Cartel Stability, Mark-Up Cyclicality and Government Spending Multipliers

Lambertini, Luca; Marattin, Luigi Working Paper Cartel Stability, Mark-Up Cyclicality and Government Spending Multipliers Quaderni - Working Paper DSE, No. 820 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Marattin, Luigi (2012) : Cartel Stability, Mark-Up Cyclicality and Government Spending Multipliers, Quaderni - Working Paper DSE, No. 820, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4196 This Version is available at: https://hdl.handle.net/10419/159659 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, Mark-Up Cyclicality and Government Spending Multipliers Luca Lambertini Luigi Marattin Quaderni - Working Paper DSE N° 820 Cartel Stability, Mark-Up Cyclicality and Government Spending Multipliers Luca Lambertini and Luigi Marattin Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected]; [email protected] March 23, 2012 Abstract Mark-up cyclical behaviour is relevant in determining the size of government spending multiplier on output. While theoretical literature priviliged the counteryclical hypothesis, empirical evidence is far from being conclusive. Based on seminal Rotemberg and Saloner (1986) contribution, we build a theoretical framework based on Bertrand duopoly, stochastic demand and product di¤erentiation, where the analysis of cartel stability under partial collusion points towards procyclical pricing. According to the intensity of marginal cost cyclicality, this can produce a procyclical mark up or - at least - render it less countercyclical than expected, with relevant e¤ects on the transmission mechanism of government spending stimuli. JEL Codes: C73, L13 Keywords: partial collusion, cyclical pricing We would like to thank Giacomo Calzolari, Paolo Manasse and Antonio Minniti per precious comments and suggestions. The usual disclaimer applies. 1 1 Introduction Recent theoretical contributions (Hall, 2009; Woodford, 2011) highlight the importance of price mark-up’s cyclical behaviour for the transmission mechanism of …scal policy. Previous literature (Galì et al., 2005; Galì, 2005) had already stressed that an exogenous reduction in the aggregate ine¢ - ciency wedge (price or wage mark-up) ampli…es the e¤ects of a government spending stimulus on aggregate demand, and vice versa. However, by relating mark-up movements to the business cycle, it is possible to investigate analytically the relationship between government spending multipliers and the degree of pro/countercyclicality of mark-up. In a stylized sticky prices macroeconomic model, Hall shows that if we de…ne (y) = y!as the price mark-up, and parameter !indicates the sensitivity to the income level y, then: sign 2 4@dy dg  @! 3 5=sign (!) Considering that the government spending multiplier is positive, equation (1) means that if the mark-up is countercyclical (! > 0) then the higher the sensitivity to aggregate demand (!"), the higher the government spending multiplier dy dg ". On the other hand, if mark-up is procyclical (! < 0), a more pronounced cycle elasticity (!")lowers the expansionary e¤ects of government purchases on output dy dg #: What does economic literature have to say about the direction of mark-up cyclicality? Theoretical literature has mainly focused on countercyclicality,1by taking two alternative roads that we could label "the macroeconomic view" and "the industrial organization view". As to the former, the traditional explanation has centered on nominal rigidities: if prices are sticky, an increase in aggregate demand - assuming 1Lindbeck and Snower (1987) and Bils (1987) achieve mark-up countercyclicality by establishing a positive relationship between aggregate demand and elasticity of demand. Edmond and Veldkamp (2009) assign the central role to income distribution: during booms, income shifts towards the lower tail of the distribution, featured by higher-elasticity consumers. 