Localized vs. Overall Competition in Monopolistic Competition

Transcription

1 研究論文 Localzed vs. Overall Competton n Monopolstc Competton Takash Omor Introducton Monopolstc competton attracts renewed attenton, not only as a model for market competton but also as a mcro-economc base for new macroeconomcs. It would also be a more realstc base, than perfect competton, for normatve analyses n the feld of envronment, servces ndustry etc. where locaton factors play an mportant role. However, ts general propertes do not seem to have been suffcently clarfed. One mportant reason for ths s that most analyses start from a specfc mcro-economc model and we can not decompose ther results nto two parts: those wth general applcablty, and those dependent on specfc mcro-economc model adopted n respectve analyses. In ths paper, I examne propertes of symmetrc monopolstc competton wth respect to three controversal respects. Frst s whether or not t approaches to perfect competton as the number of the frm ncreases. There, t s shown that the answer depends on whether competton s " localzed " or " overall ". In localzed competton, the equlbrum converges to perfect competton, whereas, n overall competton, frms may reman monopolstc even n the lmt. Ths lmt property s rather the opposte to the dfference when the number of frms s fnte. In localzed competton, frms are olgopolstc even when there are a large number frms n the whole market. On the other hand, n overall competton, olgopolstc factor quckly dmnshes as the number of frms ncreases. Ths twsted stuaton may be one of the reasons of the confused arguments on monopolstc competton. Second ssue s whether or not monopolstc competton results n too much varety, and t s shown that varety elastcty or potental heterogenety of the market s the key parameter. The thrd ssue s the sze of the welfare loss under the economes of scale and effcency of possble remedes. Here, t s shown that welfare loss can be very substantal f the demand s prce-nelastc and varety-elastc. The prescrpton depends on these two elastctes. In localzed competton, a measure of nequalty turns out to be a measure of neffcency. Although we can theoretcally thnk of monopolstc competton on homogeneous market such as Counot competton, the subject of ths paper s manly heterogeneous or dfferentated market. There are two reasons for t. One s the fact that products suppled by dfferent frms are more or less dfferentated. The other s that the propertes of heterogeneous market s qute dfferent from homogeneous market, even f potental or observed heterogenety s small, as s dscussed n ths Professor, Faculty of Envronmental and Informaton Studes, Tokyo Cty Unversty 62

2 paper. Prce Elastctes In a heterogeneous market, demand for market products can be expressed n the followng way: x x p, p,..., p,.., p, I...() 2 n where x : demand for product, p : prce of product I : ncome level Let us ntroduce an assumpton of symmetry, that all the products have the same quantty and prce at equlbrum. Ths s a restrctve assumpton, but enables us to concentrate on some essental propertes of heterogeneous markets. Our another assumpton s that each frm produces one product only. Ths assumpton wll be relaxed to a certan extent later. Let us now examne mutual relatonshp between dfferent notons of elastcty. Hcks, (956) made a dstncton between p-elastcty and q-elastcty as the followng: p-elastcty s a prce elastcty when all the other prces are kept unchanged, and q-elastcty s a prce elastcty, when quantty of partcular product s changed whle quantty of all the others are kept constant but all the prces are allowed to adjust so that an equlbrum would be restored. Here we concentrate on the two p-elastctes as the followng: log x o log p log x C log p j p x x p p x x p j : own p-elastcty : cross p-elastcty Note that the sgn of the own elastcty s reversed so that the two elastctes are both postve. There are two knds of symmetrc monopolstc competton. One s a symmetry n the sense that each frm s n an dentcal compettve stuaton, but that competton s localzed n such a way that there s a dstncton between near neghbors and remote frms, wth the former beng drect rvals. Symmetry assumptons n most of the spatal competton models are of ths type. Salop (979) and Capozza and Van Oder (978) are typcal examples. The other s an overall symmetry such that each frm has all the rest as equal compettors wth no dstncton between near neghbors and others. Ths s the case typcally dscussed by Dxt and Stgltz (977), and by Spence (976b). Ths dfference wll matter substantally as wll be shown later. In order to make the dstncton clear, let us use the word localzed for the former and overall for the latter. To be more precse, n localzed competton, j C log x log p j C for some j ( for m products ) 0 for other j ( for n m products )...(2) 63

3 where n s the total number of products n the market and m s the number of drect rvals for each product. On the other hand, n overall competton, j C log x C for any par of j log p j...(3) It goes wthout sayng that the relevant prce elastcty n monopolstc competton, s not an objectve one, but a subjectve one whch depends on the conjecture concernng the response of rvals. Conjectures can change the perceved margnal revenue and thereby mportantly affect the characterstcs of market equlbrum. In prce competton, the perceved prce elastcty of demand s O 0 m...(4) C Hereλ s a conjectural coeffcent n the sense that frms beleve that the reacton by rvals are λ tmes ther own acton, and m s the number of rvals. λs zero n a Bertland-Nash competton. Suppose there s a unform prce ncrease n the whole market. Under our symmetrc assumpton, the rate of demand decrease wll be the same for all the products, and therefore the total demand wll decrease by the same rato. The former can be decomposed nto two factors, the effect of own prce ncrease, and the effects of prce ncrease by rvals. Thus, O m...(5) c where : the prce elastcty of the total demand for the market In localzed competton, m s fxed n (2) and (4), and, m n...(6) combnng (2), (4) and (5), m( ) In contrast, n overall competton, m...(7) C n...(8) from (3), (4) and (5), ( n )( )...(9) C 2 Localzed Competton Competton can not be localzed n terms of quantty at the same tme as n terms of prce. To see ths, suppose that competton s localzed n terms of prce. In the followng, let us suppose that the number of neghbors for each frm s two, as s typcal for one dmensonal market. Then, the demand system can be expressed by a quas-dagonal matrx as the followng: 64

