A DUOPOLY MODEL OF FIXED COST CHOICE. Charles E. Hegji*

Transcription

1 004 DUOPOY MODE O IXED OST HOIE harles E Hegj* INTRODUTION omparson of frms n ournot and Stackelberg eulbrum s a subject that has receved much attenton unversally mposed assumpton n most dscussons of the ournot and Stackelberg outcomes s that partcpants n markets are confronted wth gven cost structures In some setups lke the models of Robson 990 nderson and Engers 99 and Shaffer 995 frms are assumed to have dentcal costs In others such as the Stackelberg model developed by Pal and Sarkar 00 frms have dfferent costs However the seuence of costs of frms choosng output s fxed as s the level of costs once the eulbrum s obtaned Ths assumpton s of course unrealstc snce frms nvest consderable tme and effort n cost cuttng to ether ncrease proft or market share The purpose of the present paper s to study the mpact of cost changes on frms n ournot and Stackelberg eulbrum We do ths usng a model smlar that that of Neuman Wegand Gross and Muenter 00 The model assumes that frms can reduce margnal costs by nvestng n assets thereby ncreasng fxed costs In Secton we set up the ntal revenue and cost condtons facng the frms or ease of exposton a duopoly wth lnear market demand and cost functons s employed In ths secton we present standard results for the ournot and Stackelberg duopoly models In Secton 3 we ntroduce the assumptons about the cost-changng possbltes for the frms and determne optmal fxed costs We evaluate the mpact of settng fxed costs at ther optmal levels on proft and market share for the frms n the two models Secton 4 provdes emprcal evdence on the relatonshp between fxed and varable costs chosen by frms Secton 5 contans some bref concludng remarks SIMPE MODE O OURNOT ND STKEERG EQUIIRIUM onsder a stuaton wth lnear market demand produced by two frms The nverse demand functon wll be gven by P Q where P s prce Q s uantty and parameters and are both postve Output s produced under two market settngs rst we consder a ournot model or ths model total costs for the th frm where wll be of the form T where s fxed cost and s margnal cost osts are dentcal for the Stackelberg model However for ease of exposton the subscrpt s replaced by *Department of Economcs uburn Unversty at Montgomery Montgomery

2 NEW YORK EONOMI REVIEW 45 for the Stackelberg leader and for the follower frm In the ournot model profts for frm are gven by Profts j Maxmzng profts for the two frms and smultaneously solvng the reacton functons results n the outputs: 3 3 The resultng market shares are 3 M M Profts for the two frms are [ ] [ ] Proft Profts Euatons 3 and 4 show the famlar results for a duopoly n ournot eulbrum Market shares and profts are proportonal to frm costs wth the larger market share and hgher profts gong to the lower margnal cost frm or the Stackelberg model begn wth the profts for the leader and follower frm: Pr ofts Profts 5 The follower s reacton functon s derved from ts frst-order condton: 6 Substtutng 6 nto the leader frm s profts and maxmzng wth respect to yelds the outputs for the Stackelberg model 7 4 Market shares for the Stackelberg frms are therefore 8 MS MS

3 004 Profts for the Stackelberg frms are Profts Profts 6 8 [ ] 9 [ ] Euatons 8 and 9 are smlar to 3 and 4 They show that smlar to the ournot model n Stackelberg eulbrum a frm s market share and profts wll ncrease when ts margnal costs are lower than ts rval s However n the market share euatons a larger weght s placed on the margn of prce over margnal cost for the leader frm measured by the dfferental 3 HOIE O IXED OST ND EHVIOR O THE IRM Our model of cost settng s constructed usng a model smlar to the one provded by Neumann et al 00 These authors suggest that f hgher fxed costs are a result of new and mproved producton technues such costs may result n lower margnal costs We therefore assume a relatonshp between fxed and margnal costs of the form K K K K * 0 > The parameter K 0 defnes the level of margnal costs wthout the cost-reducng nvestment whle K s the decrease n margnal costs per dollar nvestment n fxed costs Snce margnal costs must be postve euaton 0 s relevant over the range of fxed costs 0 < < K 0 /K lso note that the restrcton > K 0 must hold Ths s euvalent to assumng that the eulbrum prce must be greater than the margnal cost consstent wth zero fxed costs The assumpton assures that prce covers margnal costs under all possble stuatons ll frms wll be assumed to have access to the above technology and wll choose the proftmaxmzng level of fxed costs usng euaton 0 Ths choce can be thought of as the second stage n a two-stage game where the frst stage s the choce of a proft-maxmzng level of output lthough the Stackelberg model assumes a frst-mover advantage for the leader frm n settng output frms n both models wll engage n smultaneous play at the second stage Ths means that each frm wll choose ts proft-maxmzng level of fxed costs gven ts rval s choce The outcome of the second stage wll be a Nash eulbrum n the choce of fxed costs Substtutng 0 nto 4 and 9 and dfferentatng wth respect to results n the frm s proftmaxmzng level of fxed costs Substtutng these costs nto and 9 yelds the market shares and profts assumng the optmal levels of fxed costs are obtaned ll results are dsplayed n Table 46

