Documents de Travail du Centre d Economie de la Sorbonne


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1 Documents de Travai du Centre d Economie de a Sorbonne Rationaizabiity and Efficiency in an Asymmetric Cournot Oigopoy Gabrie DESGRANGES, Stéphane GAUTHIER Maison des Sciences Économiques, bouevard de L'Hôpita, Paris Cedex 13 ISSN : X
2 Documents de Travai du Centre d'economie de a Sorbonne Rationaizabiity and Efficiency in an Asymmetric Cournot Oigopoy Gabrie Desgranges and Stéphane Gauthier March 26, 2014 Abstract We study rationaizabe soutions in a inear asymmetric Cournot oigopoy. We show that symmetry across firms favors mutipicity of rationaizabe soutions: A merger (impying a greater asymmetry across firms) makes outofequiibrium behavior ess ikey and shoud dampen coordination voatiity. The market structure maximizing consumers surpus at a rationaizabe soution is not aways the competitive one: This may be a symmetric oigopoy with few firms. An empirica iustration to the airines industry shows that a reaocation of 1% of market share from a sma carrier to a arger one yieds a 1.3% decrease in voatiity, measured by the within carrier standard error of the number of passengers. JEL codes: D43, D84, L40. Keywords: competition poicy, Cournot oigopoy, dominance sovabiity, efficiency, rationaizabiity, stabiity, airine industry. We thank Jess Benhabib, Gaetano Gabao, Roger Guesnerie, Laurent Linnemer, Régis Renaut, Jean Phiippe Tropéano and Thibaut Vergé for usefu discussions. Aexandra Beova, Phiippe Gagnepain, Nicoas Jacquemet, and Antoine Terraco provided us generous guidance in the empirica iustration to the airine industry. The usua discaimers appy. University of CergyPontoise, France. University of Paris 1 and Paris Schoo of Economics, France. 1
3 Documents de Travai du Centre d'economie de a Sorbonne Introduction The issue of rationaizabiity in a Cournot oigopoy with identica firms has received much attention in the iterature (see, e.g., Bernheim, 1984; Basu, 1992; or Börgers and Janssen, 1995). Our paper considers the rationaizabe outcomes in a inear asymmetric Cournot setup. We interpret mutipicity of rationaizabe outcomes as favoring coordination faiures. The main resuts of our paper show that identica firms are detrimenta to the uniqueness of the rationaizabe outcome (dominance sovabiity of the equiibrium): Firm s homogeneity actuay favors efficiency in equiibrium, but it makes mutipicity of rationaizabe outcomes (and thus coordination faiures) more ikey. We consider the rationaizabe outcomes in a inear Cournot oigopoy where the overa quantity of productive assets can be controed (Perry and Porter, 1985). This aows us to separate the own effect of a change in the market size (measured by the tota number of avaiabe productive assets) from the effect of a greater asset inequaity given the overa quantity of assets. This setup naturay arises in competition poicy when a competition reguator has to choose how production faciities shoud be aocated across competitors. Exampes incude airines routes and airports (Borenstein, 1990), nucear reactors in the power industry (Davis and Wofram, 2011), hospitas in the heath insurance market (Town et a., 2006) or water sources (Compte et a., 2002). In these exampes, firms possiby differ according to the number of productive assets under their contro, but the overa quantity of assets is given. A monopoyike situation, where one firm hods most of the assets, is detrimenta to efficiency in the equiibrium: The aggregate production is ower than in the equiibrium where faciities are equay shared across severa competitors. A unique rationaizabe outcome is obtained when the spectra radius of the bestresponse map mapping a vector of individua productions to the vector of bestresponses is ess than one (Bernheim, 1984; Mouin, 1984). We show that this spectra radius increases when the tota number of avaiabe productive assets increases: A arge market size favors mutipicity of rationaizabe outcomes. Our main resuts concern reaocations of productive assets across firms. A reaocation of a given number of assets from a arge firm (a firm which owns a high number of assets) to a smaer one aso yieds a higher spectra radius: Asymmetry across firms aso favor mutipicity of rationaizabe outcomes. In addition, in the case where there are mutipe rationaizabe outcomes, this same reaocation enarges the set of rationaizabe aggregate productions. The intuition for these resuts proceeds as foows. The spectra radius of the bestresponse map is increasing in the sopes of the firms reaction functions. These sopes are increasing and concave in the firms number of assets. Concavity impies that, when productive assets are reaocated from a arge firm to a smaer one, the increase in the sope of the reaction function of the smaer firm dominates the decrease in the sope of the arger one. The overa effect then favors mutipicity. Intuitivey, this reaocation reaxes the capacity constraint of the smaer firm. The behavior of the smaer firm becomes ess predictabe. The behavior of the arger firm, reacting to its expectation about the 2
4 Documents de Travai du Centre d'economie de a Sorbonne production of the smaer firm, in turn becomes ess predictabe. Since the equiibrium aggregate production decreases when the market gets coser to the monopoy situation (where one firm hods most of the avaiabe productive assets), our resuts show that a reaocation of assets whose goa is to reach more efficiency in the Cournot equiibrium may in fact resut in a mutipicity of rationaizabe outcomes. These first resuts aow us to study the optima aocation of assets among firms considering the rationaizabe outcomes, rather than the equiibrium outcome ony. We first sove for the distribution of assets maximizing the aggregate equiibrium production subject to the constraint that this equiibrium is the ony rationaizabe outcome. Of course, the equiibrium production reaches its maximum in the competitive case (corresponding to an infinite number of sma identica firms). Thus, when the competitive equiibrium is the ony rationaizabe outcome, it is the soution we ook for. Otherwise, we find that the soution is an oigopoy