1) α+1 α .(9 qjC, qiD=α(α + 2) 2(α + 1) 2(α + 1) α (10 πi D=α(α + 2) α+1 α 2α+1 α(α + 1)2(α + 1) α (11 πj(qiD,qjC)= α(α + 2) 1 α 2(α + 1)(2α2+4α+2) 1 α (12 πj(qiD,qjC)<π j Nα>0, 3 After some simple albeit tedious calculations it can be verified that and implying that the critical discount factor is increasing and concave in The same obviously holds for and Furthermore, so that as αapproaches either zero or infinity, the critical discount factor βis indeterminate. Nevertheless,1 This results can be given an intuitive explanation. First, as αdecreases, shrinks more rapidly than Second, as αapproaches infinity, i.e., as αincreases, catches up with faster than the latter does with 2.2. Price-setting duopoly The case of Bertrand behaviour can be quickly dealt with. Single-period noncooperative profit whatever the value of α, while cooperation yields half the monopoly profit, and finally Consequently, the critical discount factor is independent of αand each firm’s discount factor must satisfy the following condition (see Tirole, 1988, p.246): βi≥β(α)=πi D−π i C πi D−π i N=(1+2α) α+1 α 21 α(α + 1) α+1 α−(α+2) α+1 α  2α+1 α(α + 1) 2α+2 α−(α+2) α+1 α(2α+1) α+1 α (13 ∂2β/∂α2<0,∂β/∂α > 0 α. πi N,π i Cπi D. lim α→0πi J=0,lim α→∞πi J=1 2,J=N,C,D,(14 lim α→0β(α) = 0,lim α→∞β(α) = 1. (15 πi D−π i C πi D−π i N.π i C−π i N<π i D−π i C, πi Nπi Cπi D. πi N=0 πi C=π M/2,π i D=π M. βi≥β=πM−π M/2 πM=1 2(16 1. The limit of βas αapproaches zero has been computed through a binomial expansion, which has not been included in the text due to its length. 4 Numerical calculation shows that when where subscripts Cand Bstand for Cournot and Bertrand behaviour, respectively. This means that a quantity-setting cartel is more stable than a price-setting cartel for 3. Conclusions The influence of the curvature of market demand on cartel stability has been analysed. Theresultspointtotheconclusionthatthedegreeofconvexity/concavityofthedemandfunction canaffectfirms’abilitytocolludeonlyiftheyactas quantity-setters. In theCournotframework, the critical discount factor increases at a decreasing rate as the parameter determining the curvature of demand increases. When firms play àlaBertrand, the nature of both competition and equilibrium payoffs is such that the critical discount factor is indipendent of the curvature of the demand function. Furthermore, if the latter is sufficiently convex, the critical discount factor associated to quantity collusion is lower than that associated to price competition, so that collusion in quantities is more stable than collusion in prices, while the opposite holds if the demand function is only slightly convex, linear, or concave. This implies that, should firms be able to coordinate the choice of the strategic variable in order to enhance cartel stability, they would choose to set quantities only when facing a rather limited class of convex demand functions. The result reached by Majerus (1988), who takes into consideration a linear demand function and shows that when products are close or perfect substitutes price collusion is more easily sustainable than quantity collusion, appears as a special case of the results drawn from the model presented here. βC=β Bα~0.4753, α∈]0,0.4753[. 5 References Anderson, S. and M. Engers (1992), Stackelberg vs Cournot Oligopoly Equilibrium, International Journal of Industrial Organization, 10, pp.127-35. Chang M. (1991), The Effects of Product Differentiation on Collusive Pricing, International Journal of Industrial Organization, 9, pp.453-69. D’Aspremont, C. and J.J. Gabszewicz (1986), On the Stability of Collusion, in J. Stiglitz and G.F.Mathewson(eds),NewDevelopmentsintheAnalysisofMarketStructure,Cambridge, MA, MIT Press, pp.243-61. D’Aspremont, C., A. Jacquemin, J.J. Gabszewicz and J. Weymark (1983), On the Stability of Collusive Price Leadership, Canadian Journal of Economics, 16, pp.17-25. Deneckere, R. (1983), Duopoly Supergames with Product Differentiation, Economics Letters, 11, pp.37-42. Donsimoni, M.-P. (1985), Stable Heterogeneous Cartels, International Journal of Industrial Organization, 3, pp.451-67. Donsimoni, M.-P., N. Economides and H. Polemarchakis (1986), Stable Cartels, International Economic Review, 27, pp.317-27. Green, E. and R. Porter (1984), Non-cooperative Collusion under Imperfect Price Information, Econometrica, 52, pp.87-100. Häckner J. (1994), Collusive Pricing in Markets for Vertically Differentiated Products, International Journal of Industrial Organization, 12, pp.155-77. Majerus, D.W. (1988), Price vs Quantity Competition in Oligopoly Supergames, Economics Letters, 27, pp.293-7. Rees, R. (1985), Cheating in a Duopoly Supergame, Journal of Industrial Economics, 33, pp.387-400. Ross, T.W. (1992), Cartel Stability and Product Differentiation, International Journal of Industrial Organization, 10, pp.1-13. 6 Rotemberg, J. and G. Saloner (1986), A Supergame-theoretic Model of Business Cycle and Price Wars during Booms, American Economic Review, 76, pp.390-407. Rothschild,R.(1992),OntheSustainabilityofCollusioninDifferentiatedDuopolies,Economics Letters, 40, pp.33-7. Tirole, J. (1988), The Theory of Industrial Organization, Cambridge, MA, MIT Press. 7