Oligopolisti and Monopolisti Competition Uni ed Noriko Ishikawa, Ken-Ihi Shimomura y and Jaques-François Thisse z Marh 28, 2005 Abstrat We study a di erentiated produt market involving both oligopolisti and monopolistially ompetitive rms. We show that the size o monopolistially ompetitive subsetor dereases as the number o oligopolisti rms rises. Soial welare level inreases as the number o oligopolisti rms rises beause the proompetitive e et assoiated with the entry o a new oligopolisti rm dominates the resulting derease in produt variety. Keywords: oligopoly, monopolisti ompetition, produt di erentiation, welare JEL Classi ation: RIEB, Kobe University, Japan, ishikawa@rieb.kobe-u.a.jp y RIEB, Kobe University, Japan, ken-ihi@rieb.kobe-u.a.jp z CORE, Université Catholique de Louvain, Belgium, thisse@ore.ul.a.be and CERAS, ENPC, Paris.
Introdution Armhair evidene shows that many industries are haraterized by the oexistene o a ew large rms, whih are able to manipulate the market, as well as o a myriad o very small rms, eah o whih has a negligible impat on the market. Examples an be ound in apparel, atering, publishers and bookstores, retailing, nane and insuranes, and IT industries. To the best o our knowledge, suh a mixed market struture has been overlooked in the literature. This is rather surprising beause this situation is airly ommon in the real world. Even more strikingly, several ountries have passed bills that restrit the entry o large rms or orbid prie disounts in order to permit small rms to remain ative. For example, in Frane the Lang Law orbids disounts on books with the aim o preserving a large network o small bookstores, whereas the Net Book Agreement in the United Kingdom between book publishers and retailers, whih also prevents prie disounting, is argued by the publishers and small book sellers to be justi able on the same grounds. In Frane again, the Royer-Ra arin Law imposes severe restritions on the entry o department stores whose surae exeeds 300 square meters, the purpose being that small shops provide various onveniene servies. More real-world examples ould be ited. The idea behind suh laws and regulations is that small rms allow or a wider array o varieties and, thus, ontribute to onsumers welare. However, suh an argument disregards the at that ompetition between strategi rms tends to oster lower pries. The purpose o this paper is to provide a uni ed approah that embodies both large/strategi and small/nonstrategi rms. We then use this ramework (i) to study how these two types o rms interat to shape the market and (ii) whether or not it is soially desirable to have large and/or small rms in business. To reah our goal, we blend two standard models, namely the oligopoly model à la Cournot with di erentiated produts and the monopolisti ompetition model à la Chamberlin. The eld o industrial organization is dominated by partial equilibrium models o oligopoly in whih strategi interations between rms appear to be the entral ingredient. They now serve as the orner-stone o many ompetition poliy studies o real-world markets (Motta, 2004). By ontrast, monopolisti ompetition has been extensively employed as the main buildingblok in the analyses o imperet ompetition within general equilibrium models developed in various eonomi elds. Examples inlude eonomi The main notieable exeption is provided by the dominant rm model in whih one large rm and a ompetitive ringe oexist (Markham, 95). Another one deals with big agents (ormally, atoms) whose role in exhange eonomies have been studied within the ontext o ooperative game theory (Gabszewiz and Shitoviz, 992). 2
poliy, growth and innovation, international trade, and eonomi geography (Matsuyama, 995). It is air to say that both approahes are useul to analyze market mehanisms and have their own merits. However, as said in the oregoing, we must also reognize the at that many industries onsist a ew large rms and many small rms. In suh industries, the large rms behave strategially, whereas small rms maximize their pro ts on the residual demand in the absene o strategi interations. This paper may then be viewed as an attempt at providing a reoniliation between suh di erent approahes to market ompetition. On the prodution side, we onsider a di erentiated produt market in whih both oligopolisti and monopolistially ompetitive rms oexist. In measure theoreti-terms, eah oligopolisti rm is an atom whereas eah monopolistially