On the sustainability of collusion in Bertrand supergames with discrete pricing and nonlinear demand

Transcription

1 PRA unih Personal RePE Arhive On the sustainability of ollusion in Bertran supergames with isrete priing an nonlinear eman Paul R. Zimmerman US Feeral Trae Commission 25. January 2010 Online at PRA Paper No , poste 26. January :56 UTC

2 On the sustainability of ollusion in Bertran supergames with isrete priing an nonlinear eman Paul R. Zimmerman Abstrat In traitional inustrial organization moels of Bertran supergames, the ritial isount fator governing the sustainability of ollusion is inepenent of key eman an supply parameters. Reent researh has emonstrate that these ounterintuitive results stem from the assumption that firms an hange pries in infinitesimally small inrements (i.e., ontinuously). This note onsiers the effets of eman urvature in the ontext of a moel of ollusion where, as in Gallie (2008), Bertran ompetitors an eviate only by lowering pries by some small, isrete amount. Two alternative eman speifiations that apture the influene of eman urvature are onsiere. In either ase, it is shown that with isrete prie hanges the ritial isount fator is etermine by the key eman parameters, inluing eman urvature. However, the iret effets of inrease onavity (or onvexity) in market eman on the sustainability of ollusion runs in opposite iretions aross the two moels. This isrepany is shown to arise from the way in whih the respetive eman urves rotate in response to a hange in the eman urvature parameter. The results support the onlusion of earlier researh that etermining the potential for ollusion in homogenous goos inustries likely requires areful ase-by-ase investigation. Keywors: Bertran supergames; artels; ollusion sustainability; isrete priing; nonlinear eman JEL Classifiation: L13, L41 January 25, 2010 Copyright 2010 by Paul R. Zimmerman. All rights reserve. The author thanks Patrik egraba an Anrea Gallie for helpful omments. The views expresse in this artile are those of the authors an o not neessarily reflet those of the Feeral Trae Commission. Eonomist, US Feeral Trae Commission, Bureau of Eonomis, Antitrust I, 601 New Jersey Avenue NW, NJ-8100, Washington, C, <paul.r.zimmerman@gmail.om>. 1

3 1 Introution Collusion is sai to be sustainable in a isrete-time, infinitely repeate Bertran-Nash oligopoly game ( supergame ) if 1 N 2 2 p pn P ( ) ³ + ( + + ), (1) where N enotes the number of perfetly symmetri firms ompeting in homogenous prouts, P the per-perio ollusive (monopoly) profits, p the one-perio profits realize by a given firm eviating from ( heating on ) the ollusive agreement, p N the per-perio Bertran-Nash equilibrium profits, an Î [0,1] the rate at whih firms isount future profits (whih is assume to be iential aross firms). The above expression an be solve for the ritial value of the isount rate,, for whih ollusion is sustainable: P ³ = 1 -, (2) N p sine t 1 å = an, by the usual assumption, p 0 (i.e., the firm t = N = s revert to playing the one-perio non-ooperative Bertran-Nash game in eah punishment perio a infinitum). 1 The traitional inustrial organization literature assumes that the heating firm an apture all market sales by utting its prie by some infinitesimal amount, i.e., firms an hange their pries in a ontinuous fashion. A ounterintuitive impliation of this latter assumption is that is unaffete by key eman (e.g., interept an slope) an supply (i.e., marginal ost) parameters that are known to affet the profitability of ollusion. However, an important an reent paper by Gallie (2008) assumes homogenous Bertran ompetitors an hange pries (i.e., eviate from the optimal ollusive prie) only in isrete (or non-infinitesimal) inrements. 1 This is the so-alle trigger strategy evelope by Frieman (1971). 2