4 x ab,, 0,..., 0, b p. bab,,, 0,..., 0.. 0, bab,,, 0,..., 0. x 0,... bab,,,..., 0 p. 0,..., b, ab,, 0.. 0,..., bab,,. x n b,..., b, a p x *.. x *.. xn * n...(0) To complete the symmetry and to avod the end-frm problem we assumed and n to be neghbors as n a crcular market. But the nverse of the matrx n (0) s not quas-dagonal as the orgnal matrx. The ntutve nterpretaton of ths s that, f quantty of product s ncreased wth the quantty of ts two neghbors unchanged, a change n prces of the two neghbors are necessary, and ths, n turn necesstate changes n prces of the next neghbors f ther quanttes are to be kept unchanged. Ths spll-over process n terms of prce contnues, although weakenng, untl the effect reaches the opposte end of the market. In what follows, we assume that, n localzed competton, what s localzed s prce competton. Ths s the case n spatal competton when buyers decde whch product to buy based on comparson of prces, not of quanttes. Now, let us see how the cross elastcty changes wth an ncrease n the number of frms. If ths approaches nfnty, from (7), the perceved prce elastcty also converges to nfnty, and the prce mark-up wll tend to zero. Indeed Capozza and Oder (978) showed that ths s the case wth spatal models wth lnear transport cost functon and lnear demand functon. Lancaster (979, p7-2) also argued that the elastcty of demand approaches nfnty as the spacng between adjacent products approaches zero. On the other hand, Chamberln (962, Appendx G) argued that the proposton that the larger the number of sellers n a market, the greater the elastcty of demand for each seller does not hold n general. In the followng secton, the behavor of the cross elastcty s examned from a more general perspectve than n the works mentoned above. Frst we wll examne a spatal model wth demand and transportaton cost functons of a general form. Then we wll go on to what we call quas-spatal model. There, a " compensaton " functon plays an equvalent role of transportaton cost functon n spatal models. For both spatal and quas-spatal model examned below, let us make the followng assumptons. Frst, we assume that the whole market area s fxed when the number of frms ncreases. By ths we do not mean that the total demand s fxed, but the catchment area of each frm becomes proportonally smaller. Of course, ths s a strong assumpton and there may be a case where new entrants add a new porton to the market. But n order to capture the pure effects of congeston, let us take ths assumpton. Second, we gnore the fact that f the market s a plane ( two dmensonal ), 65

5 the shape of each symmetrc catchment area has to be a hexagon, not a dsk, f the whole market s to be flled up by catchment areas. Smlar problem exst for hgher dmensonal market. But, n what follows, ths s neglected as a second order problem. Thrd, we assume that populaton s unformly dstrbuted n the market. Fourth, we assume the mll prcng. In other words, that transportaton costs are assumed to be pad by buyers. 3 Spatal Competton Assume that the demand functon by ndvdual consumer s dentcal and gven by x f p d...() where p d s delvery prce, as defned by p p u d ( )...(2) where p: mll prce u: dstance between consumer's locaton and the mll ψ(u): functon of u such that ( 0) 0, '( u ) 0...(3) Then, demand by a consumer at dstance u s x f( p ( u))...(4) u Integratng ths over the catchment area, we have total demand for ndvdual product. R x D f ( p ( u)) H( u) du...(5) 0 where D: densty of consumers R: radus of the catchment area H(u): functon dependent on the dmenson of the market, R such that D H u du e.g.h u ( ) s the total area of the catchment area 0 2 f ( ) 2u f 2 2 4u f 3 : : number of dmenson of the market Now, by dfferentatng (5) wth respect to R, we have, log x DR H( R) f ( p ( R))...(6) log R x But, snce DRH(R) s the number of consumers n the catchment area tmes the number of 66

6 dmenson, the average consumpton n the catchment area s, x A x DRH( R)...(7) on the other hand, the consumpton by a consumer located at the border of the catchment area s, x f ( p ( R))...(8) M Thus, (6) can be smplfed to, log x log R x x A...(9) B where : number of dmenson of the market ( nteger ) x M : demand by consumer at the border of the catchment area x A : average demand ( over the catchment area ) Now, let us see how the radus s affected by a change n compettors prce. If we denote the dstance between the neghborng mlls by u*, condton for the border s gven as p ( R) p* ( u* R)...(20) where p *: prce by neghbor By total dfferentaton, dp '( R ) dr dp * '( u * R )...(2) Snce n symmetrc equlbrum, u* 2 R, and p* p effect of a prce change by ( all the m ) rvals s, log R p * R p...(22) log p * R p * 2 '( R) R Combnng ths wth (9), we have a formula for cross p-elastcty as the followng. log x log R log x p x C m log p* m log p * log R 2m'( R) R x M A...(23) As we have assumed that total market area s fxed, R and n are nversely related. Thus, from (7), ( ) p lm m( )lm C lm...(24) n n 2 n '( R ) R It s easy to see that, f ψ(r) s a lnear functon, ths approaches to nfnty. ths s the case that Capozza and Van Oder (978) analyzed wth a lnear demand functon n two dmensonal market. Let us now examne the general property of ths. Up to certan small value of R, the cross prce elastcty can ether be an ncreasng or decreasng 67