ompetitive rm has a zero measure. Beause small rms typially exhibit more volatility than large rms in their entry behavior, we assume that the mass o monopolistially ompetitive rms is adjusted to the number oligopolisti rms until pro ts in the ompetitive ringe are zero, as in Chamberlin (933). On the onsumption side, we onsider a utility untion with a symmetri CES subutility, as in Spene (976) and Dixit and Stiglitz (977). However, unlike these ontributions, our model involves both disrete and negligible varieties. By so doing, we are able to embody within the same utility the two spei ations o the CES model that have been used in the literature (see, e.g. Anderson et al., 992; Matsuyama, 995). 2 Hene, our model may be viewed as a reoniliation o both types o market struture, whih may be used or di erent purposes. In what ollows, we restrit ourselves to the positive analysis o a mixed market as well as to its welare impliations regarding the entry o oligopolisti rms. We ous on a symmetri equilibrium in whih eah type o rms hooses the same output volume. Our main ndings are as ollows. The entry o a new oligopolisti rm extends the market share o these rms at the expense o monopolistially ompetitive rms. Hene, deregulating mixed markets is likely to lead to a progressive disappearane o small rms. 3 More surprisingly, this market expansion is su iently strong or the output o eah oligopolisti rm to rise when there is one additional oligopolisti rm. Yet, the prie at whih the large rms sell their produt is lower. However, as their pro ts are higher, large rms are better o when entry arises under the onrete orm o a new 2 Reall that Dixit and Stiglitz assumed a ontinuum o varieties in their disussion paper reprinted in Brakman and Heijdra (2004). 3 In the same spirit, there has been in the UK a sharp deline in the number o small groeries ater the passage o the Resale Pries At in 964 abolishing resale prie maintenane (Everton, 993). 3
large rm. All these results, established in an otherwise standard model, suggest that the mixed market struture model obeys di erent rules than standard oligopoly models. In terms o welare, we show the unexpeted (at least to us) result that the entry o additional oligopolisti rms is always bene ial to onsumers in a mixed market as long as these rms make positive pro ts, a ondition that is quite natural. In other words, the possible loss o welare that the ontration in the mass o monopolistially ompetitive rms ould generate is always more than ompensated by the at the whole industry beomes more ompetitive as additional oligopolisti rms operate on the market. This asts some doubt on the welare oundations o the many laws and regulations that tend to keep ative many small businesses. In addition, that result also suggests that the downsizing and rationalization that many rms experiened in the 990s may not be an unortunate, antiompetitive outome but a healthy response to new ompetitive pressure, whih have been espeially strong in retail trade, servies, and the nanial and insurane industries. To be sure, our results are obtained in the ase o a spei model, namely the CES. Being aware o its limits, we want to stress the at that this model is the workhorse o many ontributions dealing with imperet ompetition in modern eonomi theory. So our results annot be dismissed on that basis only. Although more work is alled or, we believe that our analysis provides useul insights about a topi that has been so ar negleted. In addition, even though we do not provide a ull- edged general equilibrium analysis, it is worth stressing that our analysis departs rom standard partial equilibrium models in that we allow inomes to be endogenous. The remaining o the paper is organized as ollows. The details o the model are provided in Setion 2. Setion 3 deals with the main properties o a mixed market equilibrium. The welare analysis is taken up in Setion 4, whereas Setion 5 onludes. 2 The model 2. Consumers There are two goods, two setors and one prodution ator - labor - in the eonomy. The rst good is homogenous and produed under onstant returns to sale and peret ompetition. It is hosen as the numéraire. Without loss o generality, we may assume that one unit o the homogenous good is produed by using one unit o labor, thus implying that the equilibrium wage is equal to. The other good is a horizontally di erentiated produt. 4
