Liquidation risks in the Rotemberg-Saloner implicit collusion model

Transcription

1 Econoics etters 6 (1999) iquidation risks in the Roteberg-Saloner iplicit collusion odel a Fabio C. Bagliano *, Alberto Dalazzo a, b Dipartiento di Scienze Econoiche e Finanziarie G.rato, Universita` di Torino, Corso Unione Sovietica 18bis, Torino, Italy b Dipartiento di Econoia olitica, Universita` di Siena, iazza San Francesco, 7, Siena, Italy Received 18 January 1998; accepted 17 Septeber 1998 Abstract We extend the Roteberg-Saloner s (1986) [Roteberg, J., Saloner, G., A supergae-theoretic odel of price wars during boos. Aerican Econoic Review 76: ] iplicit collusion fraework to the consideration of capital arket iperfections, captured by a non-zero probability of liquidation of firs in recessionary periods. We show that the Roteberg-Saloner result of countercyclical arkups is quite robust to the extension and, oreover, liquidation risks ay even strengthen the degree of arkup countercyclicality Elsevier Science S.A. All rights reserved. Keywords: Iplicit collusion; Countercyclical arkups; Capital arket iperfections JE classification: E30; 13; Introduction In a well-known paper, Roteberg and Saloner (1986) showed that if the rate at which oligopolistic firs discount future profits is sufficiently high, iplicit collusion over onopoly prices cannot be sustained during high-deand periods. Thus, boos ay generate price wars leading to countercyclical arkups (see also Bagwell and Staiger, 1997, and the literature quoted therein). More recently, Chevalier and Scharfstein (1996) (henceforth CS) have offered an alternative explanation of countercyclical arkups, based on capital arket iperfections. When the probability of surviving a recession is low, a financially-constrained fir will have an incentive to raise prices. CS (1996, p. 705) also conclude that when capital arket iperfections are relevant, the Roteberg-Saloner iplicit collusion odel cannot generate countercyclical arkups. We believe, however, that the iplications of liquidation risk within the Roteberg-Saloner iplicit collusion odel need to be investigated. Although we do not odel debt explicitly as in CS (1996) we capture the potential probles associated with the existence of financial constraints by assuing that oligopolistic firs survive downturns with positive probability: e.g. liquidation is stochastically enforced during recessions. For exaple, in Bolton and Scharfstein s (Bolton and Scharfstein, 1990, p. 101) agency odel with copetitive lenders, a fir will be re-financed with *Corresponding author. Tel.: ; fax ; e-ail: bagliano@econ.unito.it / 99/ $ see front atter 1999 Elsevier Science S.A. All rights reserved. II: S (98)0010-9

2 70 F.C. Bagliano, A. Dalazzo / Econoics etters 6 (1999) certainty in good states, but it will be able to borrow with a probability between zero and one in bad states. We show that the odified iplicit collusion odel proposed here can still generate countercyclical arkups for non-negligible ranges of the relevant paraeters. Furtherore, when arkups are countercyclical, the introduction of capital arket iperfections increases the degree of countercyclicality relative to the benchark Roteberg-Saloner odel, in which firs always survive downturns. The next section presents the odel and gives the ain results.. The odel We consider a siple extension of the basic Roteberg-Saloner (1986) setup, as presented in Tirole (1988), where two firs, producing an hoogeneous good, copete in a arket with stochastic deand. In every period, the deand for each fir can be either high or low, with equal probability. Each fir chooses prices so as to axiize the discounted value of profits over the entire future infinite horizon, earning profits ( ) if a good (bad) realization of deand occurs. In what follows, we solve for a pair of prices h p, pj such that: (i) both firs set the sae price ps when the state of deand is s, (ii) the pair h p, p j is sustainable as an equilibriu (i.e., deviating fro p in s state s is not privately optial), and (iii) the expected present discounted value of each fir s profit, calculated for h p, pj, is not doinated by any other pair of prices which satisfy (i) and (ii) (i.e., in case there are other pairs of prices sustainable as equilibria, both firs prefer the pair h p, pj considered; (see Tirole, 1988, p. 48)). In CS, liquidity constraints due to capital arket iperfections lead to liquidation when low deand states occur: thus, a liquidity-constrained fir never survives a recession. In the specific agency odel used by CS a recession iplies that the entrepreneur does not have enough cash to ake debt repayents and avoid liquidation. In the Roteberg-Saloner fraework, the assuption that recessions iply fir s liquidation rules out countercyclical arkups. Nevertheless, although recessions are likely to exacerbate financial difficulties, firs ay soeties avoid liquidation by raising fresh external funds. In this perspective, we assue that when a bad realization of deand occurs, each fir has a probability 0 # r # 1 of surviving to the next period and, conversely, a probability 1 r of being liquidated and cease operations. By doing so, we iick the polar cases of Roteberg-Saloner (r 5 1) and CS (1996), who assue that firs never survive recessions (r 5 0). Therefore, for a fir in period 0, the probability of being operative in period t is (1 1 r/) if in t1 1 period 0 deand is high, and r(1 1 r/ ) if deand is currently low. Given the above probabilities, we have the following expressions for the profit strea, discounted by a factor 0 # d # 1, expected at tie 0: ` 1 1 r t1 1 d V 5 1O d ts]]d S]]]D 5 1]]]] s 1 d (1) d(1 1 r) t51 t1 1 Suppose, for exaple, that at t50 the state of deand is good. The fir will survive to period t51 with probability 1. 1 Either a bad or a good state can then occur at t51, each with probability ]. In the forer case, the fir will survive to t5 with probability r; in the latter case, the fir will survive with certainty. ence the probability of being around at t5, conditional to a good state realized at t50. is equal to (11r/).

