Exit, Sunk Costs and the Selection of Firms
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Richelle, Yves; Garella, Paolo Working Paper Exit, Sunk Costs and the Selection of Firms Quaderni - Working Paper DSE, No. 214 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Richelle, Yves; Garella, Paolo (1995) : Exit, Sunk Costs and the Selection of Firms, Quaderni - Working Paper DSE, No. 214, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5112 This Version is available at: https://hdl.handle.net/10419/159057 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/
Exit, Sunk Costs and the Selection of Firms Yves Richel le Universityof Laval (Queb ec) and Paolo G. Garel la Univ ersita di Bologna march 1995 JEL Classication : L13, L41 Abstract The paper analyzes the question of whic h cost characteristics are exhibited by the rms that exit an oligop olistic market when costs are asymmetric and rms can credibly b e forced out by the remaining comp etitors. The main results are: (i) if reen try is imp ossible (due to the presence of large sunk costs), then the rm with the highest marginal cost function dtays in; if reen try is costless then the rm with the highest average cost exits. Consequenty sunk costs not only aect the numb er of rms in an industry, but they also enter the determination of the typ e of rms that resist predation. Keywords: endogenous coalition formation, exit, sunk costs. 1
1 Intro duction In mark ets where rms dier as to their cost functions is it p ossible to predict what are the cost characteristics of the rms that stay or that exit? In p erfectly comp etitive markets one can predict that the rms exiting the market are those with highest average costs. This prediction has b een extended by Ghema wat and Nalebu (1985), (1990) and Fudenberg and Tirole (1986) to the case of declining industries with few comp etitors. Their analyses show indeed that, in a war of attrition, the less ecien t rm will b e the rst to exit 1 . However many recen t works (see chapters 8 and 9 in Tirole (1988) and Wilson (1992)) show that exit can o ccur in a wide variety of circumstances. We are therefore led to ask if the abo ve prediction continue to hold in imp erfectly comp etitiv e mark ets where rms are not engaged in a war of attrition. This paper argues that the exiting rm may be the one with the lowest average cost function. To identify the basic argumen t leading to this conclusion, consider the following example. Three rms decide, at a rst stage, to stay in the mark et or to exit and, at a second stage, those that sta y decide howmuch to pro duce. All rms have identical xed costs. They also have constant marginal costs with rm i 's marginal cost b eing strictly smaller than rm j 's marginal cost whic h, in turn, is strictly smaller than rm k 's. Firms can therefore b e ranked according to their a verage cost function with rm i having the lowest one. Then suppose that if all rms sta y in the mark et, each of them will obtain a strictly negative prot at the Cournot equilibrium while, if only two rms stay in, their Cournot prot is p ositiv e and the third rm receives a zero prot. It immediately follows that for each couple of rms to stay in the market and to pro duce their Cournot quatit y is an equilibrium of the two stage game. They are therefore three equilibria and a prediction on the cost characteristics of the exiting rm cannot b e based only on this tw o stage game. Note, inciden tally, that these equilibria are all Pareto ecien t so that one cannot use coalitional pro ofness (see Bernheim, Peleg and Whinston 1 For instance, Ghema wat and Nalebuf (1985) show that in a war of attrition with complete information where rms dier according to their pro duction capacity, the biggest rm is the rst to exit. But these authors assume that rms incur only a ow maintenance cost which is prop ortional to their capacity. Accordingly, the biggest rm is the one with the highest average cost function. 2
(1987)) to select one or the other. A p ossible route to follo w for obtaining a prediction is indicated by the literature on endogenous coalition formation, as in Aumann and Myerson(1988), Gul (1989), and esp ecially Blo c h (1990a) and (1990b). These works use a non-cooperativ e sequential game to analyze the formation of coalition structures. In the same way, one can assume that a coalition formation game precedes the play of a game of the kind illustrated by the example above. We here adopt a sp ecication of the coalition formation game where each rm in turn makes a declaration consisting of (i) a set of rms that stay in, (ii) a payo vector for the three rms that can b e obtained by the play of a non-coop erativ e equilibrium of the two stage game. One can interpret these declarations as \oers", and we mo del the acceptance (refusal) of an oer as the making of an identical (dieren t) declaration. Since a declaration corresp onds to one equilibrium, if two rms make the same declaration they agree to play the same equilibrium. This determines whic h equilibrium is played and the payos to all rms irresp ectiv e of the declaration made by the third rm. It is imp ortant to realize that once two rms ha ve adopted their equilibrium strategies the third rm has no b etter alternativ e than the play of its own b est reply to those strategies, whic h coincides with the strategy sp ecied in the equilibrium chosen by the other two rms. Therefore the wa ypayos are determined has nothing to do with the application of a ma jorit y rule in collective decision making. For further reference we call this sequen tial game \the cartel formation game". The equilibrium of this game giv es a prediction of the rm that exits. For each order in whic h rms declare, there will b e a unique subgame p erfect equilibrium outcome in the cartel formation game. But, as one can exp ect, the equilibrium outcome will in general dep end on the order of declaration. We are nev ertheless able to show that, as long as a rm exits the market at the equilibrium, the cost characteristics of this rm can b e iden tied and are indep enden t of the order of declaration. The cartel formation game will therefore provide a strong prediction on the characteristics of the exiting rm. In the example the unique equilibrium is with rms j and k making the same declaration of the form (i) f j; k g , and (ii) payo zero for rm i , and Cournot payos for j and k . A proof of this statement is trivial. Indeed the Cournot prot of a rm is increasing in the marginal cost of its rival which 3
