Markovian Equilibrium in a Model of Investment Under Imperfect Competition

Transcription

1 Markovan Equlbrum n a Model of Investment Under Imperfect Competton Thomas Fagart 8th January 2014 Abstract In ths paper, we develop and analyze a classc dynamc model of rreversble nvestment under mperfect competton and stochastc demand. We characterze the markovan equlbrum when player s strateges are contnuous n the state varable. At the equlbrum, rms nvest as quckly as possble n order to jon a zone n the space of capactes where there s no competton pressure. Furthermore, the equlbrum as an e cency property: the pont of ths area whch s reach by the rms s the pont whch mnmzes the nvestment cost of the all ndustry. Keywords: Capacty nvestment and dsnvestment. Dynamc stochastc games. Markov-perfect equlbrum. Real opton games. Pars School of Economcs - Unversty of Pars 1. Centre d Econome de la Sorbonne, boulevard de l Hôptal, Pars. Emal: fagart1@hotmal.com. 1

2 Contents 1 Introducton 3 2 Investment n the one-perod game The one-shot game model Best responses Characterzaton of the equlbrum Investment n a dynamc game The dynamc model Characterzaton of the contnuous markov equlbrum Concluson 15 References 17 2

3 1 Introducton Capacty expanson or reducton under uncertanty s one of the most mportant decsons that rms can made. It mpacts ther mmedate pro t and creates some long-run commtments. In a dynamc settng, the nvestment pattern by a monopoly s well known. Because of the uncertanty, the rm has ncentves to delay a pro table project (n expectaton) n order to wat for more nformaton about the demand. Ths s the theory of real optons. What happens n mperfect competton becomes the theory of real opton games. (For recent surveys, see for nstance Boyer, Gravel and Lasserre (2004), Azevedo and Paxson (2010), or Chevaler-Rognant et al (2011)). In ths lterature several authors focused on capacty decson under uncertanty. In these models, at each tme the pro t made by a rm depends of ts sze (.e. ts capacty), the sze of ts opponents, and a parameter whch evolves randomly wth tme (whch can be a parameter of demand or cost, the mportant pont beng that ts future evoluton are unknown). As ths parameter evolves, rms wsh to adapt ther szes. Frms can ether nvest to ncrease ther capacty, dsnvest to reduce t, or let ther capacty deprecates at ts natural rate. We speak of perfectly reversble nvestment when the cost of nvestng s equal to the cost of dsnvestng, n whch case rms can perfectly adapt themselves to the stochastc evoluton of the parameter. Such repeated game framework s classc n ndustral organzaton. However, n realty, ncreasng ts sze mples hrng new employees, buldng new factores or o ce, buyng new equpment, and so on... These nvestments are usually at least partally sunk, and the rm s sze decsons are not perfectly reversble. We speak of totally rreversble nvestment when rms cannot reduce they szes, and of partally rreversble nvestment when rms can decreases ther sze by dsnvestng (but wth a scrap value nferor to the cost of nvestng) or by deprecaton. In these cases, the theory of real opton game permts to lnk the hysteress due to the rreversblty of nvestment and the mperfect competton. In ths doman, Baldursson (1998), Grenader (2002), Back and Paulsen (2009) and Chevaler-Rognant, Huchzermeer and Trgeorgs (2011) focus on the same model of nvestment. Capactes are quas-rreversble (wth lnear prce), tme s contnuous and the parameter of demand follows a stochastc d uson. Baldursson (1998) and Grenader 3