4 Uner this arguably more plausible assumption, he shows that is no longer inepenent of key eman or supply parameters. 2 This note expans upon Gallie s framework an as to the literature on ollusion sustainability in Bertran supergames by onsiering more general eman speifiations that apture the effet of eman urvature (in aition to other key eman parameters) on the sustainability of ollusion. Consiering nonlinear eman urves is warrante sine many (empirial) market eman urves are unlikely to be stritly linear, an in fat, some eonomists have opine that nonlinear eman urves are likely to be the norm rather than the exeption from a theoretial perspetive. 3 Furthermore, it has been shown that the erivation of eman urves that are linear in pries requires unerlying utility funtions (inifferene maps) that are omplex, e.g., require quarati orers of the onsumption arguments (suh as in Singh & Vives 1984), or that ontain at least a single point of isontinuity (Alperovih & Weksler 1996). It is shown that the urvature of market eman an impart sizeable effets on the ritial isount fator both iretly an iniretly (i.e., through its relation to hanges in other eman parameters), although the nature of the relationships epens on the partiular speifiation of the market eman funtion two of whih are onsiere in the analysis. For example, in the first speifiation an inrease in market eman (as affete by a positive hange in the interept parameter) an have no impat on the sustainability of ollusion for suffiiently onvex eman. It is emonstrate (numerially) that ollusion an beome easier to sustain as the eman urve beomes more onave. However, the opposite result hols uner the seon 2 Gallie onsiers two eman speifiations, linear an onstant-elastiity. For the latter ase he shows (numerially) that there is a range over whih is initially ereasing in the elastiity of eman but then jumps isontinuously to a point where ollusion is never sustainable. Earlier work by Collie (2004), in the ontext of a Cournot moel, also shows that ollusion beomes easier to sustain the more elasti is market eman. The ensuing isussion relates to the results from Gallie s linear eman moel. 3 See, e.g., Walters (1980), Formby et al. (1982), an Coughlin (1984). This boy of literature fouses on the strong potential for an impliations of upwar-sloping marginal revenue funtions that may arise out of onvex eman urves. Aoring to Fomby et al. (1982, p.306): the possibility of an upwar sloping marginal revenue funtion an never safely be assume away. Our theoretial analysis lens support to A.A. Walter s [1980] onjeture that the eman onitions [i.e., onvexity] leaing to upwar sloping marginal revenue may inee be pervasive. (emphasis in original) 3

5 eman speifiation. Thus, exatly how nonlinearity influenes the eman speifiation an lea to isparate an, in some instanes, somewhat ounterintuitive impliations for the sustainability of ollusion. The remainer of the paper proees as follows. Setion 2 presents an isusses the two eman speifiations onsiere in the analysis. It is shown that the onavity of eman an either (iretly) inrease or erease the sustainability of ollusion epening on the speifi parameterization of the eman funtion. The ifferenes in the results obtaine from the two speifiations are shown to stem from the effet that hanges in the urvature parameters have on the rotation of the respetive eman sheules. Setion 3 summarizes an offers onluing remarks. 2 The moel 2.1 eman Two speifiations of a (potentially) nonlinear market eman urve are examine. The first is similar to that onsiere, e.g., by Benson (1980) an Lambertini (1996): Qp ( ) = a - b. (3) [oel I] The seon is similar in form to that use by Tyagi (1999) an is given by p Qp ( ) = ( a - bp) y. [oel II] 4 (4) The parameters a an b in either speifiation enote the stritly positive interept an slope parameters, respetively, an Qp ( ) > 0 where p > 0 enotes prie. The parameters > 0 an y > 0 are the respetive eman urvature parameters. Notie that in oel I only market prie is raise to the power of the urvature parameter, while 4 Benson (1980), Lambertini (1996), an Tyagi (1999) exlusively or primarily onsier iret eman funtions wherein prie is a funtion of quantity, as oppose to the inverse eman funtions onsiere here. As suh, the former papers moel firms ompeting in quantities as oppose to pries. 4