with few identica firms: the optima aocation of assets now yieds a non competitive market. Finay, we show that this same distribution of assets aso maximizes the owest aggregate production in a rationaizabe outcome. In this sense, there is no room for an asset distribution which woud give rise to mutipe rationaizabe outcomes. Bernheim (1984), Basu (1992) and Börgers and Janssen (1995) study rationaizabe outcomes in symmetric Cournot games. Guesnerie (1992) studies eductive stabiity (that coincides with uniqueness of rationaizabe outcomes) in the competitive case, and Gabao (2013) considers eductive stabiity in inear symmetric Cournot games. The cosest paper to ours is Mouin (1984). Mouin (1984) provides a condition for oca Cournot stabiity and shows that this same condition aso governs eimination of nonbest responses. Our paper considers goba stabiity of the equiibrium (that amounts to take account of the entry decision of the firms). It provides a goba characterization of the set of rationaizabe outcomes in the presence of asymmetric firms. It aso studies how this set reates to the asset distribution. There are cose inks between the outcomes surviving an iterated eimitation process (dominance sovabiity, iterated weak dominance, etc.) and the outcomes stabe under adaptive earning. See, among others, Migrom and Roberts (1990), Guesnerie (1993), Marx (1999), Hommes and Wagener (2010) or Durieu, Soa and Tercieux (2011). Our paper does not adress this issue. However this iterature suggests that our resuts may have cose counterparts in adaptive earning. In practice it is we known that there is no cear evidence that concentration of production faciities is associated with higher prices (Guger et a., 2003). This ack of concusive evidence is usuay viewed as refecting a tradeoff between economies of scae and the abiity of arger firms to exercise market power (Wiiamson, 1968; Perry and Porter, 1985; Farre and Shapiro, 1990). Our theoretica anaysis suggests that this ack of evidence may refect different outofequiibrium rationaizabe behaviors: A reaocation of production faciities changes the set of rationaizabe outcomes. A merger (reducing possibe outofequiibrium rationaizabe behavior) shoud dampen market voatiity by making easier for each firm to form accurate predictions about others behavior. We iustrate this prediction by studying 3
5 Documents de Travai du Centre d'economie de a Sorbonne the impact of the merger between Deta Airines and Northwest in We use the within carriers standard error of the number of passengers as a measure of coordination voatiity. We provide evidence that the change in market power impied by this merger has reduced coordination voatiity: A 1 percent transfer of market share from a sma firm to a arger one woud decrease voatiity by 1.3 percent. The paper is organized as foows. In Section 2, we briefy present the setup, and we show that equiibrium production increases foowing a reaocation of assets from a arge firm to a smaer one. In Section 3, we give a necessary and sufficient condition for the equiibrium to be the ony rationaizabe outcome. In Section 4, we estabish the tradeoff between efficiency in equiibrium and uniqueness of the rationaizabe outcome. In Section 5, we characterize the optima distribution of assets. Finay the iustration to the airine industry is given in Section 6. 2 Setup Foowing Perry and Porter (1985), we consider a singe product mode of Cournot competition with M firms and N units of a productive asset. Firm owns N units of the asset, with N decreasing in (N R + ). The ony source of heterogeneity across firms is the (exogenous) distribution of assets. Producing q costs C (q, N ) = q 2/2σN to firm (σ > 0). A possibe interpretation of this cost function is to think of a unit of the asset as a pant, and to assume that producing q units of the good in one pant costs q 2 /2σ. By convexity, a firm minimizes its overa cost by producing the same quantity in each pant. Firm produces q which maximizes p(q +Q )q C (q, N ), where Q is the aggregate production of firms other than and p ( ) = δ 0 δq is the inverse demand function (where Q is the aggregate production, and δ, δ 0 > 0). The best response of firm is { q m b Q if Q δ 0 /δ, R (Q ) = (1) 0 if Q δ 0 /δ, where q m = b δ 0 /δ is the monopoy production of firm, and b = σδn 2δσN + 1. (2) The inear/quadratic specification impies that the sensitivity parameter b (the sope of the reaction function) is increasing and concave in N. A Cournot equiibrium is a vector (q ) such that q = R (Q ) for every. Straightforwardy, there is a unique equiibrium. Let Q be the aggregate production in equiibrium. Since the equiibrium price p(q ) is positive (otherwise no firm woud be active in equiibrium) and the margina cost tends to zero when the production tends to zero, it is aways profitabe for a firm to enter the market. Hence a the firms are active in equiibrium (q > 0 for every ). Our first resut confirms that an equa distribution of assets across the firms yieds the highest aggregate production in equiibrium, and thus the highest consumers surpus. 4
6 Documents de Travai du Centre d'economie de a Sorbonne Proposition 1. A transfer of assets from firm h to firm s increases the aggregate output Q in the Cournot equiibrium if and ony if N h > N s (firm h is arger than firm s). Proof. The equiibrium aggregate production is Q = S δ S δ, with S = M =1 b 1 b. (3) The production Q increases in S, and the ratio b /(1 b ) is increasing and concave in N. A transfer of assets from a arge to a sma firm impies a ower b h /(1 b h ) and a higher b s /(1 b s ). By concavity, S increases, and so Q increases. This resut is a particuar case of Perry and Porter (1985) or Farre and Shapiro (1990). It incudes the case of a merger (a merger between firms s and h amounts to transfer a the assets of s to h). A coroary of Proposition 1 is that, when a the firms have the same number of assets, the aggregate output Q increases in the number M of firms (at a symmetric oigopoy with M firms, the transfer of a the assets of one firm to the others resuts into a symmetric oigopoy with M 1 firms, and Q decreases). Hence, Q and the consumers surpus are maximized in a competitive equiibrium (an equiibrium with an infinite number of identica firms). 