implies that rm j mak es the highest equilibrium prot when k stays on the market and rm k makes the highest equilibrium prot when j stays on. Hence the rm exiting the market is the one with the lowest average costs, and not with the highest, as it would b e predicted in a war of attrition or in p erfect comp etition. One is led to wonder if the dierence in prediction could disapp ear if rms play a supergame instead of a one-shot game. Indeed, in a sup ergame, rms are generally able to maximize joint prots and, since joint prot maximization requires the minimization of variable cost, they will b e induced to internalize the gain made by having an ecien t partner. In what follows we shall generalize the example giv en ab ove by considering a general cost function and a pro duction game consisting of an innite rep etition of the two-stage game of the example. In this game, we sa y that atwo-rm cartel is feasible if there exists an equilibrium of the production game where these rms stay in and the third stays out along the equilibrium path. Obviously the analysis is interesting only if the pro duction game displays at least two dierent feasible cartels. The main results are that (i) if reentry is imp ossible, then the rm with the highest marginal cost function stays on at all equilibria of the cartel formation game, for any order of declaration, while xed costs only determine the set of feasible cartels. (ii) If reen try is p ossible then the rm with the highest average cost exits. These results imply that the dierence in prediction do es not dep end on the p ossibility or not to collude, but rather dep ends up on the existence or not of sunk costs for reentry . Anovel implication of the presence of sunk costs app ears here: not only, as it is already well known from the literature on entry preemption, they can determine the number of rms, but they also enter the determination of the typ e of rms that stay in a market. The paper is organised as follows: in the next Section we intro duce our assumptions relativ e to the cost and demand functions and we analyse the equilibrium outcomes of the production game. In Section 3, the cartel formation game is formally presented. Our results are stated in Section 4 for the unprotable reen try case and in Section 5 for the case of costless reentry. In Section 6 we test the robustness of the results for the case of unprotable reentry to changes in the cartel formation game. The results stated in section 4 are shown to go through. Section 7 presents some concluding considerations to relate the results to the literature on transaction cost economics. 4
2 The pro duction game We consider a sup ergame involving three rms. We shall denote this game by 0 and the set of rms by N .0 consists of the innite rep etition of the two-stage game where (i) at the rst stage each rm decides to stayin or to stay out of the mark et and (ii) at the second stage the rms whic h have decided to stay in the market, hereafter referred to as the activ e rms, play an usual Cournot game whilst an inactive rm pro duces nothing. At each stage decisions are made simultaneously and actions taken at the rst stage are p erfectly observed by all rms b efore they choose their pro duction at the second stage. The scalar , belonging to the op en interval (0 ; 1), denotes the discount factor common to all rms. The purp ose of this preparatory section is tw ofold. On the one hand, we give a precise content to the concept of a feasible cartel. On the other hand, for eac h feasible cartel s , we c haracterize the set of all pa yo vectors that rms can obtain at a subgame p erfect Nash equilibrium of 0 where along the equilibrium path only the rms in cartel s stay in the market at eac h p erio d. To simplify the exp osition we shall proceed in three steps. First, in subsection 2.1., we shall intro duce the assumptions on the cost and demand functions. In the second step, in subsection 2.2., we shall ignore the rst stage of the constituen t game and concentrate on the game 0 ( s ) consisting of the innite rep etition of the Cournot game where the set of play ers is giv en by s (i.e. rms in s decide to stay in the mark et at ev ery p erio d and the rm outside s ,if s 6 = N , decides to stay out of the mark et at every perio d). We can thereby use the results from the literature on innitely rep eated games to bring forth a c haracterization of the set of equilibrium payo vectors of 0 ( s ) . In the last step, subsection 2.3., we intro duce the p ossibilit y for each rm to exit the mark et. This will allo w us to dene what we mean by a feasible cartel and characterize, for each feasible cartel, the set of attainable payo v ectors V ( s ). 2.1 Assumptions An activ e rm has to pay a (time-invariant) xed cost F i as well as a variable cost given by the function c ( q i ; i ; q i ) where i and q i are (time-invariant) rmspecic parameters and q i stands for quantity (the time index that should b e 5
