Tacit collusion under interest rate fluctuations

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1 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) RAND Journl of Economics Vol. 38, No. 2, Summer 2007 pp. 1 8 Tcit collusion under interest rte fluctutions Pedro Dl Bó Previous literture hs shown tht demnd fluctutions ffect the scope for tcit collusion. I study whether discount fctor fluctutions cn hve similr effects. I find tht collusion depends not only on the level of the discount fctor but lso, nd more surprisingly, on its voltility. Collusive prices nd profits increse with higher discount fctor level, but decrese with its voltility. These results hve importnt implictions for empiricl studies of collusive pricing, the role tht collusive pricing my ply in economic cycles nd the study of coopertion in repeted gmes. 1. Introduction It is well known tht oligopolies cn use the thret of future price wrs to sustin prices bove competitive levels if firms cre enough bout the future (Friedmn, 1971). The extent to which firms cre bout the future depends primrily on the interest rte if the firms objective is to mximize the present vlue of profits. The firms discount fctor my lso depend on other forces, such s the probbility tht the product my become obsolete nd the time needed for cheting to be detected. Given tht the interest rte nd other vribles tht ffect the discount fctor re constntly chnging, it is importnt to study tcit collusion under discount fctor fluctutions. Ichrcterize collusive prices nd profits when the discount fctor chnges over time nd show tht collusive prices nd profits increse with both present nd future levels of the discount fctor but decrese with its voltility. These results hve importnt implictions not only for the study of collusion but lso for repeted gme theory in generl. Oligopoly gmes re one exmple mong mny of n environment in which it is nturl to ssume tht the discount fctor chnges over time. Another exmple would be tht of prtnership, where the probbility tht the prtnership might end vries over time. Thus, the voltility of the discount fctor my be n importnt determinnt of coopertion for mny kinds of repeted gmes, not just oligopoly. The environments Istudy nd the specific results Ifind re s follows. Iconsider the cse in which the discount fctor, identicl for ll firms, is rndomly nd independently drwn every period. Ichrcterize the mximum symmetric tcit collusion prices nd profits tht cn be Brown University; pdlbo@brown.edu. Im grteful to Dvid Levine for invluble guidnce nd ides nd Hongbin Ci, John Riley nd Willim Zme for their comments, support nd insights. Ilso wnt to thnk the Editor, Jcques Crémer, two nonymous referees, Ann Aizer, Willim Brock, Germán Colom, Wlter Cont, Ernesto Dl Bó, Hrold Demsetz, Hugo Hopenhyn, Philip Leslie, Herkles Polemrchkis, José Wynne nd seminr prticipnts t severl universities nd conferences for very useful comments nd discussions. Copyright 2007, RAND. 1

2 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) 2 / THE RAND JOURNAL OF ECONOMICS supported in n environment in which firms re identicl nd compete repetedly on price. The three min results derived from this chrcteriztion re s follows. 1 First nd more interestingly, Ishow tht the higher the voltility of the discount fctor, the lower the collusive prices nd profits tht cn be supported in equilibrium. The reson for this is twofold. First, given tht the combintion of the incentive comptibility nd fesibility constrints results in concve collusive profit function (s function of the discount fctor), n increse in voltility leds to decrese in expected profits. Second, this decrese in expected profits reduces the size of future punishment nd hence results in decrese in equilibrium profits nd prices. Second, the higher the discount fctor in given period, the higher the collusive prices nd profits tht cn be supported in equilibrium in tht period. The intuition behind this is strightforwrd: the higher the discount fctor, the stronger the thret of future price wrs nd the higher prices nd profits cn be without firms deviting. Third, shift in the distribution function towrd higher discount fctors would result in n increse in the profits