3 when deand is currently high, and F.C. Bagliano, A. Dalazzo / Econoics etters 6 (1999) ` t1 t 1 1 r 1 dr V5 1O drs]]d S]]]D 5 1]]]] s1 d () d(1 1 r) t51 when deand is currently low. If firs adopt a fully-collusive behaviour, prices are set at the onopoly level corresponding to each state of deand, p and p, yielding profits and in the good and bad state respectively. For the collusive outcoe to be sustainable, the future losses fro deviating fro onopoly pricing ust be larger than the (current) gains accruing to the deviating fir. Suppose that the rival firs adopt a trigger-strategy behaviour such that the deviation fro collusive (onopoly) pricing in one period deterines the reversion to the copetitive (zero-profit) pricing in all future periods ( axial-punishent principle ). Thus the gains fro deviation aount to either or, whereas the losses are given by the second ter in the right-hand side of either Eq. (1) or Eq. (). Therefore, for collusion to be sustainable in periods of high current deand we ust have: d # ]]]] s 1 d d(1 1 r) yielding the following condition on the discount factor d : (3) d $ d ;]]]] K 1 ( 1 r) (4) where K ;( / ) is the ratio between the level of onopoly profits in the low and high deand states, proxying for the aplitude of cyclical fluctuations (0#K #1). In periods of low deand collusion is sustainable if: rd #]]]] s 1 d d(1 1 r) iplying the following condition on d : (5) K d $ d ;]]]]] (1 1 r)k 1 r (6) Fro Eqs. (3) and (4) we see that a lower probability of avoiding liquidation in periods of low deand raises the critical values d and d necessary to sustain collusion: in both cases the future loss to the deviating fir is reduced by a lower r. Thus, a higher discount factor d would be needed to copensate for the resulting greater incentive to deviate. The case for r 51 (certain survival in low deand states) yields the original Roteberg-Saloner result: for d,d,d collusion is sustainable only in low-deand states and arkups display countercyclical behaviour. As shown in Fig. 1(a) the range of values for the discount factor yielding countercyclical arkups (the shaded area in the figure) is wider the lower is K : when the aplitude of cyclical fluctuations is large (K tends to 0), current profits are high in favourable states, yielding a greater incentive to deviate, whereas profits are low in bad states, aking collusion ore likely. Indeed, when the fir incurs no liquidation risk in either state, it becoes easier to enforce onopoly

4 7 F.C. Bagliano, A. Dalazzo / Econoics etters 6 (1999) Fig. 1. Cobinations of the discount rate d and K (ratio of onopoly profits in the low and high deand states) yielding counter- and procyclical arkups for different values of the survival probability r. prices in recessions, when the gain fro deviation is relatively low. Note also that, if there are no 1 cyclical fluctuations (K 51), collusion is sustainable in both high and low eand states if d $ ], asin Friedan (1971). In the above setting, the assuptions in CS lead to the terination of the firs with certainty if a low deand state occurs, corresponding to r 50. In this case, d is always greater than d, which rules out the possibility of countercyclical arkups. ence, under the extree assuption of certain liquidation in downturns, the clai put forward by CS (1996, p. 705) is correct: the Roteberg- Saloner setup is unable to rationalize countercyclical arkups; instead, it ay even generate procyclical arkups if d,d,d, as shown in Fig. 1(d). In the less extree case of a positive survival probability for firs in low deand states (0,r,1), arkups ay display counter- or procyclicality according to the agnitudes of r (capturing the relevance of financial constraints) and K (the aplitude of fluctuations). The following proposition suarizes the ain results: roposition 1. With r [(0,1), arkups are countercyclical whenever d,d,d procyclical if d,d,d. holds, and The proof goes as follows. Fro Eqs. (3) and (4), the direction of the inequality between d and d depends, for any given r, on the value of K. Denoting by K* the (adissible) value of K which solves the equation d5d, it turns out that K*5r. Then, if K,K*, d,d : as in the Roteberg- Saloner s original odel, if d,d,d collusion at onopoly prices is sustained only in low deand