assigned to the quantityvariable is omitted as long as this do es not create confusion). If a rm decides to stay out, it produces nothing and incurs no cost. Furthermore if a rm, say i , has decided to stay out at p erio d t 0 1, it must pay a reen try cost, R i , if it decides to stay in the mark et at p erio d t . We simplify the analysis by considering in turn two p olar cases namely: Assumption 1 Reentry is unprotable i.e. R i is as large as we want, for i =1 ; 2 ; 3 . Assumption 2 Reentry is costless i.e. R i =0 , for i =1 ; 2 ; 3 . The variable cost function c dep ends up on two rm-sp ecic parameters, i and q i . q i stands for the rm i 's capacity constraint whic h means that, for a giv en i , rm i cannot pro duce more than q i and accordingly c is only dened for 0 q i q i . On the other hand, i is a convenientway to rank rms according to their marginal cost function. We shall indeed suppose that for any quantity q such that the marginal cost to pro duce this quan tity is well dened for rms i and j , rm j 's marginal cost is strictly greater than the one of rm i if and only if j > i . More precisely way, let X i =[0 ; q i ], then our assumptions regarding the variable cost function of any rm are the following: Assumption 3 Let q i > 0 . The variable cost function is twicecontinuously dierentiable with respect to q i and i on X i 2 R ++ . In addition, c satises the fol lowing properties: 1. c (0; i ; q i )=0 , 8 i 2 R ++ , 2. 0 @c ( q i ; i ; q i ) =@ q i i , i 2 ]0 ; 1 [ , and @c ( q i ; i ; q i ) =@ i > 0 , 8 ( i ;q i ) 2 R ++ 2 X i , 3. @ 2 c ( q i ; ; q i ) =@ q i @ i > 0 , 8 ( i ;q i ) 2 R ++ 2 X i . Now let Q stand for aggregate output. At each p erio d, the in verse demand function for the homogeneous goo d, denoted f ( Q ), satises: Assumption 4 For al l Q 2 [0 ; P 3 i =1 q i ] , f is twicecontinuously dierentiable with f ( Q ) 0 and @f =@Q < 0 . 6
Note that we supp ose that rms pro duce p erfect substitutes in order to b e able to concentrate ourselves only up on the inuence of the cost characteristics on market structure. For an activ e rm the prot function (gross of the reen try cost) in the Cournot game will b e written as: i ( q i ;Q 0 i )= f ( q i + Q 0 i ) q i 0 c ( q i ; i ; q i ) 0 F i (1) where Q 0 i = Q 0 q i .We shall assume: Assumption 5 For al l i 2 N , i is strictly quasi-concave on X i 2 [0 ; P j 6 = i q j ] . Furthermore there exists ( q 1 ;q 2 ;q 3 ) 2 X 1 2 X 2 2 X 3 such that, for al l i , i ( q i ;Q 0 i ) > 0 . Obviously these assumptions, together with the restriction that an y active rm i must cho ose a quantity in [0 ; q i ], are sucient for the existence of a Cournot equilibrium. The second part of Assumption 5 will ensure, as we shall see later on, that there exists some \agreements" b etween the three rms with all of them remaining on the market. This could b e assumed away, in fact simplifying the analysis without changing the results, but it is kept for the sake of generality. The last restriction on the cost and demand functions is that whenever only two rms are activ e then, for any quantity its opp onent can pro duce, a rm can achieve a p ositiv e prot. Formally, let us dene w i ( s ) with s = f i; j g as the minimal pay o rm i can guarantee to itself when it faces rm j , i.e.: w i ( i; j ) = min q j 2 X j max q i 2 X i i ( q i ;q j ) Under assumption 4 i is a strictly decreasing function of q j . Therefore, dening q R i ( q j ) = arg max q i 2 X i i ( q i ; q j ), we have: w i ( i; j )= i ( q R i ( q j ) ; q j ) We shall require: Assumption 6 For al l i; j 2 N , w i ( i; j ) 0 and q R i ( q j ) < q i . This will guarantee on the one hand that the market cannot b e monopolized and on the other hand that there exist couples ( q i ;q j ) such that i ( q i ;q j ) >w i ( i; j ). Wenow turn to the c haracterization of the set of subgame p erfect equilibrium payo vectors of the innitely repeated game 0 ( s ) . 7
2.2 Equilibrium payos in 0 ( s ) The typical payo for rm i in the game 0 ( s ) is given by: P i =(1 0 ) 1 X t =0 t i ( q i ;Q 0 i ) Innitely rep eated games with discounting have b een extensively analysed in the literature. It has b een established (see, for instance, Theorem 3.2 in Sorin (1992)) that the set of subgame p erfect Nash equilibrium payo vectors of 0 ( s ) converges (with resp ect to the Haussdorf topology) to the set of individually rational and feasible payo v ectors of the constituent game as the discount factor tends to one 2 . This is one of the version of the so-called Folk Theorem. Accordingly, for our purp oses we only need to c haracterize the set of individually rational and feasible payo vectors of the Cournot one-shot game where the set of players is given by s . This set is denoted W ( s ). To b egin with let us denote the set of feasible pa yo vectors with three active rms by F ( N ) and the one with two activ e rms by F ( i; j ). Let X = X i 2 X j 2 X k , F ( N ) and F ( i; j ) are given by: F ( N ) = convex hull f ( P 1 ;P 2 ;P 3 ) j 9 ( q 1 ;q 2 ;q 3 ) 2 X such that P i = i ( q i ;Q 0 i ) for i =1 ; 2 ; 3 g : F ( i; j ) = convex hull f ( P i ;P j ) j 9 ( q i ;q j ; 0) 2 X such that P i = i ( q i ;Q 0 i ) and P j = j ( q j ;Q 0 j ) g : When the three rms are activ e, eac h activ e rm can guarantee to itself a prot giv en by: w i ( N )= min q j 2 X j ;q k 2 X k max q i 2 X i i ( q i ;q j + q k ) Accordingly the set of individually rational and feasible pa yo vectors when rms in s b eing activ e and # s 2 is simply: W ( s )= f ( P 1 ;P 2 ;P 3 ) j ( P i ) i 2 s 2F ( s ) ;P k = 0 for k 62 s and P i w i ( s ) 8 i 2 s g : (2) 2 Provided the set of individually rational and feasible p oin ts has a non-empt y in terior. This is clearly the case under Assumption 6. 8