7 functon of R. In other words, (23) may well decrease. As to whether or not t converges to nfnty as R approaches to zero, t depends on the behavor of Rψ'(R), snce p s fnte. ψ'(r)r can ncrease or decrease wth R as shown by Fgure. However, fψ'(r)r s a well behaved functon of R, ths approaches to zero as R becomes small. A suffcent condton s that ψ'(r)r s a monotonous functon of R when R s suffcently small. Appendx gves a mathematcal proof. Fg. Behavor of Rψ'(R) Thus from (7), unless s one, lm n...(25) and the spatal competton converges to perfect competton n the lmt. 4 Quas-spatal Competton Assume that all the consumers have the followng dentcal utlty functon. U x x f 0 ( u)...(26) where x 0 : numerare, x : market product f : functon such that f ' > 0, and f ''< 0...(27) u : dstance n specfcaton between the most preferred product and the best avalable product. ψ(u) : functon of u such that ψ(0)=, ψ' 0, but ψ'> 0 at the border...(28) ψ(u) can be taken as Lancaster's compensaton functon. But our specfcaton s more general than hs, snce we requre (28) only, whereas he (979, p42) requred that ψ(0)=, ψ'(0)=0, ψ'(u)>0 for u 0, ψ''>0...(29) 68

8 The frst order condton for the utlty maxmzaton s gven by or equvalently, U x p f ' ( u)...(30) x ( u) xu g p ( u) ( u)...(3) where x u : demand by consumer at dstance u g : the nverse of functon f ' then, In a smlar way to the spatal model, the demand for ndvdual product s, R 0 x D g p ( u) ( u) H( u) du...(32) log x RH( R) x M g p( R) ( R)...(33) log R x x lm R0 A log x...(34) log R Snce the followng most preferred equvalent are perfect substtutes, x ( u)...(35) the radus of the catchment area, R s a functon of p*, p as the followng: p ( R) p* ( u* R)...(36) By dfferentatng ths at symmetrc equlbrum, we get log R...(37) log p* 2 '( R) R Thus, the cross p-elastcty s log R log x x C log p * log R 2 '( R) R x M A...(38) Snce ψ(0)=, we can construct the same argument for the lmt property as n the case of spatal competton and confrm that...(39) lm n But, agan, ths does not necessarly mean that ξc monotonously ncreases as R decreases. 69

9 Note that, as nr s constant, dlog n dlog R 0...(40) Thus, (9) and (33) can be re-wrtten as log x x M 0 log n x A...(4) Now, let us summarze what we have so far seen on spatal and quas-spatal competton. Under the assumpton of unchanged ( total ) market area, (a) The cross prce elastcty may ncrease or decrease wth the radus. (b) However as the number of frms approaches nfnty, t approaches nfnty. (c) The number-of-frms elastcty of demand s related to an equalty ndex defned as the rato of margnal to average consumpton. Among the above, (b) mples that crowdng wll make a market converge to perfect competton. Of course, one can not clam that ths holds product for any localzed competton. But, gven the generalty of our two models, we may clam that ths property holds for many localzed competton. However, as (a) says, t does not necessarly follow that cross elastcty always ncrease when the number of frm ncreases. In the next secton, let us see an example wth whch, up to a certan level of n, the cross p-elastcty s constant. 5 Constant Cross Prce Elastcty Case ( CPE ) Wth the framework of the quas-spatal model, assume a set of utlty and compensaton functon as the followng. U x x 0 ( u) where 0 < r < ( u ) and r, f u u, f u...(42)...(43) The above assumpton on the compensaton functon mples that there s a certan ndfference range of specfcaton of products wthn whch people are just as satsfed as when they are offered ther most preferred products. To put t another way, ths assumpton mples that the subjectve value of product does not decrease wthn a certan dstance. Applyng (30) and (3), the demand functon by consumer at dstance u s 70

10 and x u r p ( u) r r...(44) The demand for ndvdual product s an ntegral of ths over ts catchment area. R r R r r r x / D xu H( u) du Hudu ( ) Hudu ( ) r p 0 0 u p In one dmensonal market, for example, H(u)=2, and...(45) x D r rr r r r 2 R p ( ) r r...(46) If R >> and α< (-r)/r, we can approxmate ths by 2D( r) x r r r p r R rr r...(47) As we assume that the total market area s constant, R s nversely proportonal to n. Thus, X r r r nx p R More generally, as for a dmensonal space, t can be shown that f...(48) R...(49) ( r ) r X nx p r r (r ) R...(50)...(5) (49) mples that the actual radus of the catchment area s large n comparson wth the ndfference dstance. (50) mples that when the market area of a frm s dvded nto rngs of dfferent dameter, α s small enough ( but not necessarly smaller than ) to ensure that the sales share of an outer rng s larger than that of an nner one. The boundary condton s obtaned by substtutng (43) nto (37). log R log p * 2 log x log R log x r r C ( ) log p * log p * log R 2( r)...(52)...(53) Thus, n ths case, the cross elastcty does not depend on the number of products n the market. Ths case s hereafter referred to as constant cross prce elastcty case or CPE. 7