It is supplied both by oligopolisti rms and by monopolistially ompetitive rms (in short MC- rms). Variables assoiated with oligopolisti rms are desribed by apital letters and those orresponding to MC- rms by lower ase letters (this should help the reader to remember that an MC- rm is smaller than an oligopolisti rm). Eah rm supplies a single variety, thus implying that oligopolisti rms annot ontribute to produt variety by supplying a produt line, as in Brander and Eaton (984). Let N > be the number o varieties produed by oligopolisti rms and M > 0 the mass o varieties produed by MC- rms. In other words, the di erentiated setor is mixed in that it is onstituted by two subsetors governed by distint orms o ompetition, whih interat aording to rules that will be made preise below. There exists a representative onsumer who desribes the aggregated behavior o the whole population o onsumers. This agent is endowed with L units o labor, holds the shares o all rms, and has a preerene relationship represented by the ollowing utility untion (see Anderson et al. 992, or more details): U = NX Z ( )= M Q j + [q(i)] di! X () j= 0 where Q j is the output level o oligopolisti rm j = ; :::; n, q(i) the output level o MC- rm i 2 [0; M], X the aggregate onsumption o the homogenous good, whereas and are two given parameters satisying the inequalities 0 < < and 0 < <. This spei ation o preerenes enapsulates both the oligopolisti and monopolistially ompetitive modeling strategies o the Spene-Dixit-Stiglitz model. The novel eature o preerenes () is that they inorporates both disrete varieties that have eah a positive impat on utility as well as a set o negligible varieties in that eah o them has a zero impat on U. Clearly, the proess o substitution between these two types o varieties is more involved than in standard models. To illustrate how it works, onsider the situation in whih the quantities o disrete varieties j = ; :::; N are the same and equal to Q, whereas the quantity density o negligible varieties is uniorm and equal to q over [0; M]. Let us now assume that there is a (N + )th disrete variety and onsider the variation o the total mass o negligible varieties that leaves the utility level una eted. It is readily veri ed that M must derease by a positive amount given by Q M = : q 5
Hene, or the utility level to remain the same, the entry o a new disrete variety is to be ompensated by a lower derease in the mass o negligible varieties when the degree o produt di erentiation (inversely measured by ) inreases. It is also worth noting that the value o M rises with Q and all with q. The representative onsumer solves the ollowing maximization problem: Maximize subjet to NX Z! ( )= M Q j + [q(i)] di X j= NX P j Q j + j= 0 Z M 0 p(i)q(i)di + X Y where P j is the prie o variety j = ; :::; N, p(i) the prie o variety i 2 [0; M], and Y the total inome in the eonomy. It appears to be useul to deompose this problem into two steps. In the rst one, we solve the ollowing minimization problem: Minimize Z M 0 p(i)q(i)di subjet to Z M = q(i) di Q 0 0 where we interpret Q 0 as the output index o the MC-subsetor. The rst order onditions or an interior maximum are as ollows: Z M p(i) = [q(i)] Z M ( ) 0 ( q(i) di )= 0 q(i) di = Q 0 where is the Lagrangian multiplier. =( Let R q(i)=[p(i)] ) and Z M ( )= P 0 p(i) di =( ) : (2) 0 be the prie index o the monopolistially ompetitive varieties. then rewrite Q 0 as ollows: We may Z M Q 0 = R 0 p(i) =( = ) di = RP =( ) 0 (3) 6
=( ) so that R = Q 0 =P0, whih in turns implies that q(i) = Rp(i) =( =( ) p(i) ) = Q 0 or all i 2 [0; M]: (4) P 0 Substituting (3) and (4) into the original maximization problem yields the ollowing redued maximization problem: Maximize ( )= NX Qj! X subjet to j=0 NX P j Q j + X Y: j=0 The orresponding rst order onditions imply that ( ) NX j=0 Q j! ( )= ( ) Q j X = P j j = 0; ; :::; N (5) ( )= nx Qj! X ( ) = (6) j=0 Y NX P j Q j X = 0 (7) j=0 where is the Lagrangian multiplier. Let P be the prie index o all the di erentiated varieties, whih we de ne as ollows:! ( )= NX =( ) P P : (8) j=0 j so that P inreases with any P j, j = 0; ; :::; N. It is readily veri ed that the system (5)-(7) imply that =( ) Q i =( )YPi P =( ) D(P i ; P; Y) i = ; :::; N (9) X =Y (0) q(i) =( )Y[p(i)] =( ) P =( ) d[p(i); P; Y] () where D(p i ; P; Y) is to be interpreted as the demand untion o the oligopolisti variety i = ; :::; N and d[p(i); P; Y] as that o the monopolistially ompetitive variety i 2 [0; M]. The at that the untional orms D and d are independent o i re ets the symmetry o preerenes on the varieties supplied by eah subsetor. Both D and d are also dereasing in their own prie. 7
