The Games of Cournot Sports

Transcription

1 Appled Mathematcal Scences, Vol. 7, 013, no. 4, Managers Compensaton and Colluse Behaour under Cournot Olgopoly Marco A. Marn Department of Computer, Control and Management Engneerng Unerstà d Roma La Sapenza Va Arosto, 5, Roma, Italy marn@ds.unroma1.t Abstract Ths paper shows that the exstence of a concrete outsde opton for frms' hgh-sklled workers (for nstance, the managers may nduce, under specfc crcumstances, olgopolstc frms to adopt restrcte output practses. In partcular, the paper characterzes the condtons under whch, n a Cournot olgopoly, exstng frms behae more collusely than n a standard Cournot model wthout endogenous wages. Keywords: Managers Compensaton, Olgopoly, Colluson. 1 Introducton In a economy where ether the barrers to entry are not a serous obstacle or n whch there s a way n whch nacte workforce can become acte - through the consttuton of self-employed unts, partnershps or entrepreneural frms alke - eery worker's outsde opton cannot be treated as exogenous. It should, ndeed, be consdered as part of the model arables,.e., n economc words, be ewed as endogenous to the model tself. On the other hand, although there exsts a well known stream of economc lterature (See, amongst others, [1], [3], [4]. clamng that, under olgopoly, the delegaton of sales decsons to managers can both be proftable for the frms' owners as well as boost eery frm's output wth respect to a standard Cournot model, an ncrease n managers' compensaton due to the managers' leang threat can also be thought to yeld opposte results. Whlst n prncple the ncente package for managers s responsblty of the owners, most frequently pay s set by execute compensaton commttees consstng of non-execute drectors or by senor executes themseles. Snce managers' compensaton s a cost component for the company, t may well be argued that the fat cats phenomenon wthn companes reduces competton and output n a gen market, when compared to a standard olgopoly model.

2 0 Marco A. Marn The model presented n ths paper explores ths possblty. It descrbes a noncooperate game among eery company's owner and each hred manager. The latter has to decde whether to accept the offered compensaton or just leae and setup a new busness. As a result, the model proes that, under rather general crcumstances, there may be an output restrcton wth respect to a standard Cournot model. Moreoer, equlbrum executes' compensaton turns out to be negately related to the ntal number of frms exstng n the market. Although the relatonshp between senor executes' pay and company performance has become a source of controersy n recent years, the drect lnk found n our model between company's sales and managers' compensaton s, usually, emprcally confrmed ([], [5]. The paper s organzed as follows. The next secton brefly presents the game-theoretc structure of the model. Secton 3 ntroduces a smple model specfcaton to show the man paper fndngs. Secton 4 concludes the paper.. The Model The model descrbes an olgopolstc ndustry n whch, at the begnnng, only n agents (ndcated as busness ntators possess the knowledge to produce a commodty wth a gen technology. To be effecte, the technology requres both sklled and unsklled workers. Such an excluse knowledge represents the only barrer to entry for other potental compettors (for nstance the sklled workers, henceforth labelled as managers assumed to need a specfc on-the-job tranng to start a new busness. Thus, n the ndustry, the n ntators are assumed to set up n (dentcal frms behang a la Cournot and producng a homogenous commodty y. The sequence of strateges descrbed n the model s qute smple. Frstly, eery n-th company decdes how much commodty to produce (and thus, how many dentcal managers to hre, accordng to the usual proft maxmzaton procedure. Secondly, the company has to fx each manager's remuneraton, ndcated as. Hence, a manager recruted by the frm can ether decde to stay, acceptng, or leae, to set up a competng company n the ndustry, thus earnng a proft of at most ( n 1. Eery manager that has set up a frm contnues the game exactly as before (.e., frst decdng y and then for her or hs recruted managers and the game goes on n ths way, wth a potentally nfnte number of stages. The soluton concept used to sole the game s the standard subgame perfect Nash equlbrum. The defnton that follows descrbes n detal an equlbrum of the game. Defnton An equlbrum of the game s a ector of quanttes, y n k n k, y n k, n k,.., y ( ( ( ( ( n k, n k 1, n k, where represent eery manager's equlbrum compensaton and k the number of new frms entered the market n equlbrum, such that, for eery -th frm ( = 1,,..., n k, whateer the number of entrants ( k = 0,1,...,, t must be that: [ y ( ( n k, y ( ( n k ] [ y ( ( n k, y ( ( n k ] whle, for each manager recruted by a frm, n k y n k 1, y n k 1 [ ( ( ] where n all expressons aboe, y ndcates the ector of quantty selected by all frms dfferent from