3 Dominance sovabiity of the equiibrium An equiibrium is dominant sovabe when it is the unique rationaizabe outcome of the Cournot game. To study dominance sovabiity, we first define rationaizabe strategies by the foowing eimination process. Suppose first that the strategy set of every firm is [ q inf define iterativey (for a t 1) the sequences [ q inf firm to the beief that the aggregate production of others is in [ Q inf (0), q sup (0) ) = [0, + ). Then, (t), q sup (t) ] of sets of best responses of (t 1), Qsup (t 1)], (t 1) and Qsup (t 1) = k qsup k (t 1). Strategic substi with Q inf (t 1) = k qinf k tutabiities impy that q inf (t) = R ( Q sup (t 1)), and q sup (t) = R (Q inf (t 1)). (4) These sequences are converging since (q inf they are bounded (0 q inf (t) q qsup and q sup of rationaizabe productions of firm. q inf (t)) increases in t, (q sup (t)) decreases in t, and (t) q m for a t 1). Their imits, denoted, are fixed points of the recursive system (4). The imit set [q inf, q sup ] is the set Let us first adopt a oca viewpoint. Loca dominance sovabiity is defined as the uniqueness of the rationaizabe outcome in a game where the strategy sets are restricted to a neighborhood of the equiibrium (q inf (0) and q sup (0) are cose to the equiibrium q for every firm). Loca dominance sovabiity obtains whenever the recursive system (4) is 5
7 Documents de Travai du Centre d'economie de a Sorbonne ocay contracting at the equiibrium. Since the system (4) is inear, this is equivaent to the spectra radius of the matrix 0 b 1 b 1 b B = bm 1 b M b M 0 being ess than 1. Lemma 1. The spectra radius of B is the unique positive root ρ of F (ρ) M =1 b ρ + b = 1. (5) We have ρ < 1 F (1) < 1. Proof. Let e be an eigenvaue of B, and v an associated eigenvector. Then, ev = Bv yieds M ev + b v = b v k v = b M v k for a. e + b Summing over impies that every eigenvaue e of B is such that k=1 k=1 F (e) M =1 b e + b = 1. For e 0, the function F is continuous and decreasing. Moreover, F (0) = n > 1 > 0 = F (+ ). Hence, B admits a unique positive rea eigenvaue. Since B is a positive matrix, it foows from PerronFrobenius theorem that this positive rea eigenvaue is the spectra radius ρ of B. That is, F (ρ) = 1 for ρ > 0. Finay, since F is decreasing, we have: ρ < 1 if and ony if F (1) < 1. The inequaity F (1) < 1 is the oca condition found by Mouin (1984). We show beow that it is aso the condition for goba dominance sovabiity of the equiibrium. This is done by investigating the set of rationaizabe outcomes. Characterizations of this set have been obtained by Bernheim (1984), Basu (1991), Börgers and Janssen (1995) and Gabao (2013) in the context of Cournot competition with identica firms (a the firms own the same number of assets). With identica firms, either the equiibrium is dominant sovabe = q sup = q q for a ), or [q inf, q sup ] = [0, q m ] with q m = q m for a. Indeed, by symmetry, either q inf > 0 for a or q inf = 0 for a. In the first case, the inear system (4) admits a unique fixed point which coincides with the equiibrium. In the atter case (, q inf = 0), q sup is the best response of firm to others producing 0: This is the monopoy production q m. (q inf 6
8 Documents de Travai du Centre d'economie de a Sorbonne In our setup firms are heterogenous and it is no onger true that q inf = 0 for every when the equiibrium is not dominant sovabe. The next resut shows that the vaues of q inf are ranked according to. Lemma 2. The bounds q inf and q sup are nonincreasing in. Furthermore, the owest rationaizabe production q inf is 0 if and ony if >, where 0 is the argest such that Proof. See in appendix. k b k 1 + b k + k> b k 1 + b < 1. (6) To get an intuition about the existence of the threshod, et us consider the first two steps of the iterative process of eimination of non best responses. In the first step, q sup (1) is the monopoy production q m which is decreasing in (it is increasing in the number of assets). In the second step, q inf (2) is the best response to Q sup (1) which is increasing in (sma firms face a higher aggregate production of others than arge firms). It foows that q inf (2) is decreasing in and q inf (2) is possiby 0 for arge enough. The argument extends to every further step of the eimination process. Given the threshod, we can characterize the rationaizabe outcomes of a inear Cournot game. Lemma 3. The set of rationaizabe aggregate productions is the interva [ Q inf, Q sup], where Q inf = ( 1 + ) c a δ0 a 2 c (c + e) δ, Qsup = Q inf + e δ 0 a 2 c (c + e) δ, (7) with a = 1 + b 2, c = 1 b 2 b 1 b 2 and e = > b. Proof. See in appendix. The appendix aso characterizes the set [ ] q inf, q sup of rationaizabe individua productions. Lemma 2 directy yieds a necessary and sufficient condition for dominance sovabiity of the Cournot game. On the one hand, when a the firms are active ( = M), q inf = q sup = q for a since the equiibrium is the unique fixed point of the inear system (4). On the other hand, the Cournot equiibrium is not the ony rationaizabe outcome when some firms remain inactive ( < M). Proposition 2. The Cournot equiibrium is gobay dominant sovabe (the unique rationaizabe outcome) if and ony if = M, or equivaenty Γ b 1 + b < 1. (8) 7