6 in oel II the urvature parameter applies to both the market prie an the key eman parameters. The urvature parameters an be varie to onsier a wie range of possible ownwarsloping eman funtions. oel I is linear fo r = 1, while for > 1 ( < 1 ) it is onave (onvex). oel II is linear for y = 1, while for y > 1 ( y < 1 ) it is onvex (onave). 2.2 Supply Unlike the ases Gallie (2008) onsiers, with the eman speifiation given in equations (3) an (4) it must be assume that marginal osts are zero in orer to erive lose-form solutions. At first blush the assumption of zero marginal osts may appear somewhat unappealing; however, it is a ommon one mae in the literature even when not analytially require. See, e.g., enekere (1983) an Rothshil (1992). Inee, sine marginal osts are assume to be onstant an symmetri aross firms in the moel, this treatment is analytially equivalent to simply normalizing marginal osts to zero. Alternatively, the interept term an simply be oneptualize as being measure net of marginal osts (enekere 1983). 2.2 oel I First onsier the eman urve given by Eq. (3). Iniviual eman for firm i is ì a - bp ï i if pi pj for any j ¹ i 1+ ( 1 ( ) { pj= p q p = ï i } ) í å i j ï0 otherwise ïî Uner the ollusive equilibrium the firms ooperatively set the market prie that maximizes total inustry profits. Sine the firms are symmetri this implies The joint (ollusive) profit-maximization problem is p = p = = p = p. 1 2 N max P ( p) = pq( p) p 1 = ap - b p +. 5

7 The first-orer onition ( FOC ) assoiate with the above problem is P = a - b( + 1)p = 0. p Solving the FOC for the ollusive prie yiels p é a ù = êë b ( + 1) úû 1/, where inexes ollusive items or outomes that are attaine at the monopoly equilibrium levels. Eah artel member s share of the per-perio ollusive profits equals P pq( p ) = N N æ ö a = ç è1 + ø N p. Again, if one firm hooses to eviate from the ollusive agreement, then the other nontrigger ) strategy eviating firms reat by playing the Frieman (1971) non-ooperative ( grim in all subsequent perios (so that eah firm, inluing the eviator, earns p N = 0 in eah perio of the infinite punishment phase). Following Ga llie, the heating firm s eviation from the ollusive prie (its prie unerut) is giv en by the small an positive but isrete inrement > 0, 5 whih in turn implies p <P. The firm that eviates from the artel therefore harges prie p = p - an, for a single perio, aptures all market sales. The quantity set by the eviating firm a t p is given by q = a - b, an the firm earns a one-perio profit p of 1 p b p + N p = a - > P. P p Substituting an into the inentive onstraint given by equation (1) an expressing it in term s of p an p iniates that ollusion is sustainable if 1 p ( bp - a) ³ = 1 -. (5) N p ( bp - a) 5 As in Gallie (2008), it is assume that the eviation prie is in the range of 1-2 perent beneath the ollusive prie. 6

8 The above expression iniates that, as in Gallie s moel, uner the assumption of isrete prie eviations is a funtion of the interept an slope parameters of the eman funtion. Furthermore, is also etermine by the urvature of the market eman funtion sine p an p are impliitly efine by. 6 Examination of equation (5) leas to the following intuitive results. First, if eman is linear ( = 1 ) an pries hange i sretely ( >0 ), then the ritial isount fator pertaining to Gallie s moel obtains (i.e., where = 0 in the latter): = 1 - a = 1, > 0 2 N( a b 2). 2 Seon, if eman is linear an pries hange ontinuously (regarless of the treatment of marginal osts), then the orinary ase in whih the ritial isount fator is inepenent of any eman fators is obtaine: 2 a N = 1, 0 2 N = - =. Na That is, the ritial isount fator is only a funtion of N, whih iniates that as the number of firms in the market inreases the sustainability of ollusion ereases. The key parameters of the market eman funtion play no role in etermining the sustainability of ollusion in this ase. Now onsier how, as given by equation (5), hanges with a perturbation in the interept parameter of the market eman urve: a - b a - b Na( p) ( q) p ( p )( (1 ) p ) = ³ 0. (6) a - The sign of the margin al effet epens on the sign of z = a - b(1 + ) p, whih is not immeiately apparent. 7 However, some simple numerial examples are suffiient to illustrate 6 Notie that these pries in equation (5) are also raise to the power of the urvature parameter. 7 Note that z is etermine by through its saling of b as well as through its effet on p, whih itself is raise to the power of. 7