strictly negative for any q j 2 ( q R ( q i ) ; q j ] and strictly p ositiv e for any q j 2 [0 ;q R ( q i )). Moreover, by assumption 3, marginal cost is increasing in so that @c ( q j ; j ; q j ) =@ j 0 @c ( q R ( q i ); j ; q j ) =@ j is strictly p ositive for any q j 2 ( q R ( q i ) ; q j ] and strictly negativ e for any q j 2 [0 ;q R ( q i )). It then follo ws that dq j =d j is strictly negativ e for any given q i 2 [0 ; q i ) and q j satisfying j ( q j ;q i )= w j ( i; j ). This establishes that the isoprot curv e j ( q j ;q i )= w j ( i; j ) shifts rightwards in the co ordinate ( q j ;q i )as j decreases. As shown in Figure 1 the result then follo ws. 2 As shown by the pro of, the xed cost does not matter since it aects b oth sides of the constraint in the maximization program in the same way. Hence, for any quantity produced by rm i , the quantity required to satisfy the constraint is indep enden t of the xed cost. On the other hand, the lev el of , i.e. the level of the marginal cost for a given quan tity pro duced, aects the constraint i n t wo ways. First, if the right-hand side of the constrain t were indep endent of then, in the co ordinates of Figure 1, rm k 's isoprot curve will b e en tirely belo w the rm j 's one as long as j < k . This eect reects the advantage to form a cartel with a low marginal cost rm. Second, however, the minimal payo required by a rm to participate in a cartel with rm i clearly decreases with . This translates the in tuition that a lo w marginal cost rm will be more greedy than a rm with a higher marginal cost. What the Lemma states is that the second eect dominates the rst one. According to this result, for k larger than j , rm k can always give to rm i a greater payo than the highest pay o rm i can obtain with rm j . On the other hand if cartel f i; j g forms then rm k will receiv e a zero payo while it will obtain at least w k ( i; k ) > 0 if cartel f i; k g forms. Hence, lo osely speaking, rm k has always the opportunity and the willingness to prev ent the formation of cartel f i; j g so that this cartel cannot form. Remark that we cannot exclude the formation of the grand cartel, N , for all orders of declaration. Indeed consider, for instance, the case where S = ff i; j g ; f i; k g ;N g with j < k and rm i is the rst rm to declare. If w k ( i; k ) is sucien tly large it could happ en that W i ( i; k ) is strictly smaller than the greatest payo rm i can obtain in V ( N ). Consequently, rm i will propose the formation of the grand cartel and the best either rm j or rm k (or b oth) can do is to mak e a declaration compatible with that 15
of rm i 8 . This shows that the availability of a predatory strategy is not sucien t for predation to o ccur. Finally, Part 3 of Prop osition 1 corresp onds to the example given in the Intro duction above, except for the feasibilit y of the grand cartel. However, to have the same prediction, i.e. that the lo w marginal cost rm is excluded for all orders of declarations, two additional requirements are needed. The rst is that the maximal capacity of rm j is smaller than that of rm i ; the second is that the maximal payo that j can obtain in the cartel f i; j g is larger than the one it can obtain in cartel ( N ). These conditions seem fairly unrestrictiv e: as rm i has a lower marginal cost function than j it is reasonable to assume that it has installed a higher capacit y; while it is quite plausible that a rm can obtain more in a two-rm than in a three- rm cartel. obviously, if cartel ( N ) was not feasible, as in the intro ductory example, then this second condition is trivially met. As it can be seen from the Pro of of Proposition 1 in the Appendix, these two conditions are superuous for all orders of declaration except when k is the rst to declare. In this case k in order to induce j to enter the cartel f j; k g m ust give to j apayo at least as great as the maxim um pa yo that j could obtain in the cartel f i; j g , W j ( i; j ). The same is true if k wants to induce i to enter the cartel f i; k g , that is k must giv e W i ( i; j ). Thus, k will prefer the cartel f j; k g if it gets a higher payo in it rather than in f i; k g , given the constraints imp osed by what he m ust oer to j and i . This is the case if the conditions in Part 3 of Proposition 1 are met, as it is sho wn by Lemma 4 in the App endix. 5 Equilibria of the cartel formation game with costless reentry It has b een shown in section 2.3, that the case with costless reen try diers from the one with unprotable reen try only by the fact that in the former case the minimal payo required by a rm, say j , to participate to a two- rms cartel, say f i; j g , is equal to max f 0 ;w j ( N ) g while in the latter case 8 Note that if the cartel f j; k g were also feasible then the grand cartel could form with rm i b eing the rst rm to declare and rm j (resp. rm k ) the second one provided that W j ( j; k ) (resp. W k ( j; k )) is strictly smaller than the highest payo rm j (resp. rm k ) can obtain in V ( N ). 16
it equals w j ( i; j ). Accordingly if cartels f i; j g and f i; k g are feasible, that is, if w j ( N ) and w k ( N ) are negative, then the minimal payo obtained by rms j and k in these cartels are equal to zero. Therefore, the highest payo rm i can obtain in a cartel, W i ( s ), dep ends on b oth the marginal and xed costs of its partner. More precisely, let q 0 h b e the rival's output which leads to zero prot for rm h when it plays its b est reply, q R h ( q o h ), that is, q o h is such that h ( q R h ( q 0 h ) ;q 0 h ) = 0. Furthermore, for all q 2 X 0 h with X 0 h =[0 ;q 0 h ), let ^ q h ( q ) b e the smallest quantity pro duced by h which gives it a zero prot whenever its rivals pro duce q , that is, ^ q h ( q ) is such that: h (^ q h ( q ) ;q ) 0 and @ h (^ q h ( q ) ;q ) =@ q h > 0. Wehave: Lemma 2 Suppose al l our assumptions except assumption 1 hold. There exists < 1 such that, if cartels f i; j g and f i; k g arefeasible and ( j ;F j ) and ( k ;F k ) are such that ^ q k ( q ) > ^ q j ( q ) for al l q 2 X 0 j \ X 0 k , then W i ( i; j ) > W i ( i; k ) >W i ( N ) for al l 2 ( ; 1) . Considering Figure 2, the proof of this result is clearly quite ob vious and is thus omitted. It must b e noticed that a necessary and sucien t condition for W i ( i; j ) >W i ( i; k ) to hold would involve a comparison of the cost structure of the three rms. We thus c ho ose to state our results in terms of a sucien t condition which actually requires only the comparison of rms j and k average cost function. Clearly, Lemma 2 here will play the role of Lemma 1 in the case of noreentry. It therefore follows: Prop osition 2 Suppose al l our assumptions except assumption 1 hold. There exists < 1 such that for al l 2 ( ; 1) and for any order of declaration we have: 1. A cartel forms, 2. let S = ff i; j g ; f i; k g ;N g and ^ q j ( q ) < ^ q k ( q ) for al l q 2 X 0 j \ X 0 k , then cartels f i; k g and N do not form, 3. let S = ff i; j g ; f i; k g ; f j; k g ;N g and ( i )^ q i ( q ) < ^ q j ( q ) for al l q 2 X 0 i \ X 0 j , ( ii )^ q i ( q ) < ^ q k ( q ) for al l q 2 X 0 i \ X 0 k , ( iii )^ q j ( q ) < ^ q k ( q ) for al l q 2 X 0 j \ X 0 k then cartels f i; k g ; f j; k g and N do not form. 17
