Horizontal product differentiation: Consumers have different preferences along one dimension of a good.

Transcription

1 Produc Differeiio Firms see o e uique log some dimesio h is vlued y cosumers. If he firm/roduc is uique i some resec, he firm c commd rice greer h cos. Horizol roduc differeiio: Cosumers hve differe refereces log oe dimesio of good. Vericl roduc differeiio: Cosumers hve he sme ordil refereces, u o he sme crdil refereces. e.g. ll cosumers refer eer fuel efficiecy, u heir willigess o y will differ. Three Aroch s Exogeous Produc Differeiio Vives, X.d N. Sigh "Price d Quiy Comeiio i Differeied Duooly", The Rd Jourl of Ecoomics, 5 (4), 984, Horizol Produc Differeiio Slo, S. 979, Mooolisic Comeiio wih Ouside Goods, The Bell Jourl of Ecoomics, vol., Vericl Produc Differeiio Suo, J. d Shed, A. 983 "Nurl Oligoolies,'' Ecoomeric, 983, vol 5,

2 Exogeous Produc Differeiio Vives, X.d N. Sigh "Price d Quiy Comeiio i Differeied Duooly", The Rd Jourl of Ecoomics, 5 (4), 984, Nsh Equilirium i Prices Assume firms, wih mrgil d fixed coss zero. Assume exogeously imosed roduc differeiio q + θ q + θ The rofi fucio for firm is π ( + θ ) F.O.C.: π / + θ Bes-resose * ( ) (+ θ ) / Similrly: * ( ) (+ θ ) / Thus, i (+ θ j ) / * / ( θ ) * i * / ( θ) q i * / ( θ) π i * / ( θ)

3 Nsh Equilirium i Quiies (Couro) Fid f(q,q ) d g(q,q )? Iverse Demd fucio: ( + θ ) θ ( q ) ( q ) θ θ θ To simlify, rewrie s, α βq γq Similrly, α γq βq where β >, d β > γ Mx π i (q, q ) (α βq i γq j )q i where i, j, d i j F. O. C. π i / q i α βq i γq j Bes-resose q i * R i (q j ) (α γq j ) / β Solve for he equilirium q* i α / (β + γ) * i αβ / (β + γ) π* i α β / (β + γ) 3

4 How do Berrd d Couro comre ow? Berrd Couro * i / ( θ ) αβ / (β + γ) q* i / ( θ ) α / (β + γ) π* i / ( θ) α β / (β + γ) I order o comre hem, we eed o exress {,, θ} i erms of {α, β, γ}. Re-wriig he Berrd soluio: Berrd Couro * i α(β γ) / ( β γ ) αβ / (β + γ) q* i αβ / ( β γ ) ( β + γ ) α / (β + γ) π* i α β(β γ) / ( β γ) ( β + γ ) α β / (β + γ) Le s focus o he differece i rice. Whe γ β, Berrd for homogeeous goods gives, i d c i α /3 If goods ideede, whe γ, moooly oucomes, i c i α / c i i I geerl: 4β γ α We sid β > γ, α >, so c i > i u hs (β/γ) c i coverges o i. The differece i rice cos mrgis modelled is smller whe we relx rice comeiio wih roduc differeiio. 4

5 Horizol Produc Differeiio Slo, S. 979, Mooolisic Comeiio wih Ouside Goods, The Bell Jourl of Ecoomics, vol., Lier Ciies Hoellig (99) Umrells. Cosumers uiformly disriued wih desiy log defied lie segme l.duooly 3.Cosumers icur rsor cos er ui of disce d rveled o he seller 4. * + d, cosumers uy from he seller wih lowes * 5. Le oe seller e loced A, he oher B A B l 5

6 A idiffere cosumer B Sloe of he umrells give y (exogeous) Heigh of he umrell sem give y (seller locio B hs lower h seller A) Tol cos o he cosumer from uyig from riculr seller +d Cosumers uy from seller wih lowes +d Equio of he idiffere cosumer P A +d A B + d B All cosumers loced o he lef of he idiffere cosumer will uy from seller d o he righ of he idiffere cosumer will uy from seller B 6

7 Hoellig (99) Simle Locio Model Sge Sge Locio NE Prices Choice (give locio (,),) A Idiffere cosumer B A Idiffere cosumer B No Nsh Equilirium i Pure Sregies 7

8 No rice Comeiio i Sge Sge : Firms A d B Choose locio log Secrum The Pricil of Miiml Differeiio Focus o he Cchme Are Pyoff for A(B): lie segme covered A d B re c-o-c, u o i cere: A B Iceive for A o jum i fro of B.. A d B re c-o-c i cere: A B Neiher firm hs iceive o move locio, give he locio of he oher The Midoi is Nsh Equilirium i Locios Dows Theory of Moder Voig 8