4 (2002) exhbt an equlbrum, and show that the optmal nvestment s myopc, n the sense that rms can maxmze ther pro t assumng that strateges of other players are constant. However, Back and Paulsen (2009) shows that ths equlbrum s an open-loop one, whch fals sub-game perfecton. Then, Chevaler-Rognant, Huchzermeer and Trgeorgs (2011), focuses on markov-perfect equlbrum. The authors descrbe the optmal markovan best response of the rms. However, the lnearty of the nvestment cost mples n nte value for the amount of nvestment (the captal jumps, as there s no nterest to delay purchases and sells of captal). Ths prevents them to fully characterze the equlbrum. Our paper attempts to ll ths gap. To do so, we start studyng the smplest nvestment game. In a one shot model, rm have ntal capactes and can nvest or dsnvest n a quas-rreversble way ( rms can buy or sell capactes at lnear but d erent prces). We exhbt an area n the space of capactes, that we name the no-move zone. If the capactes of the rms are nsde ths no-move zone, no rm wll nether nvest nor dsnvest. So all pont of ths no-move zone s a possble equlbrum, gven some ntal capactes. We show that for some gven ntal capactes, the equlbrum s the pont of the no-move zone whch mnmzes the costs of nvestment and dsnvestment for the all ndustry. Ths e cency result holds even f rms have a pror no nterest to coordnate ther decsons. Furthermore, as long as the prces of nvestment and dsnvestment are not equal, the no-move zone s not reduced to one pont, and an ntal asymmetry n capactes can be preserved. In a contnuous settng wth demand uncertanty, we model the d erental game n an unusual way to overcome the ssue of n nte nvestment of Chevaler-Rognant, Huchzermeer and Trgeorgs (2011). We nd the exstence of no-move zone, dependng of the demand parameter, such that rms behave as n the one shot game. At each tme rms jon the no-move zone mnmzng the ndustry costs of nvestment and dsnvestment. Furthermore, ths equlbrum s unque n the class of contnuous markovan equlbrum. The plan of ths paper s the followng. Secton 2, studes the one-shot model. Secton 3 presents the dynamc model and characterzes the markovan equlbra. Secton 4 concludes. All omtted proofs are reported n appendx. 4

5 2 Investment n the one-perod game 2.1 The one-shot game model The am of ths subsecton s to abstract from dynamcs and uncertanty ssues, n order to focus on the e ect of partal rreversblty of nvestment. To do so, we present a smple statc model of competton n capacty. More precsely, consder a market wth n rms competng à la Cournot n capactes. Each rm starts wth some amount of captal k, whch can be extended or reduced through buyng or sellng some assets. Purchases are made at a lnear prce p +, and sales at a (also lnear) prce p (wth p p + ). We name K the capacty nally nstalled by rm. Let k be the vector of ndustry s ntal capactes and K the vector of nstalled capactes. For rm, the cost of nstallng a new capacty s: 8 9 < p + (K k ) f K k = C(K ; k ) = : p (K k ) f K < k ; : (1) Frms produce and sell an homogenous good, at a prce dependng of the total quantty q = P n =1 q. Each rm s producton depends of ts capacty, accordng to the technology q = K, 1 and has a cost, c (q ). Such technology s classc n dynamc nvestment models, and has been used by Fudenberg and Trole (1983), Grenader (2002), Merh and Zervos (2007) among others 2. So, by sellng the quantty K, rm obtans a payo of: (K) = P K K c (K ); (2) 1 In the one-shot model, ths technology can be seen as the result of an endogenzed game, n whch rms buy capactes and then play a Cournot competton lmted by the capacty prevously bought. Indeed, no rms have nterest to nvest n capacty whch wll not be used to produce, as ts opponents only react to the nal quantty. (Except f the dsnvestment prce s negatve. In ths case a rm has nterest to keep ts unused capacty n order to avod a dsnvestment case. For example, ths s the case for polluted producton ste, for whch the cost of decontamnaton s more mportant than the cost of conservaton of ths asset.) 2 As t was shown n Reynolds (1987), ths technology assumpton s also the result of a dynamc games wth lmted Cournot competton, wthout uncertanty. However, when there s uncertanty, rms have an ncentve to keep ther unused capacty for a possble further uses, when demands ncreases. Assumng that quanttes are equal to capactes permts to avod such adaptablty e ects and focus on the drect e ect of uncertanty on capacty choce. 5