3 Managers compensaton and colluse behaour under Cournot olgopoly 03 frm and, for ease of notaton, y ( ( n k stands for y ( ( n k n k, for any and any k. Defnton 1 mposes three condtons on an equlbrum of the game. Frstly, that no frm must fnd proftable to change ts selected quantty; secondly, that, gen the quantty chosen, no frm must hae an ncente to change managers' equlbrum compensaton; thrdly, that eery hred manager must prefers to stay wthn the frm rather than setup a new busness, otherwse the game would contnue and an equlbrum would not be reached. It has to be notced that the soluton concept adopted here focuses on nddual players' behaour and excludes collecte deatons from equlbrum. The next secton apples the equlbrum defnton to a smple model specfcaton, n order to show ts man results. 3 A Smple Example Let us assume that n a certan ndustry n agents, ntally dsposng of the knowledge on how to produce a homogenous commodty y, decde to set up n (dentcal frms behang à la Cournot. Let also the technology aalable to them be descrbed by the followng Cobb-Douglas constant returns to scale producton functon: θ ( 1 θ y = m (1 where respectely m s the number of managers (or hghly sklled workers recruted by eery -th frm ( = 1,..., n whle s the number of unsklled workers. Let us assume, for smplcty, that θ = 1/. Let also eery frm's fxed cost be equal to zero. Wthout loss of generalty, the wage pad to unsklled workers s normalzed to one, whle denotes each manager's compensaton. Moreoer, let the market demand be lnear and equal to: n p ( Y a Y = ( where Y = y represents the total quantty of commodty delered to the market. =1 Derng by (1 eery frm's cost functon as: C ( y = y (3 t s straghtforward to get eery ntal -th frm's Cournot equlbrum quantty (for any arbtrary manager compensaton as: a y ( n = (4 n 1 and eery -th frm's equlbrum proft as: ( a n = (5 ( n 1 As explaned aboe, the basc feature of the model s that, when managers are hred by a frm, they mmedately acqure the specfc knowledge to become potental compettors of the exstng frms.

4 04 Marco A. Marn Hence, managers' compensaton must be optmally decded by a frm knowng each manager's potental threat of leang to set up, through the use of unsklled workers and other managers, a new producton unt. When a manager leaes the frm, he or she wll presumably set up a company of the same type as the one she or he s workng for. Thus, f n the preous stages of the game k frms hae entered the market, from (5, the payoff of a leang manager s at most equal to: where ( n k 1 ( n k 1 ( a ( n k 1 ( n k = (6 represents the wage that the leang manager wll pay to her or hs managers. Expresson (6 t s bult under the presumpton that eery exstng frm whose manager(s has decde to leae, can easly fnd a substtute. Now, we can apply defnton 1 to fnd managers' equlbrum compensaton and, hence, an equlbrum of the game. Frstly, we look for the leel of compensaton the frms hae to pay to make a manager ndfferent whether to stay or to leae. Ths can be done by solng expresson (6. In ths way, at any stage of the game a frm knows that, f selected manager's compensaton ( n k ( n k 1, the manager wll stay, whle, otherwse, she or he wll leae. Secondly, t has to be found the number of frms k that enter at the equlbrum, so to fully characterze the fnal compensaton ( n k and thus y ( ( n k for eery frm acte n the market. By usng a standard backward nducton procedure, we notce that under the specfcaton of the model, two dfferent cases arse. They are both llustrated below. 3.1 Infnte Number of Stages Ths descrbes the case n whch there exsts a rtual unlmted aalablty of managers ready to be hred by the frms of the ndustry and, as a consequence, the correspondng managers' reseraton wage s equal to zero. Moreoer, denotng as t the number of new entrants that makes eery frm's proft (approxmately equal to zero, under the model specfcaton, ( n t that tends to nfnty, gen that t must sole: ( n t equal to zero happens for t ( a ( n t ( a 0 = = 0 = (7 ( n t 1 ( n t 1 proded that, when ( n t = 0, also ( n t = ( n t 1 = 0.Thus, when ( 1 t frms hae already entered the market, eery manager wll be ndfferent whether to stay n the frm as employee, earnng the correspondng wage ( n t 1 = ( n t = 0or just leae. Frms at the preous stage, knowng ths, need to decde whether to pay ( n t = ( n t 1 to ther managers (where ndcates, at eery stage, the wage that respects (9 or pay less, lettng at least one manager leae to get n t 1.Thus, snce at the last stage a manager wll necessarly be pad a proft of ( n t 1 = ( n t = 0, one stage before frms wll fnd conenent to pay ( n t ( n t n t 1 ( only f: The lemma presented below proes that, f eery frm's proft s weakly poste and the next stage frm's optmal strategy s whch to fx a compensaton that respect (6, t wll be always conenent for