9 D Asremo e l (979) Assumes Qudric Trsor Coss Sge Give locios, fid NE i rices d d A Idiffere cosumer B < l > q + d d q + d idiffere cosumer + d + d 9

10 Wih qudric coss here is NE i rices + d + d d + d l solvig for mre shre, we ge ) ( ) ( ) ( ) ( * * l l d q l l d q Now solve for Nsh equilirium i rices: ) (.. R s q Mx π π A B d d d > d <

11 Mx π π q s.. R ( ) The NE i rices solves s: * ( l )( + ) 3 * ( l )( + ) 3 Thus, rices deed o locios: * (,) d * (,) Sge : Choose locios π (,) d π (,) Solve for Nsh equilirium i locios Resuls idice h π π < d < hus, igger will reduce firm s rofi iceive o miimise d move owrd ed liewise for firm so fid NE i locios occurs, where firms loce he exreme ois of he mre

12 Assumio uderlyig he locio models: disriuio of cosumers ws uiform log he lie No-uiform disriuio of cosumers: where o loce? Trde-off: how much comeiio here is i ceri iche gis he ville iches d Asremo e l:. rice comeiio srog. cocerio of cosumers is uiform oiml locio is he exremes loce i iche Hoellig:. o rice comeiio. cocerio of cosumers is uiform oiml locio is he midoi loce where he demd is Geerl Predicios: Could e ywhere log he lier ciy. Oiml locio rdes off he iesiy of rice comeiio wih mre coverge..

13 Slo (979) Circulr Rod Model Sge :. N Sellers re loced symmericlly roud he circle. Circumferece is ormlised o 3. Disce ewee ech seller is hus /N Give umer of firms N, fid NE i rices /N d d d /N d Seller i 3

14 Rereseive Seller i rofi fucio, where TC : sles d idiffere cosumer: + d + [/N d] so d _ + N _ π. d + N seller mximises i s rofi y choosig, give, π N + _ S.N.E. * i N s s N MC MC 4

15 Summry of Shor Ru relioshi Price cos mr-us d he umer of firms i he mre, i.e. he P(N) fucio, for Horizolly Differeied goods moo > > Differeied Berrd: MC Differeied Berrd: Homogeous Berrd: N P, for y give N, deeds o he iesiy of comeiio Produc Differeiio Relxes he iesiy of rice comeiio 5

16 Sge : Eer wih su cos σ? Equilirium rofis solve s: x i d i. e. where N s x i ds where s is mre N s π i i x i. N N N s size Ls firm eers where exos ery rofi σ s π i N σ Thus, solvig for equilirium umer of firms: * N s σ 6

17 Summry of Log Ru relioshi ewee mre cocerio (/N) d mre size S relive o exogeous su coss σ, for Horizolly Differeied Goods C N Homogeous Berrd ( ) > > Differeied Berrd ( ) Differeied Berrd ( ) S σ Greer roduc differeiio iduces more ery (so less cocerio) for y give s/σ For give S/σ, he equilirium level of cocerio icreses wih he iesiy of comeiio For give iesiy of comeiio, he equilirium level of cocerio flls wih icrese i Mre Size relive o su coss 7

18 Vericl Produc Differeiio Suo, J. d Shed, A. 983 "Nurl Oligoolies,'' Ecoomeric, 983, vol 5, Assume. A umer of firms offer disic susiue goods which vry i quliy. Cosumer uy ui or zero 3. Zero coss 4. lel goods,.,, firm sells roduc 5. Coiuum of cosumers of differe icomes uiformly disriued, wih desiy of oe, log lie segme o. 6. Uiliy < u < u <..< u U(, ) u ( ) U(, ) u 7. Defie C u /( u u - ) > Equio of he Idiffere Cosumer u ( ) u - ( - ) or -. ( C ) +. C NOTE: C d C 8

19 9 Mre Shre: Cosider good : if > so more h oe good survives π ( ) F.O.C π C Noe. C -. ( C ) ( ) ) C ( C -.( + < 3 > < /

20 Now ssumes h < / π ( + ) F.O.C π [ C ) + C ] + ( + Noe. C -. ( C ) (C + ) (C ) > > + > < + / Is follows h, < /, - < /4, - < /8 ec for y > / 8 ¼ ½ There exiss oud ideede of roduc quliies d cosumer desiy o he umer of firms which c survive wih osiive rices NE i rices.

21 Numer of firms deeds o lower oud o icome,.. C hve los of firms, d, u -3 firms will domie.. A firm h ses rice equl o zero will wi ll he mre. 3. Per of Mre shres will e ideede of he desiy of cosumers (Mre Size).