6 where K = P n j=1 Kj. The pro t functon of rm s thus: Note that f p (K; k) = (K) C(K ; k ): (3) = p +, the nvestment decson s totally reversble, and the ntal capacty varable has no mpact. The smaller s p, the more rreversble are the capactes of the rm, and, at the lmt, when p = 1, nvestment s totally rreversble, as n Grenader (2002), Back and Paulsen (2009), Boyer, Lasserre and Moreaux (2012), and others. In order to ensure the exstence of the equlbrum we make the followng usual hypothess: H1: For each = 1; ::; n, c (:) s a twce-d erentable postve functon such that c 0 0, c P (:) s also a twce-d erentable postve functon, wth P 0 < 0, P 00 < 0 when P s strctly postve. 3 Furthermore, for all = 1; ::; n, P 00 (q)q < c 00 (q ). 2.2 Best responses In ths part, we present the best response of rm. Assume that for all j 6=, rm j nstalls a capacty K j. Obvously, the margnal revenue of capacty for rm depends of the choces of ts opponents, and we (x) the nverse functon of the margnal revenue of capacty, 1(x) = K 1 ; ::; K = K; ::; K n By concavty of n K, such nverse functon s well de ned and ncreasng. Then, rm has three possble choces: nvests (K > k ), and makng a pro t: = (K) p + K k ; 3 These condtons are not the more restrctve one could make n order to obtan theorem 1 (presented page 8). Indeed, our proof rests on the thrd theorem of Novshek (1985), and the lnearty of cost of nvestment and dsnvestment. However, as our pont of nterest s the dynamc game, and we need some regularty for the exstence of the dynamc d erentable equatons, we place our self n the assumpton made by Szdarovszky and Yakowtz (1977). 6

7 leadng to an optmal choce of (p + k ). Hence, f k (p + k ), the monopoly has no nterest to nvest. dsnvests (K < k ), and makng a pro t: = (K) + p k K ; leadng to an optmal (p ). Thus, the rm has no nterest to dsnvest f k (p ) (whch s hgher (p + ) ). the last possblty s to do nothng (the rm nether nvest nor dsnvest). Indeed, (:) s ncreasng, the ntal capacty of the rm can be greater (p + k ), so the rm has no nterest to nvest more, but also smaller (p ), so the rm has also no nterest to Therefore there exsts two thresholds, (p ) (p + ) such that the rm does not wsh to nvest nor dsnvest f ts captal s between ths thresholds, nvest (p + ) f ts ntal capacty s small, and dsnvest (p ) f ts ntal capacty s large. Ths can be summarzed n the followng proposton: Proposton 1 : The best response of rm s: >< (p + ) f k (p + ) h KBR = k f 2 (p + (p ) (p ) f k (p ) 9 >= : (4) >; Of course, ths best response depends of the capactes of other rms, of the capacty of all rms. Graphc 1 represents the best response of rm 2 n the space of capacty, for a duopoly wth lnear demand and no producton costs. [Insert G1] In ths graphc, we can see the exstence of an area n the space of capactes, 2, lmted 2 (pk ) (p + 2 k ), such that, t s never optmal for rm 2 to be outsde ths area. Thus, f there s an equlbrum, t belongs to for all rm, so t belongs to the ntersecton of these areas. Let H be ths ntersecton. We know that all equlbra belong 7