11 6 Overall Competton Ths s a completely symmetrc case n the sense ξc s the same for any par of dfferent products. A notable dfference from the localzed competton s that ξo and the perceved prce elastcty can reman fnte even n the lmt. One such example s the CES market n Appendx 2-(). There, from (9), the perceved own prce elastcty approaches ( )( )( ) lm O n...(54) In overall competton, each frm competes drectly wth every other. Ths requres that, as the number of frms ncreases, the number of dmenson of market must ncrease suffcently rapdly, as Archbald and Rosenbluth (975) argued. Thus, every new entrant adds a new dmenson, and hence a new porton to the market. 7 Market Demand Functon wth Varety Elastcty Once the symmetrc assumpton s accepted, we can defne the market prce elastcty of demand -- the DD curve by Chamberln (962). In order to capture an mportant aspect of heterogeneous market, let us ntroduce a new parameter to a market demand functon, varety elastcty, whch s an ndex of potental heterogenety of the market demand. It s defned as the degree of the proportonal ncrease n total market demand n response to an ncrease n the number of products avalable n the market, under a gven prce level. Broadly speakng, there are two reasons why a market apprecates large varety. One s that varety s desrable n ndvdual's preference, and the other s that dfferent people have dfferent tastes and a market wth larger varety can extract more of the potental demand. As for ntra-personal apprecaton of varety, varety elastcty s small f each consumer stcks to a partcular product. On the other hand, as for nter-personal apprecaton, the varety elastcty s large f consumers are ether wdely spread or stcky to ther most preferred specfcaton. In the spatal context, a set up of a new seller wll reduce transportaton costs born by consumers and the whole market demand wll generally ncrease under a gven mll prce. Clearly, n a homogeneous market ths elastcty s zero. If ths elastcty s small, the total demand ncreases lttle wth the number of the products and the market quckly becomes satated. The other extreme s the case where ths elastcty s unty. There, no congeston occurs. Indeed, n ths case, a new entrant brngs n a new pece of market exactly as large as that of an exstng product, and ths new pece s ndependent from the rest. Thus, f varety elastcty s unty, t can be vewed as a bundle of ndependent markets rather than a heterogeneous market. From above dscusson, we requre that varety elastcty les between 0 and. Our market demand functon s wrtten as X nx F( p, n)...(55) 72

12 log X log p...(56) log X, log n 0...(57) where X : total market demand, x : demand for each product, n : number of products ( X = n x) p : prce ξ: prce elastcty δ: varety elastcty Note that, n spatal and quas-spatal competton, the varety elastcty s drectly related to the equalty n the market. Ths s because, from (4), log X log R log x x M...(58) log n log n log R x Ths holds regardless of the shape of the demand, transportaton-cost or compensaton functon. Interestngly, the varety elastcty turns out to be a measure of nequalty wth an emphass on the least-convenently-located consumers. Market apprecates varety to the extent that consumpton dffers due to locatve nequalty. 8 Inverse Problem An nterestng queston concernng the market demand functon just ntroduced s, how much, under the assumpton of symmetry, we can deduce the underlyng mcro-economc structure from a market demand functon, n partcular, the cross prce elastcty. To be more precse, A Gven the followng market demand functon : X F( p, n) : DD curve How much can one know about the underlyng demand functon for ndvdual product, namely, x f( p, p*, n) : dd curve such that X F( p, n) n f( p, p, n)? where x: demand for ndvdual product p: own prce p*: prce of all the others n: number of products n the market Note that the market demand functon, DD, s observable, at least theoretcally, whereas the ndvdual demand functon, dd, can not be observed unless p and p* move dfferently. In order to nvestgate ths nverse problem, let us examne three concrete examples, of whch detal s shown n Appendx 2, and see how the cross prce elastcty of mcro-economc demand s related to the two elastctes of market demand ξ, andδ. 73

13 () CPE market ( localzed ) In a CPE market ntroduced before, ξand δare both constant as s seen by equaton (5) and the cross elastcty s, C ( )( ) 2 m where m : number of drect rvals...(59) () CES market ( overall ) If the demand system s generated by a CES utlty functon as was assumed n Dxt and Stgltz (977), ξand δare both constant, and the cross prce elastcty s the followng : ( )( ) C n where n : number of products n the whole market () Quadratc case ( overall ) If the demand system s generated by a symmetrc quadratc utlty functon, the cross prce elastcty s the followng: C where n( n ) n ( n) From the above, followng observaton can be made:...(60)...(6) Frst, n all the above three examples, ξc s negatvely related to δ such that f δ= 0, ξc s nfnte, and f δ=, ξc s zero. Gven the defnton of δ, ths property seems to be general, although the exact relatonshp between the two s model-dependent. Moreover, as s seen n Appendx 2, f the specfcaton of the underlyng mcro-economc model s known but only the parameters are unknown, they can be deduced from the two elastctes of the market demand, ξ,δ, and n, the number of products. Secondly, there can be more than one dfferent underlyng mcro-economc models that gve the same market demand functon, snce among our three examples, () and () have the same form of market demand functon despte the fact that the mcro-economc structure and the cross prce elastcty s substantally dfferent. Thus, the nverse problem can not be unquely solved, at least n general terms. 9 Assumptons Based on the above, let us see how much can be sad about the propertes of monopolstc competton. 74

