7THE DIFFUSION OF PRODUCT INNOVATIONS AND MARKET STRUCTURE

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1 7THE DIFFUSION OF PRODUCT INNOVATIONS AND MARKET STRUCTURE Isttut of Ecoomc Forcastg Roxaa IDU Abstract I ths papr I aalyz th dffuso of a product ovato that was rctly mad avalabl for lcsd purchas wth a dustry wth tcal frms producg th sam good. Th ma assumptos ar a dcrasg yt always postv ctv to adopt th ovato, ad a xtrmly hgh cost of mmdat adopto, but whch dcrass ovr tm passd sc th ovato has bcom avalabl. Th rsultg qulbrum th dustry s a gradual adopto of th ovato rathr tha a mmdat o, wth ach frm havg a optmal tm of adopto. I th log-ru qulbrum, as th umbr of frms th dustry bcoms vry larg, t s also show that th ctv to ovat dos ot dsappar. Howvr, as th umbr of frms th dustry crass ach frm s show to hav a ctv to adopt arlr. Th assumptos hr, as wll as th rsults of ths modl, match th rsults of rct studs th mprcal ltratur. Kywords: product ovato, log-ru qulbrum, modl JEL Classfcato: D, D50 Itroducto Th ma assumpto about th adopto of a product ovato th ltratur s that t happs mmdatly aftr th prod of moopoly allowd to th ovatg frm. Howvr, vc shows that such a procss s mor gradual ad that dffuso dd happs wh t coms to product ovatos, as wll as to procss ovatos. Som of th rasoabl xplaatos for why ths happs ar th vry hgh costs of adjustmt of mmdat adopto, as wll as th dcras th profts from adopto wth th umbr of frms that hav alrady adoptd. I othr words, frms wgh th costs ad bfts of adopto vry prod aftr th lcs has b mad avalabl ad dcd th optmal tm to adopt. Currtly workg o a doctor s dgr th Ecoomcs Dpartmt at Uvrsty at Buffalo, holg a B.A. Busss Ecoomcs ad Poltcal Scc from SUNY Plattsburgh. 84 Romaa Joural of Ecoomc Forcastg /006

2 Th Dffuso of Product Iovatos ad Markt Structur I a rct mprcal study, Mullga ad Llars (003) show that th ctv to adopt a qualty-hacg ovato by skg aras dcrass wth th umbr of drct compttors that hav alrady adoptd th ovato. Thrfor, a dffuso procss s foud to occur v wh talkg about product ovatos, ot just wh cosdrg cost-rducg ovatos. Th lattr cas was dscussd by Rgaum (98b), who showd that th oly qulbrum a coctratd dustry producg a homogous good was o whr frms adoptd th procss ovato squtally. I ths papr, I wll cosdr th adopto of a product ovato alog wth th ffct of markt structur o th rsultg qulbrum. Although th assumpto rgarg th magtud of arly adjustmt costs s matad, th ffct of a product ovato o th total prst valu of profts works through dffrt chals rathr tha a cost-rducg procss ovato. Followg Klppr (996), th ma assumpto hr s that wh a frm adopts a product ovato, t ca attract mor buyrs who hav a prfrc for that product, ad t ca sll th good for a hghr prc tha th prc of th stadard good. Ths maly happs bcaus th troducto of th product ovato crats a w dmad, sc th ovato s a w product, ad ths dmad usually rprsts a submarkt of th orgal markt, but to whch oly thos frms who hav purchasd th lcs hav accss. Thrfor, t ca b cocvd that tally th umbr of frms wllg to tr ths submarkt s smallr tha th total umbr of frms gv th rlatvly hgh costs of adopto ad adjustmt that ths procss would rqur. Ths frms wll th mak profts hghr tha th orgal markt, at last for a whl, utl all th frms vtually adopt th ovato. It s show ths papr that v th cas of a product ovato, th qulbrum outcom s a dffuso procss rathr tha th smultaous adopto by all th frms th markt. Furthrmor, all frms hav a postv ctv to adopt th ovato, albt a dcrasg o wth th umbr of adoptrs. It s also show that th largr th sz of th orgal markt wll b, th arlr ach frm wll dcd to adopt th ovato, ad that as th umbr of frms bcoms vry larg th ctvs drvg th dffuso procss do ot dsappar. Th Modl Lt us cosdr a dustry wth tcal frms producg ad sllg a homogous good. Ths frms ar orgally a Courot-Nash qulbrum, producg a o-gatv output ad makg o-gatv profts. Wh facd wth a lar vrs dmad, p(q) a b Q ad havg costat margal cost c, ach frm wll dcd to produc q ad wll mak profts Ð, gv by th quatos blow: q ( ) a b c ( + ) ( a c) b ( + ) () ad Π ( ) At tm t 0 a product ovato bcoms avalabl to b lcsd by ay frm th markt. Ths ovato, f adoptd, provds accss to th frm to a submarkt, whr th vrs dmad s gv by p(q) a b Q. W ca safly assum that sc ths s a submarkt of th orgal markt, a a ad b < b, thus mphaszg th Romaa Joural of Ecoomc Forcastg / ()

