SUNK COST EFFICIENCY WITH DISCRETE COMPETITORS

Transcription

1 Sunk Cost Efficincy with Discrt Comptitors SUNK COST EFFICIENCY WITH DISCRETE COMPETITORS Linus Wilson, Univrsity of Louisiana at Lafaytt ABSTRACT Whn ntrants only diffr in thir xognous ntry costs, th ordr in which potntial firms ntr dos not affct industry siz. With discrt comptitors, ntry ordrings can affct total sunk costs and th idntity of ntrants. A ncssary and sufficint condition is stablishd for sunk ntry costs in th industry to b minimizd, rgardlss of ntry ordring. JEL Classification: L INTRODUCTION This papr xplors th rol of path dpndnc, th ordr in which firms ntr, in th fficincy of production. In particular, it xplors undr what conditions lowr sunk cost rivals driv away highr sunk cost comptitors. Ncssary and sufficint conditions for fr ntry to lad to sunk cost fficincy ar dvlopd. Nvrthlss, thr will likly b many circumstancs whr ths conditions ar not mt, and highr sunk cost comptitors will driv away mor fficint firms. An xampl may illustrat why squntial ntry and diffrnt sunk costs ar important and may lad to infficintly high ntry costs. Considr a town in Kansas that has on gnral practic physician, and th markt could support on mor. A physician from Orgon is rcruitd to start up what would b a vry profitabl practic. H has som family and frinds in Orgon and prfrs th landscap and wathr in Orgon to that in Kansas. Nvrthlss, h would happily liv and work in Kansas, and this practic is his bst opportunity. Suppos that thr is anothr doctor in Kansas, who is just about to bgin hr third yar of a thr yar rsidncy prior to bing abl to go into privat practic. Had this mdical rsidnt bn in a position to start a practic in that Kansas town, sh would hav bn happir than th doctor from Orgon. Social wlfar would b highr. Yt, whn th doctor from Kansas finishs hr rsidncy, th doctor from Orgon is sttld in, and thr is no room for a third gnral practitionr in th town. Yt, th nw physician from Kansas dos hav a good opportunity in Tnnss Ths physicians had diffrnt sunk costs of moving to th Kansas town, but th Orgon doctor with highr costs movd first and prvntd th Kansas doctor from ntring whn sh was rady. Diffrnt ntrprnurs may hav mor or lss familiarity with stting up a storfront or complying with local rgulations that prtain to thir businss. Ths ntry cost advantags do not ncssarily lad to variabl cost advantags. Managrs of rival firms fac th sam costs of labor, raw matrials, and nrgy as thir comptitors. Thrfor, it sms that htrognity in sunk ntry costs and narly idntical variabl costs ar common in many businss sctors. Nw businsss form in dribs and drabs for th most part. On ntrprnur or firm dcids to bring a 37

