Bertrand versus Cournot with Convex Variable Costs

Delbono, Flavio; Lambertini, Luca Working Paper Bertrand versus Cournot with Convex Variable Costs Quaderni - Working Paper DSE, No. 994 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Delbono, Flavio; Lambertini, Luca (2015) : Bertrand versus Cournot with Convex Variable Costs, Quaderni - Working Paper DSE, No. 994, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4168 This Version is available at: https://hdl.handle.net/10419/159832 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/ ISSN 2282-6483 Bertrand versus Cournot with Convex Variable Costs Flavio Delbono Luca Lambertini Quaderni - Working Paper DSE N°994 Bertrand versus Cournot with Convex Variable Costs Flavio Delbono#and Luca Lambertinix # Department of Economics, University of Bologna Piazza Scaravilli 2, 40126 Bologna, Italy ‡[email protected] § Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected] February 26, 2015 Abstract Within a simple model of homogeneous oligopoly, we show that the traditional ranking between Bertrand and Cournot equilibria may be reversed. For price setting entails a continuum of price equilibria under convex variable costs, departure from marginal cost pricing may be observed. As a consequence, Bertrand-Nash equilibrium pro…ts (welfare) may be higher (lower) than Cournot-Nash ones. The reversal of the standard rankings occurs when pricing strategies mimic collusive behaviour. JEL Codes: D43, L13 Keywords: oligopoly, pricing strategy, multiple equilibria 1 1 Introduction A classical issue in modern industrial organization deals with ranking Nash equilibria generated by price or quantity competition. Absent externalities, the standard conclusion emerging from such comparison states the social superiority of Bertrand competition w.r.t. Cournot competition. This has been proved in a broad class of static games.1However, in a homogeneous product oligopoly, the comparison between the two types of equilibria has been long limited by the strict assumptions about technology needed to ensure the existence of a pure strategy equilibrium under Bertrand rules. Such a limitation has been bypassed by Dastidar (1995), proving that, under concave demand and convex costs, price competition in a homogeneous oligopoly yields a continuum of Bertrand-Nash equilibria in pure strategies. This result may then allow one to challenge the alleged greater e¢ ciency of Bertrand-Nash equilibria w.r.t. the Cournot-Nash equilibrium. As long as Bertrand-Nash behaviour doesn’t need to coincide with marginal cost pricing, the standard ranking between Bertrand-Nash and Cournot-Nash pro…ts and social welfare may be reversed. In this note, indeed, we show that, in the continuum of price equilibria under convex variable costs, departure from marginal cost pricing may be observed. As a result, in a broad range of the parameter constellation, Bertrand-Nash equilibrium pro…ts (welfare) may be higher (lower) than Cournot-Nash ones. It’s worth noting that the reversal of the standard rankings occurs when pricing strategies mimic collusive behaviour. The remainder of the paper is organised as follows. In section 2 we set up the model and solve the two games. In section 3, we perform some comparative statics, instrumental to our main results illustrated in section 4. Section 5 concludes. 1See, fon instance, Singh and Vives (1984), Vives (1985), Okuguchi (1987) and Dastidar (1997). With cost asymmetry and a small degree of product di¤erentiation, Zanchettin (2006) shows that the opposite can occur. 