3 Isttut of Ecoomc Forcastg upward prssur o prc whch allows oly a small umbr of w buyrs to bft from th ovato. Th Courot-Nash qulbrum output of a frm ths submarkt wll b a fucto of th umbr of adoptrs, m : q ( m) a b c ( m + ) Howvr, t s oft also th cas that th margal cost of producg th w product s hghr tha th margal cost of th stadard good, c > c, but smply assumg that a c > 0 surs that thr ar mportat ctvs to adopt th ovato. Wth ths codtos, w ca coclud that th qulbrum prc th submarkt s hghr tha that for th stadard good, whl th qulbrum output s lowr (s Fgur ). W furthr assum that th frm matas th sam lvl of output of th stadard good, whch s ot affctd by th umbr of adoptrs of th ovato. Thrfor, a postv proft th submarkt rprsts th ma ctv for th frms to adopt. I addto, th frm s also abl to sll ths w product to a fracto α of ts currt customrs at th w prc. Thrfor, th cras profts to frm f t adopts th ovato, gross of adjustmt costs, ca b wrtt as a fucto of ad m: Π (, m) [α q () + q (m)][ p (m) c ] (4) Ths proft s o-gatv for all valus of m, ad a strctly dcrasg fucto of m ad, thus llustratg th dcrasg ctv to adopt a ovato as th umbr of adoptrs grows. Howvr, th ctv to adopt vr dsappars. Lt th fucto y(t), dfd for all o-gatv valus of t, rprst th prst valu of all adopto ad adjustmt costs that a frm adoptg th ovato at tm t has to cur. W assum that ths s a covx ad dcrasg fucto of adopto tm, but that aftr a crta tm t starts to cras, thus mphaszg that costsavg from postpog adopto tm caot cotu dftly. Ths two codtos ca b xprssd as follows: y (t) < 0, y (t) > 0 for all t є [o, ) ad lm t?8 y (t) > 0 (5) Also, w would lk to clud a codto o y(t), whch spcfs that mmdat adopto s too costly, xcpt for th frst adoptr: y (0) Π (, ) (6) A fal codto o y(t) surs that th objctv fucto th maxmzato problm that wll follow s strctly cocav for all valus of t: y (3) rt ( t ) > Π ) r (7) Equlbrum profl of adopto dats Th prst valu of all costs ad bfts to frm wh adoptg th ovato at tm ca b wrtt as a fucto of th adopto dats of all frms as follows: V m+ γt m+ γt (, Κ, ) Π ( ) dt + Π m) dt y( ) (8) m m m 0 m whr ad + 8. Sc all frms ar 86 Romaa Joural of Ecoomc Forcastg /006

4 Th Dffuso of Product Iovatos ad Markt Structur tcal, th xact ordrg s ot rlvat. Furthrmor, ths wak ordrg of adopto dats dos ot lmat th possblty that mor tha o or v all frms adopt at th sam tm. It rmas to vstgat what profl of adopto dats costtuts a qulbrum udr ths stup. Accorg to th abov spcfcato, th prst valu of th profts of th frm crass f th prst valu of adopto ad adjustmt costs ar coutrwghtd by th total bfts rprstd by th profts th w submarkt. Each frm wll choos ts optmal adopto dat,, so that th abov fucto may b maxmzd. Gv th codto (7) o y(t), V ( ) s strctly cocav, ad so th frst ordr codtos ar cssary ad suffct for fg th maxmzg valus of ts argumts. Usg Lbz s rul, w obta: V 0 Π ) y ( ) 0 (9) for all,,. Gv that V ( ) s strctly cocav ad cotuous all ts argumts, ths maxma xst ad ar uqu. Evaluatg (9) for ad 0 w obta: V 0 Π ) y ( 0) (0) V But gv (6), w coclud that 0 0 whch mas that 0. I addto, V lm lm ( ) < 0 y usg (5), ad so <. I ordr to stablsh that dd s btw - ad +, w wll valuat (9) at ach of ths two valus. V Π ) y ( ) Π ) + Π ) [ Π ) Π )] > 0 () Th abov statmt s tru, sc th xpotal fucto s always postv ad th fucto Π (, m) s strctly dcrasg ts scod argumt. Sc V s cocav, () shows that - <, wth a strct qualty sg du to th strctly dcrasg submarkt proft fucto th umbr of adoptrs. A smlar argumt shows that < +. Sc ths aalyss s arbtrary, t follows that th ordrg of all optmal adopto dats s strct: 0 < < < - < < + < < - < < () I ordr to show that th abov profl of optmal adopto dats s a Nash Equlbrum, w must frst df what w ma by a Nash Equlbrum ths cotxt. Dfto: A profl of adopto dats, (,, ) s a Nash Equlbrum f V () V (,, -,, +,, ) for all є [o, ) ad,,. To show that from () s a Nash Equlbrum w d to rcosdr th maxmzato problm of th frm. I fg th optmal adopto dat for frm, w Romaa Joural of Ecoomc Forcastg /006 87