5 Managers compensaton and colluse behaour under Cournot olgopoly 05 a frm at the preous stage to do the same, payng eery manager just enough to keep her or hm wthn the frm. Lemma 1 Under the model specfcaton, f p Y ( n k the followng nequalty holds for eery frm Proof. (See Appendx. AC y ( n k = 1,..., n k :, ( ( n k ( n k 1 for any possble k =1,,.., n, Snce Lemma 1 holds at any stage of the game, eery frm's optmal strategy wll be whch of payng the mnmal remuneraton suffcent to nduce ts managers to stay wthn the frm,.e., a wage that respects (6. By backward nducton, at the frst stage of the game, when the n ntators decde whether to let ther managers stay (payng them accordngly or leae (thus ncreasng the exstng number of frms, they wll certanly set ( n equlbrum remuneraton ( n = and the equlbrum number of entrants wll be k = 0. The s thus obtanable by solng the followng equaton: ( n = ( n 1 ( a ( n 1 = (8 ( n Expresson (8 s a non lnear dfference equaton that can be soled by teraton on the potental number of entrants k. A straghtforward substtuton process yelds: k 1 k 1 a( 3 ( 4 ( k a( 4 ( k 4a( 5 ( k 8a ( 1 ( k 1 = ( n ( ( 3 ( k Now, from (8 we know that, for k =, ( n t 1 = 0 nto expresson (9 and rearrangng, we obtan: ( n t 3 t ( n t 1 = ( n t = 0. t t [( 1 ( a ] ( n 3 j ( 1 ( a = 0 j= t 3 = lm t t = 0 ( n. (9 Hence, puttng. (10 Interestng propertes of expresson (10 are that t s unque for eery set of parameters alues and that takes fnte alues, een for a fnte and ery low number of stage t. Moreoer, the alue of ( n s monotoncally decreasng n the number n of ntators assumed to exsts at the begnnng of the game. Now, by nsertng expresson (8 nto the equlbrum output, t ensues that, accordng to Defnton 1, the unque equlbrum of the game s ( y1 ( ( n, n, y( ( n, n,..., yn ( ( n, n. In fgure below, ( n s plotted aganst dfferent number of busness ntators and compared to a gen market clearng wage. Managers' compensaton s decreasng wth the number of frms assumed to exst n the market. Notce also that, snce for a range of n equlbrum managers' compensaton s hgher than the market clearng leel, there wll always be, wthn ths range, managers aalable to be hred by the frms. Ths endogenous aalablty of managers allows for ther substtuton when they decde to leae the frm and ges consstency to the game descrbed aboe.

6 06 Marco A. Marn. and w= for a=350 and n=1,..0. In ths example ( n s equal to =10 for n=8. Fg 3.-Values of n Proposton 1 Under the model assumptons, when the number of ntators s no greater than n the equlbrum number of managers m selected by eery frm s less than t would be under standard neoclasscal assumptons n the market for managers. Proof. Snce from (1 an effcent choce of managers mples m =, and, from (4 Cournot y s monotoncally decreasng n, t ensues that m ( n m ( n, where ndcates the neoclasscal market clearng wage. Snce ( n = ( n 1 y equlbrum output < wheneer >, and, by lemma, the second member s monotoncally decreasng n n, there wll certanly be a alue of n = n for whch ( n =. Thus, for < m n < m and the result follows. ( n n n, It can be notced that equlbrum quantty y ( n, s, for eery frm, dependent on equlbrum managers' compensaton. Thus, snce ( n s hgher than for n < n, t turns out that, when n < n, y ( n, n < y, n Ths means that each frm s, wthn a gen range of n, more colluse n terms. of output than under usual market clearng condtons. There also exsts an ntal number of frms for whch y ( ( n, n exactly concdes wth the perfectly colluse output choce under market clearng wage,.e., that obtaned when all frms cooperately maxmze ther jont proft. These results are descrbed n the next proposton. Proposton The output selected by eery frm under Cournot equlbrum and managers' threat to leae s more colluse than under Cournot equlbrum and managers' compette market for n n, y n, n < y n for n < n. Moreoer, there exsts a leel of n = n for whch, (, c ( ( n, n = y (, n, that s, y where c y s the output resultng by cooperate agreement among frms.