2 ‡exibility of some elements of marginal costs - results in a mark-up reduction (Rotemberg and Woodford 1999, Woodford 2003). The industrial organization view focuses instead on …rms’strategic interaction in a non-competitive environment. Rotemberg and Saloner (1986) argue that oligopolies are likely to behave more competitively when demand rises, especially when price is the strategic variable. Under these circumstances, in fact, the bene…t from deviation is larger, and the punishment is diminished because it will be implemented when the expansionary demand shock will have already been absorbed. As a results, price/marginal cost ratio declines as aggregate demand increases. Haltiwanger and Harrington (1991) extend the analysis to allow for time-varying …rms’expectations on future demand, by relaxing the assumption of i.i.d. demand shocks so to induce serial correlation in the cycle. They highlight potential asymmetries in collusive pricing behavior across di¤erent state of the business cycle, as their …ndings show that collusion is more di¢ cult during recessions than during booms. In fact, establishing a relation between future and current demand induces asymmetries between the opportunity costs of engaing in price wars according to the direction of demand. Under falling demand, the forgone collusive pro…ts are on a intertemporally decreasing path, so the incentive to collude is lower and likely to remain so. On the other hand, in a period of increasing demand, the traditional Rotemberg and Saloner result is mitigated by the fact that joint-maximizing pro…ts are going to be higher in the future. Along this path, Bagwell and Staiger (1997) develop a theory of collusive pricing in a framework where aggregate demand alternates stochastically between slow and fast growth states, and where the transition is governed by a Markov process. They …nd that the cyclical behaviour of collusive prices depends crucially on correlation of demand growth rates through time and the expected duration of boom and recessions. Particularly, collusive prices are procyclical in presence of positive demand correlation through time, and countercyclical otherwise. Furthermore, the amplitude of the collusive pricing is larger when the recession has a longer expected duration or - converselywhen the boom has a lower lenght. Those two contributions stress that the qualitative and quantitative dimensions of collusive pricing can di¤er according to the state of the business cycle. Such an asymmetry in the intensity of the collusion is directly related to the mark-up cyclical behaviour and thus - as we will argue - to the size of government spending multipliers. Therefore, those results might provide an explanation for multipliers’asymmetries over the business 3 cycle (Canzoneri et al., 2011)2. More recently, a new strand of literature combines traditional general equilibrium macro models with an industrial organization approach, emphasizing the procyclicality of entry in determining mark-up countercyclicality, through the competition e¤ect (Ghironi and Melitz 2005, Jaimovich and Floetotto 2008, Etro and Colciago 2010) However, how empirically robust is the evidence about mark-up countercyclicality? Although a considerable number of contributions points towards countercyclicality,3empirical literature on mark-up cyclical behaviour is not unambiguous. Donowitz et al. (1986,1988) …nd evidence on procyclicality in the US; Chirinko and Fazzari (1994) use a dynamic factor model to estimate markups, …nding that they are procyclical in nine of the eleven 4-digit industries they analyze. Updating Bils (1987) analysis - in favor of countercyclicality - with more recent and richer data, Nekarda and Ramey (2010) …nd that all measures of markups are either procylical or acyclical. Hall (2009) provides a simple …rst-cut test for cyclicality by noting that the mark-up can be expressed as the ratio between the elasticity of output with respect to labor input @Y @L L Yand the share of labor compensation over nominal income W L P Y =s. In fact, since by the envelope theorem property, a cost-minimizing …rm equalizes the marginal cost of increasing output across all possible margins for varying production, we can express marginal cost as: MC =W @Y @L (1) As gross mark-up (= )is de…ned as the ratio between price index and marginal costs,and multiplying and dividing by Y L;then: = @Y @L L Y s(2) 2Rotemberg and Woodford (1992) develop the idea of mark-up countercyclicality in a dynamic general equilibrium setting, …nding that the model’s empirical performances are closer to actual postwar US data than the corresponding predictions of the perfectly competitive model. 3See Martins (1996) on OECD, Chevalier et al. (2003) on the US, Portier (1995) on France. 4 If the production process is approximated by a Cobb-Douglas Y=LK1; then the numerator of (2) is and can be considered relatively stable over time. Thus, the countercyclicality of the mark-up requires the procyclicality of labor share s: Figure 1 shows the dynamics of labor share and output in …ve major OECD economies from 1990 to 2009. 