14 We assume the followng : A. constant prce elastcty of market demand B. constant varety elastcty of market demand C. fxed set up cost F D. constant margnal cost c E. one frm produces one product unless otherwse noted A and B are partcularly restrctve. But there are cases that these hold good as s already dscussed. Moreover, what we really requre s not necessarly the strct constancy over the whole regon, but suffcent constancy n the relevant regon. The market demand functon s: X nxk p n...(62) where K: constant...(63) 0...(64) C and D mples economes of scale. We assume ths for two reasons. One s that, n most ndustres, there are economes of scale at least to a certan extent. The other reason s that, f there were no economes of scale, every consumer can be served wth hs most preferred product and there s no trade-off between producton effcency and varety. C and D leads to a smple cost functon as the followng: Cx ( )F cx...(65) In ths regard, let us defne an ndex, h, by the followng: h Fc / K As h s the rato of the fxed cost to the sales at margnal cost prcng, t s an ndex of economes of scale, or degree of scale mert. Note, however, that the total cost ncreases wth h. 0 Structure of the Analyss The market regmes to be compared are the followng: --- Frst best (F) Ths s the case where everythng s managed so as to maxmze socal surplus. The socal surplus s defned as the sum of consumer surplus and profts of frms. Although the socal surplus approach has two weak ponts: neglect of dstrbuton aspects and ncome effects, t does enable us to solate the effcency aspect of the problem. As Wllg (976) showed, ths approxmaton s not bad f ncome elastcty s small or f the prce and quantty change s small n relaton to the consumers' whole ncome. We assume that the approxmaton s good enough to show us the order of magntude of changes n welfare. Indeed, we wll see later that welfare dfference can be so substantal that a rough measure s suffcent for our purpose. 75

15 Gven the demand and cost functons, the socal surplus s: X SS( p, n) pdx c X nf p X c X nf 0...(66) In the frst best, ths s maxmzed wth respect to p and n. Ths results n the famlar margnal cost prcng. But under the assumpton of economes of scale, ths mples defcts for frms, and requres some lump sum transfer, whch may not be practcal n real world. --- Second best (S) If the lump-sum transfer s not practcal, we have to seek a constraned optmum. Here, the socal surplus s maxmzed subject to the constrant that the sales must cover the producton cost. px cx F 0...(67) If ths s combned wth the demand functon (62), the break-even condton can be rewrtten as, ( k) k n h...(68) where k: prce cost rato ( k = p/c ) As s shown by Fgure 2, ths s an nverse-u shaped curve n the k-n plane, and the second best s gven by ts tangency wth a socal surplus soquant. --- Monopolstc Competton (MP) Here, we confne ourselves to competton wth free entry. Ths does not necessarly mply that the break-even condton s bndng. Especally n the case of localzed competton, f product locaton does not change, t may happen that a vacant part of market s not large enough to support a new product, and excess proft exsts even under free entry. However, n the followng, we take up the cases where excess proft s soaked up by free entry. Even under ths smplfcaton, we can not have a unque soluton for monopolstc competton. As s already seen, the equlbrum depends on, among other thngs, whether or not competton s localzed, whch varable ( prce or quantty ) s used as a mode of competton, and how much colluson the frms expect of ther rvals. To cope wth ths dffculty, we wll do two thngs. Frst, we wll examne two extreme cases whch wll gve upper or lower bounds to the varables we are concerned. One s the second best market ntroduced above, whch gves an upper-bound of socal surplus. The other s the collusve market, whch gves an upper-bound for the number of products and lower-bound for the socal surplus as wll be seen shortly. Second, where useful, we wll use an example of mcro-economc model. --- Collusve market (C) Here we assume that frms have complete colluson under free entry. In other words, the conjectural coeffcent s assumed to be unty. Then, the dstncton between the prce and quantty competton does not matter. The perceved own prce elastcty of demand s the same as prce elastcty of market demand, and we do not worry about the mcro-economc detal. Note that ths 76

16 case s what s called Löshan n spatal competton. The equlbrum s characterzed by the break-even condton and the monopolstc prcng. Snce ths corresponds the peak of the break-even curve n the k-n plane ( see Fgure 2 ), the number of products n collusve market s largest among monopolstc competton. As for the socal surplus, combnaton of (66) and (68) enables us to wrte the socal surplus as a functon of k. Snce ths s a decreasng functon of k, when k s larger than ts second best level, the socal surplus n the collusve market s lowest among monopolstc competton. --- Monopoly (M) For the sake of comparson, we nclude monopoly, n whch all the products are served by a sngle monopolst n the conventonal sense. Ths s characterzed by monopolstc prcng and the optmal number of products from the vewpont proft maxmzaton. Fg.2 Equlbra n heterogeneous market Homogeneous Market and Comparson Heterogeneous market s dfferent from homogeneous market n a number of mportant respects. As for the welfare loss due to the break-even condton ( the rato of socal surplus n the second best to that n the frst best ) n heterogeneous market, A: the welfare loss s ndependent from h, economes of scale B: the welfare loss s negatvely related to prce elastcty C: the welfare loss can be very substantal even f the market s served D: the number of products s not a good proxy measure 77