nd prices tht cn be supported in equilibrium for ech discount fctor. Agin, the intuition is strightforwrd. From the previous result, we know tht the higher the reliztion of the discount fctor, the higher collusive prices nd profits will be. Hence, shift in the distribution function towrd higher discount fctors would result in n increse in the expected vlue of collusive profits nd n increse in the thret of future punishment, llowing higher equilibrium prices nd profits. The rest of the rticle is orgnized s follows. In Section 2, I relte this rticle to the previous literture. In Section 3, I study the optiml tcit collusion solution nd provide comprtive sttics results. In Section 4, I conclude. 2. Relted literture To my knowledge, thereis only one rticle tht considers fluctuting discount fctor. This is Bye nd Jnsen (1996), which provides folk theorem results for repeted gmes with stochstic discount fctors. The well-known rticle by Rotemberg nd Sloner (1986) offers interesting results with respect to tcit collusion tht follow, s do the results in this rticle, from chnges in the reltive importnce of present nd future profits. In their rticle, however, those chnges re driven by chnges in demnd, not the discount fctor. This difference is not trivil nd leds to significntly different results. First, in this rticle, n increse in the discount fctor lwys hs nonnegtive effect on the equilibrium price, while in Rotemberg nd Sloner, n increse in demnd my result in either decrese or n increse in price (depending on whether the incentive comptibility restriction is binding or not). In ddition, the effect of n increse of demnd on prices my not be robust to ssuming quntity competition insted of price competition, s Rotemberg nd Sloner note, or to the existence of cpcity constrints, s Stiger nd Wolk (1992) note. The effect of n increse in the discount fctor is robust. Second, while in this rticle n increse in the voltility of the discount fctor lwys results in decrese in profits nd prices, in Rotemberg nd Sloner s model, n increse in the voltility of demnd is gin mbiguous. Third, under discount fctor fluctutions, the issue of seril positive correltion of the shocks is less importnt thn under demnd fluctutions. The fct tht high demnd tody mkes it difficult to support collusion, while high demnd in the future mkes it esy hs led number of uthors to study the consequences of demnd correltion on collusion (see Kndori, 1991; Hltiwnger nd Hrrington, 1991; Bgwell nd Stiger, 1997). In contrst, both high discount fctors tody nd in the future fcilitte collusion. Therefore, positive correltion per se does not ffect the positive effect of n increse of the discount fctor on collusive prices. Finlly, the literture on customer mrkets lso reltes oligopoly prices with the discount fctor (see Phelps nd Winter, 1970; Gottfries, 1991; Klemperer, 1995; Chevlier nd Schrfstein, 1996). In those models, n increse in the discount fctor increses the incentives to invest in new customers nd results in lower prices, contrry to the results of this rticle. 1 The sme results hold under quntity competition, see Dl Bó (2002).

3 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) 3. Model nd results DAL BÓ / 3 The model is s follows. Consider mrket with N identicl firms with constnt mrginl cost of c nd fcing demnd function D(p) for p R. Assume tht D(p) is bounded, continuous nd decresing in p nd tht there exists price p > c such tht D( p) = 0 for ny p > p. Firms compete repetedly on price nd, in ech period, demnd is divided eqully mong those firms chrging the lowest price. Firms cre only bout profits nd re risk neutrl nd, hence, their objective is to mximize the discounted strem of profits. The distinctive feture of this model is tht the discount fctor, t, which discounts ernings from t +1tot, is continuous, independent nd identiclly distributed rndom vrible between nd b, with p.d.f. f ( t ) nd c.d.f. F( t ). The timing of the gme in given period t is s follows: the firms observe the reliztion of the discount fctor, t, then they choose the price for tht period nd finlly they observe ll the chosen prices, quntities nd pyoffs. All chrcteristics of the environment re common knowledge. Given tht