9 that it is non-negative. As suh, an outwar shift of the eman urve will ten to make ollusion harer to sustain sine all firms will have a greater inentive to eviate an apture the entirety of the larger market. Consier { a = 100; b = 50 ; = 0.01 ; N = 10 } an { = 0.01 } ( whih orrespons to a highly onvex market eman urve). Then z = 0 an a = 0 i.e., an inrease in the size of the market has no effet on the sustainability of ollusion. Keeping all parameter values the same but setting = 1000 (whih orrespons to a highly onave eman urve) results in z = 100 ; learly a > 0 in this ase. 8 Other examples an be shown to emonstrate that, eteris paribus, an outwar shift of the eman urve will have a greater (smaller) effet on the ritial isount fator the more onave (onvex) the market eman urve. Turning to the slope parameter b, it an be shown that a =- 0. b b a (7) This result also mirrors Gallie (2008). 9 All else equal, an inrease in the slope oeffiient of the market eman funtion makes ollusion easier to sustain. The impliations of how the urvature of eman will affet the extent to whih a hang e in b will affet the ritial isount fator are straightforwar given that the sign of equation (7) is simply the negative of equation (6) times a/ b > 0. As the eman urve beomes relatively more onave a marginal hange in b results in a proportionally smaller (negative) hange in. Gallie fins that the sustainability of ollu sion is inreasing in a lower eviation prie effetively lowers the opportunity ost of heating. This same result an the intuition 8 Reognize that lim 0 z = 0 an lim z = a, whih are the funamental results that unersore the relationship between the urvature of the market eman funtion an the impat of a hange in eman on the ritial isount fator. 9 Gallie shows that hanges in the slope parameter affet the sustainability of ollusion through their impat on the interept term of the inverse market eman urve. 8

10 behin it are also borne out in the present ontext. The effet of a marginal hange in the prie tik on the ritial isount fator is given by a =- 0. (8) a Thus, the greater the sprea between the ollusive an eviation pries the easier it is to sustain ollusion. Furthermore, the magnitue of the (negative) marginal effet rises as the market eman urve beomes more onave. This result hols sine a higher raises both the parenthetial expression given in equation (8) as well as a, as isusse above. It is also possible to onsier the iret effet of a hange in eman urvature in oel I on the ritial isount fator uner the assumption of isrete prie hanges, whih is given by æ 1 ö p{ bpp ln( p) + 2 [ - a( a - bp)( (1 + ) - p) (1 ) + ç æ a ö ç+1+ ( ){( abp)( p p) ( z) q}ln b(1 ) ç è + ø bp( ap z (1 ) qp ln( p))]} ç çè ø =, (9) N( -p ) ( q ) where z = a - b(1 + ) (whih, like z, is stritly non-negative). The sign of equation p (9) is not apparent ue to the various (potentially) positive an negative terms that interrelate aitively or multipliatively in the numerator. 2 2 However, a numerial example is suffiient to emonstrate that the sign of equation (9) is negative, whih implies that as eman beomes inreasingly onave the sustainability of ollusion inreases (albeit at a ereasing rate). Again, onsier the parameter values { a = 100; b = 50 ; = 0.01 ; N = 10}. Figures 1.a an 1.b graph an, respetively, at these values while allowing to vary over the range [0.0001, ]. Figure 1.a learly shows that the slope of monotonially ereases over the relevant range of. Figure 1.b shows that the rate of hange in the partial erivative falls the higher the 9