2 Southwstrn Economic Rviw businss into bing on a diffrnt day, month, or yar than th nxt comptitor. In most thr is not a starting lin with twnty ntrprnurs racing whn th starting gun gos off to build th most succssful businss. Instad, th mor common cas is that first ntrprnur crosss th starting lin on day on, th scond crosss th starting lin six months latr, and th third crosss th starting lin in on yar and ight months aftr th first bgan hr businss. Thrfor, a modl of ntry into an industry is bst modld as a squntial gam. This papr attmpts to find out undr what circumstancs path dpndnc th ordring of ntry can affct th fficincy of th industry (th industry s aggrgat sunk costs.) This papr is mor gnral than th standard approach of assuming that sunk and variabl costs ar idntical. Instad, this papr assums that thr is som xognous htrognity in th sunk costs of potntial ntrants. In this papr, potntial comptitors play a fr ntry gam whr th squnc of potntial ntrants is common knowldg. Each firm is givn th opportunity to pay thir ntry cost or stay out of th industry forvr. Thn th firms compt and collct thir profits bfor invstmnt costs in th last stag. With idntical variabl costs, th xognous htrognity in sunk costs of ntry dos not improv a firm s ability to compt in th final stag of th gam. Instad, lowr stup costs mak ntry mor attractiv (and socially fficint) for som firms. W will s that th squncing of ntry dcisions dos not affct th siz of th industry. Yt, th ordring of ntry potntially affcts th magnitud of ntry costs incurrd. This study finds th ncssary and sufficint condition for fixd ntry costs to b minimizd for a givn industry siz, rgardlss of ntry ordring. Whn th ncssary and sufficint condition is satisfid, only th lowst sunk cost firms will ntr. This is th first and only study to driv sufficint conditions for sunk costs to b minimizd rgardlss of ntry ordrings whn thr ar discrt comptitors with idntical variabl cost functions. Th author knows of no othr modl of this typ. Th modl is primarily distinguishd by its xognous htrognity in firms ntry costs. Studis ithr assum that ntry costs ar idntical, as in Mankiw and Whinston (986), or that sunk costs ar ndognous. Exampls of th lattr approach ar Spnc (977) and Dixit (980) whr th first movr chooss its variabl costs to dtr or accommodat furthr ntry. A mor rcnt papr by Robrts (007) ndognizs th opportunity cost of ntry. This distinguishs it from th prsnt approach. Furthr, th modl of Robrts (007) assums that ntry dcisions ar takn simultanously. In contrast, this papr assums that ntry dcisions ar squntial. Many of th infficincis in Robrts (007) disappar whn squntial ntry dcisions ar mad in this papr. Th rlativ fficincy of th squntial rsults hr whn compard to th rsults in Robrts (007) simultanous mov gam is not surprising. Mixd stratgy quilibria found in simultanous mov gams oftn lad to infficincis vn whn playrs wish to bhav cooprativly. Ghmwhat and Nalbuff (985) hav a modl of xit in a dclining industry whr firms must pay rnt on thir capital to rmain in th industry. Dspit this obvious diffrnc, thr ar som analogis btwn modls of ntry and xit. Th rnt on capital in Ghmwhat and Nalbuff s study can b viwd as a sunk cost of ntring th industry. Ghmwhat and Nalbuff find that firms which pay highr costs of staying in th industry ar th first firms that xit in a dclining industry. This is 38

3 Sunk Cost Efficincy with Discrt Comptitors analogous to this papr s rsult that, undr som circumstancs, rgardlss of ntry ordr, only th firms with th lowst sunk costs will ntr th industry. Yt, thr is an important diffrnc btwn this study and Ghmwhat and Nalbuff (985). That study assums that highr sunk costs of staying in th industry ar associatd with gratr capacity and markt shar. Hr w assum that highr sunk ntry costs ar xognous and unrlatd to variabl costs and thus th firm s markt shar. Ghmwhat and Nalbuff, as do many othr authors discussing ntry and xit, assum for a givn unit of capacity (or markt shar) that sunk costs ar idntical. Unlik Ghmwhat and Nalbuff s study, in th prsnt papr, th sunk cost of ntry pr unit of markt shar is diffrnt for vry potntial ntrant. As dos this papr, Wilson (00) assums that comptitors diffr in thir ntry costs, but potntial ntrants hav idntical variabl cost functions. Yt, it looks at ntry with a continuum of comptitors. That papr, unlik this on, finds that fixd costs ar always minimizd with a continuum of ntrants that mak squntial ntry dcisions. Th papr procds as follows. In sction two, w dscrib th modl. In sction thr, w show industry siz is invariant to ntry ordring, and a ncssary and sufficint condition for all ntry ordrings to b fficint is provd. In sction four, w illustrat th rsults with thr xampls. Finally, sction fiv concluds th papr. FIGURE SEQUENCE OF EVENTS Priod - Priod 0 Priod Ordring of potntial ntrants is dtrmind and bcoms common knowldg. Firms squntially choos to ntr. Entrants pay th sunk costs of ntry. N ntrants ach arn a payoff of π(n ). MODEL Lt us bgin by dscribing th gam outlind in figur abov. Th ordr by which firms of a givn rank ar allowd to ntr is known by all firms in priod. Furthr, all prvious ntry dcisions ar common knowldg as thy occur in priod 0. In priod, ntrants compt without rgard to thir xognous sunk costs. In priod, all ntrants rciv an idntical payoff bfor ntry costs, which is only a function of th N firms that ntr. Lt us dfin π(n ) as th payoff bfor ntry costs for a unit-sizd firm. A firm that ntrs pays a sunk ntry cost, K i, which dpnds on th firm s rank i, whr i F = {,, 3,, n-, n}. 3 Th st of firms is a propr subst of th st of natural numbrs. Th numbr of lmnts of this st is n(f) = n, whr n is containd in th st of natural numbrs. Th invstmnt costs, Κ i, ar incrasing in th firm s indx numbr, i. That is, K < K < K 3 < < K n. Lt us assum that pr firm 39

