The different consumption functions of products and product differentiation

Transcription

1 Th diffrnt consumption functions of products and product diffrntiation Waka Chung Dpartmnt of Economics, Monash Univrsity, Clayton, Vic 368, Australia Yw-Kwang Ng Dpartmnt of Economics, Monash Univrsity, Clayton, Vic 368, Australia Abstract: This papr dvlops a monopolistic comptition modl to study th charactristics of products, such as th diffrnt consumption functions of products, vrtical diffrntiation (uality and horizontal diffrntiation (individuality, and th division of labor in production. Our modl contributs in svral aspcts: first, in contrast to many vrtical and horizontal product diffrntiation modls which assum that ach consumr dmands on unit of th product, th uantity dmandd in our modl is dtrmind by th markt. Scond, modls of gnral purpos products suggst that, as an tnsion of Hotlling s modl, products with mor functions will hav highr production costs but lowr pr-unit-distanc transaction costs, but no clar planation is givn. n our modl, th consumption functions srvd by a product ar rprsntd by a functional intrval, with th usag cost and production cost both variabl. Third, w gav a simpl production function which includs ths product s charactristics and has th conomis of functional spcialization. Last, ths factors ar analyzd systmatically in a gnral uilibrium modl. Morovr, through comparativ static analysis, w amin th corrlations btwn th products charactristics and th ognous variabls such as population and popl s prfrncs, transaction costs and managmnt fficincy, and production tchnology. EL classification: L, L, Kywords: product diffrntiation, gnral purpos products, division of labor, monopolistic comptition,. ntroduction n rality, markts not only dtrmin th uantity and pric of products, but also th charactristics of products. On might summariz ths charactristics into thr aspcts. Th first is th diffrnt consumption functions of products. For ampl, as functional gnralization, computr provids mor and mor functions of mdia play (such as TV, CD, VCD and DVD and communication (such as tlphon, fa and . On th othr hand, thirty yars ago, popl usd to war a kind of canvas shos for many sports itms. As athltic shos bcoming mor and mor functionally spcializd, you can buy any spcializd athltic shos for most sports itms. Th scond is horizontal diffrntiation (Lancastr, 979. Diffrnt popl will prfr diffrnt products charactristics, such as styl, dsign, siz, color, and othrs which mak no obvious cost

2 diffrnc to th producrs. Th third is vrtical diffrntiation (uality. Products with highr uality usually nd mor production costs. n ordr to modl th variation (th varity of diffrntiatd products, two prvalnt rsarch strams volvd. Th first is th class of spatial modls in th spirit of Hotlling (99 and Lancastr (979. Thy us an intrval or a circl to rprsnt th charactristics spac of products in which firms can compt in pric, location (horizontal diffrntiation, and uality (vrtical diffrntiation. t has bn tnsivly tratd in a sris of paprs, such as D Asprmont, Gabszwicz and Thiss (979, Gabszwicz and Thiss (979 and 986, Shakd and Sutton (98, 983 and 987, Frrira and Thiss (996, D Frutos, Hamoudi and aru (999 and othrs. Th scond is th class of non-spatial modls in th spirit of Chambrlin (933, Spnc (976, and Diit and Stiglitz (977, Krugman (980, 98, Yang and Shi (99 and othrs. Thy hav ndognizd th diffrntiation by formalizing a trad off btwn th conomis of scal and th prfrnc for divrs consumption. Comparing th spatial with th non-spatial modls, ach framwork has strong points. First, in th spatial modls, consumrs and products can b dscribd intuitivly by th charactristics spac, whil th dgr of diffrntiation th non-spatial modls is rprsntd ognously by th lasticity cofficint in a CES utility function. As a rsult of this diffrnc, a consumr in spatial modls will choos th bst varity in charactristic spac; whil in non-spatial modls, consumr may choos all th varitis. Scond, for th convninc of computation, Linr and Singr (937 assum that th dmand function in spatial modls is rctangular: ach consumr dmands on or zro unit of th product. And it has bcom a dfault assumption in many spatial modls, whil thr is no such a constraint in non-spatial modls. As an tnsion of Hotlling s modl, Von Ungrn-Strnbrg (988 and Witzman (994 study th gnral purpos products. Thy assum that th products with mor gnral purpos will hav highr production costs but lss pr-unit-distanc transaction costs, th trad off btwn ths two typs of costs dtrmins th dgr of spcialization of th product. Howvr, no planation is givn on th rasons why thr is a tradoff btwn ths two typs of costs. Th purpos of this papr is to formaliz ths charactristics into a monopolistic comptition modl. Clarly, th planation is not nw as it coms from th Hotlling- Lancastr spatial modl. What w do is to mak som improvmnts and tnsions, and synthsiz thm into a gnral uilibrium framwork. First, in contrast to th rctangular dmand function assumption in many of th spatial modls which misss th rsult that popl s trad off btwn th varity and uantity of products, th uantity dmandd in our modl is dtrmind by th markt. Scond, to kp th intuition in Hottling s modl, w us anothr circl with unit circumfrnc to rprsnt th products functions. Th consumption functions srvd by a product ar rprsntd by a functional intrval, with th usag costs and production costs both variabl. Products with a widr rang of functional intrval will nd mor usag costs and variabl costs but lss fid costs, and vic vrsa. This balanc dcids th boundary of products. Third, w gav a simpl production function which includs ths charactristics of products, and rvals th conomis of functional spcialization and conomis of scal. Last, ths factors ar analyzd systmatically in a gnral uilibrium modl. Morovr, through comparativ static analysis, w amin th ffcts of changs in th ognous variabls such as