5 F.C. Bagliano, A. Dalazzo / Econoics etters 6 (1999) states, whereas in high deand states the price is p *, lower than the corresponding onopoly level p, so that the following condition is satisfied: d 5]]]]]] ( p ) ]]] 1 ( 1 r) *( p *) (7) On the other hand, if K.K*, d.d : thus, when d,d,d, collusion occurs only in high deand states ( p5p ), whereas in low deand states the price is p *, lower than the corresponding onopoly level p, such that: * ( p * ) ]]] ( p ) d 5]]]]]]] * ( p * ) (1 1 r) ]]] 1 r ( p ) (8) Fig. 1(b) and 1(c) illustrate exaples with r,1, showing the ranges of d iplying counter- and procyclicality of arkups. (Recall also that, if d.(d,d ), firs always collude on onopoly prices ( p, p ). On the contrary, when d,(d,d ), firs collude in neither state of deand.) The relevant iplication of roposition 1 is that, in contrast with the arguent put forward by CS, the introduction of a survival probability in the Roteberg-Saloner setup does not destroy in general the possibility that arkups reain countercyclical. As the graph shows, the possibility of countercyclical arkups is crucially related to the agnitude of the survival probability r, easuring the rate at which oligopolistic firs escape liquidation during recessions. One ay also argue that, since oligopolistic firs are in general relatively big, their liquidation risk is rather sall ( deeppockets ): thus, the Roteberg-Saloner explanation of arkup countercyclicality ay look as a plausible alternative to the liquidity-constraint explanation of CS. Interestingly, survival probabilities ay play a specific role also in the iplicit-collusion odel. In fact, it can be shown that when countercyclical behaviour occurs, the degree of arkup countercyclicality is even agnified with respect to the standard Roteberg-Saloner case, holding for r 51. The following proposition holds: roposition. Consider the case with countercyclical arkups (d,d,d ). It holds that: (i) The price set in low deand states is equal to p * (the onopoly price), independently of r. (ii) Denoting as p* the price set in a high deand state when r,1, and as p* 9 the price set when r 51 (the standard Roteberg-Saloner case), it follows that p *,p* 9. The proof of part (i) of roposition is straightforward, since p axiises current profits in low deand states (recall that the current period profit function is independent of r). As for part (ii), the arguent goes as follows. In a high deand state, expression Eq. (7) ust hold: thus, the lower r, the higher the ratio ( / * ). As a consequence, given,ar saller than one iplies a lower * and, hence, a p* lower than p* 9 (the positive relation between p and is ensured by the fact that prices higher than the onopoly level, p, would always ake undercutting profitable: see Tirole (1988, note 17, p. 49)).

6 74 F.C. Bagliano, A. Dalazzo / Econoics etters 6 (1999) The rationale for this result can be found by recalling that, according to Eq. (3), uncertain survival decreases the potential future loss for the deviating fir, enhancing the incentive to deviate in high deand states. Therefore, prices ust be relatively lower in equilibriu. roposition has an epirically relevant iplication. Since liquidation risks can offer a specific contribution to the extent of arkup countercyclicality also in the Roteberg-Saloner odel, it becoes quite difficult to sort out the iplicit-collusion approach fro the CS approach on the basis of regressions that ainly test the significance of liquidity-constraint variables on pricing behavior. Acknowledgeents Financial support fro Consiglio Nazionale delle Ricerche is gratefully acknowledged. References Bagwell, K., Staiger, R., Collusion over the business cycle. Rand Journal of Econoics 8, Bolton,., Scharfstein, D.S., A theory of predation based on agency probles in financial contracting. Aerican Econoic Review 80, Chevalier, J., Scharfstein, D.S., Capital-arket iperfections and countercyclical arkups: theory and evidence. Aerican Econoic Review 86, Friedan, J., A non-cooperative equilibriu for supergaes. Review of Econoic Studies 38, 1 1. Roteberg, J., Saloner, G., A supergae-theoretic odel of price wars during boos. Aerican Econoic Review 76, Tirole, J., The Theory of Industrial Organization. Mit ress, Cabridge.