Finally, using (2) and (9), it is easy to show that @D(P i ; P; Y)=@P j > 0 or all i; j = ; ; N and j 6= i, implying that the oligopolistially provided varieties are strong gross substitutes. As the same holds or j = 0, we may onlude that the output index o the MC-subsetor plays the same role in onsumption as any variety o the oligopolisti subsetor. 2.2 Oligopolisti rms Eah oligopolisti rm selets its output level to maximize its pro t. Hene, the solution to the interative pro t-maximizing problem is given by a Nash equilibrium o the ollowing strategi-orm game: (i) the players are the N oligopolisti rms; (ii) the strategy o rm i = ; :::; N is its output level Q i ; and (iii) the payo or player i is given by its pro t untion i (Q ; ; Q N ; Y; Q 0 ) = i(q ; ; Q N ; Y; Q 0 )Q i CQ i F where i() is the inverse demand untion or the produt o oligopolisti rm i, C > 0 the onstant marginal ost and F > 0 the xed ost o an oligopolisti rm, both expressed in terms o labor units. The demand untions (9) and () allow us to desribe the market behavior o both types o rms. First, an oligopolisti rm is aware that its output level a ets the prie index P and is, thereore,involved in a strategially interdependent environment. It also understands that the prie index P is in uened by the aggregate behavior o the MC- rms, as expressed by Q 0. Finally, eah oligopolisti rm should aount or the inome e et that its strategi hoie generates through pro t distribution. However, or reasons disussed below, we assume that these rms ignore the impat that their output poliy has on the total inome. By ontrast, being negligible to the market, eah MC- rm may aurately treats the prie index P and the total inome Y as parameters when seleting its pro t-maximizing output. Hene, unlike the oligopolisti rms, the MC- rms do not behave strategially. ( ) i Solving (9) or P i yields P i = [( )Y] Q P. Substituting this expression into (8) yields the prie index as untion o the onsumption levels: P = ( )Y Q i + X j6=i Q j! = : (2) ( Plugging (2) into P i = [( )Y] ) Q i P then yields the inverse demand untion or variety i = ; :::; N: i(q ; ; Q N ; Y; Q 0 ) = ( )Y Q ( ) i Q i + P : (3) j6=i Q j 8
Consequently, the pro t untion o rm i may be written as ollows: i (Q ; ; Q N ; Y; Q 0 ) = ( Q i )Y Q i + P j6=i Q j CQ i F: (4) Let Q i (Q ; :::[Q i ]; :::; Q N ) be the vetor o all outputs but that o rm i. Then, we have (the proo is given in Appendix A): Lemma. For any i = ; :::; N and any given Q i, i is stritly onave with respet to Q i over [0; ). Hene, the best reply untion Q i (Q i ; Y; Q 0 ) o rm i is the unique solution to the ollowing rst order ondition: [( )Y] = [( )Y] 2 C P (Q i ) + P (Q i ) : (5) Beause the oligopolisti varieties are substitutes, one may expet the variables Q i and Q j to be strategi omplements (Vives, 999). However, we show below that this need not be the ase. Indeed, Thus, @Q i @Q j = @ i = @ i(q i (Q i ; Y; Q 0 ); Q i ; Y; Q 0 ) = 0: @Q i @Q i @ 2 i @Q i @Q j = @2 i @Q 2 i or all i; j = ; :::N and j 6= i: Sine @ 2 i =@Q 2 i is negative by the lemma, the sign o @Q i =@Q j is the same as the sign o @ 2 i =@Q i @Q j. Computing this expression yields @ 2 i = ( ) 2 ( ) ( ) Q i Q j Q i @Q i @Q + X! 3! X Q k Q i Q k : j k6=i Aordingly, or all i; j = ; :::N and j 6= i, the output levels Q i and Q j are strategi omplements when Q i > P k6=i Q k but they beome strategi substitutes as long as Q i < P k6=i Q k. Thus, no general haraterization arises. Furthermore, even though Q 0 is not hosen by a player per se, the output index o the MC-subsetor may be viewed either a strategi substitute or a strategi omplement o oligopolisti rms output beause the same inequalities hold or j = 0. Among other things, this implies that an expansion o the MC-subsetor (through o an inrease o M) does not neessarily imply that oligopolisti rms lower their output. However, we will see below that we have a well-behaved model in that an inrease in the number o rms o eah type leads to lower pries or these rms as well as a derease o the overall prie index. 9 k6=i