3 population siz, prfrncs, transaction costs and managmnt fficincy, and production tchnology on th charactristics of products. This papr is organizd as follows. n sction, w study th modl and its main rsults. Fiv stps wr takn in th analysis of th modl. First, w introduc th utility function. Thn, w introduc th production function. Third, through gnral uilibrium analysis, w find th uilibrium charactristics of products. Fourth, comparativ-static analysis is undrtakn. Sction 3 is th discussion of rsults. This papr concluds in sction 4, and sction 5 contains all th proofs of propositions as an appndi.. Th Modl Suppos in an conomy thr ar M idntical individuals ach with on unit of tim ndowmnt, and ach kind of product has thr charactristic inds dfind blow. Suppos that thr ar two circls, C and C, whr C is unit circumfrnc and C is l in circumfrnc; ach kind can b rprsntd as (s, s ] t, with (s, s ] C, t C and R [0,. Whr C is th function spac and ach lmnt of C is th ind of th product function, and (s, s ] is th functional intrval; C is th individuality spac and t th ind of individuality. l rprsnts th tnt of th divrsity of individuality in th markt; R is th uality spac and is th ind of uality. (s, s ] t rprsnts th product which provids th function of (s, s ], individuality ind t and uality ind. For simplicity w suppos that diffrnt prsons hav diffrnt t valus which uniformly distribut in C. For convninc, w us t to rprsnt individuals latr. Dnot m (M/l as th dnsity of population in individuality spac. Thr ar som diffrncs btwn th product of this papr with that of th othrs. n most conomics modl, th charactristics of a product is (partially ognous; whil in our modl, th charactristics of a product is ndognous by som charactristics variabls, such as th function intrval, individuality and uality inds. Morovr, bcaus ths thr inds ar rprsntd by continuums, that will b mor appropriat to analys th variability of product. Furthrmor, w us diffrnt function intrvals in C to rprsnt diffrnt kinds of products (diffring in functions, and us th sam function intrval but diffrnt individuality or uality ind in C to rprsnt diffrnt varitis of products (diffring in individuality.. Utility functions Suppos that thr ar kinds of product and i varitis for ach kind i, for i and j i, product rprsntd as (s i-, s i ] t, whr (s i-, s i ] is product s functional intrval; t is its individuality ind and is its uality ind. For simplicity, w also Ns suppos that U i (s i-, s i ]C and (s i-, s i ] (s h-, s h ]φ, for i h. That mans all products consumd will covr th whol function spac, and diffrnt products will not ovrlap in function. For any t C, suppos prson t s utility function is -ρ t -t i ρ si si i i j u (, i, j ( (. Whr is th uantity of product (s i-, s i ] t, ρ (ρ >0 is th individuality cofficint, and ρ (ρ >0 is th uality lasticity cofficint. Suppos prson t s budgt constraint is 3

4 Σ p k r (. i c ( si si i j t Whr p is th mill pric of product (s i-, s i ] t which cluds transaction cost; thr ar transaction costs in th markt, whr k (k> is th transaction cost cofficint (whn thr is no transaction costs, k and p k is th pric of product for which consumrs nd to pay; thr ar also usag costs whn popl consum products, c ( s' i si whr c (c >0 is th functional intrval cofficint and rprsnts th usag costs. That mans that consumrs nd to pay mor usag costs for a product with a longr lngth of functional intrval. Prson t will choos th products through: -ρ t -t i ρ si si i j ma ( st : Σ p k r i c ( si si i j t (.3 (.3 implis svral proprtis.. Th function spac rprsnts all th ncssary function in consumption. From (. w know that, as a basic proprty of CD utility function, if th union of function intrvals of product from which w consum dosn t covr th whol function spac, no matr how much w consum, th utility lvl is always zro. That mans in this cas w miss out on som ssntial products.. n ach kind of product, consumrs (cpt marginal consumrs will choos only on varity of products according to thir prfrnc. From (.3 w know that for ach functional intrval, consumr t will choos ths i varitis by ma j i -ρ t -t p ρ (.4 t implis that consumr t will just choos on varity of products unlss mor than on varity of products gts th maimum at th sam tim, a cas whr consumr t is a marginal consumr. 3. t is assumd that th consumr is indiffrnt to how th whol function spac is covrd by diffrnt combinations of diffrnt kinds of products, providd that th individuality and uality inds ar th sam. For ampl, suppos that consumr t has two products, (s, s ] t and (s, s 3 ] t. Thy hav th sam individuality and uality inds but diffrnt function intrvals. Th combination of ths two kinds is (s, -ρ t-t ρ s s -ρ t-t ρ s3 s -ρ t-t ρ s3 s s 3 ] t. Bcaus ( ( (, From utility function (. w know that Consumr t will gt th sam utility. 4. Th usag costs ar conomis of functional spcialization. From budgt lin (. w know that th usag cost in consuming pic of products with functional intrval (s, s 3 ] c ( s3 s c ( s s c ( s3 s c ( s3 s is (. Suppos that s<s <s 3, from <, w hav ( c ( s3 s c ( s3 s ( c s s ( < (. That mans th usag costs ar conomis of functional spcialization. 5. Each prson has a particular prfrnc for th styl or individuality (which could b colour, siz, tc. of a product. Sinc ρ >0, th lowr is t i -t, th distanc of individuality ind btwn th product and consumr, th highr th utility lvl. For ampl, suppos somon wars L siz shirt, so h/sh will gt mor satisfaction if th shirt h/sh wars has th siz closr to L. 4