9 Documents de Travai du Centre d'economie de a Sorbonne Dominance sovabiity is obtained whenever the tota sensitivity of firms is ow enough, an intuition simiar to the one found in the competitive case (Guesnerie, 1992). Proposition 2 generaizes the oca anaysis by taking into account the decision of entry anayzed in Lemma 2. It shows that condition (8) aso governs goba dominance sovabiity, and thus it governs rationaizabiity of entry. 4 Rationaizabiity and asset distribution We now reate the asset distribution to the set of rationaizabe outcomes. A first approach consists in studying the variations of the spectra radius ρ w.r.t. the distribution of assets. The underying idea is that a ower spectra radius favors dominance sovabiity. The spectra radius indeed refects the speed of convergence of the sequences q inf (t) and q sup (t) toward their imits q inf and q sup. Desgranges and Ghosa (2010) provides a forma justification of the interpretation of ρ as a pausibiity index for dominance sovabiity. Proposition 3. A transfer of assets from firm h to firm s increases the spectra radius ρ of B if and ony if firm h is arger than firm s (N h > N s ). Proof. Consider a transfer of dn > 0 assets from firm h to firm s, i.e., N s increases by dn and N h decreases by dn (N s < N h ). The resuting change dρ in the spectra radius is obtained by differentiating (5): [ ( ) F bs (ρ)dρ + ( )] bh dn = 0. N s 1 + b s N h 1 + b h Since the ratio b /(1+b ) is increasing and concave in N, the term into brackets is positive. Since F (ρ) < 0 for ρ > 0, we have dρ > 0. Proposition 3 shows that introducing asymmetries across firms favors dominance sovabiity: For a given number of firms, the equiibrium is more ikey to be dominant sovabe when there are both arge and sma firms, rather than identica firms. The effect of a transfer of assets from h to s is a priori ambiguous. Dominance sovabiity is favored by firms inertia to changes in the (expected) production of others, i.e., a sope of the reaction functions cose to 0. The (absoute vaue of the) sope is aways increasing in the number of assets. Therefore the transfer of assets considered in Proposition 3 impies that the sope b s of the reaction function of the smaer firm increases (which is detrimenta to dominance sovabiity) whie the sope b h for the arge firm decreases (which favors dominance sovabiity). The overa effect is made unambiguous by appeaing to concavity of the ratio b / (1 + b ) in the number of assets (so that the increase in b s has a greater effect than the decrease in b h ). When every sope b is increasing in the number of assets N, this ratio is concave when b is concave in the number of assets N. 1 1 In a genera noninear setup, Proposition 3 hods in the neighborhood of the equiibrium when these monotonicity and concavity properties are satisfied ocay. Such properties rey on assumptions on 4th derivatives of cost and demand functions (the sope b of the reaction function being characterized by second derivatives of these functions). 8
10 Documents de Travai du Centre d'economie de a Sorbonne Propositions 1 and 3 highight that any reaocation of assets which improves consumers surpus in equiibrium makes ess ikey that this equiibrium is the unique rationaizabe outcome. 2 A second approach to the impact of assets reaocations bears on the set of rationaizabe aggregate productions [ Q inf, Q sup]. The difference Q sup Q inf can be viewed as a measure of the strategic uncertainty in the market. When the equiibrium is not dominantsovabe, mutipicity of rationaizabe outcomes corresponds to the possibe occurrence of fuctuations due to outofequiibrium beiefs: some coordination voatiity occurs foowing the process of eimination of nonbest response strategies. The magnitude of this voatiity can be measured by the size of the interva [ Q inf, Q sup] of rationaizabe aggregate productions. Voatiity is dampened when [ Q inf, Q sup] is a narrow interva around the Cournot equiibrium Q. The next resut is another version of the tradeoff between efficiency and dominance sovabiity: A reaocation of assets which yieds a higher aggregate production in equiibrium aso enarges the set of rationaizabe outcomes. Proposition 4. Assume that the equiibrium is not dominant sovabe (Q inf < Q sup ). Consider an infinitesima reaocation of assets from a arge firm h to a smaer one s, dn h = dn s < 0 (N s < N h ). We have: Proof. See in appendix. d ( Q sup Q inf) > 0. In order to grasp some intuition, consider again the iterative process (4). There, the production q sup (1) is the monopoy production q m which is increasing and concave in N. The reaocation of assets from firm h to firm s impies that qh m decreases whie qm s increases. The tota effect is not ambiguous because of concavity: qs m + qh m increases. Hence, for each firm s, the reaocation of assets impies an increase in Q sup (1) so that qinf (2) decreases (firm s faces a smaer production Q sup s (1) but we show that this effect is dominated by the aggregate effect on a the other firms). The argument then extends to every further step of the iterative process. 5 Optima asset distribution This section characterizes the distribution of assets which maximizes consumers surpus. When one considers the issue of mutipicity of rationaizabe soutions, there are two ways of assessing consumers surpus. One way is to maximize consumers surpus in equiibrium under the additiona constraint that the equiibrium is the ony rationaizabe soution. The other way is to maximize consumers surpus at some (nonequiibrium) rationaizabe 2 An aternative assessment coud refer to Γ characterized in Proposition 2. Proposition 3 actuay hods true when the spectra radius ρ is repaced by Γ. 9
11 Documents de Travai du Centre d'economie de a Sorbonne soution. Assuming risk aversion of the competition reguator whose goa is to maximize consumers surpus eads to consider rationaizabe soutions invoving ow production eves. We consider the poar case of the worst scenario where the owest rationaizabe aggregate production Q inf occurs (this corresponds to an arbitrariy arge risk aversion towards strategic uncertainty). First, we characterize the distribution of assets which maximizes the aggregate production in a Cournot equiibrium subject to the constraint that this equiibrium is the unique rationaizabe outcome. Second, we show that this same distribution aso maximizes the owest rationaizabe aggregate production Q inf. Consider first the distribution of assets (N ) and the number M of firms which maximize the equiibrium aggregate production Q defined in (3) subject to two constraints: the constraint of dominant sovabiity (F (1) < 1 in (8)), and the feasibiity constraint M N N. (9) =1 Proposition 5. Any ((N ), M) maximizing the aggregate equiibrium production Q given by (3) subject to the constraints (8) and (9) invoves an equa sharing of productive assets: N = N/M for a. Furthermore, if σδn < 1, then the soution invoves an infinite number of firms (competitive market), and (9) is the ony binding constraint; if σδn 1, then the soution is a symmetric oigopoy with firms. 3 The aggregate production is M = 3σδN σδn 1 Q = 1 3σδN σδN δ 0 δ (10) Both constraints (8) and (9) are binding at the optimum. Proof. See in appendix. By Proposition 5 the competitive equiibrium is the ony rationaizabe outcome when σδn < 1. This is the condition found by Guesnerie (1992) in a setup where there is a continuum of size N = 1 of competitive firms. The competitive case with a unique rationaizabe outcome can be viewed as a situation where a arge number of identica firms share a sma capacity of production. 3 When M is not an integer, the soution is the argest integer beow M. 10