17 where the opposte s the case n homogeneous market. As regards the varety elastcty, E: the welfare loss ncreases wth varety elastcty A s because, the number of product s endogenously determned, reflectng economes of scale. Thus, even a very small degree of scale economes can cause a sgnfcant welfare loss. As for B, there are two reasons. Frst, f prce elastcty s small, consumer surplus s large relatve to the sales. If the market s under suppled, relatvely large consumer surplus s lost. To put t another way, one suffers a lot f necesstes are under-consumed. Second, B s rather complcated. In homogeneous market, n order to cover the gven set up cost, the prce has to be set above the margnal cost. If the prce elastcty s large, ths leads to a relatvely large decrease n demand, whch n turn pushes prce even hgher, resultng n a large decrease n consumer surplus. Thus, the prce n the second best market s an ncreasng functon of the prce elastcty. Ths s the reason why the loss s an ncreasng functon of the prce elastcty. In contrast, n heterogeneous market, the fxed costs are endogeneous as well as the number of products, and the prce n the second best market turns to a decreasng functon of the prce elastcty. The fact that products of small prce elastcty s necesstes, becomes domnant. Consumer surplus on such products s large relatve to the sales, and one suffers a lot f such commodtes are under-suppled. To see C and D, a numercal example would be helpful. If ξ=. and δ= 0.9, the socal surplus under the second best s almost non-exstent as compared to that n the frst best, despte the fact that a large number of products are suppled to the market. In contrast, n homogeneous market, the second best s not so bad so long as the market s served. For the welfare loss n homogeneous market, there have been a number of analyses on ts magntude, and ther conclusons are dfferent from one another. For example, Harberger (954) estmated that t was of the order of 0.% of the natonal ncome for the US economy n the 920s, based on the assumpton of unt prce elastcty. Bergson (973) argued that, f one takes account of the factors such as larger prce elastcty, nteracton among markets and ncome effect, t may be much larger. Although he dd not show hs own emprcal estmates, hs llustratve calculaton seems to suggest the order up to 0 to 30% of the natonal ncome. But ths concluson s crtczed by Carson (975) and by Worcester (975) for an nconsstency between the proft maxmzng prce mark-up rato and the prce elastcty of demand. If ths pont s modfed, accordng to Carson, the welfare loss would be at maxmum 3.2% of the natonal ncome. These arguments are not very dfferent from what we see for homogeneous markets based on our model, but qute dfferent from the order of magntude of possble welfare loss our model suggests for heterogeneous markets. E s related to the tendency of monopolstc competton to supply too few products when the varety elastcty s large. Ths wll be shown later. It follows that, we should be concerned wth markets wth a small number of frms and a large prce elastcty, snce there the performance of the monopolstc competton s serously unsatsfactory. 78

18 2 Number of products n monopolstc competton Let us examne whether or not monopolstc competton leads to too many products or too few products n comparson to the frst best. Ths s the queston to whch most dversfed answers have been gven. Snce the number s largest n collusve market, we have a followng suffcent condton: n MP n f n F C nf h( ) h( )...(69) Ths mples that f potental heterogenety s suffcently large n comparson to the market prce elastcty, monopolstc competton leads to too few products regardless of the nner structure of the market. On the other hand, f the potental heterogenety s small, monopolstc competton may lead to suffcent or too many number of products dependng on the nner compettve structure of the market. The stuaton s llustrated n Fgure 3. ξ collusve market too many collusve market too few monopolstc competton may or may not supply too few regon any monopolstc competton too few ndetermnate 0 regon /e δ Fg.3 Varety n monopolstc competton 3 What can be done? If the welfare loss can be very substantal, two mportant questons would naturally arse: How can we dstngush markets wth large welfare loss from others, and what should be done for them? For the frst queston, we have seen that the welfare loss s substantal n markets wth a small prce elastcty and a large varety elastcty. The prce elastcty can be measured wth reasonable accuracy. As for the varety elastcty, t s one mnus the rato of the margnal to average consumpton, n localzed competton. It s nterestng that t s an ndex of nequalty as we have already seen, but can also be used as an ndex for market neffcency. Subsdes to facltes n local areas are often justfed on the equty ground, but seldom on the effcency ground. Indeed, t s generally beleved that there s an mportant trade-off between equty and effcency. But what the forgong analyss suggests s that the two consderatons can be qute consstent n the world of 79