firms cnnot commit to chrge given price, or sign contrcts mong themselves or with third prties regrding prices, ny equilibrium of the model must be subgme perfect equilibrium of the infinitely repeted oligopoly gme. Irestrict my ttention to equilibri in which ll the firms chrge the sme price, p. In this symmetric cse, I cn write the profits of ech firm s π(p) =(p c)d(p)/n. Given the ssumptions regrding the demnd function, it is strightforwrd to show tht π(p) is continuous nd ttins mximum. Denote s π m the mximum (monopoly or perfect collusion) profit per firm. Assume tht there is unique price, p m, tht results in profits π m ( sufficient condition is for the demnd function not to be too convex reltive to its slope: d 2 D(p)/dp 2 < [2/(p c)](dd(p)/dp) for every p). Optiml tcit collusion with rndom discount fctor. In this section, I chrcterize the optiml symmetric tcit collusion solution. First, Ichrcterize the profits tht cn be supported by subgme perfect equilibri. Second, Idefine nd prove existence nd uniqueness of the optiml tcit collusion solution. Finlly, Ichrcterize this solution. Istrt by clculting the expected vlue of profit per firm if firms gree to set prices to chieve profits of π() when the discount fctor is. Using the recursiveness of the problem, the present vlue t t of strem of profits to firm cn be written s V ( t )=π( t )+ t V ( t+1 ) f ( t+1 )d t+1, (1) where π( t ) denotes the profits tht the firms receive t time t if the discount fctor is t. Integrting both sides of eqution (1) nd rerrnging, we hve 1 V ( t ) f ( t )d t = 1 π( t ) f ( t )d t, where is the expected vlue of t. Plugging this into (1), the present vlue of profits cn be written s V ( t )=π( t )+ t π( t+1 ) f ( t+1 )d t+1. (2) 1 The following lemm chrcterizes the profit functions tht cn be supported by subgme perfect equilibri. Lemm 1. A profit function π() cn be supported by subgme perfect equilibrium if nd only if π() π( ) f ( )d for ll (3) (1 )

4 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) 4 / THE RAND JOURNAL OF ECONOMICS nd b } π() min π( ) f ( )d,π m for ll. (4) (N 1)(1 ) Proof. Firms cn secure zero profit in every period by setting p = c; this implies tht, for ny, V () is nonnegtive, which proves (3). By just undercutting the collusive price, firm cn increse its profits in the current period by (N 1)π(); in ny future period, its profits will be equl to zero nd therefore we must hve (N 1)π() (1 ) π( ) f ( )d for ll. (5) In ddition, the profits per firm cnnot be lrger thn the profits under monopoly. Together with (5), this proves (4). We hve proven the only if prt of the lemm. To prove the if prt, ssume tht function π() stisfies (3) nd (4) nd let φ(π) be the smllest price tht yields profit equl to π, so tht ny price p <φ(π) yields profits smller thn π. Then, π() cn be supported by the following strtegy: if there hs been no devition, choose p() =φ(π()); otherwise, choose p = c. Q.E.D. Now Iturn my ttention to optiml tcit collusion profit functions defined s follows. Definition 1. An optiml symmetric tcit collusion profit function π () mximizes the expected present vlue of profits V () for every mong the subgme perfect equilibrium profit functions. Proposition 1. There exists unique optiml tcit collusion profit function, π (). It stisfies π () = min (N 1)(1 ) π ( ) f ( )d,π m }. (6) Proof. For ny, the set of profits tht cn be supported by subgme perfect equilibrium is bounded bove by π m. It therefore hs lest upper bound, which we cll π (). It is cler tht π stisfies (3), s replcing ny profit function tht is supported by subgme perfect equilibrium by π increses the left-hnd side of (3) nd decreses its right-hnd side. To show tht π stisfies (4), note tht, for ny nd ny ε>0, there exists supported π such tht π () π()+ε min (N 1) ( 1 ) min (N 1) ( 1 ) π( ) f ( )d,π m } + ε (by (4 )) π ( ) f ( )d,π m } + ε, (s π( ) π ( ) for ll ). We now must prove tht π stisfies (6). Assume tht this is not true nd tht, for some nd some η>0, we hve π ( ) = min (N 1)(1 ) π ( ) f ( )d,π m } η. Then the function equl to π + η for nd to π everywhere else stisfies (3) nd (4) nd hence cn be supported by subgme perfect equilibrium, which contrdict the definition of π. Q.E.D.