This Proposition 9 contrasts with our previous results in two ways: First, the grand cartel, N , does not form, so that if seeing that there exists a predatory strategy it will b e played i.e. predation occurs . This comes from the fact that, as long as cartels f i; j g and f i; k g are feasible, the minimal payo rms j and k will obtain in b oth a two-rm cartel and in the grand cartel is equal to zero. It then follows that rm i can always obtain a larger payo in a two-rm cartel than in the grand cartel (see Lemma 3). Consequently if rm i is the rst rm to declare it will never prop ose the formation of the grand cartel. On the other hand if it is rm j (resp. rm k ) whic h is the rst to declare then it will never propose the formation of the grand cartel. Indeed if it do es so then b oth rm i and rm k (resp. rm j ) can obtain a higher payo than the one proposed in rm j 's (resp. rm k 's) declaration by making compatible declarations which propose the formation of the cartel f i; k g (resp. f i; j g ). The second dierence b etween the results with costless reentry and the ones with unprotable reentry can b e illustrated if we suppose that rms have iden tical xed costs 10 . In this case ^ q j ( q ) < ^ q k ( q ) for all q 2 X 0 j \ X 0 k will hold if and only if j < k . Then Proposition 2 states simply that the rm with the highest marginal cost function will b e predated. Therefore with costless reen try, contrary to what happ ens in the unproptable reentry case, a low marginal cost constitues a strong adv antage to face predation. On the other hand, if we suppose that i = j = k then the conditions used in Prop osition 2 will b e satised if and only if F i <F j <F k . Hence we nd back a result stated rst by Ghema w at and Nalebu (1985) for declining industries according to which the rm with the largest capacities i.e. with the highest xed cost lev el is the rst rm to exit the mark et. Suc h conclusion has also b e drawn by Fudenb erg and Tirole (1986) from the analysis of an incomplete information game. To conclude with, if we are able to rank the rms with resp ect to their average cost function then Prop osition 2 states that the exiting rm is the one with the highest average cost function. 9 The pro of of this Proposition follo ws so closely that of Prop osition 1 that it s omitted. 10 Recall that our results in the case of unprotable reentry do not dep end on the rms xed costs. 18
6 Robustness of the results with unprotable reentry One sp ecic feature of the cartel formation game presented ab ove is that each rm in its declaration prop oses simultaneously a particular cartel and the payos that each mem b er of the cartel will receiv e. As a consequence, the cartel formation game gives to al l rms a strong inuence on the way payos are allo cated among cartel mem b ers. This seems reasonable when reen try costs are negligible. In this case indeed the production game remains a three players game ev en if a rm exits the market. However when reen try costs are large, the pro duction game b ecomes a two players game once a rm decides to stay out of the mark et. In this case one can ask the question if the cartel formation game does not giv e to the exiting rm an unrealistically excessiv e inuence on the equilibrium of the resulting two rms pro duction game which shall b e play ed. In order to provide an answer, we shall analyze the sensitivit y of the ineciencies stated in Prop osition 1 to the way rms are supposed to co ordinate. Toinvestigate this issue we lo ok at a t wo step co ordination pro cess where the exiting rm has no inuence on the wa y the remaining rms will share the gains from co operation in the production game. This co ordination pro cess constitutes a game: its rst step is a substitute for the cartel formation game presented before. The only dierence is that it is no w supposed that a rm declaration only consists of a feasible cartel, s . If all declarations dier the game ends and each rm receiv es its reserv ation payo g i . Otherwise one mo ves to the second step. The second step consists of a negotiation b etween the members of the cartel given in the identical declarations of the rst step ,say s , to determine apay o vector, p , belonging to V ( s ). If a rm does not b elong to s then its action set in this step is simply f do nothing g . We shall not sp ecify explicitely the bargaining game pro cedure. W e assume instead that, the gains from co operation (i.e. the actual payo min us the sum of appropriately discounted Cournot prots of the one-shot quantity game) are shared according to a bargaining solution. The bargaining solution we adopt here b elongs to the family of egalitarian (also called proportional) 19