5 6. Sinc ρ >0, from (. w know that a product with highr uality will provid mor utility.. production function n this modl, w suppos that th factoris ar privatly own firms. Th ownr s goal is maimum profit. For th convninc of computation, w assum that labor tim is th only input; th ownr s and all workrs tim is usd in production; and th managmnt cost is assumd to b proportional to th labor tim. Suppos that a factory which producs product (s, s ] t has a pair of output, uantity and uality, and its production functions is c (s'-s c (s'-sa a (s'-sl (.5 whr and ar th uantity and uality of th product. Th marginal cost of uantity and uality, c (s -s/ and c (s -s/, ar assumd to vary in accordanc to th functional intrval, and c and c ar calld as marginal cost cofficints of uantity and uality, rspctivly; is th ntry cost for ach kind of product is takn to b [a a (s -s]/, whr positiv numbrs a and a ar th ntry cost cofficints; labor tim L is th only input in production, and is th managmnt fficincy cofficint, whr 0<<. For L units of labor tim, th factory nd (-L units of managmnt tim. Th dfinition of production function (.5 implis two proprtis.. Th production functions rval conomis of functional spcialization. From (.5 w know that a a (s -s is th ntry cost in producing uantity and uality. A product with lss lngth of functional intrval (s -s will hav lss ntry cost, so that functional spcialization can improv productivity.. Th production functions display conomis of scal. From (.5 w know that whn k, th output (c c with kl input is mor than th k tims of output with L..3 Euilibrium analysis Th goal of this sction is to find th charactristics of products, such as uality, th rang of functional intrval and th distribution of individuality inds, in a symmtric uilibrium stat. Although w us static uilibrium analysis to dal with our modl, stratgis ar arrangd in thr stps. First, ach individual dcids to bcom th ownr of a factory or an mploy. Scond, ach mploy s stratgy is to work with on unit tim for incom, whil an ownr has four stratgy variabls to maimiz profit: pric (or uivalntly th uantity and th uality of th product; th individuality ind (location in individuality spac C ; th last stratgy variabl is th functional intrval. Third, individuals trad, distribut and consum. For computational simplicity, our modl only dals with th spcial cas in which thr is no ovrlapping of functional intrval. Nt, w suppos that ach factory producs only on kind of products, (s, s ] t, that mans ach factory chooss only on functional intrval (s, s ], on individuality ind t, and on uality ind. Bcaus all individuals ar assumd idntical and ar fr to dcid to bcom th ownr of a firm or bcom an mploy, and production functions ar uniu, without loosing any gnrality, w assum that th wag rat is on, so that ach mploy will gt on unit of incom, and ach ownr will gt on unit of profit in uilibrium stats. 5

6 Bcaus all ownrs and mploys hav th sam incom in th uilibrium stat, lmma rvals a ncssary condition of uilibrium solution. Lmma Suppos that thr ar factoris in an industry, say industry i, and thir positions in C is {t i, t i t i t i t i0 0}. For j, suppos that factory maimizs profit in (p,, t, whr p is th pric and is th uality of product. For j, if t -t - don t hav th sam valu, thn (p, will not b th sam, and th maimizd profit π(p,, t will not hav th sam valu. Actually, thr ar infinit symmtric solutions in our modl, but all ths solutions hav th sam structur. For this rason, two rstrictions ar addd to our modl so that w hav only to considr on symmtric solution. n th symmtric stat whr thr ar kinds of goods and thr ar varitis in ach kind of goods, w suppos that, for i, and 0 j -, th kind of products producd by i factory can b dscribd as ( j, i ] l. This assumption adds two rstrictions to th symmtric stat. First, th functional intrval of th first kinds of products bgins at point 0 in C. Nt, in ach C, th first factory always stays at point 0 in C. f w rla ths two rstrictions, w will hav infinit symmtric solutions. For ampl, for i, and 0 j -, w assum that th kind of products producd by factory i can b dscribd as ( f, i ( j h f ] i l. That mans that th functional intrval of th first kinds of products bgins at point f in C, and in industry i, th first factory stays at point h i in C. t s asy to s that all ths symmtric solutions will hav th sam structur. Proposition Thr is a uniu solution in symmtric uilibrium stat of monopolistic comptition, which is dcidd by (.6, (.7, (.8, (.9 and (.0. jl t, whr ti0 ti (.6 a a [ ( a 4 d a a a c ] (.7 ρ ρ 4n a d [( n ( n ], n ρ ρ ρ m k (.8 ( ρd p (.9 ρmd ck ( ρd (.0 md f ck ( ρ d (. c f M ck ( ρd (. 6