5 6 We can notice that labour share is far from showing an unambigous procyclical behaviour ; table 1 reports a simple correlation analysis showing that - with the exception of US - there seems to be a negative rather than positive correlation between labor share and the business cycle. Table 1: Correlation between detrendend GDP and labour share COUNTRY CORRELATION US 0.61 UK 0 FRANCE -0.12 GERMANY -0.52 ITALY -0.49 In this paper, we present a theoretical framework based on strategic interaction able to rationalize the existence of pro-cyclical pricing. Our benchmark model is the one put forward by Rotemberg and Saloner (1986). We indeed set out by o¤ering a brief summary of their analysis, reconstructing the countercyclical behavour of prices in a simple repeated duopoly game with homogeneous goods, in which demand is subject to random shocks a¤ecting the reservation price. In addition to their result, we show that increasing the probability of a positive shock brings about an increase in cartel stability, which could be large enough to more than counterbalance the e¤ect identi- …ed by Rotemberg and Saloner. Then, we review the established IO debate on the behaviour of …rms involved in an implicitly collusive price supergame under product di¤erentiation and perfect certainty, producing well de…ned procyclical conclusions. Our e¤ective contribution consists in bridging the two approaches, pursuing two distinct but related goals: (i) to characterise the maximum degree of collusion (i.e., the highest collusive price) that can be sustained in a stochastic environment, given time preferences and product di¤erentiation; and (ii) to check whether the counterciclity emerging under stochastic demand and perfect product substitutability can indeed be compatible with the seemingly opposite result produced by the traditional cartel theory belonging to IO, in which much emphasis is posed on product di¤erentiation but this is accompanied by perfect certainty. Our analysis indeed shows that, provided unilateral deviations grant monopoly power, Rotemberg 7 Now look at the deviation period, and suppose the cheated …rms remains on the market selling a positive output. The optimal deviation is the best reply to pM;i.e.: pDpM=a(2 s) 4(17) which is viable as long as the resulting sales volume of the …rm remaining loyal to the cartel is positive, which requires s2h0;p31:(18) The deviation pro…ts are D=a2(2 s)2 16 (1 s2):(19) If instead s2p31;1;then the cheated …rm is out of business and the deviator stands alone on the market place. In this range, the deviation price solves qch =a 1 + spM 1s2+spD 1s2= 0;(20) yielding pD0pM=a(2s1) 2s(21) delivering deviation pro…ts equal to D0pM=a2(2s1) 4s2:(22) Then one can easily verify that pDpM=pD0pMand DpM=D0pM(23) in correspondence of s=p31: We can now turn to the derivation of the critical thresholds of the discount factor , above which full collusion is sustainable in the two cases. For all s20;p31, (19) is the relevant deviation pro…t, and the critical threshold of the discount factor above which full collusion in prices is stable is  B=DpMM D(pM)BN =(2 s)2 8 (1 s) + s2(24) 14 while for all s2p31;1;i.e., in the region where (22) applies, it is 0 B=D0pMM D0(pM)BN =(2 s)2[s(1 + s)1] (2 s)2[s(1 + s)1] + s4(25) with 0 B= 1=2if s= 1: 2.2.2 Partial collusion Here we deal with the performance of a cartel whose members’time preferences are below the thresholds identi…ed above. The issue, in such a case, is to …nd the highest collusive price p2pN; pMsustainable over time, given . If all …rms set the collusive output p;the per-period pro…ts of each cartel member are C=(ap)p 1 + s:(26) Again, the optimal deviation against the cartel price will take two different forms, depending on the degree of substitutability s. If the latter is su¢ ciently low (i.e., product di¤erentiation between the two varieties is high enough), the deviator will adopt its best reply to cartel pricing solving (12) in which one has to plug pj=p;to get the optimal deviation price pD(p) = a(1 s) + sp 2:(27) This generates the following one-o¤ deviation pro…ts D(p) = [a(1 s) + sp]2 4 (1 s2);(28) provided that the cheated …rm is still active, i.e., it must be selling a positive quantity qch. This happens if qch =1 2app 2 (1 s)+2 (ap) + p 2 (1 + s)>0(29) which requires p2pN;a(1 s) [2 + s(6 s)] 2s2:(30) 15 In this price range, condition (11) yields the following highest collusive price p=a(1 s) [4 s(1 ) (1 s)] (2 s) [4 (1 s) + s2(1 )] (31) which is monotonically increasing in aand lower than pMfor all admissible values of sand . Now we turn our attention to the collusive price range wherein any unilateral deviation makes the cheating …rm a monopolist, driving all loyal cartel members out of business. This happens if expression (29) is non positive, i.e., for all p2a(1 s) [2 + s(6 s)] 2s2;a+c 2(32) with a+c 2a(1 s) [2 + s(6 s)] 2s2 for all s2hp31;1i:(33) Imposing qCh = 0 yields the deviation price pD0(p) = pa(1 s) s(34) which in turn delivers the deviation pro…ts D0pM=(ap) [pa(1 s)] 4s2:(35) It can be easily checked that pD0(p) = pD(p)in correspondence of p=a(1 s) [2 + s(6 s)] 2s2:(36) Plugging the above expression in the stability condition (11) and solving w.r.t. p;one obtains p0 = ah4(5 2s)s2(2 s)2(1 + s)spi 2 (2 s) [(1 ) (1 + s)s2](37) 16 in which  = 2(4 + s)s(2 s)s3+(2s1) s24 (1 s)(4 + s)s+(2 s)2: (38) One can verify that >0;by solving =0w.r.t. and checking that the resulting solutions 6=Rfor all s2p31;1. Therefore, has the same sign as the coe¢ cient of 2in (38), which is positive, as can be easily ascertained. Moreover, the denominator of p0 is positive for all sand in the unit interval. The next step consists in observing that a(1 s) [2 + s(6 s)] 2s2< p0 <a 2< p0 +(39) for all  < 0 B:Accordingly, we select p0 as the optimal collusive price in the admissible parameter range identi…ed by 2;and s2hp31;1i:(40) The partial derivative of p0 +w.r.t. ais: @p0  @a =(2 s) [(1 ) (2 + s)s2(2 )] sp 2 (2 s) [(1 ) (1 + s)s2](41) Therefore, relying on the fact that >0, we may evaluate the sign of the numerator of @p0 =@a by evaluating the sign of (2 s)2(1 ) (2 + s)s2(2 )2s4(42) which turns out to be positive for all 2h0;e :The latter result, together with Lemma 3, implies that @p0 =@a > 0in the entire admissible range (40). Hence, the foregoing discussion allows us to formulate: Lemma 2 In the Bertrand supergame under perfect certainty, any increase (resp., decreases) in consumers’reservation price increases (resp., decreases) the intensity of partial collusion, for any degree of product di¤erentiation and irrespective of whether unilaterial deviation from the collusive path grants the cheating …rm monopoly power or not. 17 3 Bridging two visions What we have reviewed so far boils down to the following two synthetic and seemingly antithetic messages: i] if goods are undi¤erentiated and the market is subject to stochastic shoks a¤ecting consumers’ reservation price, then …rms’ pricing behaviour exhibits a de…nite countercyclical pattern; ii] if goods are di¤erentiated and the demand level is deterministic, then optimal cartel prices are always monotonically related to the reservation price, heedless of …rms’time preferences. Our aim in this section is to develop anew a model in which product di¤erentiation and uncertainty operate together, so as to see whether the above conclusions may indeed be compatible with each other. As a …rst step, we will examine the case in which duopolistic competition survives unilateral deviations from cartel pricing. The second step will be to look at the opposite case where the defecting …rm attains a monopolistic position. On the supply side, the setup is the same as above. On the demand side, the demand function for …rm iwill be: qit =t 1 + spit 1s2+spjt 1s2;(43) with t2 fa; bg; a > b > 0and probabilities p(a) = mand p(b)=1m; respectively, with m2[0;1] : 3.1 Best reply deviation and the persistence of duopoly Here, the deviation price maximises Dand qch >0:We set out by taking a quick look at the stability condition for full collusion. Monopoly price in state tis pM t=t=2;delivering expected per-…rm cartel pro…ts EC=ma2+ (1 m)b2 4 (1 + s):(44) The deviation price and pro…ts in correspondence of the best demand state correspond to (17) and (19), respectively, and apply for all s20;p31. 