4 Southwstrn Economic Rviw payoff, π(n ), dclins in th numbr of ntrants. That is, π() > π() > > π(n ) > π(n). 4 W will assum that all firms that xpct to mak non-ngativ profits will ntr. To nsur that it is subgam prfct Nash quilibrium (SPE) that at last on firm ntrs, lt us assum that it is profitabl for th lowst cost firm to always ntr. π () K 0 () Furthr, lt us assum that if all firms ntr, th highst cost firm in th st F will find ntry unprofitabl. π ( n) K n < 0. () Suppos that th ordring of ntry is a prmutation of th st F. Th ntry st, E, consists of n(e) N firms. If thr is a potntial ntrant of rank i, it must b th cas that i F. Furthr, if th firm of rank i ntrs, i E. Sinc both N and n ar intgrs, th st E can consist of any ordring of th firms in st F such that th E F. Thr is an uppr bound of n! M( n, N ) = ( n N )! (3) possibl prmutations of ntry ordrings. Nvrthlss, thr will only b at most C(n, N ) diffrnt combinations of quilibrium ntry sts for any random ordring of ntry. n! CnN (, ) = N!( n N )! (4) Yt, thr will only b on (unordrd) ntry st that is part of th uniqu subgam prfct Nash quilibrium (SPE) for any ntry squnc. ANALYSIS In this sction w will prov two propositions rgarding fr ntry and sunk cost fficincy whn firms ar of idntical but discrt siz. First, w dmonstrat that th industry siz dos not dpnd on th ntry ordring. Thn, w will prov th ncssary and sufficint conditions to achiv sunk cost fficincy in th industry. Th implication of th inqualitis in () and () is that som firms ntr and som firms stay out of th industry. Lt us dfin th fr ntry numbr of firms by th following st of conditions: 40 π ( N ) K 0 N π N K N + ( + ) < 0 (5)

5 Proposition Th ordring of potntial firms will not affct th numbr of ntrants. Sunk Cost Efficincy with Discrt Comptitors From quation (5), w know that all firms with indx numbrs 0 < i < N will ntr if no on ls dos. Furthr, ths low indx numbr firms, i < N, might vn ntr if th industry has gratr than N firms. To gt mor than N ntrants, it is ncssary that th firm of rank N + or som highr rankd firm ntrs also. Th fr ntry condition in (5) prcluds this possibility bcaus π(n + ) < K. N + If th firm with th indx numbr N + will not ntr an industry which will hav N + ntrants, thn no othr highr indx numbr firm with an indx numbr N > N + will ntr undr such circumstancs. Q.E.D. It is wll known that consumr surplus incrass with industry output. Th ordring of ntry dos not affct industry siz with idntical comptitors. If aggrgat industry output dpnds on th numbr of idntical comptitors, ntry ordrings cannot affct consumr surplus. Yt, w will s that th ordring of ntry can affct industry profits and thus ovrall wlfar whn ths othrwis idntical rivals diffr in thir sunk costs. For any givn quilibrium industry siz, N, fficincy minimizs th ntry costs incurrd to rach that industry siz. Thrfor, fficincy undr fr ntry dpnds on th idntity of who ntrs not just on th numbr of firms ntring, N. (Th lattr problm was th only concrn of Mankiw and Whinston (986)). Total sunk cost of ntry ar Ki. Sinc sunk costs ar incrasing in firms indx i E numbrs, sunk cost fficincy mans that only th N firms with th lowst indx numbrs, i < N, ntr for a givn industry siz, N. Proposition Th ncssary and sufficint conditions for all possibl ntry ordrings to minimiz ntry costs for any quilibrium siz, N, is j > N ithr A) K j > π ( N ), or B) i N, which could ntr aftr j, π ( N + ) K i It is not hard to s that if w rplacd any firm with rank i < N with a firm of rank j > N, th total ntry costs would unambiguously ris bcaus ntry costs ar strictly incrasing in a firm s rank. W will bgin by proving th sufficincy of A and B. Sufficint conditions imply that a statmnt is tru. Th proof for part A of proposition is as follows. K j > π(n ) implis that all firms of rank j > N will stay out, laving only th potntial ntrants of rank i < N. W know from th first inquality that all ths firms will ntr if that mans that th siz of th industry is qual to or lss than N. This must b th cas, and all firms of indx numbrs of i, whr i < N, will ntr. Q.E.D. Now, w can considr part B in proposition. Suppos that π(n ) K j > 0 for som firms of rank j > N. If all th i rankd firms ntr bfor a firm j > N, thn th j-th firm will not ntr bcaus π(n + ) K j < 0, according to th scond inquality in th fr ntry conditions in quation (5). Yt, if th a firm of rank i ntrs aftr a firm or rank j, thn th j-th firm will only find ntry profitabl if it can dtr firm i from ntring. Considr all th firms of rank i < N that firm rankd j * = 4