7 whr d (/ is th lngth of functional intrval and d (l / is th distanc of two nighbor inds of individuality in C, and f and c is rspctivly th uantity producd by ach factory and th uantity consumd by ach individual. Proposition shows that, although thr ar infinit asymmtric solutions for th uilibrium stats, from th standard of social wlfar, th symmtric solution is bttr than th asymmtric solutions mntiond. Proposition whn is an vn numbr, thr is infinit asymmtric solutions. For ampl, for any -<h<, (.3, (.7, (.8, (.9 and (.0 dcid a solution of monopolistic comptition. Whn h0, it s th symmtric solution of proposition ; whil h 0, it s th asymmtric solution. Morovr, th social avrag utility Eu(h is a dcrasing function of h. kl t whn j k, whr ti0 ti (k h l t whn j k, < h< (.3.4 Comparativ static analysis Through comparativ static analysis, w will find th corrlations btwn th products charactristics and th ognous variabls such as population siz and individual prfrncs, transaction cost and managmnt fficincy, and production tchnology. To rduc complity, w us som simpl nots to rprsnt th corrlations. For ampl, if z is an incrasing in but dcrasing in y, w will rprsnt ths rlation with formula z(, y. Nt ampl, in uation F(, y0, if function F(, y is incrasing in but dcrasing in y, w will dnot as F(, y 0. This basic tchniu is oftn usd in th proof of th propositions blow. Lmma ( if F(z,, y 0, thn z(, y (.4 ( if F(z,, y 0, thn z(, y Corollary th uilibrium lngth of functional intrval products satisfis: d ( a, a, c (.5 ( a, a, c (.6 Corollary th uilibrium varitis satisfy: l d ( m,, k, c, ρ, ρ, a, a (.7 ρ d ( m,, k, c, ρ, ρ, a, a (.8 Corollary 3 th uilibrium uality satisfis: ( m,, k, c, ρ, ρ, a, a, c (.9 7

8 Corollary 4 th uilibrium uantity producd by ach factory ( f satisfis: ( m,, k, c, ρ, ρ, a, a, c f (.0 Corollary 5 th uilibrium uantity consumd by ach consumr ( c satisfis: ( ( m,, k, c, ρ, ρ, a, c (. ( f ρ 4 c ( a ( ( ( nρ nρ > < (. ac W hav c (a whn m k is larg nough, whil ρ is larg nough w hav c (a. Th main conclusions from corollary to 5 ar summarizd in tabl, whr ndognous variabls ar st in column on whil ognous variabls ar st in row on. n tabl, sign mans th rlvant variabls hav positiv corrlation, whil mans ngativ corrlation; sign 0 mans no corrlation btwn thm and sign /- mans thr ar both positiv and ngativ corrlations btwn thm. 4a Tabl : corrlation summary m k c ρ ρ a a c c d l c /- 0 - f Discussion of rsults Th advantag of th synthtic analysis in our modl is that it rvals how popl tradoff btwn th uantity and th charactristics of product, such as th rang functional intrval, th ind of individuality and uality. n this sction, w will plain th rsults in tabl intuitivly and also giv ampls in our daily lif. For ndognous variabls, th rang of functional intrval, th varity, th uality and th uantity consumd by ach prson (d, /l,, c ar four choic variabls of consumption, which ar dirctly affctd by th factors from th dmand sid; whil th formr thr variabls and th uantity producd by ach factory (d, /l,, f ar four choic variabls of production, which ar dirctly affctd by th factors of th supply sid. Th ognous variabls ar dividd into two groups. Th first four variabls ar (m, k,, c. Bcaus thy induc th ntir ndognous variabls to chang in th sam dirction, w call ths ffcts as growth ffcts. On th othr hand, th last si variabls (ρ, ρ, a, a, c, c will bring about mid ffcts, th factor may b positivly corrlatd with som variabls but ngativly corrlatd with th othrs. 8

9 3. Growth ffcts Column m, k,, and c show th growth ffcts. Th incras of dnsity of population and managmnt fficincy, and th dcras of transaction cost and usag cost will incras th uantity and improv all thr charactristics inds. ntuitivly, with th condition of incras rturns in production, th incras in th dnsity of population, th dcras in transaction costs or usag costs will incras th dmand and thus improv th production fficincy, whil th incras of managmnt fficincy will dirctly improv th production fficincy. Consuntly, positiv profit will attracts mor producrs to ntr into th markt and induc thm to incras output and improv th charactristics inds. This conclusion throws som light on th intra-industry trad thory. Krugman (980, 98 concluds that, with th condition of th conomis of scal and idntical ndowmnt, thr is no intr-industry trad, but lots of intra-industry trad which bring mor varity of goods and bnfit all incom-arnrs. Nvrthlss, our modl points out that thr is anothr factor, th individuality spac, which will distort Krugman s conclusion. For ampl, suppos that thr ar two idntity countris, say A and C; thr is only on kind of products, say cloths; thy hav th sam production function which nds on kind of input, say labor and hibit incrasing rturns; th only diffrnc in ths two countris is that th popl of A hav biggr statur, thy war cloths of XXL siz; whil th popl of C war M siz. According Krugman s conclusions, th common markt of ths two countris bnfit all popl with two varitis of cloths. But in rality, th popl of A don t nd siz M and th popl of C don t nd siz XXL, so th common markt will not chang th conomic structurs. n our modl, it s th spcial cas whr two countris hav th sam siz but compltly diffrnt individuality spac. Th common markt will doubl both population and circumfrnc and hnc dosn t chang th dnsity of population. As a rsult, thr is no chang in conomic structurs. Mor gnrally, if th common markt is formd btwn two similar countris but with partially diffrnt individuality spac, th common markt will incras both th population and circumfrnc which has th opposit ffcts to th dnsity of population, and thus changs th ffcts of th common markt. 3. Othr ffcts Whil th variabls abov bring th growth ffcts, th thr pairs of variabls blow hav mid ffcts on th uantity and th thr charactristics inds.. For variabls ρ and ρ, a largr ρ mans that individuals hav a highr dgr of prfrnc on individual styl of th product; a largr ρ mans that thir prfrnc for highr uality incrass, As shown in Tabl, ithr undr th sam production capability of th conomy (gnral uilibrium analysis or th sam incom lvl of th individuals (partial uilibrium analysis, thy will accommodat thir highr prfrnc by rducing th othr ruirmnts on products.. As c and c rprsnt rspctivly th marginal cost cofficint of uantity and uality, and hnc th dcras of c and c, say as th dynamic ffct of larning by doing, will incras th uantity and uality rspctivly. 3. As a a (s -s rprsnt th ntry cost of products with functional rang (s, s ], a chang in a and a has svral ffcts. 9