8 to H. Furthermore, assume that the ntal dstrbuton of capacty belongs to H. Then, f all players except keep ther capacty constant, then the best response of s to mantan ts ntal capacty. In the case of duopoly wth lnear demand and no producton cost, ths can be seen n graphc 2. We called H the no-move zone. [Insert G2] 2.3 Characterzaton of the equlbrum In the last subsecton, we show the exstence of an area n the set of space capactes, the no-move zone H, such that all equlbra belong to H. If the ntal capactes of the rms are n H, the equlbrum s to keep the same capactes. Ths no-move zone s de ned by, whch can rewrtten as H = K 2 R n + : 8 = 1; 2 [p ; p+ ], (5) H = K 2 R n + : 8 = 1; ::; n; P K + P 0 K K c (K ) 2 [p ; p + ] (6) However, we stll do not know what the equlbrum s when the ntal capactes does not belongs to H. Theorem 1 solve ths pont. Theorem 1: Assume H1. ver es: Then, there exsts only one Nash equlbrum K, whch K = arg mn K2H Ths condton s equvalent to the dstance condton, nx C(K ; k ): (7) =1 K = arg mn K2H nx K =1 k. (8) Theorem 1 provdes exstence and unqueness for the equlbrum, and ts characterzaton. To understand ths characterzaton, let focus on the no-move zone. In ths area, 8

9 accordng to (5), the margnal revenue of each rm s nferor to the prce of addng a new capacty, but superor to the prce of sellng some capacty, so no rms wsh to nvest nor dsnvest. The no-move zone s thus the set n the space of capacty such that no rms change ts capacty at the equlbrum. By theorem 1, H can also be vew as the set of all possble equlbra, n the meanng that all equlbrum belongs to H and all pont of H can be an equlbrum for some ntal value. There exsts another way to thnk of H. If we de ne the compettve pressure as the fact to react (for a rm) to a change n opponents strategy, then the no-move zone s the set of capactes wthout any compettve pressure. Indeed, assume that rms decde to nstall a capacty whch belongs to H and that one rm changes ts strategy to mplement another capacty. Then, f the new vector of capacty belongs to the no-move zone, the other rms have no new ncentves to change ther capactes. It does not mean that ther strategy s optmal, but f for one rm the best strategy was to dsnvest at the rst place, ts best strategy wll stll be to dsnvest after the change of capacty. In ths lght, the best response (4) can be renterpret as the pont of H whch mnmzes ts cost of mplementaton,.e. the cheapest vector of capacty wthout compettve pressure. In theorem 1, (7) establshes that the equlbrum s the vector of capacty n the no-move zone whch mnmzes ts total cost of nstallaton, for the all ndustry. So there exsts of some e cency n the market, n the meanng that the equlbrum wll coordnate the ndvdual cost mnmzaton of each rm to a global cost mnmzaton to reach the no-move zone, a poston where there s no competton pressure. Of course, n case of totally reversble nvestment (when p + = p ) the no-move zone s reduced to a unque pont, and no ndustry e cency appears. Ths s the usual Cournot competton 4. When nvestment s not totally reversble, the no-move zone s a set, none reduced to a sngleton, and each pont of ths set can be an equlbrum for some ntal values. Graphc 3 presents whch ponts of the no-move zone wll be an equlbrum, n functon of the ntal capactes, for a duopoly wth lnear demand and no producton costs. As t can be seen on the graphc, rms wth d erent ntal capactes can stll be asymmetrc at the equlbrum, and there are d erent possble symmetrc equlbra, even when rms has the same pro t functon. 4 Wth a cost c (K ) + p + K. 9