2.3 Monopolistially ompetitive rms The pro t maximization problem o MC- rm i 2 [0; M] is given by Maximize (i) = p(i)q(i) q(i) subjet to q(i) = d[p(i); P; Y] where > 0 is the onstant marginal ost and > 0 the xed ost o a MC- rm, both expressed in terms o labor units. Note that the resoure onstraint implies that L > NF + M otherwise the eonomy does not supply the homogenous good. It ollows rom () that the inverse demand untion is given by p(i) = [( )Y] q(i) ( ) P : As a result, the pro t untion o rm i is (i) = [q(i); P; Y] = [( )Y] [q(i)] P q(i) where eah MC- rm aurately treats the prie index P and the total inome Y as parameters. Sine <, (i) is stritly onave in q(i). The rst order ondition or pro t maximization leads to q(i) = ( )Y =( ) P =( ) : Aordingly, we may determine the equilibrium prie p and output q ommon to all MC- rms as ollows: p = =( ) and q = ( )Y P =( ) : (6) This equilibrium is thus unique and symmetri. Whereas the equilibrium prie is onstant, the equilibrium output o an MC- rm is a untion o the prie index P and, thereore, depends on the quantities hosen by the oligopolisti rms. When the mass o rms is M, the equilibrium pro t is then given by =( ) = ( )( )YP =( ) : (7) Finally, the mass M o MC- rms is determined by the zero-pro t ondition = 0 in whih P ; :::; P N and Y are treated parametrially. 0
3 Equilibrium A mixed market equilibrium is de ned as a state in whih the ollowing onditions simultaneously hold: (i) the representative onsumer maximizes her utility subjet to the budget onstraint, (ii) both oligopolisti and MC- rms maximize their own pro ts, and (iii) the mass o MC- ms is positive and suh the pro ts o these rms are zero. In other words, or any given number N o oligopolisti rms, we assume that the mass M o MC- rms is adjusted until their pro ts are zero. Even though the output o eah oligopolisti rm, the output index o the MC-subsetor and the total inome in the eonomy are endogenous, when hoosing its own output level, eah oligopolisti rm treats the other rms output as well as the output index o the MCsubsetor and the total inome as parameters. This implies that these rms behave as inome-takers in that they neglet the at that the total inome in the eonomy is positively a eted by pro ts, thus hanging their demand level. Handling suh an e et is ormally very hard and not neessarily empirially meaningul (Bonanno, 990). Yet, eah oligopolisti rm is aware that a higher/lower inome in uenes positively/negatively the level o its demand. Aordingly, even though our model is not a omplete general equilibrium model, it is a losed, general equilibrium model in whih in whih oligopolisti rms aount or both strategi interations and endogenous total inome. These rms ignore the impat o their poliy on the total inome in the eonomy beause, perhaps, the industry under onsideration represents a small share o the whole eonomy. Admittedly, suh an approah has a partial equilibrium avor (Hart, 985). The di erene lies in the at that, in a typial partial equilibrium setting, the total inome would be exogenous. We may haraterize our equilibrium onept by means o the ollowing our onditions or some M > 0 and N : (a) the demand untions, (b) the pro t-maximization onditions o MC- rms, () the pro t-maximization onditions o oligopolisti rms, and (d) the zero-pro t ondition o MC- rms. In this way, we an view 0 as a pseudo-player who would hoose the mass o MC- rms in order to make zero pro ts. Let us stress that the oligopolisti rms do not behave here as the leaders o a sequential game in whih the MC- rms (or the pseudo-player 0) would be the ollowers. The size o the MC-subsetor is determined simultaneously with the variables o the oligopolisti subsetor. In what ollows, we ous on a symmetri mixed market equilibrium in whih all oligopolisti rms hoose the same output Q whereas all MC- rms have the same prodution poliy q given by (6). Our rst proposition is proven in Appendix B.
Proposition There exists a unique symmetri mixed market equilibrium. This result is important beause it implies that we stay on the same equilibrium path (i any), when studying the impat o the entry o a new oligopolisti rm. The equilibrium pro t o an oligopolisti rm at suh an equilibrium is then as ollows: The total inome is given by whih is the unique solution to = [( )Y] (Q ) P CQ F: (8) Y = L + N + M Y = L + N [( )Y] (Q ) P CQ F =( ) +M ( )( )YP =( ) : (9) Observe that the equilibrium value o Y expliitly aounts or the level o xed osts in eah subsetor as well as or their respetive size. It ollows immediately rom (2) that P = (P0 ) =( ) + N(P ) =( ) ( )= = ( )Y [(Q 0 ) + N(Q ) ] (20) Substituting (6) into (2) and (4), we