11 value of the slope parameter, approahing a horizontal asymptote at 0 =. Therefore, while a marginal inrease in unambiguously make ollusion eas ier to sustain, the magnitue of the effet is smaller the higher the (initial) onavity of the market eman urve. Arguably, these results may appear somewhat ounterintuitive. For instane, Hayrapetyan et al. (2007) note that: A onave eman urve is appliable for moeling eman for a servie with a omparable alternative (at a high enough prie all users will swith to the alternate servie). One might therefore expet, a priori, that the ritial isount fator woul be inreasing in the onavity of eman (i.e., hiner the sustainability of ollusion) sine onsumers oul more reaily substitute towars alternative goos/servies at suffiiently high pries. Inee, this is preisely the result obtaine by Lambertini (1996, p. 332) with respet to a homogenous Cournot moel. 10 However, in the present ontext it is straightforwar to emonstrate why the same intuition oes not apply. efine p ( ) = p -( p / ) > 0, or the ifferene between the N eviation profits an the ollusive profi ts that the heating firm woul have otherwise obtaine in eah one-shot (or single-pe rio) game over the range Î [0.0001,10 000]. Figure 2 presents the graph of this funtion for the above parameter values. Notie that p / > 0 (iniating that evi ation profits are monotonially inreasing in eman urvature) while p/ 0 as. Thus, the more onave the market eman urve of oel I the lower the relative gain from heating, whih implies that ollusion will be easier to sustain as 2.3 oel II inreases. The ollusive profit-maximization problem for the eman speifiation given by equation (4) is with the FOC: max P ( p) = p( a - bp) y, p 10 Lambertini (1996) takes the eviation quantity set by the heating firm as set uner the assumption that eah of the non-eviating firms ontinues to proue its joint profit-maximizing output level. 10

12 P( p) 1 = Qp ( )- bp( a - bp) - +y = 0. p The equilibrium ollusive an eviation pries are given by p = a b(1 + y) p = a - b (1 + y), b(1 + y) respetively, with the assoiate equilibrium profits P = y æ ay ö a ç è1 + y ø b(1 + y), æ ay ö ç b + ( a b (1 1 y - + y)) è + ø P =. b(1 + y) Plugging the latter two expressions into equation (2) gives the ollusion sustainability onition: y y æ ay a ö æ ö ç b 1 y ç + 1 y è + ø è + ø ³ = 1 - (10) N( a - b (1 + y)) ay -y Equation (10) also iniates that uner the assumption of isrete prie hanges the ritial isount fator is a funtion of key eman parameters, inluing the urvature of eman. Again, the sustainability of ollusion is ereasing in the size of the market uner this framework: 2 2æ ay ö 2 ay (1 y) b (1 + y) ( b ) ç è1+ y ø 1+ y = > (11) a 2 N( a - b (1 + y)) The omparative statis results with respet to the slope an prie-tik parameters are qualitatively similar to those foun in oel I: 11 Note that for P > P / N it must be the ase that a - b (1 + y) > 0, an as suh, equations (11)- (12) hol as a strit inequalities. 11

13 a 0 b =- b a < =-a < 0. a (12) Now onsier the iret effet of eman urvature, whih is given by æ ay ay (1 ) ( )( ) - +y ö b+ ( ab(1 + y) 1+ y 1+ y ç ay ay ç+ ( a - b (1 + y)(ln( ) - ln( b + ))) ç çè 1+ y 1+ y ø =-. (13) 2 N ya ( - b (1 + y)) Again, numerial simulations must be use in orer to gain insight into the sign of the numerator in equation (13). Figure 3 graphs equation (13) for the parameter values eman urve beomes relatively more onvex (onave), the ritial isount fator iminishes (rises) the exat opposite of the result obtaine from oel I. Thus, oel II generates the seemingly more intuitive result that to the extent onavity reflets substitution options (i.e., at least at the higher range of market prie) ollusion is more iffiult to sustain the more easily onsumers an swith to alternatives besies the artel s output. { a = 100 ; b = 50 ; N = 10 ; = 0.01 } where y varies over the interval [0.01,100]. In this ase, as the market Given that both oels I an II appear to be reasonable speifiations in whih to apture the influene of eman urvature (an, again, both have been use in previous stuies), the question remains why the orresponing results are so similar with respet to the (iniret) effets of perturbations in the eman parameters, but iametrially in opposition with respet to the (iret) effet of eman urvature. The eonomi intuition for the latter stems from the way in whih the respetive (iret) eman speifiations rotate about their interepts in (,) pq spae in response to a perturbation in the relevant urvature parameter. For oel I, a eteris paribus inrease in the urvature parameter results in the market eman urve rotating inwars about its horizontal (quantity) interept (i.e., a ). That is, as this eman urve beomes more onave it also rotates inwars, whih (all else equal) results in a 12