2 2 Setup and Nash equilibria Consider a market supplied by a set N= 1;2;3; :::; n of identical …rms producing a homogeneous good whose demand function is p= 1 Q; where Q= n i=1qiis aggregate output and pis price. All …rms share the same technology, summarised by the convex cost function Ci=cq2 i=2. Accordingly, the pro…t function of …rm iis i=pcqi 2qi=1qiQicqi 2qi(1) where Qi= j6=iqj. Firms play simultaneously a non-cooperative one-shot game under complete, symmetric and imperfect information. The solution concept is the Nash equilibrium. 2.1 The quantity-setting game If …rms are Cournot players, the relevant …rst order condition for …rm iis: @i @qi = 1 2qiQicqi= 0 (2) which, under the symmetry condition qj=qi=qfor all iand j, yields the Cournot-Nash (CN) equilibrium output qCN =1 n+1+c(3) for each individual …rm. The resulting equilibrium pro…ts are CN =2 + c 2 (n+1+c)2(4) and social welfare is SWCN =nCN +CSCN =n(n+2+c) 2 (n+1+c)2(5) where CSCN =nqCN 2=2is consumer surplus. 3 2.2 The price-setting game Here, we follow Dastidar (1995), where it is shown that, if costs are strictly convex in output levels, Bertrand competition yields a continuum of Nash equilibria. The Nash equilibrium in pure strategies involves indeed all …rms setting the same price p2[pavc; pu]:At the lower bound pavc;equilibrium price equals average variable costs, so that …rms would be indi¤erent between producing or not. At the upper bound pu;the equilibrium price is such that …rms would be indi¤erent between playing puor marginally undercutting it in order to capture the entire market demand. The range of equilibrium prices is identi…ed by:2 pBN =c c+ 2 (n)(6) where BN mnemonics for Bertrand-Nash, and is a non-negative parameter whose range, to be speci…ed below, determines the continuum of equilibrium prices. The associated individual output and pro…ts are qBN =2 (n) n[c+ 2 (n)] (7) BN =2c (n) n2[c+ 2 (n)]2(8) and social welfare is SWBN =nBN +CSBN =2 (n) [n(n) + c] n[2 (n) + c]2(9) The admissible range is 2[0; n2=(1 + n)] :This is because in = 0;the equilibrium price equals average variable cost; at =n=2;marginal cost pricing obtains; if =n2=(1 + n); pBN reaches the highest level above which undercutting takes place. 2See Dastidar (1995, pp. 27-28); and Gori et al. (2014, pp. 373-75). 4 3 Comparative statics The very fact of the existence of a continuum of price equilibria ranging well above marginal cost pricing raises two related questions. The …rst deals with the monotonicity (or the lack thereof) of equilibrium pro…ts w.r.t. the number of …rms under Bertrand competition. The second issue is whether the pro…t ranking across the two regimes is robust to variations in industry structure as measured by the number of …rms, and/or the price mark-up determined by the value of . In this section we tackle the …rst question, while the second is postponed to the next section. For completeness, we set out by summarising the e¤ect of an increase in non Cournot-Nash equilibrium pro…ts. This is captured by the following derivative:3 @CN @n =c+ 2 (n+1+c)3<0(10) everywhere. This is the standard result we are well accustomed with, telling that individual pro…ts are monotonically decreasing in the number of quantitysetting …rms. Now we examine the behaviour of Bertrand pro…ts w.r.t. nin our setting, where there exists a continuum of equilibria. We are going to prove the following: Lemma 1 @BN =@n > 0for all n2nB ; nB +and negative elsewhere, with nB =10cpc2+ 28c + 42 12 Proof. The partial derivative of Bertrand-Nash pro…ts w.r.t. nis: @BN @n =2c [2(5n2)6n2c(n2)] n3[c+ 2 (n)]3(11) 3As usual, we are treating nas a continuous magnitude when performing comparative statics. As soon as we will be looking at numerical examples, we will con…ne our attention to integers. 