solutions as axiomatized by Kalai (1977) and Kalai and Samet (1985) 11 12 . To b e precise, let us rst assume that: Assumption 7 For any feasible cartel, s , the Cournot equilibrium in the quantity game is unique. Then, let c i ( i; j ) denote the rm i 's Cournot equilibrium prot when only rms i and j are activ e on the market. Furthermore denote by ( q e i ( i; j ) ;q e j ( i; j )) the quantity vector whic h maximizes P i sub ject to P i 0 c i ( i; j )= P j 0 c j ( i; j ) and let P e i ( i; j ) (resp. P e j ( i; j )) b e giv en by (1 0 ) P 1 t =0 t i ( q e i ( i; j ) ;q e j ( i; j )) (resp. (1 0 ) P 1 t =0 t j ( q e j ( i; j ) ;q e i ( i; j )) ). Obviously ( P e i ( i; j ) ;P e j ( i; j )) is the symmetric egalitarian solution 13 to the co op erativ e bargaining game de- ned by a set of outcomes given by F ( i; j ) and a statu-quo p oint given by ( c i ( i; j ) ; c j ( i; j )). We can immediately state: Lemma 3 Let al l our assumptions except 2 be satised. Furthermore, for any feasible two-rms cartel, say f h; l g , suppose that ( q e h ( h; l ) ;q e l ( h; l )) belongs to ]0 ; q h [ 2 ]0 ; q l [ and that there exits ( q h ;q l ) 0 which maximizes P h + P l . Then there exists < 1 such that, for al l > , P e i ( i; k ) >P e i ( i; j ) if and only if k > j . 11 This kind of structure has already b een used in the literature. F or instance, in Grossman and Hart (1986), two agents rst c ho ose non-co op eratively and sim ultaneously a level of investment and then, giv en these in vestments, take actions such that the gains from renegotiation, which correspond to the gains from co op eration in our framework, is shared equally. In their context, this corresponds also to the Nash bargaining solution. The Gro osman and Hart's analysis has b een extended by Hart and Mo ore (1990) to many agents and the bargaining solution adopted there to share the gain from trade is the Shapley value. We adopt here an egalitarian solution one the one hand because it is m uch more tractable than the other ones (in particular the Nash bargaining solution), and on the other hand b ecause the egalitarian solutions are the only ones which, in the presence of other standard requirements, satisfy the monotonicity prop ert y (see Kalai and Samet (1985)). This condition simply states that if the feasible set of one coalition increases and the feasible sets of all other coalitions remain the same, then none of the mem b ers of this coalition should b ecome worse o b ecause of this change. 12 Note that similar results could b e obtained by using the symmetric Nsah bargaining solution. 13 It will b e obvious to v erify that the results presented b elow will hold if we take an asymmetric egalitarian solution provided the weight of rm i in the solution dep ends negatively on i and is indep endent on the xed costs lev el. 20
Notice that the rst additional assumption in this Lemma will simply guarantee that there exists a feasible payo vector strictly greater than the Cournot equilibrium prots vector. Again an increase in j will havetwo eects on the co op erativ e bargaining game involving rms i and j : On the one hand, it leads to a mo dication in the set of feasible outcomes whic h aects negatively the payo of rm i at the egalitarian solution while, on the other hand, it increases (resp. decreases) rm i 's (resp. rm j 's) statu-quo payo whic h will rise the rm i 's payo at the egalitarian solution. The Lemma 14 states simply that the positiv e eect arising from the move in the statu-quo payo dominates the negative eect coming from the reduction in the set of feasible outcomes. This result will play, for Proposition 3 b elow, the role play ed by Lemma 1 and 2 for Proposition 1 and 2 resp ectiv ely .To see this it suces to realize that the set of subgame p erfect equilibria of the game deriving from the two step pro cedure here considered coincides with the one of the cartel formation game where a rm i 's declaration consists of a feasible cartel, s i , to whic h rm i b elongs and of a payo v ector whic h gives to each rm in s i the symmetric egalitarian payo dened above 15 and a zero payo to a rm (if any) which does not b elong to s i .Formally the set of rm i 's declarations is now D i = f ( s; p ) j i 2 s; s 2S ; for all h 2 sp h = p e h ( s ) and, for l 62 s; p l =0 g . Therefore we have: Prop osition 3 Let al l assumptions in Lemma 3 hold. Then there exists < 1 such that for al l 2 ( ; 1) and whatever the order of declaration we have: 1. A cartel forms, 2. if S = ff i; j g ; f i; k g ;N g and j < k , then cartel f i; j g does not form, 3. if S = ff i; j g ; f i; k g ; f j; k g ;N g and i < j < k , then cartels f i; j g and f i; k g do not form. 14 The pro of of this result comes quite straightforw ardly from the application of the envelop e theorem as well as the Folk theorem. Hence it will b e omitted. 15 To save space we do not dene formally the egalitarian pay o when the three rms are active. However this can easily be done even if one wants to consider a coalition form game instead of a co op erative bargaining game. Anyway this do es not matter for our analysis. 21
This shows the robustness of our conclusions with resp ect to the inuence of the exiting rm on the waypay os are allo cated in the production game. 7 Concluding remarks Wehave considered in this paper a dynamic pro duction game in volving three rms whic h are dierentiated according to their cost function. More precisely we h ave assumed that rms can b e ranked unam biguously according to their marginal cost function and that their xed cost ma y dier. Furthermore we suppose that one rm can b e credibly forced to stay out of the market by the two others and that at least tw o rms can b e put under such a threat. We then investigate the cost characteristics of the exiting rm under two alternativ e hyp othesis concerning the p ossibility of reen try namely the case where reen try is unprotable in any circumstances due to the presence of large sunk costs, and the one where reentry is costless. Wehave obtained two predictions (whic h appears quite robust to the specication of the cartel formation game). First if reen try is alwa ys unprofitable then the exiting rm has the lowest marginal cost function as compared with the marginal cost function of the rms which can credibly be predated. Furthermore this result do es not dep end on the lev el of xed costs 16 . Accordingly, in this case, cost ineciencies will arise since the exiting rm is the one which uses the most ecien t technology . A second result is that when reen try is costless and when we can rank rms according to their average cost function then the exiting rm has the larger average cost function as compared to the average cost function of the rms whic h can b e put under the threat of predation. Therefore in this case cost ineciencies do not appear. The result obtained in the no-reentry case lo oks strange since it goes against the common belief that the most ecien t rm will remain on the market. But this b elief has b een dev elopped in the context of \neo-classical economics". If instead we look at this result from the p oint of view of \transaction cost economics" (as dev elopped in Williamson (1985) for instance) then they app ear rather unsurprising. Indeed in this context such kind of ineciencies are frequen tly obtained. It is worthwhile emphasizing the deep 16 However the set of rms which can b e predated depend obviously on the level of xed costs. 22