4 NEW YORK EONOMI REVIEW Table : The Impact of Optmal xed ost hoce by Type of rm rm Type Optmal xed ost Resultng Market Share Resultng Proft ournot K0 9 K 4K K K 0 7 6K Stackelberg eader K0 3 K 6K 3 5 K K 0 5 3K Stackelberg ollower K0 7 K 3K 5 K K 0 7 9K The optmal fxed costs for the three frms contan the common factors K 0 /K and /K Snce 7/3 > 9/4 > 3/6 optmal fxed costs are hghest for the Stackelberg follower and lowest for the Stackelberg leader wth fxed costs for the ournot frm fallng between these two extremes Ths result s ntutvely appealng It suggests that snce nvestment n captal expendtures reduces margnal costs wth the potental of ncreasng proft margns the ncentve to do so s greatest for the frm n the most dsadvantageous market poston smlar lnk between frm captalzaton and frm postonng has been noted by Porter 980 hapter 5 Ths choce n fxed cost results n eual market share for the two frms n ournot eulbrum Ths result s the same as the market shares obtaned from euaton 3 assumng whch s the standard result for a ournot duopoly wth eual margnal costs for the two frms and no choce of fxed costs The result s a drect conseuence of the frms havng access to the same margnal cost reducng technology and s the expected outcome of the smultaneous play at the second stage of the game It suggests that no matter what the ntal margnal costs smultaneous play and access to the same margnal cost reducng technology at the second stage of the game wll always allow the frm n a dsadvantageous market poston to catch up n market share 3 Ths fndng supports the apparent 47

5 004 wllngness for some frms to compete for market share as well as profts snce hgher market share may be a more easly attanable goal than hgher profts In the Stackelberg model the resultng market shares are 3/5 gong to the leader and /5 of the market output gong to the follower Ths can be compared to the market shares for the Stackelberg model wth eual ntal margnal costs and no margnal cost reducng choce of fxed costs 4 In ths alternatve stuaton the leader s market share wll be /3 of the market wth the follower s share beng /3 Snce /5 > /3 n terms of market share the Stackelberg follower frm prefers the stuaton n whch t can reduce margnal cost by nvestng n fxed costs to the stuaton n whch ts margnal costs are the same as the leader frm s However any potental market share dsadvantage cannot be completely made up by choce of fxed cost as n the ournot model In essence some of the frst-mover advantage of the Stackelberg leader n choosng output remans wth the leader n the second stage of the game Profts assumng fxed costs are at the optmal level are smlar to fxed costs both havng the common factors -K 0 /K and /K However snce 5/3 < 7/6 < 7/9 the orderng of profts s opposte that of fxed costs Profts for the Stackelberg leader are largest Profts for the Stackelberg follower are smallest The ournot profts le between these two values The results for profts n the Stackelberg eulbrum are lke those comparng market share They show that the reduced profts resultng from movng second n a Stackelberg game cannot be elmnated by nvestng n fxed assets once eulbrum s obtaned The result s n the sprt of observatons made by Porter 980 hapter 5 who argues that n attemptng to catch up wth market leaders follower frms have a tendency to over captalze thereby decreasng profts 4 EMPIRI EVIDENE O OST HOIE Y IRMS The key assumpton of the above analyss s euaton 0 whch postulates a negatve relatonshp between fxed costs and margnal cost We tested ths relatonshp as follows Data on 73 md-captalzaton frms were collected for the perod from the ompustat Data tapes ccountng costs for these frms were categorzed nto fxed and varable costs The varable cost proxy used was ost of Goods Sold These costs ncluded labor heat power freght and other costs drectly nvolved wth producng and dstrbutng a product Some admnstratve costs such as plant nsurance were also reported under ths headng The fxed cost measure ncluded non tangble fxed costs such as advertsng and marketng costs as well as fxed costs assocated wth tangble property Examples of such costs would be tools and des software and arcraft xed costs were measured relatve to total assets to avod the mpact of frm sze on cost measurement Two functonal forms were estmated t α β t β t β 3 t3 t \ t α β t β t β 3 t3 a b 48