12 Documents de Travai du Centre d'economie de a Sorbonne Q Q sup ( M ) 0 B A inf Q 0 ( M 0) * Q sym ( M ) ** Q inf Q 1 ( M 0) ** M M 0 M Figure 1: Optima distribution of assets When σδn 1, the competitive equiibrium is not dominant sovabe. Uniqueness coud be restored using two different kinds of poicies. The first kind consists in aocating ony a part of the existing production assets: the Cournot equiibrium becomes dominant sovabe when ony 1/σδ assets (1/σδ < N) are aocated to a arge number of firms, so that one goes back to the previous competitive situation. High competition then goes with production inefficiency, since some productive assets are not used. The second poicy is to aocate the whoe stock of assets to few firms, so that the market gets coser to a monopoyike situation. Proposition 5 shows that this ast poicy is better for consumers surpus. At this stage, however, Proposition 5 ony gives partia insights into the optima distribution of assets. Indeed, when σδn 1, there might exist some distributions of assets such that the associated owest rationaizabe production Q inf is greater than Q exhibited in Proposition 5. With such distributions, the consumers surpus at a rationaizabe outcome is necessariy greater than the highest surpus achievabe at a dominant sovabe equiibrium. An iustration is given in Figure 1. The soid curve represents the equiibrium aggregate quantity when the productive assets are equay shared across firms. This quantity increases in the number M of firms and eventuay coincides with the competitive equiibrium. In Figure 1 the competitive equiibrium is not dominant sovabe: By Proposition 5 there exists a finite threshod M such that the equiibrium is dominant sovabe if and ony if the number M of firms satisfies M M. For a M, the highest equiibrium production obtains in the case of identica firms. Suppose that there are M 0 firms in the market, with M 0 M. The equiibrium production in the case of identica firms is at point B. When the market structure is cose to the monopoy situation, the equiibrium is the ony rationaizabe outcome. By Proposition 5 the aggregate production then stands beow Q. Figure 1 depicts the case where the market structure is cose the the symmetric 11
13 Documents de Travai du Centre d'economie de a Sorbonne case, with a set of rationaizabe outcome being not reduced to the Cournot equiibrium. Point A corresponds to an equiibrium aggregate production where firms differ according to the number of productive assets they hod. There are two possibe cases for the set of rationaizabe aggregate productions. In the first case rationaizabe production is aways higher than Q inf 0 (M 0 ) satisfying Q inf 0 (M 0 ) > Q. In this case any rationaizabe outcome in a market with M 0 firms dominates (in terms of consumers surpus) the dominant sovabe symmetric equiibrium with M. The optima distribution of assets then invoves a mutipicity of rationaizabe outcomes. The other possibiity where the owest rationaizabe aggregate production is Q inf 1 (M 0 ) is however theoreticay possibe too. Then, Q > Q inf 1 (M 0 ), so that a mutipicity of rationaizabe outcomes can be detrimenta to consumers surpus. The next resut shows that the ast configuration aways prevais: for a M > M, the owest rationaizabe aggregate production is aways beow Q, as is Q inf 1 (M 0 ) in Figure 1. Proposition 6. There is no distribution of assets such that the owest aggregate production Q inf is greater than Q. Proof. See in appendix. This resut impies that there is no asset distribution such that the competition reguator can hod for sure that one can achieve a surpus higher than in the CournotNash equiibrium where the aggregate production is Q defined in Proposition 6. In this sense, one can not justify a poicy which woud give rise to mutipe rationaizabe outcomes. In summary, this section raises the question of the choice by a competition reguator of the distribution of assets. If one reaxes the Nash equiibrium assumption and one considers rationaizabe outcomes as possibe outcomes, then one must wonder whether outofequiibrium outcomes may improve consumers surpus. Looking at Q inf as in Proposition 6 amounts to have a competition reguator whose risk aversion toward the strategic uncertainty is infinite (the reguator puts a high probabiity on worst aggregate productions in the case of mutipicity of rationaizabe outcomes). Proposition 6 shows that sufficient risk aversion eads to seect a distribution of assets which yieds dominance sovabiity of the Cournot equiibrium. But sufficient risk aversion does not aways recommend to pick out the competitive outcome: The optima distribution of assets is an oigopoistic one when the production capacity is arge (σδn > 1). 6 An Iustration from the U.S. Airine Industry Our anaysis predicts that a merger dampens coordination voatiity by making outofequiibrium behavior ess ikey. We assess this prediction by considering the merger between Deta Air Lines (DL) and Northwest Airines (NW) in the U.S. airine industry. This merger was announced on Apri 2008, approved by the Department of Justice on October 2008, and competed on January It took pace over a period of high voatiity, with the goba recession, soaring fue prices and H1N1 fu pandemic. 12