obtain the equilibrium values o the prie and output indies o the MC- rms: = : P 0 = M ( )= (2) and =( ) Q 0 = ( )Y M = P =( ) : (22) The unknown variables Y, P, Q and Q 0 are thus determined in terms o M by using the our equations (5), (9), (20), and (22). This gives us the market outome when the size M o the MC-subsetor is xed. It is worth studying how this outome hanges with M beause this will shed light on the way the two subsetors interat at the market equilibrium. Unortunately, the at that the outputs Q i may be either strategi omplements or strategi substitutes does not allow us the determine how Q and Q 0 are a eted when M rises. However, we are able to haraterize the impat on equilibrium pries. Clearly, (2) implies that the prie index o the MCsubsetor dereases as the size o this subsetor rises. Furthermore, as proven in Appendix C, inreasing M has a similar impat upon P and P. 2
Proposition 2 Consider a symmetri mixed market equilibrium in whih oligopolisti rms earn positive pro ts. Then, both the prie index o the di erentiated industry and the prie at whih oligopolisti rms sell their output derease when the mass o MC- rms inreases. To provide a ull haraterization o the market equilibrium, we still have to determine the size o the MC-subsetor. Using (7), the zero-pro t ondition = 0 is equivalent to ( )Y = Substituting (23) into (22), we obtain Q 0 = =( ) P =( ) : (23) (M ) = : (24) Hene, both the equilibrium mass and output index o the MC-subsetor move in the same diretion. The market equilibrium is then desribed by the ve simultaneous equations (5), (9), (20), (23) and (24) whose unknowns are Y, P, Q, Q 0, and M. Our objetive is now to identiy two onditions that will allow us to study the behavior o Q and M. Let [( )=](=) =( ). First, substituting (23) and (24) into (20) leads to P =( ) = M =( ) + N(Q ) : (25) Seond, using (20), (23), (24) and the oligopolisti rms rst order ondition (5), we have 4 P =( ) = (Q ) Finally, substituting (23) into (9) yields C (Q ) : (26) P =( ) = ( ) L NF + NQ (Q ) ( ) C : (27) 4 For this expression to be meaningul, its RHS must be positive. We show below that this amounts to assuming that oligopolisti rms earn positive equilibrium pro ts. 3
Equating, respetively, (26) and (27) as well as (25) and (27) give us the equilibrium output o oligopolisti rms and the equilibrium mass o MC- rms: (Q ) = ( ) L + N [(Q ) CQ F ]g M = ( ) Finally, it ollows rom (20) and (25) that C (Q ) L NF NQ (Q ) ( ) + C (28) : (29) P = (Q ) ( ) : (30) In words, any ore induing oligopolisti rms to expand their output leads to a lower prie index or these rms. In partiular, a larger number o oligopolisti rms leads to a lower equilibrium prie or these rms. We assume through the rest o the paper that, in equilibrium, oligopolisti rms earn stritly positive pro ts. Substituting (23) into (8), it is readily veri ed that this assumption is equivalent to the ollowing inequality: = (Q ) CQ F > 0: (3) This expression together with Q > 0 and (28) derived below then implies that (C=)(Q ) > 0: (32) Proposition 3 Consider a symmetri mixed market equilibrium in whih oligopolisti rms earn positive pro ts. Then, the equilibrium output o an oligopolisti rm inreases when the number o oligopolisti rms rises. Proo: Di erentiating (28) with respet to N yields dq dn = A [ (Q ) CQ F ] ( ) C 2 (Q ) B + (Q ) ( ) ( ) NA 2 where A (C=) (Q ) and B L + N [(Q ) CQ F ]. Both A and B are positive beause o (32) and (3). The rst ator in the expression above is positive by (3). In the seond, urly braketed ator, the rst term is positive beause B > 0. For the proo to be omplete, it remains to show that its seond term is positive. Note that (28) may be rewritten as ollows: (Q ) = ( ) AB 4
so that (Q ) ( ) ( ) NA 2 = (Q ) AB (Q ) ( ) NA 2 = (Q ) A [B (Q ) NA] : Replaing A and B by their respetive expression, we get (Q ) ( ) ( ) NA 2 = (L NF ) + CNQ whih is positive beause (3) implies that L > NF. Q.E.D. Proposition 4 Consider a symmetri mixed market equilibrium in whih oligopolisti rms earn positive pro ts. Then, both the equilibrium mass o MC- rms and quantity index o this subsetor derease when the number o oligopolisti rms inreases. Proo: Di erentiating (29) with respet to N yields Q dm dn = ( ) + (Q ) ( ) + C (Q ) ( ) + C + F N dq < 0: dn The seond part o the statement ollows immediately rom (24). Q.E.D. This proposition has an important impliation: the MC-subsetor may disappear when the number N o oligopolisti rms is su iently large. Indeed, using (29), we see that the ritial value N O or whih M = 0 must be a solution to: N = L ( ) [Q (N)] + CQ (N) + F : This equation has a single and positive solution beause the LHS is inreasing in N and equal to zero at N = 0, whereas the RHS is dereasing by Proposition 3 and always positive. When N > is an integer suh that N N O, we have M = 0 so that the market is entirely oligopolisti. The equilibrium values o the remaining variables are then given as below: Q O = ( )(N ) (L NF ) CN [N + ( )(N )] N (2 )= P O = P O = C N Y O N = (L NF ): (33) N + ( )(N ) 5