relationship b etween our analysis and the transaction cost approach. Indeed although the latter approach fo cuses mainly on the in ternal organisation of the rm the present study shows that the basic p oints whic h distinguish transaction cost economics from other economic appro c hes are also well suited to study the comp osition of an industry and more generally to mak e substantial progresses in the understanding of the formation and comp osition of groups or coalitions on a market. Roughly speaking transaction cost economics seeks to analyse situations involving agents characterized by opportunism and boundedrationality where ( i ) agents will meet frequently ,( ii ) agents do not rely on courts for settling disputes among them i.e. private ordering prevails, ( iii ) agents have the opportunity to mak e asset specic investments and ( iv ) agents ev olvein an uncertain environment. In the present analysis we have ruled out b oth uncertainty and b ounded rationality since these characteristics appear unessential for our results. Note furthermore that frequency will not b e relevant here as the example given in the Intro duction p oints out. The dierence b et ween opportunism and self-in terested b ehavior does not matter here b ecause the set of subgame p erfect Nash equilibria and the set of Nash equilibria of the production game coincides for a discount factor sucien tly close to zero. We shall however argue that if we mak e abstraction of the presence of either priv ate ordering or asset specic investments then the cost ineciencies obtained in the pap er disappear. Let us b egin with private ordering. Many exchange analysis suppose that ecacious rules of law are in place so that any disagreemen t regarding the execution of a contract is settled by courts in a fully informed and low-cost way. This assumption of court ordering is very convenient since it allows to disregard the ex-p ost side of a contract. In our context, rms cannot rely on court since the kind of contract they are willing to do is simply illegal. An immediate consequence of private ordering is that we cannot disregard the execution phase of the contract since the latter m ust b e self enforcing. This entails that rms, as is supposed in the cartel formation game, will only consider payo vectors which can b e associated with a subgame p erfect Nash equilibrium of the pro duction game. But suppose to the contrary that rms can rely costlessly on court to enforce an agreement. This implies that the set of payo v ectors that m ust now b e considered in the cartel formation game coincides with the one cor23
resp onding to the costless reentry case. Indeed, a rm whic h can b e forced to exist can commit to obey an agreement in whic h it receiv es a zero payo. Without this p ossibilit y of commitmen t, such an agreemen t is not credible in the no-reentry case while it is in the case of costless reen try. Consequently, the result stated in Lemma 2 will hold ev en if reen try is unprotable and the exiting rm is the one with the highest average cost function (see Proposition 2 for a more precise statemen t). Therefore the cost ineciencies disappear once court ordering is allowed for. Remark that this clearly shows that considering tacit co op eration b etween rms as illegal is p ossibly costly. Let us now turn to the asset sp ecic character of in vestments. Investments are said wholly asset sp ecic if they are unredeplo yable. Accordingly investment costs are sunk for wholly asset sp ecic in v estments while they are xed when investment lo oses its asset sp ecic character. The main consequence of the presence of asset specic in v estment is the occurence of the fundamental transformation . The latter concept refers to the transformation in the nature of the comp etition prevailing b efore and after the adoption of the contract. In our context, the sunk reen try cost we ha ve in troduced can simply b e interpreted as the cost of unredeploy able investments. More precisely, the unprotable reen try case corresp onds to the situation where large asset specic in vestments m ust b e achieved b efore b eing activ e on the market while in the costless reentry case such in v estments are negligible. When reentry is unprotable the fundamental transformation o ccurs since, once a rm exits, the production game b ecomes a tw o players game. If instead reentry is costless this transformation does not o ccur. Indeed, in this case even if a rm exits it can participate to the punishmen t of a deviation from the equilibrium path by one of the two rms which remain on the mark et. In other words the pro duction game still in volves three players ev en if a rm exits the mark et. As we hav e shown, cost ineciencies app ear only in the case of unprotable reen try which means that the presence of large asset specic investments is a necessary condition for such cost ineciencies to occur. 24