10 First, formula (.7 shows that th incras of ratio a /a will nlarg th rang of functional intrval. Consumrs nd to buy all kinds of products. n thy choos th situation of mor kinds of products but lss lngth of functional intrval in ach kind of products, thy will pay mor tims of a. Hnc, whn a is largr, popl will prfr lss kinds of goods but a largr lngth of functional intrval. ntuitivly, a rprsnts th common componnt and a rprsnt th diffrnt componnts in diffrnt products. Th conclusion shows that diffrnt products which hav largr portion of common componnt sm asir to b unifid into on product, whil a product with two indpndnt componnts to a crtain tnt sms asir to b spcializd into diffrnt products. For ampl, as functional gnralization, a computr provids mor and mor functions of mdia play (such as TV, CD, VCD and DVD and communication (such as tlphon and fa. t s du to th fact that computrs alrady hav som uipmnt of mdia play and communication. To provid ths functions, it just nds to add som softwar and improv th uipmnt. On th othr hand, if computrs kp in th narrow function rang, mor basic uipmnt will b usd in diffrnt products. This tradoff dcids th boundary of th function of th computr. Nt ampl, athltic shos is bcoming mor and mor functional spcializd. Thirty yars ago, popl usd to war th gnral athltic shos, th canvas shos, for many sports itms. Nvrthlss, ach sports itm mphasizs diffrnt part of th shos. For ampl, bcaus basktball playrs oftn jump, so th basktball shos hav thick and soft bottom with gap in it; football playrs oftn run in wt ground and control th ball with th bottom of shos, so that th football shos hav a hard bottom with rubbr nails on it. This diffrnc, along with othrs, mak it difficulty to produc a kind of gnral shos which kp th advantag of both basktball and football, and hnc mak th gnral athltic shos asir to b spcializd. Now you can buy any spcializd athltic shos for most sports itms. Scond, whn a and a chang in opposit dirction, thr is a tradoff btwn th rang of functional intrval and th uality of products. As mntiond abov, say whn a incras and a dcras, popl will prfr mor rang of functional intrval, for cost saving, thy accpt th lowr uality. A larg numbr of ampls show that th tradoff btwn th rangs of function intrvals and th uality lvls is ubiuitous. Hr ar svral ampls in our daily lif. As mntiond abov, th spcialization of athltic shos will improv thir uality; th ric cookr offrs popl much mor convninc in cooking ric than th gnral purpos cookr; thr ar many varitis of spcializd automobils, such as truck, bus, sdan and othrs, which prform bttr thn th gnral purpos automobil. Convrsly, although computr may offr th function as a mdia playr, but its ffct is not as good as th spcial audio and vido uipmnt; today th mobil phon can tak photo but its ffct is not as good as th camra; suprmarkts lik Wal-Mart sll almost vrything at low prics, but provids lss srvic than th spcializd stors; comparing with playing in outdoor court, popl can play basktball, vollyball, badminton and othr gams in an indoor hall, but th ovrlapping ara distorts playrs utility. Third, as shown in column a and a, th dcras of ntry cost will incras th varity of products, and vic vrsa. Th dcras of ntry cost will incras th profit and hnc induc mor producrs to ntr th markt, so that mor varitis of products will b 0

11 providd. This mans th chapr products will hav mor varitis. For ampl, w hav a lot of varitis of cloths or shos, but comparativly, w hav fwr varitis of cars or airplans. 3.3 Consistnt tndncy rul Bsids th opposit tndncy of functional rang and uality, rows and f in Tabls show that producrs tnd to chang thir uality and uantity in th sam dirction, cpt for factors c and c which affct uality and uantity diffrntly. From th analysis abov, w know that growth ffcts mak producrs chang th uantity and uality in th sam dirction. For othr ffcts, lt s s th ndognous variabls on by on. First, bcaus ρ is positivly corrlatd only with th varity of products, so it has th sam ualitativ ffcts on uantity and uality. Scond, although ρ is ngativly corrlatd with th uantity consumd by ach prson ( c and th varity of products (, but from th formulas (.8 and (. w know that th ffct of is dominating. From th formula f M c / w know that th uantity producd by ach factory is positivly corrlatd to ρ and hnc tnd to chang in th sam dirction as uality. Third, bcaus th rang of functional intrval, uantity ( f and uality ar thr substitutabl variabls within a factory, th chang of a and a which nlargs th rang of functional intrval will rduc th uantity and uality. Last, it s asy to undrstand why c and c will affct th uantity and uality rspctivly. Form th analysis of incom and othr ffcts, w conclud that th uantity and uality on of a producr tnd to chang in th sam dirction. 4. Conclusion This papr shows th corrlations btwn th uantity and th charactristics of products, such as th functional rang, th uality and varity of products, and th ognous variabls such as population and popl s prfrncs, transaction costs and managmnt fficincy, and production tchnology. As its applications, w rval how th markt dcids th boundary of functional rang of th products, how th markt dcids th varity of diffrntiatd products, and how diffrncs in th prfrnc for individual styls affct th rsults of a common markt. Bsids, bnfiting from th multifactor analysis, this papr also offrs a map to dscrib how popl tradoff among th uantity and th charactristics of products. For ampl, our modl shows that th functional rang and uality tnd to mov in opposit dirction as ognous variabls chang, and th uantity and uality of ach producr tnd to chang in th sam dirction. Nvrthlss, as th cost of an intgrativ modl, w hav to simplify som assumptions which induc invitably th gap btwn th modl and rality. Hr ar two aspcts. First, in our modl, thr is only on kind of goods for any givn function, but in rality, popl may us diffrnt kinds of goods for th sam function. For ampl, popl may us mobil phon to tak photo for convninc or us camra for highr uality. For mor ralism, on may wish to rla th assumption that th functional intrval of diffrnt kinds of goods do not intract. Nt, on may wish to rla th symmtry assumption in our modl, or to chck whthr thr is any asymmtric solution in our modl. Thr may b asymmtric solutions vn in a symmtric systm. n our modl w