5 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) DAL BÓ / 5 Define the optiml tcit collusion price function p () s the equilibrium prices tht result in π (). Proposition 2. The optiml symmetric tcit collusion price function, p (), exists nd is unique. Proof. By Lemm 1, π () [0,π m ] for every.ifπ () =π m, then p () =p m.ifπ () <π m, then the incentive-comptibility constrint (5) is binding nd the price must be equl to φ(π ()) for firms not to devite, where φ is defined s in the proof of Lemm 1. Q.E.D. Note tht it is strightforwrd to prove tht the function φ is incresing nd, therefore, equilibrium profits nd prices re positively relted. The following proposition chrcterizes the optiml symmetric tcit collusion profit function. Proposition 3. The optiml symmetric tcit collusion profit function π () depends on f () nd N in the following wy: (i) if 1 [/(N 1)], π () =π m ; (ii) if (N 1)/N <1 [/(N 1)], π () =π m for nd π () =(/ )π m for <, for number (, b] tht solves the following eqution: =(N 1)(1 )+ F()d; (7) (iii) if <(N 1)/N, π () =0. Proof. Tking expecttions over both sides of eqution (6) nd denoting the expected vlue of profits s π, eqution (6) implies tht π [/(N 1)(1 )]π. Ifπ>0, this is possible only if (N 1)/N.If<(N 1)/N, then π () =0. If 1 [/(N 1)], π m [/(N 1)(1 )]π m [/(N 1)(1 )]π m for ll nd, by eqution (6), perfect collusion, π () =π m, cn be supported for every discount fctor. Finlly, Ishow tht, for (N 1)/N <1 /(N 1), there is unique solution to eqution (6) with positive expected profits. In this solution, the two terms inside the brckets in eqution (6) re binding for different rnges of. Given tht the first term is incresing in, this term is binding for <, the second term is binding for > nd both terms re equl nd binding for =, where [, b]. This implies tht π stisfies By (6), we lso hve π = π = π m (N 1)(1 ). (N 1)(1 ) π f ()d +(1 F( ))π m. From these two equtions (nd integrting by prts), we obtin eqution (7). The difference between the right-hnd side nd the left-hnd side of eqution (7) is continuous, decresing function of, which is strictly positive for = nd negtive for =, which proves the result. Q.E.D. Two chrcteristics of the optiml tcit collusion solution cn be esily derived from Proposition 3. First, collusive profits nd prices re decresing in the number of firms. In fct, if the number of firms is lrge enough, the optiml tcit collusion solution coincides with Bertrnd s outcome. Second, the chrcteriztion of optiml tcit collusion under discount fctor fluctutions includes the cse of fixed discount fctor. For the fixed discount fctor cse, = b, Proposition 3 coincides with the textbook solution: perfect collusion if (N 1)/N nd no collusion otherwise.

6 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) 6 / THE RAND JOURNAL OF ECONOMICS Finlly, from Proposition 3, it is strightforwrd to show tht n increse in the discount fctor results in n increse in equilibrium profits nd prices. Corollry 1. Optiml collusive period profits π () nd prices p () re incresing in. The effects of chnges in f(). In this section, I study how chnges in the distribution of the discount fctor ffect collusive profits nd prices. Istudy, s first step, how chnges in the distribution function modify the rnge of perfect collusion under cse 2 of Proposition 3. For cumultive distribution function F, define F s the expected discount fctor nd F s the limit to perfect collusion under cse 2 of Proposition 3. Lemm 2. Consider two cumultive distribution functions, F nd G. If(N 1)/N i < 1 [/(N 1)], i = F, G, nd F second-order stochsticlly domintes 2 G, then F G. Proof. By second-order dominnce, F()d G()d for ll [, b] nd, furthermore, F G ; hence, by (7), F F =(N 1)(1 F )+ F (N 1)(1 G )+ F = G + G()d. G F()d G()d If we hd F > G, becuse G() < 1 for <b, this would imply F < G +( F G )= F, contrdiction. Q.E.D. Denote πf (), π F nd p F () s the optiml tcit collusion profit, its expected vlue nd optiml collusion prices under F, respectively. Proposition 4. Consider two cumultive distribution functions, F nd G. IfF second-order stochsticlly domintes G, then πf () π G () nd p F () p G () for every. In ddition, π F π G. Proof. By second-order stochstic dominnce, F G. It follows tht, if the solution under F belongs to cse 1 from Proposition 3, the solution under G cn belong to ny of the three cses; if the solution under F belongs to cse 2, the solution under G cn belong to cse 2 or 3; nd if F belongs to cse 3, the solution under G lso belongs to cse 3. If the solutions under both F nd G belong to cse 2, then F G, by Lemm 2. Thus, πf () =(/ F )π m πg () =(/ G )π m if F ; πf () =π m >πg () =(/ G )π m if F << G ; nd πf () =π G () =π m if G. It