14 Documents de Travai du Centre d'economie de a Sorbonne In this industry a route inking two cities can be viewed as a separate market, and carriers as producing an amount of passengers transported. The number of passengers transported in a given route by a given airine is imited by the time aocated by the airports to this airine under the form of anding sots. In the short/medium run, the distribution of the sots across airines can be considered as given. These sots, or the corresponding seat capacity, are used as a proxy for the productive assets of our Cournot setup. 6.1 Data Description Our data comes from the Airine Origin and Destination Survey, a quartery 10% sampe of airine tickets coected by the U.S. Department of Transportation. The data gives the origin and the destination airports, the ticketing carrier, the number of passengers and the airfare for about 5 miions observations per quarter. A market (route) is defined as a fights between two U.S. cities, irrespective of the serving airports, intermediate transfer points and the direction of the fight path. The period of anaysis comprises 18 quarters from 2006:3 to 2010:4, i.e., 9 quarters before and after the merger approva. 4 There are 1, 154 (resp. 3, 305) routes from which DL (resp. NW) is absent every quarter during the whoe period of anaysis. The intersection of these two sets of routes yieds a group of 1, 099 routes. These routes are considered to be not affected by the merger, and they are used as contro group. The treatment group consists of a the routes where both DL and NW are active every quarter before the merger approva. This is the case in 2, 934 (resp. 1, 702) routes for DL (resp. NW). DL and NW were not competing directy on most routes: The intersection of these two sets of routes ony comprises 839 routes. The data is aggregated so that one observation gives the number of tickets per carrier, quarter and route. The fina sampe comprises 12, 639 observations, corresponding to 1, 939 routes inking 235 cities. The market share of DL (resp. NW) computed from 2006:3 to 2008:3 equas 11.3% (resp. 6.7%). Tabe 1 shows that routes in the treatment group invove a ower traffic voume and greater competition, with a ower Herfindah index and more carriers active in 2006: Variabe Definitions The main variabe of interest is the variance within firms of the number of passengers transported. It measures how much a firm changes its output with respect to its own average output. It provides a proxy for coordination voatiity by refecting the difficuties 4 Data ceaning proceeds as in Kim and Singa (1993). We remove observations with missing carrier, with a zero fare or abnorma fares in the bottom and top 1% of observations, and tickets with a fare higher than 3 USD per mie. Foowing Ciiberto and Wiiams (2010) we ony consider carriers with at east 20 reported passengers per quarter. This corresponds to an airine using at east a 20seat pane at fu capacity every week. Finay we negect routes with ess than 30 reported passengers per quarter, and routes where information is missing for some quarter during the period of anaysis. 13
15 Documents de Travai du Centre d'economie de a Sorbonne Tabe 1: Descriptive statistics Variabe Contro Treatment Number of routes 1, Number of cities Number of passengers by route (2006:3) 3, 071 1, 194 Mean fare by route (2006:3) Number of carriers by route (2006:3) Herfindah by route (2006:3) faced by a carrier to set a suitabe output eve, i.e., to adjust suitaby its production to others behavior. This variance is our endogenous variabe. Let carriers be indexed by i, routes by j, quarters by t = 1,..., 18 (2006:3 2010:4), and periods by p, with p = 0 before the merger announcement (t 7) and p = 1 after the announcement. Let P 0 (resp. P 1 ) the set of quarters such that p = 0 (resp. p = 1). The average production of firm i in market j over period P {P 0, P 1 } is q ij = 1 (#P) and our measure of coordination voatiity is σ 2 w(i, j) = 1 (#P) 1 q ij (t), t P (q ij (t) q ij ) 2. The set of regressors incudes route and airine variabes. A the route variabes are computed from the first quarter of the period of anaysis (2006:3). They consist of the number of passengers transported and the average fare for each route, the number of active carriers, and the Herfindah index of the route. We aso use the owest distance between the two market endpoints and the tota number of airports in the corresponding cities. Airine variabes incude a company fixed effect, the fue price per gaon paid by the company (the ratio between its domestic fue cost in USD and its domestic fue consumption in gaons) and the distance between the market endpoints and the cosest hub of the company. The ast two variabes vary over time to contro for changes in the production cost in a context of sharp increase in oi prices. 6.3 Merger and Market Voatiity Figure 2 shows the evoution of the (og of the) tota number of passengers transported in the contro and treatment group. Both foow a reguar seasona pattern, with a sight (not t P 14
16 Documents de Travai du Centre d'economie de a Sorbonne Nb of passengers (in og) 15.0 Contro Treatment Quarter Figure 2: Number of passengers by group statisticay significant) decrease in the contro group. The two vertica ines respectivey indicate the time of the merger announcement (2008:2) and approva (2008:4). We consider the mode og(σ ij (p)) = cste + r j +c 0 i +c 1 i (p)+period+treatment+(period Treatment)+ɛ ij (p), (11) where r j comprises the route variabes, c 0 i is a company fixed effect, and c 1 i (p) incudes the two company variabes which change over time (fue price and the distance to the cosest hub of the company). The variabe Period is 0 before the merger announcement, and 1 after the announcement. Treatment is a dummy which takes vaue 0 if the route beongs to the contro group, and 1 otherwise. The interaction term (Period Treatment) gives the impact of the merger onto the within standard error of the number of passengers. Tabe 2 reports the resuts for three variants of the mode (11). In each variant the period before the merger covers a the quarters from 2006:3 to the quarter preceding the announcement of the merger (2008:1), and the period after the merger begins on the merger announcement (2008:2). In the variant reported in Coumn 1 (resp., 2 and 3) the period after the merger ends at the merger approva in 2008:4 (resp., the merger competion in 2010:1, and the ast quarter of 2010). Route and company variabes affect within voatiity in a simiar way in the three variants. Routes with a ow traffic voume and a ow fare dispay ow voatiity. This tends to be aso the case for routes with a high number of competitors dominated by few arge airines. The company fixed effects indicate that arge companies typicay have a stabiizing effect, though the two merging companies DL and NW seem to encounter greater difficuties to set a stabe output. 15