7 Managers compensaton and colluse behaour under Cournot olgopoly 07 Proof. By proposton 1, for < n n, ( n monotoncally decreasng n t follows that, for < manpulatons show that t s equal to: Substtutng for ( n nto s greater than. Snce frm's equlbrum output s n n, y ( n, ( n y ( n, a ( n 3n 05n 881n 1818n 144 n = y yelds: ( n ( n 3 ( n 4 ( n 5 ( n 6 ( n 7 ( ( n, n ( n <. Moreoer, straghtforward a y = (11 n 1 whle, the colluse quantty under market clearng managers' compensaton s: (, n a y c = (1 n Thus, expresson (11 s equal to (1, for n = n, where n s the only poste soluton of an equaton that, for ease of brety, s not presented here. It can be notced that the hgher the managers' market c y n, n = y, n. clearng wage and the lower wll be the number of frms for whch. Fg Equlbrum quantty both for the manageral frm y m (n and for the perfectly colluse frm (yc(n (a=1000, =10, n=0, A partcular example of the result aboe s presented n fgure 3.3. Gen, for n = 0 colluse y c, n concdes wth y ( ( n,n. Moreoer, for n < n, y ( ( n, n turns out to equlbrum quantty be een more colluse than y ( c,n. y n, n s also a Nash equlbrum quantty, t wll be stable aganst each frm's temptaton to deate from the equlbrum choce of output, dfferently to what normally happens under colluse agreement. It can be notced that such a partcular example of non-cooperate colluse soluton can take place ether through mergers among frms (when the number of ntators s greater than n or through controlled departure of managers nduced by a frm (when n s lower than n. The latter can specally be the case when the Moreoer, snce eery frm's quantty

8 08 Marco A. Marn ntators mantan a share of the new companes' control, a relately wdespread practse n hgh-tech ndustres. Anyway, the basc result of the model s that, whether the behaour of exstng frms n a market s less or more colluse than n usual Cournot models depends upon the number of busness ntators that dspose of the basc know-how to set up a frm. 4. Concludng remarks The paper has descrbed, through a smple model, that companes owners' need to fx a leel of compensaton hgh enough to keep managers wthn the frm can ge rse to a colluse choce of output stable aganst nddual frm's deatons. The result holds when the depresse effect of leang managers on frms' proft preals on the poste effect due to a reducton of ther compensaton, and the exstng number of frms s not ery hgh. Furthermore, the model generates the emprcally appealng property (see, for nstance, [] and [5] that managers' compensaton s decreasng wth the number of frms exstng n the market and, consequently, wth ther sze. References [1] C. Fershtman and K. Judd, (1987, ''Equlbrum Incentes n Olgopoly'', Amercan Economc Reew, 77, pp ; [] K. Murphy Corporate Performance and Manageral remuneraton, An Emprcal Analyss', Journal of Accountng and Economcs, 7, (1985, [3] S. Sklas, The Strategc Choce of Manageral Incentes', Rand Journal of Economcs, 4, (1987, [4] J. Vckers, Delegaton and the Theory of the Frm', Economc Journal, 95, (1985, [5] R.Watson, D.J. Storey, P. Wynarczyk, K. Keasey, and H. Short, 'The Remuneraton of Non-owner Managers n Small and Medum-szed UK Enterprses, Journal of Management Studes, 31, (1994, Appendx Proof of Lemma 1. As long as p Y ( n k = 1,..., n k AC y ( n k, ( ( n k n k ( n k 1, the followng nequalty holds for eery ( n k 1, The meanng of ths expresson s that, under poste proftablty of exstng frms, eery company fnds conenent to pay each manager the equlbrum wage ( n k rather than let her or hm go, startng a new negotaton wth another manager. Let us proe the lemma by contradcton. Suppose that the nequalty aboe does not hold, that s: [ p( Y ( n k ( n k ] y ( n k< Y ( n k [ p( 1 ( n k 1 ] y ( n k 1

9 Managers compensaton and colluse behaour under Cournot olgopoly 09 Ths expresson can be soled by teraton and, for each frm under the potental market entry of t frms, the followng result ensues: ( n k ( ( n k p Y ( n t p Y [ y ( n t ( n t 4y ( n t ] 4( y ( n > 4 t where t ndcates the number of frms that can enter before eery frm's proft s equal to zero and the game ends. Snce n ths case there are no entry costs, room potentally exsts for an nfnte number of entrants. Hence, takng the lmt of expresson aboe for t that tends to nfnte, we get: ( n k that can be rewrtten as: and then, > ( ( n k p Y ( n t p Y [ y ( n t ( n t 4y ( n t ] 4( y ( n lm 4 t k ( n k y ( n k > p Y ( n k y ( n k lm ( ( n t t ( n k y ( n k > p Y ( n k y ( n k 0 from whch: ( n k = AC y ( n k > p( Y ( n that contradcts the assumpton of eery frm's weak poste proftablty. Ths concludes the proof. Receed: September, 01