8.2.4 Supp ose that rm k is the last rm to declare: 1. Let rm j b e the second rm to declare. For the cartel f i; j g to form it must b e the case that d i = d j , d i 2D j nD k . But if rm i makes a declaration b elonging to D j nD k , rm j will obtain at most W j ( i; j ) by declaring d j = d i while it will receiv e W j ( j; k ) if it declares d k j = ( f j; k g ; (0 ;W j ( j; k ) ;w k ( j; k ))), since for such declarations ( d i ;d k j ) rm k will maximize its payo by declaring d k = d k j . By Lemma 1 we know that, for sucien tly close to one, W j ( j; k ) >W j ( i; j ) and therefore cartel f i; j g does not form. For cartel f i; k g to form it must be the case that d i 62 D i \D j . If rm j makes a declaration whic h induces rm k to declare d k = d i then it will receiv e a zero payo. However for any p ik 2 [ w k ( i; k ) ;W k ( i; k )] we know by Lemma 1 that, for sucien tly close to one, there exists d j 2D j \D k such that p jk >p ik and p jj >w j ( j; k ) > 0. Therefore cartel f i; k g does not form. 2. Let rm i b e the second rm to declare. For the cartel f i; j g to form it m ust b e the case that d i = d j , d j 2D i nD k . But if rm j mak es such a declaration rm i will receiv e at most W i ( i; j )by declaring d i = d j while it will receiv e W i ( i; k ) if it declares d k i = ( f i; k g ; ( W i ( i; k ) ; 0 ;w k ( i; k ))) since for such ( d j ;d k i ) rm k will maximize its payo by declaring d k = d k i . By Lemma 1, for sucien tly close to one, W i ( i; k ) is stricly greater than W i ( i; j ) and therefore cartel f i; j g does not form. For cartel f i; k g to form it must b e the case that d i 2D k nD j and d i 62 D j \D k and d j is such that p ik p jk (the equality b etween p ik and p jk is allo wed only if rm k , facing two indieren t alternativ es, cho oses to declare d k = d j ). In this situation rm j will obtain a zero payo while rm k will obtain at most W k ( i; k ). However, by Lemma 1, we know that, for suciently close to one, there exists d j 2D j \D k such that p jk >W k ( i; k ) and p jj >w j ( j; k ) > 0. Consequently cartel f i; k g does not form. Summing up, if k is the last rm to declare, Lemma 1 is sucien t to ensure that the cartel which form is either f j; k g or N . 31
8.2.5 Supp ose that rm j is the last rm to declare: 1. Let rm k b e the second rm to declare. For the cartel f i; j g to form it m ust b e the case that d i 2D j nD k and d k is such that p kj p ij (the equalit y b etween p kj and p ij is allowed only if rm j , facing two indieren t alternativ es, c ho oses to declare d j = d i ). In this case rm j will obtain at most W j ( i; j ) while rm k will obtain a zero payo. However, by Lemma 1, if is sucien tly close to one then for any d i which do es not b elong to D i \D k there exists d k 2D j \D k such that p kj >p ij and p kk >w k ( j; k ) > 0. Therefore cartel f i; j g does not form. For cartel f i; k g to form it must b e the case that d i 2D k nD j and d i = d k . But since d i 62 D j rm j will maximize its payo by declaring d j = d k as long as d k 2D k and p kj > 0. Therefore once d i 62 D i \D j rm k can obtain a payo of at least W k ( j; k ) while it obtains at most W k ( i; k ) by declaring d k = d i . By Lemma 1, for sucien tly close to one, W k ( i; k ) <W k ( j; k ) and consequen tly cartel f i; k g does not form. 2. Let rm i b e the second rm to declare. Dene D R i ( d k )= f d i 2 D i j p ii p ki and ( d i ;d k ) is such that rm j 's b est-resp onse is d j = d i : g .For cartel f i; j g to form a necessary condition is that rm k makes a declaration such that D R i ( d k ) nD k 6 = ; . But if rm k makes such a declaration it will obtain a zero payo while we know by Lemma 1 that, for sucien tly close to one, there exists d k 2D k nD i such that it obtains a payo strictly greater than w k ( j; k ) > 0 and for which D R i ( d k ) nD k = ; . Consequently cartel f i; j g does not form. Now for cartel f i; k g to form it is necessary that rm k makes a declaration such that d k 62 D j and p ki max f W i ( i; j ) ;W i ( N ) g . Indeed if d k 62 D j then rm j will declare d j = d i as long as d i 2D j and p ij > 0. Accordingly for d k such that d k 62 D j rm i can obtain either W i ( i; j ) by declaring: d j i =( f i; j g ; ( W i ( i; j ) ;w j ( i; j ) ; 0)) or the maximal payo, denoted by P N i ( ) ; that rm i can obtain in V ( N ) when P j = > 0 and P k = 0, by declaring d N i =( N; ( P N i ( ) ;; 0). P N i ( ) tends to W i ( N ) when tends to zero. It follows that if d k 62 D j and p ki < max f W i ( i; j ) ;W i ( N ) g then rm i will maximize its payo by declaring either d j i or d N i and cartel f i; k g does not form. Con32
sequently if W i ( i; k ) <W i ( N ) cartel f i; k g does not form while if W i ( i; k ) W i ( N ) the maximal payo rm k can obtain when cartel f i; k g forms is equal or smaller than ~ w k ( i; k ). On the other hand for cartel f j; k g to form it is sucien t that rm k 's declaration b e such that d k 62 D i and p kj > max f W j ( i; j ) ;W j ( N ) g . Indeed for such rm k 's declaration there does not exist d i 2D i such that p ij p kj and thus rm j will declare d j = d k . The assumption that W j ( i; j ) W j ( N ) together with Lemma 1 ensure that, for sucien tly close to one, we have W j ( j; k ) > max f W j ( i; j ) ;W j ( N ) g = W j ( i; j ). Hence there exists d k 2D k nD i such that p kj > max f W j ( i; j ) ;W j ( N ) g . Futhermore, for suciently close to one, we also have by Lemma 4 that ~ w k ( i; k ) < ~ w k ( j; k ). This implies the existence of d k 2D k nD i such that p kj > max f W j ( i; j ) ;W j ( N ) g and p kk > ~ w k ( i; k ). Consequently cartel f i; k g does not form. 8.2.6 Supp ose that rm i is the last rm to declare: The arguments to prove the results stated in the proposition are so close than those used in the previous case that we omit them here. 9 Lemma 4 Let the maximal payo that rm k can obtain in V ( i; k ) sub ject to P i = W i ( i; j ) b e denoted by ~ w k ( i; k ).Similarly,~ w k ( j; k ) stands for the maximal payo that rm k can obtain in 2 sub ject to P j = W j ( i; j ). Lemma 4: Suppose all our assumptions except assumption 2 hold. Furthermore let S = ff i; j g ; f i; k g ; f j; k g ;N g , i < j < k , q j q i and W j ( i; j ) W j ( N ). Then there exists < 1 such that for all 2 ( ; 1) ~ w k ( i; k ) < ~ w k ( j; k ). The proof is available up on request. 33