14 lower eviation prie relative to the ollusive prie at any given level of output, hene making ollusion easier to sustain. 12 On the other han, in oel II an inrease in y, whih makes the market eman urve relatively more onvex, results an upwar rotation of the urve about its vertial (prie) interept. Thus, all else equal, at any given level of prie, greater onvexity of eman is assoiate with a higher level of output an, orresponingly, a lower prie that a firm oul obtain from eviating. Therefore, inreasing onvexity in this ase makes ollusion easier to sustain; onversely, more onave eman makes ollusion more iffiult to sustain Summary an onluing remarks This paper onsiers the effet of eman urvature on the sustainability of ollusion in Bertran supergames in whih firms an only eviate from the optimum artel prie by unerutting in small, isrete inrements. epening on how the market eman urve is parameterize, a hange in eman urvature an have very ifferent iret impliations for the sustainability of ollusion. The empirial literature on artel uration generally fins that an inrease in market eman tens to hiner ollusion (Levenstein & Suslow 2006), although this nee not be the ase uner the eman speifiation in oel I an suffiiently onvex eman. Overall, the results support Tyagi s (1999, p. 298) onlusion (rawn in the ontext of a repeate quantity-setting game) that: [W]ithout oing a ase-by-ase analysis, it is iffiult to say whether firms in more homogenous prout markets fin it easier or harer to taitly ollue. 12 This inwar rotation of the oel I market eman urve is what also rives the onavity of p. 13 It is important to keep in min that these results arise beause of the assumption of isrete prie hanges in the priing supergame, whih it turn results in the ritial isount fator being a funtion of the respetive urvature parameters. 13

15 Referenes Alperovih, G. an I. Weksler (1996), A lass of utility funtions yieling linear eman funtions, Amerian Eonomist 40: Benson, B.L. (1980), Löshian ompetition uner alternative eman onitions, Amerian Eonomi Review 70: Collie,. (2004), Collusion an the elastiity of eman, Eonomis Bulletin 12: 1-6. Coughlin, P.J. (1984), Changes in marginal revenue an relate elastiities, Southern Eonomi Journal 51: enekere. R. (1983), uopoly supergames with prout ifferentiation, Eonomis Letters 11: Formby, J.P., S. Layson, an W.J. Smith (1982), The law of eman, positive sloping marginal revenue, an multiple profit equilibrium, Eonomi Inquiry 20: Frieman, J.W. (1971), A non-ooperative equilibrium for supergames, Review of Eonomi Stuies 38: Gallie, A. (2008), The neglete effets of eman harateristis on the sustainability of ollusion, CEPR isussion Paper No Hayrapetyan, A., É. Taos, an T. Wexler (2007), A network priing game for selfish traffi, istribute Computing 19: Lambertini, L. (1996), Cartel stability an the urvature of market eman, Bulletin of Eonomi Researh 48: Levenstein,.C. an V.Y. Suslow (2006), What etermines artel suess?, Journal of Eonomi Literature 44: Rothshil, R. (1992), On the sustainability of ollusion in ifferentiate uopolies, Eonomis Letters 40: Singh, N. an X. Vives (1984), Prie an quantity ompetition in a ifferentiate uopoly, Ran Journal of Eonomis 10: Tyagi, R.K. (1999), On the relationship between prout substitutability an tait ollusion, anagerial an eision Eonomis 20: Walters, A.A. (1980), A note on monopoly equilibrium, Eonomi Journal 90:

16 FIGURE 1.A: THE EFFECT OF ON FOR a = 100, b = 50, N = 10, AN = FIGURE 1.B: THE EFFECT OF ON FOR a = 100, b = 50, N = 10, AN = μ μ μ μ μ μ μ

17 FIGURE 2: THE EFFECT OF ON p FOR a = 100, b = 50, N = 10, AN = FIGURE 3: THE EFFECT OF y ON FOR a = 100, b = 50, N = 10, AN = y 16