6 NEW YORK EONOMI REVIEW Euaton a assumes a lnear relatonshp between the current change n varable costs and lagged changes n fxed costs The lags are ncluded to allow for adjustment tme Euaton b expresses the change n varable cost n the form of the rato of current varable cost to the prevous perod s varable cost If euaton 0 s a reasonable assumpton we should fnd sgnfcant negatve values for the β n Table dsplays these estmated coeffcents The t-statstcs are n parentheses The results are consstent wth euaton 0 The table shows a sgnfcant negatve relatonshp between fxed and varable costs at the one perod lag for the years 00 and 000 and a mld negatve relatonshp at the three perod lag for 00 or 00 the one perod lag coeffcent was sgnfcant when the dependent varable was expressed n both change and rato form whle only the rato form was sgnfcant for 000 When estmated over the entre tme perod a sgnfcant negatve relatonshp between changes n fxed and varable cost was dsplayed at the one perod lag Ths relatonshp held when changes n varable costs were measured n both absolute and relatve terms Table : hanges n Varable osts Regressed ganst xed ost Independent Varable t t/t- t t/t- t t/t- t t/t- / t- -786*** -059**** *** -6648** -06**** / t ** / t * N -06* R-suared *Sgnfcant at the 0% level **Sgnfcant at 5% level ***Sgnfcant at % level ****Sgnfcant at % level In sum ncreases n fxed costs enable frms to decrease varable costs nd ths relatonshp seems to occur rather rapdly that s wthn one to two years 5 ONUSIONS Most comparsons of frms n ournot and Stackelberg eulbrum assume gven cost structures The current paper analyzes the mpact of margnal cost reducng fxed nvestment on market share and profts of frms n ournot and Stackelberg eulbrum The setup can be conceptualzed as a two-stage game t the frst stage frms choose ther proft-maxmzng output ournot and Stackelberg duopoly 49

7 004 models were used at ths stage In the second stage frms were gven the opton of reducng margnal costs by nvestng n fxed costs or both models frms engaged n smultaneous play at ths stage We found the greatest nvestment n fxed cost was for the Stackelberg follower and the smallest for the Stackelberg leader The ournot frm s nvestment fell between the two extremes Resultng profts mrrored these nvestments wth the smallest accrung to the Stackelberg follower and largest to the Stackelberg leader Perhaps the most nterestng result was the dfference n market shares resultng from the two models In the ournot model f a frm started from a smaller market share due to hgher margnal costs ts nvestment n fxed costs at the second stage allowed t to eualze ts market share relatve to the lower-cost frm Ths was not true n the Stackelberg model No amount of fxed nvestment allowed the follower to eualze ts market share wth the leader s market share The same held true for profts Ths suggested that the frst-mover advantage of the Stackelberg model was carred by the leader nto the second stage of the game These conclusons assume that frms can reduce margnal costs by nvestng n fxed assets Emprcal evdence supportng ths assumpton was provded ENDNOTES Proofs of these results are avalable from the author upon reuest The set of feasble solutons reures that fxed costs are greater than zero These parameter restrctons are also avalable from the author on reuest 3 We are not certan the extent to whch ths concluson s dependent on our use of a lnear functon lnkng fxed and margnal costs 4 ssume n euaton 8 REERENES nderson S and M Engers Stackelberg versus ournot Olgopoly Eulbrum Internatonal Journal of Industral Organzaton Neumann M Wegand J Gross and M T Muenter Market Sze xed osts and Horzontal oncentraton Internatonal Journal of Industral Organzaton Pal Debashs and Jyotrmoy Sarkar Stackelberg Olgopoly wth Nondentcal rms ulletn of Economc Research prl Porter M E ompettve Strategy ree Press New York 980 Robson rthur J Stackelberg and Marshall mercan Economc Revew 80 March Shaffer Sherrll Stable artels wth a ournot rnge Southern Economc Journal January