Capacity and Reaction Functions of Joint-Stock Firms

Transcription

1 Intenatonal Jounal of Management, Accountng and Economcs ISSN (Onlne) Authos, All Rghts Reseed Capacty and Reacton Functons of Jont-Stock Fms Kazuho Ohnsh 1 Insttute fo Basc Economc Scence, Osaka, Japan Abstact Ths pape nestgates two-stage competton wth two ont-stock fms. In the fst stage, each fm ndependently chooses a leel of capacty. In the second stage, each fm ndependently chooses an output. The pape shows that each fm s eacton functon has a knk at ts nstalled capacty leel. In addton, the pape consdes nteactons between a ont-stock ncumbent and a potental ont-stock entant as well as a pa of establshed ont-stock fms. The pape demonstates that thee exst cases n whch the ont-stock ncumbent can dete enty by the potental ont-stock entant. Keywods: Capacty, ont-stock fm, eacton functon, enty. Cte ths atcle: Ohnsh, K. (2016). Capacty and Reacton Functons of Jont-Stock Fms. Intenatonal Jounal of Management, Accountng and Economcs, 3(3), Intoducton The possblty of fms usng excess capacty as a stategc dece n duopolstc competton has been studed by many economsts. Fo nstance, Dxt (1980) demonstates that an ncumbent nstallng excess capacty n stage one s able to dete enty by a potental entant n stage two. Wae (1984) consdes thee-stage competton n whch an ncumbent nstalls capacty n stage one, an entant nstalls capacty n stage two, and a Counot duopoly equlbum s acheed n stage thee. He concludes that although hs thee-stage equlbum s qualtately smla to Dxt s two-stage equlbum, t dffes n that the stategc adantage aalable to the fst moe s lessened. In addton, Podda (2003) examnes a two-stage model of stategc enty deteence (a la Dxt 1980) unde demand uncetanty, and demonstates that to mpoe ts stategc poston n the poduct maket competton an ncumbent wll choose a leel of capacty that may eman dle n a low state of demand. Howee, these studes examne the behaous of poft-maxmzng fms. 1 Coespondng autho s emal: ohnsh@e.people.o.p 166

2 Intenatonal Jounal of Management, Accountng and Economcs ISSN (Onlne) Authos, All Rghts Reseed We examne the behaous of ont-stock fms. Mead (1972) shows the dffeences n ncentes, shot-un adustment and so foth among ont-stock, labou-managed and poft-maxmzng fms. Hey (1981) estcts attenton to the case of a pefectly compette fm poducng a homogeneous fnal poduct wth nputs of labou and captal, and nestgates the behaous of ont-stock, labou-managed and poftmaxmzng fms. In addton, Ohnsh (2010) pesents the equlbum outcome of a quantty-settng model compsng a ont-stock fm and a poft-maxmzng fm, demonstates that ntoducng lfetme employment nto the model of Counot duopoly competton s benefcal only fo the ont-stock fm. In ths pape, we fst consde a two-stage model wth two ont-stock fms. In the fst stage, each fm non-coopeately chooses a leel of capacty. In the second stage, each fm non-coopeately chooses an output. We show that each fm s eacton functon s downwad slopng and has a knk at ts nstalled capacty leel. Next, we consde nteactons between a ont-stock ncumbent and a potental ontstock entant athe than a pa of establshed ont-stock fms. We demonstate that thee ae cases n whch the ont-stock ncumbent can dete enty by the potental ont-stock entant. The man pupose of ths pape s to show the eacton functons of the ont-stock fms n the duopoly model wth capacty. The Model Let us consde a maket wth two ont-stock poft-pe-capacty-maxmzng fms, fm 1 and fm 2. We do not consde the possblty of enty o ext. In the emande of ths pape, subscpts 1 and 2 denote fm 1 and fm 2, espectely. In addton, when and ae used, they should be undestood to denote 1 and 2 wth. The maket pce s detemned by the nese demand functon pq ( ), whee Q q1 q2. We assume that p ' 0 and p '' 0. The two stages of the game un as follows. In the fst stage, each fm smultaneously k 0 and ndependently chooses a leel of capacty. Nethe fm can educe o dspose of capacty. In the second stage, each fm smultaneously and ndependently chooses and q 0 sells an actual output. Theefoe, fm s poft pe capacty s gen by p( Q) q q f k p( Q) q ( ) q k f k f q k f q k (1) 167

3 Intenatonal Jounal of Management, Accountng and Economcs ISSN (Onlne) Authos, All Rghts Reseed whee 0 denotes the total cost of each unt of output, pe unt of capacty, and kq ( ) f 0 (0, ) s the constant cost s the fxed cost. Ohnsh (2015) adopts the captal nput q functon as a functon of output. Thoughout ths pape, we use subgame pefecton as an equlbum concept. The fact that nese demand s defned only fo non-negate outputs ensues that all outputs obtaned n equlbum ae non-negate. Reacton Functons We state both fms eacton functons n quanttes. We dee fm s best eacton q functon fom (1). If, then ts eacton functon s defned by p( Q) q q f R ( q) ag max q k (2) and f q k k, then ts eacton functon s defned by p( Q) q ( ) q k f R ( q) ag max q k (3) Theefoe, f fm nstalls k, then ts best esponse s shown as follows: R ( q ) f q k R ( q ) k f q k R ( q ) f q k (4) The Counot equlbum occus whee each fm maxmzes ts obecte wth espect to ts own output, gen ts al s output. The equlbum must satsfy the followng condtons: Fm ams to maxmze ts poft pe captal wth espect to ts own output, gen fm s output. The Counot equlbum must satsfy the followng condtons: If q k, then the fst-ode condton s p' q p 0 (5) and the second-ode condton s p" q 2 p ' 0 (6) If fm nstalls condton s k and educes ts magnal cost of poducton, then the fst-ode p ' q p 0 (7) and the second-ode condton s 168