5 Since n>n2=(n+ 1) ;which is the upper bound of the admissible interval for , the denominator of (11) is strictly positive. Hence, the sign of @BN =@n is the sign of the numerator. The roots of 2(5n2)6n2c(n2) = 0 (12) are nB =10cpc2+ 28c + 42 12 (13) and, given the concavity of the numerator w.r.t. n, this implies that @BN =@n > 0for all n2nB ; nB +:Outside this range, @BN =@n < 0: Notice that, in order for the interval nB ; nB +to be economically meaningful, it must be that at least nB +2;i.e., nB +2 = 1024 c+pc2+ 28c + 42 12 0(14) The existence of a range of industry structures wherein an increase in the number of …rms yields an increase in the Bertrand-Nash pro…ts suggests that Bertand-Nash pro…ts might overcome those generated by those associated to the Cournot-Nash equilibrium. In the next section we show that this can indeed happen in an admissible portion of the parameter space. 4 Ranking equilibrium pro…ts and welfare Under marginal cost pricing, it would be true that CN > BN for all n 2. However, since we follow Dastidar’s (1995) approach to model Bertrand competition, we have to admit the possibility for BN to increase in ndue to the presence of a mark-up exceeding its competitive level as increases above n=2: To investigate whether this brings about a reversal of fortune across equilibria, it is appropriate to rede…ne the upper bound of in terms of a lower 6 bound to n: This trivially requires solving the following inequality: (n+ 1) n2(15) w.r.t. n, which delivers the equivalent condition n+p(+ 4) 2n(16) If one compares nagainst nB ;it turns out that nnB =c4+ 6p(+ 4) + pc2+ 28c + 42 12 >0(17) everywhere, because 3p(+ 4) >2. Therefore, n > nB always. The comparison between nand nB involves evaluating the sign of the following expression: nnB +=c4+ 6p(+ 4) pc2+ 28c + 42 12 (18) Again, c4+ 6p(+ 4) >0since 3p(+ 4) >2. Consequently, the sign of nnB +is the sign of hc4+ 6p(+ 4)i2 c2+ 28c + 42(19) which can be usefully rewritten as 483 + p(+ 4)+ 12cp(+ 4) 3(20) where 483 + p(+ 4)>0for all admissible values of ; while p(+ 4) 3R0for all S1=2. Thus, if 2[0;1=2] ; n > nB +everywhere; if  > 1=2;(i) n > nB +for all c2(0;ec); (ii) nnB +for all cec; with ec(2+ 5) (23) p(+ 4) 21:(21) 7 BN CN =1 450;SWCN SWBN =2 45 (28) ec=413 5p2 7= 3:38; c1=2 3;c2= 2 1+2r7 3!= 8:11 (29) nB +=17 + p193 6= 5:14; n= 2 1 + p2= 4:83 (30) 5 Concluding remarks In this paper we have proved that the traditional ranking between Bertrand and Cournot equilibria may be reversed under convex variable costs. The presence of a continuum of price equilibria allows for departures from marginal cost pricing, to such an extent that Bertrand-Nash equilibrium pro…ts become higher than Cournot-Nash ones. In the same region of parameters where this happens, the opposite occurs to the sequence of social welfare levels. These reversals are driven by the fact that equilibrium pricing mimics collusion. 14 References [1] Dastidar, K.G. (1995), “On the Existence of Pure Strategy Bertrand Equilibrium”, Economic Theory,5, 9-32. [2] Dastidar, K.G. (1997), “Comparing Cournot and Bertrand in a Homogeneous Product Market”, Journal of Economic Theory,75, 205-12. [3] Gori, G., L. Lambertini and A. Tampieri (2014), “Trade Costs, FDI Incentives, and the Intensity of Price Competition”, International Journal of Economic Theory,10, 371-85. [4] Okuguchi, K. (1987), “Equilibrium Prices in the Bertrand and Cournot Oligopolies”, Journal of Economic Theory,42, 128-39. [5] Singh, N. and X. Vives (1984), “Price and Quantity Competition in a Di¤erentiated Duopoly”, RAND Journal of Economics,15, 546-54. [6] Vives, X. (1985), “On the E¢ ciency of Bertrand and Cournot Equilibria with Product Di¤erentiation”, Journal of Economic Theory,36, 166-75. [7] Zanchettin, P. (2006), “Di¤erentiated Duopoly with Asymmetric Costs”, Journal of Economics and Management Strategy,15, 999-1015. 15 