6 Southwstrn Economic Rviw (N + ) could discourag from ntring th industry. Firm N + will hav th bst chanc of dtrring th N -th firm from ntry, bcaus lowr rankd firms hav lowr costs of ntry. Yt, if firm N + ntrs bfor firm N, it cannot dtr firm N from ntring by its ntry alon if firm N is profitabl in an industry of N + firms. That is, if π(n + ) K > 0, th ntry of th j * -th firm will not discourag any firm of N rank i from ntring. Thrfor, th (N + )-th firm will stay out if K < π(n + ), N rgardlss of th ntry prmutation. Sinc lowr rankd firms hav lowr ntry costs, w can b sur that all firms of rank i will ntr if firm j ntrs. That is, K i < π(n + ) for all i < N. Furthr, all highr j firms, of rank highr than j * = N +, will also stay out, rgardlss of th ntry ordring bcaus th π(n + ) < K < K <... N + N + Q.E.D. A ncssary condition must b satisfid for a proposition to b tru. Bcaus thr ar no othr circumstancs than A or B in proposition in which only th lowst cost firms ntr undr fr ntry rgardlss of ntry ordring, thn minimizd sunk costs rgardlss of ntry ordring implis that ithr A or B is tru. Q.E.D. If th ncssary and sufficint conditions for proposition ar mt, it is a uniqu subgam prfct Nash quilibrium (SPE) that all firms of rank i < N will ntr and all firms of rank j > N will stay out rgardlss of ntry ordring. THREE EXAMPLES Hr w will xplor thr numrical xampls. Th Nash quilibrium outputs, producr surplus, consumr surplus and wlfar for a givn numbr of ntrants ar drivd in th appndix. Firms ar assumd to play a simultanous Cournot gam in priod. For th xampls in this sction, th invrs dmand intrcpt, a, invrs dmand slop, b, and th marginal cost, c, paramtrs for both xampls ar a = 0 b = c = 0. (6) Undr th paramtrs suggstd in quation (6) abov and th formula for th pr firm payoff bfor ntry costs drivd in th appndix quation (), w know that pr firm producr surplus as a function of th numbr of ntrants will b π () = 5 π () = 9 (7) π (3) = 6.5. Insrting th paramtr valus in (6) into th quation for quilibrium consumr surplus, which is drivd in quation (5) in th appndix, consumr surplus is th following, dpnding on th numbr of ntrants: 4

7 Sunk Cost Efficincy with Discrt Comptitors CS() =.5 CS() = 9 (8) CS(3) = 8.5 Exampl K = 7, K = 8, and K 3 = 9. Hr fr ntry dictats that only two firms ntr. N = as dfind by quation (5). That is, K = 8 < π() =., and K 3 = 9 > π(3) = 6.5. Yt, th most fficint ntry st is not th only possibl SPE whn all ntry ordrings ar qually likly. W know this bcaus K 3 = 9 < π() =., and π(3) = 6.5 < K = 8. Thrfor, nithr part of proposition is satisfid. Indd, thr is no ordring whr th 3 rd rankd firm is givn th first or scond opportunity to ntr whr th last firm, of ithr rank or, will crdibly ntr. That is, both K = 7 > π(3) = 6.5 and K = 8 > π(3) = 6.5. Thrfor, if ntry squncs ar indpndnt and idntically distributd (i.i.d.) on-third of th tim th first-bst fr ntry st, E = {, }, will ntr, gnrating social wlfar of 9.4; on-third of th tim th scond-bst st of firms will ntr, E 3 = {, 3}, gnrating social wlfar of 8.4; and on-third of th tim th worst st of firms will ntr, E = {, 3}, gnrating social wlfar of 7.4. Thrfor, if ntry ordrings ar i.i.d., thn xpctd wlfar is 8.4, which is lowr than 9.4, th social wlfar if th lowst rankd firms ntrd first. 5 Th xampl of th physician from Orgon, who prvntd th ntry of th lowr sunk cost physician from Kansas, is consistnt with xampl s lss fficint ntry sts, whr at last on highr sunk cost ntrant blocks a lowr sunk cost ntrant. Exampl K = 5, K = 6, and K 3 = 0. Hr, too, fr ntry dictats that only two firms ntr. That is, K = 6 < π() =., and K 3 = 0 > π(3) = 6.5. Th 3 rd rankd firm with fixd costs K 3 = 0 would want to ntr if thr would only b two ntrants, givn that it ntrd. Yt, it would los mony if thr firms ntrd. Th 3 rd rankd firm cannot dtr ntry by th lowr rankd firms. With ths fixd costs for potntial ntrants, only th most fficint fr ntry quilibrium is possibl. That is, th ntry st E = {, } is th uniqu SPE, rgardlss of ntry ordring. W know this is th cas bcaus th sufficint condition from th part B of proposition is mt. Namly, K 3 = 0 < π() =.. Yt th 3 rd firm cannot dtr ntry of ithr th scond or first firm bcaus π(3) = 6.5 > K = 6 > K = 5. Thrfor, thr is no ntry prmutation in which th lowr rankd firms will not ntr if th third rankd firm ntrd. Furthrmor, th 3 rd 43