10 [Insert G3] 3 Investment n a dynamc game 3.1 The dynamc model In ths secton, we use a varaton of a classc model, as used n Baldursson (1998), Grenader (2002), Back and Paulsen (2009), and Chevaler-Rognant, Huchzermeer and Trgeorgs (2011), to study the e ect of dynamc and uncertanty on the prevous results. Baldursson (1998) and Grenader (2002) exhbt open-loop equlbrum for olgopoly. However, Back and Paulsen (2009) shows that ths equlbrum fals sub-game perfecton. Thus, Chevaler-Rognant, Huchzermeer and Trgeorgs (2011) focus on markovan equlbrum, and nd nterestng propertes. In ths part, we take a further step, and fully characterze the markovan equlbrum. Let Kt be the captal of rm at tme t, tme s contnuous and captal s partally reversble. Let (A t ; Kt 1 ; ::; Kt n ) be the nstantaneous payo of rm, A t beng the parameter of uncertanty, followng a d uson process: da t = (A t )dt + (A t )dw t ; (9) where W t s a standard Wener process. We assume Cournot competton. Let P At (:) be the nverse demand functon, dependng of the level of demand A t, and some producton cost for each rm, c (:) such that (A t ; Kt 1 ; ::; Kt n ) = P At Kt K t c (Kt). (10) In the followng, we assume that the prce s a contnuous functon of A t, and H1. The nterest rate s r and, as prevously, the purchase prce of captal s p +, and the sellng prce p. The total expected pro t of rm at tme 0 s thus: = E Z +1 0 e rt (A t ; K 1 t ; ::; K n t )dt p + Z e rt dk + t + p Z +1 0 e rt dk t ja 0. (11)

11 The objectve of each rm s to maxmze ts own expected pro t, gven the ntal levels of captal and demand 5. In ths framework, the usual method s to ntroduced I t, the nvestment done by rm at date t, so that the captal of each rm s determned by the followng d erental = I t. (12) Chevaler-Rognant, Huchzermeer and Trgeorgs (2011) assume that nvestment s markovan, so rm chooses a functon of the demand level and the captal of each rm, and at each tme, t nvests accordng to the value of the state varable (demand level, captal), I t = ~ I (A t ; K 1 t ; ::; K n t ). In ths case, the Bellman formula gves: r (A; K) = sup I t 8 < P At K K p + (It) + p (It) + : + P n j=1 I j (A; + P n P j6= j j=1 j6= j6= (@A) 2 I j (A; K)I h h The optmal nvestment I t p + (I t) + p (I t). 2 [p ; p + ], the rm has no nterest to nvest nor dsnvest. Otherwse, the optmal ow of nvestment It s n nte: the rm nstalls ts optmal captal (captal n the regon 1 ([p ; p + ])) nstantly. The optmal captal polcy of the rm s thus to jump @K 1 pk ; p + K, and to do nothng as long as the captal stays n ths area. So the optmal strategy cannot be de ned by the nvestment varable, as the lnearty of the cost of nvestment mples non-contnuous captal strateges. To avod such d culty, we focus on the choce of capacty nstead of nvestment, assumng that K t s the markovan control of rm. If tme was dscrete, then the choce of capacty at tme t s just a functon of the level of captal at tme t optmal control at tme t (K t) depends on the state varable at tme t (whch s K t 9 = ;. (13) 1. In ths case, the 1), whch s the markovan control of the prevous perod. To mmc ths constructon n contnuous tme, the choce of captal at tme t should depend on a state varable representng the level 5 As usual, ths maxmzaton s done n the set of left-contnuous A t -adapted stochastc process. Furthermore, n order to ensure the exstence of (11) we also assume that the process has nte varaton. Ths assumpton s natural wth our cost functon. Indeed, f a rm have a n nte varaton of ts captal, t wll pay an n nte cost of nvestment and dsnvestment (as p + > p ). However, ts future revenue s nte (due to hypothess H4), whch leads to a negatve and n nte pro t! 11