12 just hav provd that thr is only on symmtric solution, but hav not shown whthr thr ar asymmtric solutions. Th scond aspct is th multi-products problm in Hotlling s framwork. Our modl assums that ach factory producs only on varity of products, whil in rality ach factory may produc svral varitis of products. As our modl shows, thr is a tradoff btwn functional gnralization and th uality of products. Similarly, th ffort to mor ralism is blockd by computation complity; modl spcialization is an asir way out. 5. Appndis Lmma Suppos that thr ar factoris/producrs in an industry, say industry i, thir position in C is {t i, t i t i t i t i0 0}. For j, suppos that factory maimizs profit in (p,, t, whr p is th pric and is th uality of product. For j, if t -t - don t hav th sam valu, thn (p, will not b th sam, and th maimizd profit π(p,, t will not hav th sam valu. Proof: From (.4 w know that consumr t chooss product (s, s ] t iff j ' -ρ l -t -ρ t -t ρ ρ,for j' (5. p p t rvals that t is blonging to an intrval ( t r, t r and th radius satisfis: -ρ t -t p ρ -ρ t -t (5. p ρ r p ρ r ρ ρ ρ(t ln( (t - ln( (5.3 p ρ r ρ ( t t ρ[ln( ln( ] [ln( p ln( p ] (5.4 r ( [ln( ln( ] [ln( ln( ] ρ ρ ρ (5.5 For simpl computation blow, dnot zik p, whr kj, j, w hav: r ( t t ρ [ln( ln( ] [ln( z ln( z ] (5.6 ρ ρ For th sam rason, w hav r ( [ln( ln( ] [ln( ln( ] t t ρ ρ ρ z z (5.7 t t ρ ln( ln( ln( ln( z z r r [ln( ] [ln( z ] (5.8 ρ ρ Bcaus vryon has on unit of incom whn popl bgin to trad in stp thr, from utility function (.4 w know th dmand function of factory is ik

13 ( r r Md z ( r r md z D(z,, t, d lk k md z ( t t ρ md z ln( ln( [ln( ] (5.9 k k ρ md z ln( z ln( z k [ln( z ] ρ whr d (s -s is th lngth of th functional intrval of th products, ts rvnu function is ( r r Md ( R(z,, t, d r r md lk k md ( t t ρ md ln( ln( [ln( ] (5.0 k k ρ md ln( z ln( z [ln( z ] k ρ From (.5 w know that a ad L (5. Bcaus th wag rat is on, so that w hav: a ad C(, L- (5. cd(z,, t, d d a ad C(z,, t, d L- (5.3 π(z,, t, d R(z,, t, d -C(z,, t, d md( t t z ρ ln( ln( md z ( ( [ln( ] k k ρ (5.4 md z ln( z ln( z a ad ( [ln( z ] k ρ For simpl computation, dnot k k a ad, ˆ (, k c z ik ik π ρπ a ρ ik ρ, z ik (5.5 md md m whr kj-, j, j, thn w hav: ln( ˆ ˆ ln( π( z,, t, d ρ[ln( ]( z (5.6 ln( zˆ ln( zˆ ρ ( t t ( z [ln( z ]( z ˆ a For maimum profit π(z,, t, d, w hav th first and scond-ordr conditions: ρ ( ˆ z ˆ π ˆ (5.7 ˆ 3