17 Documents de Travai du Centre d'economie de a Sorbonne The merger occurs over a period of steep rise in the price of oi in 2008, and incuding the begining of the goba crisis in 2008:3. The overa period of anaysis has indeed seen a huge increase in the within standard error of the number of passengers. This effect dampens in 2010 but remains 58.3% higher than before the merger announcement. One coud have expected the merger to magnify market voatiity: The market coud have reached an equiibrium that is perturbed by the merger. The merger woud then cause an increase in the standard error as the market transitions to the new equiibrium. This is not what happens: The shortrun (threequarter) impact of the merger is to reduce significanty the within standard error of the number of passengers transported. The merger is found to reduce coordination voatiity by 23.2% in the shortrun. This arge impact dampens over time: Voatiity is ony reduced by 8.6% during the whoe competion period (Variant 2), which covers eight quarters after the announcement. The impact entirey ooses significance after two years. There is consequenty a strong stabiizing effect of the merger in the short/medium run, in a context of high voatiity. Foowing Kim and Singa (1993) one can interpret the resut in Variant 2 as refecting the own impact of a change in market power. Indeed they argue that the anticipation of the merger make DL and NW more cooperative from the announcement quarter, and that efficiency gains (synergies) are absent unti reaching the competion: Ony the change in market power matters during the period from announcement to competion. The impact measured in Variant 3 woud instead mix changes in market power and synergies by covering about one year after the competion. A possibe interpretation of our resuts is therefore that the change in market power due the merger stabiizes coordination voatiity by 8.6% whie synergies eventuay offset most of the effect of market power. From the mode in Variant 2, NW having 6.7% market share before the annoucement, we find that the own impact of a change in market power corresponding to a 1% transfer of market share is to reduce the within standard error of the output by about 1.3% ( 8.6/6.7). 16
18 Documents de Travai du Centre d'economie de a Sorbonne Tabe 2: Impact of the merger og(σ w(i, j)) 2006:3 2008:1 vs 2008:2 2008:4 2006:3 2008:1 vs 2008:2 2010:1 2006:3 2008:1 vs 2008:2 2010:4 Constant (1.063) (1.168) (1.282) Route variabes (2006:3) og(nb of passengers) (0.008) (0.008) (0.008) og(fare) (0.035) (0.035) (0.035) og(nb of carriers) (0.032) (0.032) (0.032) og(herfindah index) (0.015) (0.016) (0.016) og(market distance) (0.015) (0.016) (0.016) og(nb of airports) (0.015) (0.015) (0.015) Company (ref: American Airines) AS (Aaska Airines) (0.042) (0.043) (0.042) CO (Continenta Airines) (0.025) (0.025) (0.025) DL (Deta Air Lines) (0.031) (0.028) (0.028) FL (AirTran) (0.041) (0.041) (0.040) G4 (Aegiant Air) (0.111) (0.112) (0.111) HA (Hawaiian Airines) (0.113) (0.118) (0.120) NW (Northwest Airines) (0.032) (0.033) (0.034) UA (United Airines) (0.024) (0.024) (0.024) US (US Airways) (0.024) (0.025) (0.025) WN (Southwest Airines) (0.033) (0.032) (0.032) YX (Midwest Airines) (0.066) (0.064) (0.061) og(distance to the cosest hub) (0.003) (0.003) (0.003) og(fue price) (0.081) (0.089) (0.098) Impact of the merger Period (0.023) (0.027) (0.026) Treatment (0.028) (0.029) (0.029) Period Treatment (0.029) (0.029) (0.029) Observations 11,770 12,397 12,639 R Adjusted R Residua Std. Error (df = 11747) (df = 12374) (df = 12616) F Statistic 1, (df = 22; 11747) 1, (df = 22; 12374) (df = 22; 12616) Notes: Significant at the 1 percent eve. Significant at the 5 percent eve. Significant at the 10 percent eve.
19 Documents de Travai du Centre d'economie de a Sorbonne References [1] Basu, K., 1992, A characterization of the cass of rationaizabe equiibria of oigopoy games, Economics Letters 40, [2] Bernheim, B., 1984, Rationaizabe strategic behavior, Econometrica 52, [3] Borenstein, S., 1990, Airine mergers, airport dominance, and market power, American Economic Review 80(2), [4] Börgers, T. and M. Janssen, 1995, On the dominance sovabiity of arge Cournot games, Games and Economic Behavior 8, [5] Ciiberto, F. and J. Wiiams, 2010, Limited Access to Airport Faciities and Market Power in the Airine Industry, Journa of Law and Economics 53 (2010), [6] Compte, O., F. Jenny and P. Rey, 2002, Cousion, mergers and capacity constraints, European Economic Review 46, [7] Davis, L. and C. Wofram, 2011, Dereguation, consoidation, and efficiency: evidence from U.S. nucear industry, NBER working paper [8] Desgranges, G. and S. Ghosa, 2010, PStabe Equiibrium : Definition and Some Properties, The Warwick Economics Research Paper Series (TWERPS) 952, University of Warwick, Department of Economics. [9] Durieu, P., P. Soa and O. Tercieux, 2011, Adaptive earning and pbest response sets, Internationa Journa of Game Theory 40, [10] Farre, J. and C. Shapiro, 1990, Asset ownership and market structure in oigopoy, Rand Journa of Economics 21, [11] Gabao, G., 2013, Market Power, Expectationa Instabiity and Wefare, mimeo. [12] Guger, K., D.C. Mueer, B.B. Yurtogu and C. Zuehner, 2003, The effects of mergers: an internationa comparison, Internationa Journa of Industria Organization 21, [13] Guesnerie, R., 1992, An exporation of the eductive justifications of the rationaexpectations hypothesis, American Economic Review 82, [14] Guesnerie, R., 1993, Theoretica tests of the rationa expectations hypothesis in economic dynamica modes, Journa of Economic Dynamics and Contro 17, [15] Hommes, C. and F. Wagener, 2010, Does eductive stabiity impy evoutionary stabiity?, Journa of Economic Behavior and Organization 75,