8 Southwstrn Economic Rviw firm will nvr ntr in quilibrium rgardlss of ntry ordring. In this cas, social wlfar is 33.4 undr fr ntry or whn ntry is rgulatd by a social plannr. Suppos that w chang th physician xampl slightly to fit th paramtrs of xampl. Suppos that th mdical rsidnt from Kansas would stablish a third practic in th town upon compltion of hr rsidncy, rgardlss of whthr or not th doctor from Orgon stablishs a practic. Thus, if th physician from Orgon contmplatd that a lowr sunk cost rival would ntr aftr h did, thn h would look for anothr town in which to st up his practic. That is bcaus h would find th Kansas practic unprofitabl with thr physicians in town. Thus, sunk cost fficincy somtims dpnds on th willingnss of lowr cost rivals to ntr if high cost rivals ntr first as proposition B argus. Th potntial of highr lvls of comptition scars away th infficint ntrant. Exampl 3 K = 0, K = 6, and K 3 = 7. In this xampl only on firm can profitably ntr bcaus both K = 0 < π() = 5 and K = 6 > π() =. Indd, only th lowst sunk cost firm can 9 profitably ntr bcaus K = 6 > π() = 5 satisfying proposition A. Thus, this distribution of sunk costs lads to sunk cost fficincy rgardlss of ntry ordring. Discussion of th Exampls In ach xampl, th st of potntial ntrants is F = {,, 3}. Thrfor, w hav thr firms, n(f) = n = 3, that ar contmplating ntry. In th fist two xampls, w find that two firms, N =, ntr th industry in quilibrium. Nvrthlss, wlfar would always b wakly highr undr fr ntry if th ntry ordr was such that th lowst fixd cost firms movd first. In th first xampl, thr ar C(n, N ) = C(3, ) = 3 potntial subgam prfct Nash quilibrium (SPE) ntry combinations E {, }, E {, 3}, and E 3 {, 3}. In contrast, in th scond xampl, th 3 rd rankd firm cannot dtr th ntry of th nd rankd firm. Thrfor, as in proposition, th most fficint fr ntry quilibrium combination E {, } is th SPE rgardlss of ntry ordring. In xampl 3, proposition A is illustratd. If only th lowst cost firms will b profitabl in an industry of siz N, firm in this xampl, sunk cost fficincy is achivd rgardlss of ntry ordring. All ths xampls illustrat th concpt that ordring dos not affct th numbr of ntrants whn comptitors hav idntical payoffs aftr ntry, π(n ), proposition. Nvrthlss, ordring dos potntially affct wlfar whn firms ar discrt as highr fixd cost comptitors may ntr and prclud lowr fixd cost firms from ntring. Th ncssary and sufficint conditions for ntry ordring irrlvanc in proposition ar illustratd by ths xampls as wll. 44