14 ln( ˆ ln( ˆ ln( zˆ ln( zˆ ˆ π ρ ˆ zˆ ρ ˆ ( t t ( z zˆ zˆ [ln( ] [ln( ] (5.8 ˆ π ˆ 0 ˆ ρ ( zˆ (5.9 ˆ π 0 zˆ ( zˆ ln( ˆ ln( ˆ ln( zˆ ln( zˆ ρ ( t t ρ [ln( ˆ ] [ln( zˆ ] (5.0 zˆ W hav th scond-ordr drivation matri: ρ ( ˆ z ρ ˆ ˆ ˆ ˆ π π z ˆ π ˆ π ˆ z π zz ρ ( zˆ ˆ ˆ z (5. π < 0 zˆ < ρ (5. That mans if zˆ <, th solution ( zˆ, ˆ from (5.9 and (5.0 will maimiz ρ profit; whil zˆ >, th ( zˆ, ˆ will not maimiz profit. ρ From (5.6 and (5.9 w hav ( zˆ π( z π( z,, t, ( ˆ ˆ d ρ z a (5.3 zˆ From (5.3 w know that π ( z is a dcrasing function whn zˆ <, it covrs th ρ rang of th solution ( z, ˆ which satisfis th first and scond-ordr conditions ˆ So w can conclud that: for j, if t -t - don t hav th sam valu, thn th maimization solutions ( zˆ, ˆ will not hav th sam valu. t s asy to prov. For j, suppos ( zˆ, ˆ ar th sam, thn form (5.9 and (5.0 w hav all t-t - ar th sam. From (5.3, bcaus π ( z is a dcrasing function, so that diffrnt ( zˆ, ˆ implis diffrnt π ( z. Translat ( zˆ, ˆ to (p,, and π to profit π, w gt th Lmma. Proposition Thr is a uniu solution in symmtric uilibrium stat of monopolistic comptition, which is dcidd by (.6, (.7, (.8, (.9 and (.0. jl t, whr ti0 ti (.6 4

15 d a a a [ ( 4 a a a c ] (.7 ρ ρ 4n a d [( n ( n ], n ρ ρ ρ m k (.8 ( ρd p (.9 ρmd ck ( ρd (.0 md f ck ( ρ d (. f c (. M ck ( ρd Proof: Suppos in a symmtric stat, thr ar kinds of products and varitis in ach kind of product. For i, and j, suppos that factory producs product i j (, i ] l, and all factoris hav pric p. Th ida to prov this symmtric stat is a uilibrium solution is that, for ach factory, givn th variabls of othr factoris, whn any factory taks th stratgy variabls of th assumd symmtric stat, it will gts th maimum profit, and this profit lvl uals th incom of all workrs and ownrs. Bcaus of ma π ( z,, t, d ma{ma π ( z,, t, d }, thr stps will b takn. z,, t, d d z,, t First, look for th uniu solution ofπ ( d ma π ( z,,, d. Scond, look for th z,, t t uniu solution of ma π ( d. Third, prov that thr is a uniu solution of th d monopolistic comptition uilibrium. First, w will prov that thr is a uniu solution for ma π ( z,, t, d with th z,, t mthod that thr is a uniu solution of th first-ordr conditions which has ngativ scond-ordr drivation matri. From (5.4 w know that t dosn t affct profit π(z,, t, d, but dos affct th profit π(z ik, ik, t ik, d, whr kj- and j. As a ruirmnt of a symmtric solution, w st jl t, w hrti0 ti (5.4 From th first-ordr condition (5.0 w hav zˆ zˆ ρ d, hnc w hav ( ρd p z (5.5 ρρd And from th first-ordr condition (5.9 w hav ˆ ˆ ρ d, hnc w hav: 5

16 ρmd (5.6 ck ( ρd From th scond-ordr condition (5. w hav: π < 0 zˆ < ( ρ d > ρ (5.7 ρ From (5.3 w know that ρ d -ρ >0 which implis (ρ d >ρ, so that w hav ngativ dfinit scond-ordr drivation matri, it implis that th symmtric solution (p, of (5.5 and (5.6 maimizs th profit. From (5.9 and (5.5 w hav mddz md f D(, z,, t d (5.8 k c k ( ρ d jl π( d ma π( z,, t, d π( z,,, d z,, t mdd z a ad ( (5.9 k mdd( ρd ρ a ad ck ( ρ d Scond, w also us th first and scond-ordr conditions to look for th ma π ( d m( d( ρd ρ a π d ck ( ρ d (5.30 W hav th first-ordr condition: md( ρd ρ a π d 0 (5.3 k( ρ d From (5.30 w hav th ngativ scond-ordr condition, π d < 0. So that from (5.3 w gt th maimiz solution. Sinc vry individual can choos to bcom th workr or th ownr frly, at th uilibrium stat, th ownr and workr will hav th on unit of incom ach. So w hav md ( ρd ρ ( a ad π ( d (5.3 k( ρd d Third, from (.6, (.7, (.8, (.9 and (.0 w gt th solution of th symmtric uilibrium, now w will prov that th solution is uniu. From (5.3 and (5.3 w hav a a ad (5.33 d ad ( a ad( ( a a a (5.35 d a a a [ ( 4 a a a c ] (5.36 d 6