20 Documents de Travai du Centre d'economie de a Sorbonne [16] Kim, E. and V. Singa, 1993, Mergers and Market Power: Evidence from the Airine Industry, American Economic Review 83, [17] Marx, L., 1999, Adaptive Learning and Iterated Weak Dominance, Games and Economic Behavior 26, [18] Migrom, P., and J. Roberts, 1990, Rationaizabiity, earning, and equiibrium in games with strategic compementarities, Econometrica, 58, [19] Mouin, H., 1984, Dominance sovabiity and Cournot stabiity, Mathematica Socia Science 7, [20] Perry, M. and M. Porter, 1985, Oigopoy and the incentive for horizonta merger, American Economic Review 75, [21] Town, R., D. Whoey, R. Fedman, L.R. Burns, 2006, The wefare consequences of hospita mergers, NBER working paper [22] Wiiamson, O., 1968, Economies as and antitrust defense: The wefare tradeoffs, American Economic Review, 58(1),
21 Documents de Travai du Centre d'economie de a Sorbonne Proof of Lemmas 2 and 3 Let L 0 be the set of vaues of such that q inf Soving for q inf and L 0, q inf is (7), namey: with and q sup / L 0, q inf L 0, q inf = 0, gives, q sup / L 0, q inf = b2 1 b 2 / L 0, q sup = b 1 b 2 L 0, q sup Q inf = = 0. The reations (4) give: ), = b ( δ0 δ (Qsup q sup ) ( δ0 = b δ ( Q inf q inf ( δ0 ) δ Qinf + b 1 b 2 ) δ Qinf + b2 1 b 2 ( δ0 = b ( δ0 δ Qinf ) ). ( ) δ0 δ Qsup, (12) ( ) δ0 δ Qsup, (13) ), (14) = 0. Summing over gives a inear system in Q inf and Q sup whose soution ( 1 + ) c a δ0 a 2 c (c + e) δ, Qsup = Q inf + b 2 a = b 2 / L 0, c = / L 0 b 1 b 2 e δ 0 a 2 c (c + e) δ, and e = L 0 b. For every L 0, q inf = 0 so that k qsup k > δ 0 /δ. Using (13), (14) and the expressions of Q inf and Q sup, this atter inequaity is equivaent to: Since e 0 and (c a) (b m + 1) + e a 2 c (e + c) Q sup Q inf = e δ 0 a 2 c (e + c) δ > 0. (15) 0, (16) it foows that a 2 c 2 ce > 0, so that the inequaity (15) is equivaent to which is equivaent to (6) since (a c) (b m + 1) < e, (17) c a = k / L 0 b k 1 + b k 1. Hence, q inf = 0 if and ony if (6) does not hod true. Since the LHS of (6) is increasing in, there is a vaue such that q inf = 0 if and ony if >. 20
22 Documents de Travai du Centre d'economie de a Sorbonne Proof of Proposition 4 Let dn h = dn s < 0. By (2), b is increasing and concave in N. Hence we have db h < 0 < db s and db h + db s > 0. Differentiating (16) gives: dq sup dq inf = δ ( ) 0 δ d e. (18) a 2 c (c + e) We distinguish between 3 cases for the computation of dq sup dq inf. Case 1: < h < s. a and c remain constant and (18) writes: dq sup dq inf = δ 0 δ a 2 c 2 (a 2 c (c + e)) 2 de, where de = db h + db s > 0. Since a 2 c 2 = (a c) (a + c), simpe agebra aows us to check that the above numerator is positive so that dq sup dq inf > 0. Case 2: h < s. e remains constant and (18) writes: dq sup dq inf = δ 0 δ e (a 2 2 (2ada (2c + e) dc). c (c + e)) This has the same sign as ((2c + e) dc 2ada). It is positive if and ony if ( c + 1 ) ( ) 2 e b d > a ( ) b 2 d. (19) 1 b 2 1 b 2 On the one hand, b 1+b is increasing and concave in N, which impies ( ) b d > b This atter inequaity rewrites The LHS is positive since cannot be signed because b 1 b 2 b2 1 b 2 ( ) b d > ( ) b 2 d, (20) 1 b 2 1 b 2 is shown to be increasing and concave in N (but the RHS is neither concave nor convex in N ). If the RHS is negative, then (20) impies that (19) hods true. If the RHS is positive, then rewriting (17) for gives a c > e, (21) 1 + b which impies c + 1 e < a. Combining this atter inequaity with (20) proves that (19) hods 2 true. This shows that dq sup dq inf > 0. 21
23 Documents de Travai du Centre d'economie de a Sorbonne Case 3: h < s. (18) writes: dq sup dq inf = δ 0 δ dq sup dq inf has the same sign as + δ 0 δ e (a 2 2 (2ada (2c + e) dc) c (c + e)) a 2 c 2 (a 2 c (c + e)) 2 de e (2ada (2c + e) dc) + ( a 2 c 2) de = e (2c + e) (1 + b2 h ) 4ab h (1 b h ) 2 (1 + b h ) 2 db h + ( a 2 c 2) db s Since db s > 0 > db h and db h + db s > 0, the above expression is positive if e (2c + e) (1 + b2 h ) 4ab h (1 b h ) 2 (1 + b h ) 2 < ( a 2 c 2) (22) Inequaity (21) impies (h ): a c > e 1 + b e 1 + b h > 0. (23) Using (a 2 c 2 ) = (a c) (a + c) a sufficient condition for Inequaity (22) is This rewrites: e (2c + e) (1 + b2 h ) 4ab h 1 (1 b h ) 2 (1 + b h ) 2 < (a + c) e. 1 + b h (2c + e) ( 1 + b 2 h) 4abh < (a + c) (1 b h ) 2 (1 + b h ). Using again Inequaity (23) (e < (a c) (1 + b h )), a sufficient condition for the above inequaity is This rewrites (2c + (a c) (1 + b h )) ( 1 + b 2 h) 4abh < (a + c) (1 b h ) 2 (1 + b h ). 2ab h (b h 1) < c2b 2 h (1 b h ), a > cb h. Since a c > 0, a > c > cb h. This shows dq sup dq inf > 0. 22
24 Documents de Travai du Centre d'economie de a Sorbonne Proof of Proposition 5 We maximize Q subject to (8) and (9) in three steps. Since Q is increasing in S (see (3)), the optimization probem is to maximize S subject to (8) and (9). Step 1. Consider the Lagrangian: ( ) ( ) M M M 1 + η N N. =1 σδn σδn µ =1 σδn 3σδN + 1 It is the Lagrangian associated with the maximization probem for a given vaue of M. Any soution to the initia optimization probem satisfies the firstorder conditions in N associated with this Lagrangian. The firstorder conditions in N are: where P (x) = σδp (σδn ) = 0, for every, 1 (1 + x) 2 µ 1 (1 + 3x) 2 η σδ. Hence, the number of different firms (different vaues of N ) at a soution of the optimization probem equas the number of positive roots of P. Observe that Since P (x) 0 rewrites P 2 (x) = (1 + x) 3 + µ 6 (1 + 3x) 3. (3 (3µ) 1/3 )x (3µ) 1/3 1, P can change its sign at most once. Hence, either P is monotonic or P admits one oca extremum. It foows that P admits at most 2 positive roots: the soution to the optimization probem invoves at most two types of firms. Denote i = 1, 2 the type of a firm. Let M i the number of firms of type i (i = 1, 2). Every type i firm uses N i assets (0 N 1 N 2 w..o.g.). Step 2. We maximize S for given N 1 and N 2 under the 2 constraints (8) and (9). S is inear in M 1 and M 2 : σδn 1 S = M 1 σδn M σδn 2 2 σδn The stabiity constraint (8) is inear: M 1 3σδN σδn 1 N 2 N 1 3σδN σδN M 2, and the feasibiity constraint (9) is inear too: M 1 N N 1 N 2 N 1 M =1
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