That Q O dereases when the level o xed osts inreases stems rom the at that the total inome Y dereases with F. The values (33) slightly di er rom those derived in partial equilibrium models in whih the total inome is xed and exogenous beause pro ts are redistributed here (Anderson et al., 992). In the oregoing, we have unovered the existene o a trade-o between the two subsetors: as the oligopolisti subsetor expands, the MC-subsetor shrinks and vie-versa. This in turn allows us to determine the impat o an inrease in the number o oligopolisti rms on market pries. Indeed, as N rises, it ollows rom (30) that P dereases. However, by (2), the derease in the mass o MC- rms leads to an inrease o P 0. Thus, the total impat on P is a priori undetermined. Yet, we have: Proposition 5 Consider a symmetri mixed market equilibrium in whih oligopolisti rms earn positive pro ts. Then, the prie index o the di erentiated industry dereases when the number o oligopolisti rms inreases. Proo: Using (26) leads to dp dn = P ( 2)=( ) (Q ) (+) C (Q ) + ( ) C dq (Q ) dn < 0: whih is negative by (32). Q.E.D. In other words, despite the at that the entry o a new oligopolisti rm triggers the exit o some MC- rms, the entry o a new oligopolisti rm makes the global market more ompetitive. Thus, even though the market might involve less variety, ompetition beomes erer and pries are lower. 4 Welare The soial welare is given by the utility o the representative onsumer: W = NX Z! ( )= M Q j + [q(i)] di X j= 0 Introduing (9)-() into W, we the indiret utility: 5 W = (Y) [( )Y] P ( ) : (34) 5 Note that it is legitimate to assume the existene o a representative onsumer beause preerenes satisy the Gorman polar orm. 6
Reall that Y takes into aount both the number and the xed osts o oligopolisti rms. We may thus onsider the impat o inreasing N upon both P and Y to determine how it a ets welare. We already know rom Proposition 5 that P goes down. It remains to onsider how Y is a eted. Using (23), we see immediately that a lower value P leads to a higher value o Y. Proposition 5 thus implies: 6 Proposition 6 Consider a symmetri mixed market equilibrium in whih oligopolisti rms earn positive pro ts. Then, the total inome inreases when the number o oligopolisti rms inreases. We are now ready to show: Proposition 7 Consider a symmetri mixed market equilibrium in whih oligopolisti rms earn positive pro ts. Then, the soial welare inreases when the number o oligopolisti rms rises. Proo: Di erentiating (34) leads to dw dn = ( ) P ( ) dy dn ( ) 2 YP (2 ) dp dn : The result then ollows rom Propositions 5 and 6. Q.E.D. In words, this result has the ollowing major impliation: a di erentiated market with several large rms and a small number o small rms is more e ient than a market with ewer large rms and a larger number o small rms. Given that Y = L + N (N), the proposition above implies that total pro ts in the eonomy rise with the number o oligopolisti rms. However, this does not mean that individual pro ts inrease. To hek it, we di erentiate (3) and get d (N) dn = (Q ) ( ) C (Q ) dq dn > 0 by (32) and Proposition 3. Unlike what we observe in partial equilibrium models o oligopoly where individual pro ts derease with the number o rms (see, e.g. Anderson et al., 992), suh pro ts inrease here as long as there exists an MC-subsetor. This is beause the entry o a new oligopolisti rm leads to an expansion o the market supplied by these rms at the expense o the MC-subsetor, the size o whih shrinks as shown by Proposition. Indeed, as the MC-subsetor vanishes, (N) evaluated at the purely 6 Di erentiating (9) with respet to N yields a similar result. 7