19 heterogenety. To sum up, f a market of necessty ( wth large potental consumer surplus ) s unevenly served, t s hghly desrable to ncrease the supply varety, both from equty and effcency pont of vew. One way of addressng such a market s to try to acheve the frst best drectly, through such means as natonalzaton. However, as s emphaszed by Vckers and Yarrow(988), lack of ncentve for effcency mprovement makes such an opton unattractve, and ths s the man reason why prvatzaton of natonalzed frms n many developed countres s called for as an mportant method of structural reform and of economc revtalzaton. In order to guarantee ncentves for mprovement, some element of competton has to be ntroduced. On the other hand, competton n heterogeneous market s subject to monopolstc factor to a certan extent. Thus there s a trade-off. However, f the welfare loss due to nsuffcent varety s much lager than the loss due to monopolstc factor, we can gan by decreasng the former loss at the cost of the latter. There are two market-conformng optons. One s prvatzed competton and the other s subsdzed competton. Of course, both are nferor to the frst best, as some monopolstc factor s nduced, but can be much superor to the second best. Let us examne exactly how such operatons can brng a welfare gan over the second best or over the free market competton. We wll frst examne the constant cross prce elastcty case ( CPE ), whch we dscussed before, and then consder other cases Bertland -Nash free competton Frst let us see what free market competton looks lke n the CPE market. We assume that frms beleve that ther rvals do not change prces n response to there prce change. In other words, frms have Bertland conjecture, and λ, the conjectural coeffcent s assumed to be zero. Then, snce the prce cost rato s dependent on the perceved prce elastcty of demand, from (7), we have, mc k m C If we combne ths and (59) for CPE market, we have the prce-cost rato, ( )( ) 2 k B ( )( )...(70)...(7) Snce ths does not depend on n, ths s a perpendcular lne n the k-n plane as n Fgure 2. Snce ths lne s based on the structure of competton, let us call t competton curve. The equlbrum s determned by ts ntersecton wth the break-even curve, whch s expressed by pont B n Fgure Prvatzed competton Ths operaton nvolves (a) publc provson of supply dversfcaton ( large n ) whch seeks to realze more equalty among people than the prvate competton would acheve, and (b) prvatzaton of producton unts by compettve bddng. The fnancal burden for the government s not as large as n the frst best. Ths s because, n a compettve bd for the producton unts, ther prce would be bd up to elmnate the excess profts. Snce the cross prce elastcty s ndependent from the number of products n CPE market, the prce cost rato s the same as (7). Subject to ths constrant, the government chooses the optmal n. Ths mples that the resultng equlbrum s pont P n Fgure 2, whch s a tangent pont between the perpendcular lne and one of socal-surplus 80

20 soquants Subsdzed competton Alternatvely, government can subsdze for the producton. We assume lnear subsdy such that the amount of subsdy s n proporton to the producton quantty. For frms, ths s dentcal to a reducton n margnal cost, and the prce s lowered proportonally, resultng n a leftward shft of the competton curve n Fgure 2. At the same tme the break-even curve s pushed up. Thus, for a gven subsdy rate, the equlbrum s characterzed, n the k-n plane, by an ntersecton of the new competton curve and the new break-even curve. If we change the subsdy rate, such a pont moves, and ts locus s presented by the subsdy curve n Fgure 2. The optmal subsdy rate s determned by the tangency of ths curve to one of socal surplus soquants, as pont U. As to whch of the above two operatons s preferable depends on the propertes of the market. For CPE market, Fgure 4 llustrates the stuaton. If the varety elastcty s large and the prce elastcty s small, whch s the case that the performance of the second best s worst, prvatzed competton s the best regme. The reason why subsdzed competton s nferor s that, f demand s prce n-elastc, the prce reducton due to the subsdzaton leads to only a lmted ncrease n producton quantty, and does not help much to cover the fxed cost. Thus, n order to ncrease the varety to a gven level, a large subsdy s needed, and ths wll lead to a prcng under the margnal cost, whch mpnges effcency. The low prce elastcty also mples that consumers do not change the consumpton pattern much, and the subsdy s rather lke a transfer to the consumers. In comparson, f the prce elastcty s large, subsdzed competton s recommendable. The second best s superor to these operatons, only when the varety elastcty s small. Fg.4 Best regme non-cpe market What f the cross prce elastcty ncreases wth the number of products? Then the competton curve s not perpendcular n the k-n plane but downward slopng. The welfare gan of the prvatzed competton wll be larger than the constant prce elastcty case. Ths s because the ncrease n the 8

21 number of products wll have a benefcal effect through strengthenng compettve pressure. Ths s llustrated by P' n Fgure 2. On the other hand, subsdzed competton becomes less attractve as s llustrated by U n Fgure 2. Thus, we can say that, f the cross prce elastcty ncreases wth the number of products, prvatzed competton s relatvely more advantageous than subsdzed competton. To summarze the property of these operatons, there are three factors. Frst, an ncrease n number requres addtonal resources. Second, there s a gan from ncreased varety, to an extent whch s dependent on varety and prce elastctes. Thrd, there s a gan f compettve pressure ncreases and the prce cost rato s lowered as the number of products ncreases. In heterogeneous markets, all the three factors can be at work, and a welfare gan s expected by these market-conformng operatons, f the second or the thrd s large. Especally, the second can be very sgnfcant. In a homogeneous market, n contrast, the second factor can not be at work f the market s already served, and the thrd factor may fal to outwegh the frst factor. 4 Concluson If we assume symmetry, we can express the perceved own prce elastcty by the market prce elastcty and the cross prce elastcty. Ths enables us to examne the propertes of competton n a heterogeneous markets, whch are qute dfferent between localzed and overall competton In localzed competton, because t s a chan of local olgopoles, frms have only a few drect rvals, and they tend to be olgopolstc even when there are a large number of compettors n the whole market. On the other hand, n overall competton, olgopolstc factor quckly dmnshes as the number of compettors ncreases However, the lmt property s rather opposte. In localzed competton, as the number of frms approaches nfnty, the cross prce elastcty almost always converges to nfnty and so does the perceved own prce elastcty, makng the equlbrum converge to perfect competton. In overall competton, on the other hand, frms may reman monopolstc even n the lmt. Ths twsted stuaton may be one of the reasons of the confused arguments on monopolstc competton The welfare mplcatons of the market competton under economes of scale are substantally dfferent, and often opposte between homogeneous and heterogeneous market. Conventonal economc theores based on homogeneous market may be msleadng In a market of large heterogenety and small prce elastcty, the welfare loss due to the break-even condton s so large that market-conformng nterventon, such as prvatzed competton or subsdzed competton, s desrable. To dentfy such a market, a measure of nequalty wll help. Effcency and equty s no trade-off n such a market. References Archbald, G.C. and Rosenbluth, G. The New Theory of Consumer Demand and Monopolstc Competton, Q.J.E., vol.89, p569-90, 975 Bergson, A. On Monopoly Welfare Losses, A.E.R., vol.63, p853-70, 973 Blanchard. O. and Kyotak, N. Monopolstc Competton and the Effects of 82