18 The individual expected Bertrand-Nash pro…ts in each period of the punishment phase are given by EN=[ma2+ (1 m)b2] (1 s) (1 + s) (2 s)2:(45) As a result, collusive stability now requires a2(2 s)2 a24m(1 s)(2 s)2+b2(1 m)2(1 s)2;(46) with @ @a /(1 m) (1 s)>08m2[0;1) ; s2h0;p31(47) and @ @m / (1 s)<08s2h0;p31:(48) As for partial collusion, de…ne EC=m(ap(a)) p(a) + (1 m) (bp(b)) p(b) 1 + s(49) with p(b) = b=2, in such a way that the only unknown is the partially collusive price in the best state, p(a). That is, we assume …rms will charge the best collusive price in the worst state, and appropriately tune p(a)so as to satisfy the stability condition (11), which we are about to construct step by step. Accordingly, the best deviation against p(a)along the reaction function is pD(p(a)) = a(1 s) + sp(a) 2:(50) Then the expected payo¤ in each period of the punishment phase is (45). From the usual stability condition, we get the pair of solutions: p(a) =a(2 s) (1 s)  2(1 s)sp (2 s) [s2(1 ) + 4 (1 s)] (2 s)24m(1 s)(2 s)2 (51) where  = [2m + (2 s) (1 )] s2(1 ) + 4 (1 s)>0(52) 19 and  = a2m2s2(1 ) + 4 (1 s)2+(53) 4b2(2 s)2(1 ) (1 m)(2 s)24m(1 s)(2 s)2>0: It is then easily checked that lim m!1p(a) =a(1 s) [4 s(1 ) (1 s)] (2 s) [4 (1 s) + s2(1 )] =p(54) i.e., the same price as in (31). Finally, taking the partial derivative of p(a)  w.r.t. a; one can verify that @p  @a /a2m2s2(1 ) + 4 (1 s)(2 s)2(1 )+4m+ 4b2(2 s)4(1 ) (1 m) [2m + (2 s) (1 )]2>0(55) everywhere. Therefore, we can state Proposition 3 If deviation from the collusive path does not grant monopoly power, then the maximum collusive price sustainable under stochastic demand conditions is monotonically increasing in the level of the best demand state. 3.2 Defecting to monopoly The last step consists in investigating the case in which a unilateral deviation from the cartel price turns the deviator into a monopolist. Under full collusion, the only detail that has to be modi…ed is the deviation price in correspondence of the best state, which causes the cheated …rm’s output to drop to zero. This is (21), ensuring the deviation pro…ts (22) for all s2p31;1. The resulting stability condition is a2(2 s)2[(1 + s)s1] a2[s4(m+ 1) + 4 (4s1) s2(1 + 3s)] + b2(1 m)s40;(56) with @0=@a > 0and @0=@m < 0for all s2p31;1, so that the picture remains much the same as we already know it, along the frontier of industry pro…ts. 20 Now suppose …rms’time preferences fall short of (56). If so, they may activate the highest sustainable degree of partial collusion. In such a case, they set p(b) = b=2whenever demand is low, and solve the stability condition w.r.t. p(a);obtaining: p0 (a) = a(2 s) [(1 ) (2 s(2s1)) ms2]sp 2 (2 s) [(1 ) (1 + s(1 s)) ms2](57) in which  = a2s2(1 (1 ms))2+(58) 4 (1 ) ((1 ) (1 s)m (2s1))] b2s2(1 m)(1 ) (1 + s(1 s)) ms2 with (1 ) (2 s(2s1)) ms2>0;(59) s2(1 (1 ms))2+ 4 (1 ) ((1 ) (1 s)m (2s1)) >0(60) and s2(1 m)(1 ) (1 + s(1 s)) ms2<0(61) for all and min the unit interval and all s2p31;1:In the same parameter region, one also has that (i) >for all a2 b2>s2(1 m) [(1 ) (1 + s(1 s)) ms2] s2(1 (1 ms))2+ 4 (1 ) ((1 ) (1 s)m (2s1)) (62) with the threshold on the r.h.s. of (62) being always lower than one (hence, both p0 (a)and p0 +(a)are real), and (ii) p0 (a): To identify the correct solution, it su¢ ces to verify that lim m!1p0 (a) = ah4(5 2s)s2(2 s)2(1 + s)spi 2 (2 s) [(1 ) (1 + s)s2]=p0 :(63) The last step consist in di¤erentiating p0 (a)w.r.t. a; de…ning a=rb with r > 1and then solving @p0 (a)=@a = 0 w.r.t. r; getting6 r=s(2 s) [(1 ) (2 s(2s1)) ms2]p(1 m) [(1 ) (1 + s(1 s)) ms2] 2p(1 s)  =r (64) 6The second solution can be disregarded as it is always negative. 21 with  = (1 ) (1 + s(1 s)) ms2ms2(1 ) (2 s)2(1 + s) s2(1 (1 ms))2+ 4 (1 ) ((1 ) (1 s) + m (2s1)):(65) Now, it can be shown analytically that r2 R+;moreover, lim m!1r= 0 and (66) lim m!0r=s[2 (1 s2) + s]p (2 s)p(1 ) (1 s2)>1(67) for all  > (2 s)2(1 s2) s2+ 4 [1 s(1 + s(1 s)) (1 s) (1 s2)] =(68) which is decreasing and concave in s, with = 3 39 + 38p3=937 '0:336 in s=p31and = 0 in s= 1:Consequently, we have that @p0 (a) @a >0for all r > max f1; rg(69) and conversely between one and r, if indeed r > 1:In general, ris decreasing and concave in m(this can ve ascertained numerically), and will give rise to a picture like the one reported in Figure 2, in which "+" and "-" signs indicate the sign of @p0 (a)=@a: 22 Figure 2: Collusive pricing in the space (m; r). 6 - r m 1+ (0;0) 1 To sum up, above the upper envelope @p0 (a)=@a > 0; if r > 1for at least some acceptable parameter values, then in such a region @p0 (a)=@a < 0:7 An analogous exercise can be carried out in the space (s; a)as well as in (; a):This is done in Figures 3 and 4, respectively. In particular, Figure 3 shows that countercyclical pricing emerges in the region in which r2(1; r) and the two product varieties are su¢ ciently close substitutes. It is also worth observing that rasymptotically increases to in…nity as sapproaches 1 in the limit, in which case pricing is countercyclical irrespective of r; as in Rotemberg and Saloner’s (1986) original formulation. 7Recall that in state b, by assumption, …rms collude on the monopoly frontier. 23 