12 of captal of the ndustry just before tme t. As for all s there exsts another s closer to t, the state varable wll be the left lmt of ndustry capactes. Ths permts to de ne a d erent modelzaton. De nton: The nvestment game prevously consdered s n ts markovan state-control form f: () the state varable at tme t s k t = (k 1 t ; ::; k n t ), as de ned by where k 0 s the gven ntal level of captal; kt = lm Kt, (14) s!t s<t () for each player, the strategc varable s ts capacty, and the strategy s markovan,.e. the rm choose a functon, ~ K, of the state varable (ndustry s capactes and level of demand) and K t = ~ K (A t ; k t ). In such framework, a markov perfect equlbrum s de ned as usually, by the vector of functons K ~ (:; :) = ~K 1 (:; :) ; ::; K ~ n (:; :) such that, for all (A; k) 2 R n+1 +, 8 2 f1; ::; ng; K ~ (A; k) 2 arg max E[ (A; k; K ; ( K ~ j ) j6= ) j A]. (15) K (:;:) Furthermore, a contnuous markovan equlbrum s de ned as a markovan equlbrum n whch the functons ~ K 1 (:; :),.., ~ K n (:; :) are contnuous. To our knowledge, ths s a new way to model markovan strategy. In our problem, such de nton allows to properly de ne the best responses of the rms. In the next secton, proposton 2 ver es that the best responses are the same n both model. In addton, ths de nton allows us to characterze the markov perfect equlbrum when we assume that the strateges are contnuous functons of the state varable. Theorem 2 presents the parallel wth the one-shot game n a general framework. 3.2 Characterzaton of the contnuous markov equlbrum In ths subsecton, we characterze the contnuous markovan equlbra. We start by ntroducng techncal assumptons. H2 s needed to prove proposton 2 (n order to use Ito s Lemma, to nverse the Ito s Lemma results and to apply theorem 1). H3 s classc to ensure 12

13 the exstence of strong soluton to (9). H4 ensures the exstence of the stochastc ntegral determnng the pro t of the rms. H2: For each = 1; ::; n, c (:) s a four tmes d erentable postve functon such that c 0 0, c 00 0, and for all A 2 R +, P (:; :) s also four tmes d erentable postve and strctly concave functon n each varable. Furthermore, for all = 1; ::; n, P 00 (A; q)q < c 00 (q ). H3: (A) and (A) are contnuous functons, and verfy the Lpschtz condtons. H4: There exsts a functon G : R +! R +, such that, 8(A; x) 2 R 2 +, xp (A; x) < G(A), and R +1 0 e rt G(A t )dt < +1. These assumptons allow to state proposton 2, whch gves the form of the best response n the markovan state-control form of the game. Proposton 2: Assume H2, H3 and H4. Let 2 f1; ::; ng. In the markovan statecontrol form of the game, assume that for all j 6=, the strategy of rm j, ~ K j (:; :) s a contnuous functon of the state varable. Then, for all (A; k) 2 R n+1 + there exsts some 6 contnuous decreasng functon : R +! R + such that the best response of rm s: 8 >< ~K (A; k) = >: A;k (p ) f k > A;k (p ) k f k 2 A;k (p + ); A;k (p ) A;k (p + ) f k < A;k (p + ) Furthermore, the best response of rm s contnuous n the state varable. 9 >= : (16) >; Ths proposton shows that the optmal capacty of rm can jump: f at some tme t, k t s strctly smaller than A;k (p + ), then the rm has nterest n nvestng nstantly to A;k (p + ). In ths case the nvestment n perod t s n nte, so the markovan statecontrol form gves the same result as the regular form. However t also allows to go a step further and to characterze the equlbra, as presented n theorem 2. In fact, at each tme, everythng happens as n the one-shot game presented n the last secton (wth of course some mod caton of the no-move zone H n order to take nto account the future pro t). Frms always want to nvest or dsnvest forthwth n order to reach the no-move zone. As 6 The mplct de nton of s gven n the proof n appendx but, for smplcty, s not presented here. In partcular, we have the property than s contnuous and d erentable n a and k. 13