4 Intenatonal Jounal of Management, Accountng and Economcs ISSN (Onlne) Authos, All Rghts Reseed p" q 2 p ' 0 (8) Moeoe, we hae p" q p ' R '( q ) R '( q ) p" q 2 p ' (9) Snce poposton: p ' 0 and p" 0, p" q p' L s negate. We now state the followng Poposton 1: Unde Counot competton, both slopng. R ( q ) and R ( q ) ae downwad Both fms eacton cues ae llustated n fgue 1. s the eacton cue epesentng the best quantty choce of fm n the esponse to the quantty sold by fm, f t has to ncu the full magnal costs of poducng any gen quantty. s the eacton cue of fm, f capacty has been sunk up to an amount at least equal to the desed quantty. R R R2 k2 B R2 0 k1 q1 Fgue 1 Capacty and eacton cues 169

5 Intenatonal Jounal of Management, Accountng and Economcs ISSN (Onlne) Authos, All Rghts Reseed Enty Suppose that fms 1 and 2 hae nstalled leels of capacty equal to and. Fom (4), fm 1 s eacton cue s the knked bold lne, and fm 2 s eacton cue s the knked boken bold lne. Each fm s eacton cue wll hae a flat segment at ts nstalled capacty leel. In ths secton, the al s taken to be a potental entant athe than an establshed fm. That s, the followng stuaton s consdeed. In the fst stage, fm 1 (establshed fm) chooses a leel of capacty, and fm 2 (potental entant) obsees the behaou of fm 1. In the second stage, fm 2 decdes whethe to ente the maket. If fm 2 entes, a Counot duopoly equlbum s acheed, wheeas f fm 2 does not ente, fm 1 keeps ts monopoly poston n the maket. We suppose that fm 2 s eacton cue meets C C at C ( q1, q2 ) E E E ( q, q ) D D D ( q, q ) k 1 k 2 and 1 2 as shown n fgue 2. Futhemoe, we suppose 1 2 as a pont on fm 2 s eacton cue, whee ts post-enty poft pe capacty s zeo and t does not ente. Fm 2 s eacton cue s dscontnuous at D, made up of the two segments llustated by the bold lne n fgue 2. at R2 C D E 0 A q1 Thee ae thee cases to consde. Fgue 2 Enty deteence 170

6 Intenatonal Jounal of Management, Accountng and Economcs ISSN (Onlne) Authos, All Rghts Reseed D () q1 q1 C Fm 2 does not ty to ente the maket at all. Fm 2 s enty s blocked. Enty beng eleant, fm 1 wll enoy a pue monopoly. () q1 q1 q1 C D E Fm 2 s ethe accommodated nto the maket o deteed. In accommodated, the postenty equlbum occus at the most appopate pont fo fm 1 between C and D on fm 2 s eacton cue. In deteed, fm 1 nstalls coespondng to D. Hence, f C D E q1 q1 q1, then the enty-deteence equlbum occus at A. k D 1 () q q E D 1 1 Fm 1 cannot hope to peent enty, so t can only seek the best aalable duopoly k 1 poston. Dependng on fm 1 s choce of, the post-enty equlbum can be at any pont between C and E on fm 2 s eacton cue. The pont C coesponds to fm 1 s smallest output that can be sustaned as a Counot equlbum. The pont E s fm 1 s lagest output that can be sustaned as a Counot equlbum. Theefoe, the post-enty equlbum occus at the most appopate pont fo fm 1 between C and E. Concluson We hae examned duopoly competton n whch each ont-stock fm chooses a leel of capacty. We hae establshed that the ont-stock fm s eacton functon s downwad slopng and has a knk at the leel of capacty. Next we hae consdeed nteactons between a ont-stock ncumbent and a potental ont-stock entant athe than a pa of establshed ont-stock fms. The ont-stock ncumbent does not always dete enty by choosng excess capacty, but we hae demonstated that t s a possblty. In ths pape, we hae examned a two-stage model. In the futue, we wll study aous long-un makets whee ont-stock fms exst. Refeences Bonn, J. P., Jones, D. C., & Putteman, L. (1993). Theoetcal and empcal studes of poduce coopeates: Wll ee the twan meet? Jounal of Economc Lteatue, 31, Bulow, J., Geanakoplos, J., & Klempee, P. (1985). Holdng dle capacty to dete enty. Economc Jounal, 95, Dxt, A. (1980). The ole of nestment n enty-deteence. Economc Jounal, 90,

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