follows tht πf () π G (). For the other combintions of solution cses, the result is strightforwrd. The result with respect to prices follows directly from the positive reltionship between profits nd prices. Note tht πf () is incresing nd concve; hence, by second-order stochstic dominnce nd πf () π G () for every, we hve tht π F π F ()g()d π G. Q.E.D. The intuition of this result becomes cler if we consider two prticulr cses of secondorder stochstic dominnce: when F first-order stochsticlly domintes 3 G nd when G is men-preserving spred of F. 2 For two cumultive distribution functions F() nd G(), F second-order stochstic domintes G if for ny r, r b, r F()d r G()d, nd the inequlity is strict in some rnge. Then, F G nd u() f ()d u()g()d, for ny incresing concve twice-piecewise-differentible function u(). See Hirshleifer nd Riley (1992). 3 For two cumultive distribution functions F() nd G(), F first-order stochsticlly domintes G if, for ll r, r b, F(r) G(r), nd the inequlity is strict in some rnge. Then F second-order stochsticlly domintes G

7 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) DAL BÓ / 7 From Corollry 1, we know tht, given distribution of the discount fctor, sy G, equilibrium prices nd profits re incresing in the reliztion of the discount fctor. Then shift in the distribution function to higher vlues (which yields cumultive distribution function F tht first-order stochsticlly domintes G), would result in n increse in expected profits. This, in turn, increses the thret of future punishments nd increses equilibrium prices nd profits. Corollry 2. IfFfirst-order stochsticlly domintes G, then πf () π G () nd p F () p G () for every. In ddition, π F π G. From Proposition 3, it follows tht given distribution function, sy F, the optiml tcit collusion profit function is concve in the discount fctor. Therefore, men-preserving spred (which yields G) would result in reduction in expected profits. This, in turn, reduces the thret of future punishment nd results in lower equilibrium prices nd profits. Corollry 3. IfGis men-preserving spred of F, then πf () π G () nd p F () p G () for every. In ddition, π F π G. Therefore, the voltility of the discount fctor is inversely relted to the firms profits. The combintion of the incentive-comptibility constrint with the fesibility constrint yields profit function tht is concve in the discount fctor even when firms re risk neutrl. Hence, n increse in voltility reduces expected profits, reducing the thret of future punishment nd lowering equilibrium prices nd profits. Note tht this result does not depend on the firms not hving ccess to insurnce ginst discount-fctor fluctutions. Even if they could buy cturilly fir insurnce, n increse in the voltility of the discount fctor would reduce the preinsurnce expecttion of profits nd, hence, the fixed mount tht firm could ern with insurnce. 4. Conclusions In repeted oligopoly, I chrcterized the optiml symmetric collusion under discount fctor fluctutions nd found tht collusive prices nd profits increse with both present nd future discount fctor levels nd decrese with discount fctor voltility. These results show the importnce of discount fctor levels in repeted gmes nd introduce new element to the literture: the voltility of the discount fctor. This work hs severl importnt implictions for future study. While most of the existing empiricl literture on collusive pricing hs lrgely ignored the role of the interest rte (see, for exmple, Porter, 1983; Rotemberg nd Sloner, 1986; Domowitz, Hubbrd, nd Petersen, 1986; Slde, 1987; Ellison, 1994; Borenstein nd Sheprd, 1996), this rticle suggests tht both the level nd the voltility of the interest rte re importnt determinnts of collusive pricing nd should be considered in future empiricl work. This rticle lso hs implictions for the study of ggregte fluctutions. 4 Ishow tht ny chnge in policy, preferences or technology my hve n impct on the ggregte level of ctivity through chnges in collusive behvior, not only by ffecting the rel interest rte but lso by ffecting its voltility. It is importnt to note tht the results of this rticle my not necessrily hold under generl stochstic processes of the discount fctor see Dl Bó (2002) for n exmple. Determining more generl conditions under which the results presented here hold remins for future work. References Bgwell, K. nd Stiger, R.W. Collusion Over the Business Cycle. RANDJournl of Economics, Vol. 28 (1997), pp nd u() f ()d u()g()d, for ny incresing piecewise differentil function u(). See Hirshleifer nd Riley (1992). 4 See Rotemberg nd Sloner (1986) nd Rotemberg nd Woodford (1992) for the role of tcit collusion under demnd fluctution in ggregte fluctutions.