17 W know that thr is only on positiv d. From (5.35 w know that a a (5.37 a k Dnot n (5.38 m From (5.3 (5.37 and (5.38 w hav: ρ d -(ρ nρ d -n0 (5.39 Thr is only on positiv solution: ρ [( ρ ( 4n d n n ] (5.40 ρ ρ ρ Aftr proving th uniunss of d and d, from (5.36 and (5.40 w can gt th uniunss of p and and othr variabls in th symmtric uilibrium. Morovr, w hav th uantity consumd by ach consumr: f f c M md (5.4 From (5.8 w hav: c (5.4 ck ( ρ d Proposition : Whn is an vn numbr, and for any -<h<, (.3, (.7, (.8, (.9 and (.0 dcid a solution of monopolistic comptition. Whn h0, it s th symmtric solution of proposition ; whil h 0, it s th asymmtric solution. Morovr, th social avrag utility Eu(h is a dcrasing function of h. kl t whn j k, whr ti0 ti (.3 (k h l t whn j k, < h< Proof: whn h0, (.6 and (.3 ar th sam, so that th solution is th symmtric solution in proposition ; whil in th cas of h 0, although th st of t from (.3 distribut asymmtrically in C, but thy kp t -t - th sam valu for j. consuntly. From th proof of proposition w know that, th rst variabls will b th sam as that in th symmtric uilibrium stat in proposition, so that (.3, (.7, (.8, (.9 and (.0 will dtrmin an asymmtric uilibrium solution. n th uilibrium stat with -<h<, t distribut according (.3, and hnc w know kl that, whn jk, factory locats in and its markt rang is (4k h l (4k h l ( hl [, ]; whn jk, factory locats in and its markt (4k h l (4k 3 h l rang is [, ]. Bcaus th markt rang of ach factory distributs rgularly on C, so that th social avrag utility uals th avrag utility in 7

18 th markt rang of on factory, say factory i.whn t blong to this markt rang, consumr t has th utility ρ ( hl ( t u(t ρ c l Eu(h ( l dt Eu(h( (3 hl (3 hl ( hl ( ( ρ t ρ hl u t dt ( hl c h ρ ρs c h ds (5.43 (5.44 From (5.45 w hav th rsult that Eu(h is a dcrasing function of h. (5.45 Lmma ( if F(z,, y 0, thn z(, y (.4 ( if F(z,, y 0, thn z(, y Proof: it s asy to prov. Corollary th uilibrium lngth of functional intrval products satisfis: d ( a, a, c (.5 ( a, a, c (.6 Proof: thy com from (.7. Corollary th uilibrium varitis satisfy: l d ( m,, k, c, ρ, ρ, a, a (.7 ρd ( m,, k, c, ρ, ρ, a, a (.8 l Proof: From (.8 w hav d ( ρ, ρ, n (5.46 a k ac k From (.8 n m ( m (5.47 Bcaus 0<c d <, so w hav ( (5.48 ac k m From (.6 w hav n ( a, (5.49 ( m k a k From (.8 and (5.37 w hav n ( ( m From (.6 and (5.50 w hav n ( c, a ( 5.5 From (5.49 and (5.5 w hav n ( m,, k, c, a, a (5.5 From (5.46 and (5.5 w hav l d ( m,, k, c, ρ, ρ, a, a (5.53 (5.50 8

19 From (.8 w hav ρd [( ρ ρn ( ρ ρn 4 ρn ], and it is asy to gt ρ d ( m,, k, c, ρ, ρ, a, a (5.54 Corollary 3 th uilibrium uality satisfis: ( m,, k, c, ρ, ρ, a, a, c ρmd Proof: From (.0 w hav ; from (.0 and (.7 w know: ck ( ρ d (.9 ( ρ, ρ, a, c (5.55 ρ( a ad From (.0 and (5.3 w know ( ρd ρ (5.56 From (.5 and (.8 w know ( m k (5.57 From (.8 w hav: ( nρ ρ ( nρ ρ 4nρ d n ρ( ρ ρ d ( ρ nρ ( ρ nρ 4 n ρ(ρ From (.8 and (5.37 w hav: a k a k n m ( m ρ ( ρ ( ρ d ( ( ( 4 akρ akρ a k ρ ρ ρ m m ( m (5.58 (5.59 (5.60 (5.6 From (.6 w hav (, c (5.6 From (.6 w hav a ac ac 4a ( (5.63 ak ρ So that ( a, c m (5.64 a k( ρ From (.6 w hav ( ( m ( a, c From (5.6, (5.64 and (5.65 w hav: ( ρ ( a, c d (5.65 (5.66 From (.0 and (5.66 w know that ( c, a (5.67 From (5.55, (5.57 and (5.67 w hav 9

20 ( m,, k, c, ρ, ρ, a, a, c (5.68 Corollary 4 th uilibrium uantity producd by ach factory ( f satisfis: ( m,, k, c, ρ, ρ, a, a, c (.0 Proof: from (. w know f f ck md ( ρ d as, cpt th chang from c to c, so that w hav: ( m,, k, c, ρ, ρ, a, a, c (5.69 f Corollary 5 th uilibrium uantity consumd by ach consumr ( c satisfis: ( ( m,, k, c, ρ, ρ, a, c (. c, it has th sam proof and conclusion ρ 4 4a ( c ( a ( ( ( nρ nρ > < (. ac W hav c (a whn m k is larg nough; if ρ is larg nough, w gt c (a. Proof: ( From (. w hav c ck ( ρ d ( m,, k, c, ρ, ρ, a, c (5.70 c ( From (.8 w hav:, from (.6 and (.8 w hav: ( nρ ρ ( nρ ρ 4nρ ρd (5.7 ( ρd (( nρ ρ ( nρ ρ 4nρ (5.7 Dnot function f c d ln(( nρ ρ ( nρ ρ 4nρ (5.73 ρ n c ρ n f c c d d ( nρ ρ 4 nρ ( nρ ρ 4nρ (5.74 a From (.7 w know that 4 (5.75 ac 4a f > ( < 0 ( nρ ρ 4 nρ > ( < nρ d ac ρ 4 4a ( ( nρ nρ a > < (5.76 c From (.5 w know that d (a, so that 4 4 c (a ( f d > ( < 0 ( ρ a ( nρ nρ > < a (5.77 c 0

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