22 Aggregate Demand, A.E.R. Sep., pp , 987 Capozza, D. R. and Van Oder, R. A Generalzed Model of Spatal Competton, A.E.R., vol.68, p , 978 Carson, R. On Monopoly Walfare Loss: Comment, A.E.R., vol.65, p008-4, 975 Chamberln, E. H. The Theory of Monopolstc Competton, ( 8th ed.) Cambrdge Mass. Harvard Unversty Press, 962 Dxt, A. K. and Stgltz, J.E. Monopolstc Competton and Optmum Product Dversty, A.E.R., vol.67, p , 977 Greenhut, M. L., Norman, G., and Hung, C. The economcs of mperfect competton, A spatal approach, Cambrdge Unversty Press, 987 Harberger, A. Monopoly and Resource Allocaton, A.E.R., vol.44( p & p ), pp 77-87, 954 Hcks, J.R. A Revson of Demand Theory, Oxford Unversty Press, 956 Kyotak, N. Multple Expectatonal Equlbra Under Monopolstc Competton, Q.J.E., pp , 988 Lancaster, K. Varety, Equty and Effcency, Columba Unversty Press, 979 Omor T. Generalzed Models of Heterogeneous Markets and the Propertes of Monopolstc Competton, D.Phl thess, Oxford Unversty, 990 Salop, S. C. Monopolstc Competton wth Outsde Goods, Bell Journal of Economcs, vol.0, p4-56, 979 Schmalensee, R. Product Dfferentaton Advantages of Poneerng Brands, A.E.R., vol.72, p349-65, 982 Spence, A. M. Product Selecton, Fxed Costs and Monopolstc Competton, R. E. Stud., vol.43, p27-35, 976 Vckers, J. and Yarrow, G. Prvatzaton: An Economc Analyss, The MIT Press, 988 Wllg, R. D. Consumer's Surplus wthout Apology, A.E.R., vol.66, p589-97, 976 Worcester, D. On Monopoly Welfare Loss: Comment, A.E.R., vol.65, p05-23, 975 Yarrow, G. Welfare Losses n Olgopoly and Monopolstc Competton, Journal of Industral Economcs, vol.33, p55-29, 985 Appendx : Proof of the lmt property of spatal competton Snce we have assumed that ψ'(r)r s monotonous wth respect to R where R s suffcently small ( R < R*), t must be the case that, as R decreases etherψ'(r)r ncreases decreases and converges to a postve value decreases and converges to zero Suppose the last s not the case. Then, for suffcently small R, ψ'(r)r >ε> 0 and f we defne ε* = Mn( ε, R*) we have 83

23 ψ'(r)r > ε*> 0, for R ε* Now, defne a functon φ(r) as φ(r) = ε*lnr - ε*lnε*+ ψ(ε*) Then, snce φ(ε*) = ψ(ε*) and φ'(r) = ε*r- < ψ'(r) for R < ε* It has to be the case that ψ(r) < φ(r) for R < ε* But as R 0, φ(r) -, thus ψ(r) - too. But on the other hand, ψ(0) = 0 at (30). Ths nconsstency shows that such a functon does not exst. Q.E.D. Appendx 2: Three Models of Heterogeneous Market () CPE The market demand functon s gven by X nx K p r n r (r )...() Thus, the varety elastcty s postvely related to the stckness of people, and nversely proportonal to the number of dmenson. Parameters are related n the followng way:, r...(2) ( )( ), C 2 2 o and...(3) ().CES market Suppose a utlty functon of the form of U ( x n r 0, x,..., xn ) x0 ( a x )...(4) where 0<β<, r<, x 0 : numerare x : market goods, a : constant The market demand functon when m goods are suppled s r r r r r r X n x ( r a ) p n...(5) Snce r<, the varety elastcty s a decreasng functon of the elastcty of substtuton, /(-β), but ndependent of m. β and r can be expressed by the followng., r and...(6) 84

24 Usng these, all the prce elastctes are expressed as the followng: ( ) ( ) ( ) ( ) o, and C...(7) n n () Quadratc case Wth a utlty functon of the form of U x n n n n 0 a ) 2 2 j The nverse demand functon s 2 x a ( x x j p a x a ( ) x n j j x...(8)...(9) The market demand when n goods are suppled symmetrcally s n ( p ) X x...(0) a n ( n ) Thus, the varety elastcty s gven by n n ( n ) n...() In partcular, f β=0, then δ=0, and goods are homogeneous each other wth a demand functon as the followng: p a x a X...(2) If β=, then δ=, and goods are ndependent from each other, wth the ndvdual demand functon same as when δ=0,.e., p a x...(3) As for ξ, dx p p...(4) dp X p Wth these, a, α and β are expressed by the followng: n p n( n ) a, p, X n ( n ) The prce elastctes are as follows: o ( n )( ), and C...(5)...(6) 85