8 mss # Dl Bó; rt. # 14; RAND Journl of Economics vol. 38(2) 8 / THE RAND JOURNAL OF ECONOMICS Bye, M.R. nd Jnsen, D.W. Repeted Gmes with Stochstic Discounting. Economic, Vol. 63 (1996), pp Borenstein, S. nd Sheprd, A. Dynmic Pricing in Retil Gsoline Mrkets. RANDJournl of Economics, Vol. 27 (1996), pp Chevlier, J.A. nd Schrfstein, D.S. Cpitl-Mrket Imperfections nd Countercyclicl Mrkups: Theory nd Evidence. Americn Economic Review, Vol. 86 (1996), pp Dl Bó, P. Tcit Collusion Under Interest Rte Fluctutions. Mnuscript, Brown University, Domowitz, I., Hubbrd, G.R., nd Petersen, B.C. Business Cycles nd the Reltionship Between Concentrtion nd Price Cost Mrgins. RANDJournl of Economics, Vol. 17 (1986), pp Ellison, G. Theories of Crtel Stbility nd the Joint Executive Committee. RANDJournl of Economics, Vol. 25 (1994), pp Friedmn, J.W. A Non-Coopertive Equilibrium for Supergmes. Review of Economic Studies, Vol. 38 (1971), pp Gottfries, N. Customer Mrkets, Credit Mrket Imperfections nd Rel Price Rigidity. Economic, Vol. 58 (1991), pp Hltiwnger, J. nd Hrrington, J.E. The Impct of Cyclicl Demnd Movements on Collusive Behvior. RAND Journl of Economics, Vol. 22 (1991), pp Hirshleifer, J. nd Riley, J.G. The Anlytics of Uncertinty nd Informtion. Cmbridge University Press, Kndori, M. Correlted Demnd Shocks nd Price Wrs During Booms. Review of Economic Studies, Vol. 58 (1991), pp Klemperer, P. Competition When Consumers Hve Switching Costs: An Overview with Applictions to Industril Orgniztion, Mcroeconomics, nd Interntionl Trde. Review of Economic Studies, Vol. 62 (1995), pp Phelps, E. nd Winter, S. Optiml Price Policy Under Atomistic Competition. In E. Phelps, ed., Microeconomic Foundtions of Employment nd Infltion Theory. New York: W.W. Norton, Porter, R.H. A Study of Crtel Stbility: the Joint Executive Committee, Bell Journl of Economics, Vol. 14 (1983), pp Rotemberg, J.J. nd Sloner, G. A Supergme-Theoretic Model of Price Wrs During Booms. Americn Economic Review, Vol. 76 (1986), pp nd Woodford, M. Oligopolistic Pricing nd the Effects of Aggregte Demnd on Economic Activity. Journl of Politicl Economy, Vol. 100 (1992), pp Slde, M.E. Interfirm Rivlry in Repeted Gme: An Empiricl Test of Tcit Collusion. The Journl of Industril Economics, Vol. 35 (1987), pp Stiger, R.W. nd Wolk, F.A. Collusive Pricing with Cpcity Constrints in the Presence of Demnd Uncertinty. RANDJournl of Economics, Vol. 23 (1992), pp