oligopolisti outome (33) appears to be a dereasing untion o N. By ontrast, when there is an MC-subsetor, the market expansion e et generated by the entry o a new oligopolisti rm dominates the ompetitive e et assoiated with the presene o more oligopolisti rms. 5 Conluding remarks The mixed market model seems to di er signi antly rom standard oligopoly theory. This is worth noting beause we oten enountered suh markets in the real world and beause keeping a ompetitive ringe in quite a ew setors seems to be a onern in several ountries. To be typed or to be done.. The rst best outome 2. Is welare ontinuous at N O? 3. How is welare between N O and L=F? 4. The onditions or N oligopolisti rms to earn positive pro ts and the ondition or M (N) > 0. 8
Appendix A It ollows rom (4) that @ i @Q i = ( )YQ i Q i + X j6=i Q j! 2 X j6=i Q j! C whih, in turn, implies that lim Qi!0 @ i =@Q i =. Note that i (0) = F. Beause we have @ 2 i =( )Y X! Q @Q 2 j Q 2 i Q i + X! 3 Q j i j6=i j6=i " # ( + )Q i ( ) X j6=i Q j < 0 i is strongly onave with respet to Q i. Appendix B. Existene. Beause P i = [( )Y] Q ( ) i P, we have P i Q i CQ i F = P i = [( )Y] Q i P CQ i F (B.) whereas (23) leads to P = [( )Y] ( ) : (B.2) Plugging (B.2) into (B.) and using symmetry yield the equilibrium pro t o an oligopolisti rm Q = CQ F so that the equilibrium value o the total inome is as ollows: " # Q Y = L + N CQ F : Plugging (B.2) into (5) in whih Q i = Q and simpliying lead to = C (Q ) + [( )Y] (Q ) 9
whih implies Y = C ( ) (Q ) (Q ) : Hene, we have " L + N = C ( ) # Q CQ F (Q ) (Q ) : (B.3) Beause the numerator o the RHS o (B.3) is always positive, its denominator must also be positive or Y > 0 so that Q must be lower than Q " C # =( ) : h(q) Let " C Q Q : " Q 0 = ( )Y # ( L + N " # = N(Q ) #) Q CQ F Note that Q = Q i and only i Q is a solution o h(q) = 0. Sine h(0) = L NF > 0 and sine h(q) < 0, the intermediate value theorem implies that Q 2]0; Q[ exists suh that h(q) = 0. 20
2. Uniqueness. Standard algebra shows that dh=dq = ( ) C ( Q L + N " N Q C Q < 0: Hene, h(q) intersets the Q-axis only one. " 2 Q # #) Q CQ F Appendix C (i) Consider rst the impat o a larger M on P. Substituting (22) into (20) and simpliying, we obtain Q = ( )YN = P " = ( )YN = P E = # =( ) = MP =( ) (C.) where E (=) =( ) MP =( ) is positive provided that Q > 0. Substituting (C.) into (5), we have NP = C N (2 )= E ( )= + P =( ) MP =( ) : (C.2) Di erentiating (C.2) with respet to M yields dp dm = =( ) P =( ) P+ " N + C N (2 )= ( 2)= E =( ) MP (2 )=( ) P + C N (2 )= ( 2)= E whih is negative. (ii) Let us now study the variation o P. Substituting (2) in (20), di erentiating the resulting expression with respet to M and simpliying # 2
leads to dp dm = N (P ) =( ) 8" < N(P ) =( : ) + M =( ) # = dp dm 9 =( ) = ; whih is negative beause o dp=dm < 0. Reerenes [] Anderson, S.P., A. de Palma and J.-F. Thisse (992) Disrete Choie Theory o Produt Di erentiation, Cambridge, MA, The MIT Press. [2] Bonanno, G. (990) General equilibrium theory with imperet ompetition, Journal o Eonomi Surveys 4, 297-328. [3] Brakman, S. and B.J. Heijdra (2004) The Monopolisti Competition Revolution in Retrospet, Cambridge, Cambridge University Press. [4] Brander, J.A. and J. Eaton (984) Produt line rivalry, Amerian Eonomi Review 74, 323-334. [5] Chamberlin, E. (933) The Theory o Monopolisti Competition, Cambridge, MA, Harvard University Press. [6] Dixit, A.K. and J.E. Stiglitz (977) Monopolisti ompetition and optimum produt diversity, Amerian Eonomi Review 67, 297-308. [7] Everton, A.R. (993) Disrimination and predation in the United Kingdom: small groers and small bus ompanies - a deade o domesti ompetition poliy, European Competition Law Review, 7-4. [8] Gabszewiz, J.J. and B. Shitovitz (992) The ore o imperetly ompetitive eonomies. In R.E. Aumann and S. Hart (eds.), Handbook o Game Theory with Eonomi Appliations, Amsterdam, North-Holland, 460-483. [9] Hart, O. (985) Imperet ompetition in general equilibrium: an overview o reent work. In K. Arrow and S. Honkapohja (eds.), Frontiers in Eonomis, Oxord, Basil Blakwell, 00-49. [0] Markham, J.W. (95) The nature and signi ane o prie leadership, Amerian Eonomi Review 4, 89-905. 22
[] Matsuyama, K. (995) Complementarities and umulative proess in models o monopolisti ompetition, Journal o Eonomi Literature 33, 70-729. [2] Motta, M. (2004) Competition Poliy, Cambridge, Cambridge University Press. [3] Spene, M. (976) Produt seletion, xed osts, and monopolisti ompetition, Review o Eonomi Studies 43, 27-235. [4] Vives, X. (999) Oligopoly Priing. Old ideas and new tools, Cambridge, MA, The MIT Press. 23