9 Sunk Cost Efficincy with Discrt Comptitors CONCLUSION This papr has considrd an ntry gam in which firms hav idntical payoffs upon ntry but diffr in thir sunk ntry costs. In this gam, both th xognously givn cost functions and ntry ordrings ar common knowldg. Undr ths conditions, ntry ordring cannot affct th siz of th industry or th numbr of comptitors. Yt, ntry ordrings can somtims affct th sunk costs in an industry. With discrt comptitors, highr fixd cost ntrants can potntially block lowr fixd cost potntial comptitors from ntring. A ncssary and sufficint condition for all ntry ordrings to minimiz fixd costs is drivd. APPENDIX DERIVING THE LINEAR COURNOT MODEL Hr w driv th linar Cournot modl which is usd to analyz th xampls in sction 0. q(n ) is pr firm output, which is a function of th numbr of ntrants, N. Total industry output is N q(n ) Q. Invrs dmand is dfind as pric as a function of industry output, P(Q). Suppos that all firms ar idntical Cournot comptitors who fac a linar invrs dmand curv P(Q) = a bq. Furthr, all comptitors hav idntical cost functions c(q) = c q(n ), whr c > 0 is th marginal cost paramtr. Firms ar assumd to play a simultanous mov Cournot gam in priod. Th Nash quilibrium pr firm output for an industry with N idntical comptitors is a c qn ( ) =. bn ( + ) (9) Total industry output in quilibrium is N a c QN ( ) NqN ( ) =. N + b (0) Th quilibrium pric is a+ N c PN ( ) =. N + () Pr firm producr surplus for all ntrants is a c π ( N ) =. N + b () If th i-th firm ntrs, its profits aftr sunk costs ar 45

10 Southwstrn Economic Rviw a c π ( N ) Ki = Ki. N + b (3) Total profits for th industry ar industry producr surplus, or payoff bfor sunk costs, ( N ) N π ( N ), lss total invstmnt costs, Ki. That is, industry i E profits in quation (4) blow ar obtaind by summing th pr-firm profits in quation (3) for all firms in th ntry st, E: a c N Π( N ) Ki N π ( N ) Ki = Ki. i E i E N + b i E (4) Total consumr surplus, CS(N ), gnratd by this industry is bq N ( a c) CS( N ) = =. N + b (5) Thrfor total wlfar, W(N ; E), which is a function of both th siz of th industry, N, and th ntry st, E, is obtaind by adding quations (4) and (5). It is th following: W( N ; E) Π ( N ) + CS( N ) K = N π ( N ) + CS( N ) K i E i E a c N N = Ki. + N + b i E i i (6) REFERENCES Baumol, William J. 98. Contstabl Markts: An Uprising in th Thory of Industrial Structur. Amrican Economic Rviw 7(): -5. Brsnahan, Timothy F. and Ptr C. Riss. 99. Entry and Comptition in Concntratd Markts. Journal of Political Economy 99(5): Dixit, Avinish Th Rol of Invstmnt in Entry Dtrrnc. Economic Journal 90(): Frank, Charls R., Jr Entry in a Cournot Markt. Rviw of Economic Studis 3(3): Ghmawat, Pankaj, and Barry Nalbuff Exit. Th Rand Journal of Economics 6(): Mankiw, Grgory, and Michal D. Whinston Fr Entry and Social Infficincy. Rand Journal of Economics 7(): Robrts, Kvin Th Participant's Curs and th Prcption of Unqual Tratmnt. Economics Lttrs 97():

11 Sunk Cost Efficincy with Discrt Comptitors Spnc, Michal Entry, Capacity, Invstmnt and Oligopolistic Pricing. Bll Journal of Economics (): Sutton, John. 99. Sunk Costs and Markt Structur: Pric Comptition, Advrtising, and th Evolution of Concntration Cambridg, Massachustts: MIT Prss. Wilson, Linus. 00. Fixd Cost Efficincy with Infinitsimal Comptitors. Applid Economics Lttrs 7(7): ENDNOTES. Physicians hav bn sn as a good xampl of vry simpl firms in mpirical studis of ntry. Brsnahan and Riss (99), for xampl, analyzd ntry into 0 gographically isolatd markts, studying th comptition of physicians, dntists, druggists, plumbrs, and tir dalrs.. Th xampls ar too numrous to do justic to bcaus idntical sunk costs ar a standard simplifying assumption. A sampling of studis that prsnt thortic modls of ntry with idntical sunk costs ar Frank (965), Mankiw and Whinston (986), and Sutton (99). 3. Bcaus ntry costs ar sunk, this papr prcluds th possibility of th hit-andrun quilibria of contstabl markts dscribd by Baumol (98). 4. An xampl of a gam with ths proprtis would b a Cournot gam whr all ntrants had idntical variabl cost functions. This xampl is pursud in sction 4 and is drivd in th appndix. 5. Mankiw and Whinston (986) argu that somtims fr ntry lads to xcssiv ntry. This is th cas hr. If ntry was rgulatd by a social plannr in this xampl, thn only th st rankd firm would b allowd to ntr and social wlfar would b This is highr than th bst fr ntry quilibrium in trms of wlfar, which gnratd wlfar of

12 Southwstrn Economic Rviw 48