14 long as they are n the no-move zone, no rms change ts capacty. The shape of the no-move zone depends on the expected amount of money earned over tme by unt of capacty when rms keep ther capacty constant, 8x > 0; v(a; x) = E Z +1 0 P (A t ; x) e rt dtja. (17) Ths value v s the nte soluton to the followng d erental equaton 7 : 8x > 0; rv(a; x) = P (A; x) (A; x) v 2 (A; x): (18) 2 (@A) The no-move zone s then de ned by: H v (A) = k 2 R n +j8 2 f1; ::; ng : v k + va 0 k k 1 r c0 (k ) 2 [p ; p + ]. (19) Now, we can present theorem 2. Theorem 2: Assume H2, H3 and H4. Then, there exsts at most one contnuous markov perfect equlbrum, K ~ (:; :) = ~K 1 (:; :) ; ::; K ~ n (:; :). Furthermore, for all (A; k) 2 R n+1 + ; 2 f1; ::; ng, ths equlbrum ver es: ~K (A; k) = arg mn K2H v(a) Ths condton s equvalent to the dstance condton, nx C(K ; k ). (20) =1 ~K (A; k) = arg mn K2H v(a) nx K =1 k. (21) Theorem 2 s the analog of theorem 1, but n a contnuous tme settng. It ensures the unqueness of the contnuous markovan equlbrum, and characterzes t. At each tme, rms nvest (or dsnvest) n order to jon the no-move zone at the smallest possble cost for the ndustry. However, ths e cency result s tme-myopc. When demand evolves, rms face nvestment and dsnvestment perod, leadng to a costly path of nvestment. Ths path 7 Notes that (18) s not the classc Bellman d erental equaton. Indeed, n our proof, we just need to characterze the pro t nsde the no-move zone, as we already know what happens outsde the no-move zone. So (18) s just due to Ito s lemma applcaton to the evoluton of uncertanty. 14

15 of nvestment verfy the dynamc compettve pressure, and belongs to H v (A t ) at each tme t. But there s no reason to thnk that another path compatble wth the dynamc compettve pressure, less senstve to the evoluton of demand, whch can not be less costly on the all perod. Ths theorem focuses on the contnuous markovan equlbrum. Nevertheless, one can ask about other markovan equlbrum. Indeed, there s a pror no reason that colluson can not be sustan by markovan strategy n a d erental game. In a companon paper on the same nvestment game, (Fagart (2013)), we exhbt a markovan equlbrum wth tt-for-tat strategy mplementng the monopoly pro t. 8 4 Concluson In ths work, we characterze the contnuous markovan equlbrum of a classc model of nvestment under uncertanty whth Cournot competton. We establsh the exstence of an area n the space of rms capactes, the no-move zone, such that rms nvest or dsnvest n order to jon ths area as soon as possble, and keep ther capacty constant when they are nsde ths area. At the equlbrum, the rms reach the pont of the no-move zone whch mnmzes the cost of nvestment of the all ndustry. The ntuton on ths result s that the no-move s the area where other rms actons do not mpact the acton of a rm, so the e cency of the equlbrum comes from the absence of compettve pressure nsde the no-move zone. The exstence of ths area s due to the rreversblty of nvestment, and when nvestment s perfectly reversble, the no-move zone s reduces to a unque pont, as n usual Cournot competton. In the other case, the no-move zone s a set of vector of capacty, each 8 Such collusve equlbra arse because we assume that the strategy of the players rests on the state varable. In the classc modelzaton of d erental games, the strategy of the players leads on the dervatve of the state varable, whch mposes that the state varable wll evolves contnuously wth tme. As shown by theorem 2 n our case, ths contnuty assumpton provdes the ndng of collusve equlbra when the strategy of the player rests on the dervatve of the state varable. If the ndng of collusve equlbra s an argument n favor of our modelzaton, t also brngs the queston to know why the monopoly pro t can not be mplemented by contnuous strategy. 15

16 pont of t can be an equlbrum for some ntal value. It mples that the asymmetry of an ndustry can be preserved at the equlbrum, even f rms have the same pro t functon. However, when the tme goes by, the parameter of uncertanty evolves, and the no-move zone evolves wth t. The ssue of the evoluton of an ndustry wll be study n our companon paper (Fagart (2013)). In partcular, we show that even f rm s asymmetry s preserved n the short run, t dsappears n the long run, and that shocks of demand have an mpact whch depends of the sze of the rm consdered. 16

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