5 Isttut of Ecoomc Forcastg hav assumd that all othr frms do ot chag thr adopto dats as a rsult of chagg. Also, th maxmzg valu of ô dd ot dpd o th adopto dats of othr frms. Thrfor, from (9) was foud for ay gv valus of th othr frms adopto dats. Ths also appls th cas wh all othr frms adopt accorg to, lag us to coclud that th optmal adopto dat for frm s also th Nash Equlbrum adopto dat for that frm. Ad sc ths s vald for ay,,, w ca coclud that ô s a Nash Equlbrum profl as dfd abov. Howvr, sc all frms ar tcal, thr wll b! as may Nash Equlbra, sc thr ar! possbl prmutatos of umbrs from to, but whr th optmal adopto dats ar always strctly ordrd. Hr t s trstg to commt o th coomc rasos why a qulbrum whr all frms adopt at th sam tm s ot possbl. Frst of all, f thr s such a qulbrum, th t wll b at a dat wh t s optmal for all frms to adopt th product ovato. Thrfor, ths dat would b th maxmzg valu of for th followg objctv fucto: V rt γt () Π () dt + Π ) dt y() (3) 0 As abov, th frst ordr codto s cssary ad suffct to fd th maxmzg valu,, whch wll b uquly dtrmd du to th strct cocavty of V : V 0 Π ) y ( ) 0 (4) I words, ô wll b th arlst dat at whch th prst valu of xtra profts ard by a frm by adoptg th ovato covrs just th prst valu of all adjustmt costs. W kow ths dat wll b strctly gratr tha zro, sc codto (6) spcfs that mmdat adopto s too costly for mor tha o frm. Furthrmor, ths dat s lkly to b a rlatvly lat dat sc ) Π s th smallst lvl of proft a frm ca mak th w submarkt. W ca ow cosdr th dcso of frm at ths pot, gv that all othr frms mata thr adopto dats at. If frm chooss to adopt at a arlr dat tha, say, t wll cras ts prst valu of th total proft flows by Π ) > 0 ' γ dt t. Thrfor, a profl of adopto dats wh all frms adopt at th sam tm s ot a Nash Equlbrum. Also, a smlar argumt shows that ay mor tha o frm adoptg at th sam tm s ot a Nash Equlbrum. Th ffct of th markt structur o qulbrum adopto dats To s th ffct of th markt structur as dtrmd by th valu of o th qulbrum valus of adopto dats, w should rcosdr th frst ordr codtos from th abov maxmzato problm: V 0 Π ) y ( ) 0 (9) 88 Romaa Joural of Ecoomc Forcastg /006

6 Th Dffuso of Product Iovatos ad Markt Structur Totally dffrtatg (9) wth rspct to w obta: dπ ) d + Π d ) r y ( ) dπ [ ( ) ( )] ) rπ, y d 0 Th trm o th rght sd of th qual sg s gatv bcaus profts th w submarkt ar dcrasg th sz of th orgal markt. Usg codto (7), th d sg of s gatv (th trm th brackts s postv du to th cocavty of V ). Thrfor, a cras wll dcras th valu of, mag that as th sz of th orgal markt crass, frms wll choos to adopt th ovato arlr. Th cas wh also lads to th dffuso of th product ovato. Th qulbrum adopto dats wll b gv by: () p ( ) bcaus lm ) [ c ] y ( ) 0 (5) q Π lm [α q () + q ()][ p () c ] q ()[ p () c ] ad lm q a c () lm 0. Th sam codtos as abov apply hr to b( + ) show that th frms wll adopt th product ovato squtally. Th oly dffrc s that as profts th orgal markt bcom ormal, th ctv to adopt th ovato wll b rprstd by oly th postv profts mad by adoptg. Howvr, as m, profts ths submarkt wll also go to zro. < 0 Coclusos Gv th atur of ctvs assocatd wth a product ovato, t has b show that a dffuso procss of adopto s th oly optmal rsult a coctratd dustry producg a homogous good. Th ma coomc rasos for a mmdat ad/or a smultaous adopto of th product ovato ar that th prst valu of adjustmt costs s vry hgh arly, ad that th prst valu of profts from adopto s dcrasg wth th umbr of adoptrs. It has also b show that as th umbr of frms th markt crass, ach frm has a ctv to adopt th ovato arlr, ad that as th umbr of frms bcoms vry larg, th ctvs drvg th dffuso procss do ot dsappar. As a xtso of ths modl, a aalyss of th outcoms th markt wh both a product ad a procss ovato ar avalabl would b trstg. O could look at th prfrc of th frms for thr typ of ovato ad whch typ wll b adoptd soor. Romaa Joural of Ecoomc Forcastg /006 89

7 Isttut of Ecoomc Forcastg Bblography Klppr, S., Etry, Ext, Growth, ad Iovato ovr th Product Lf Cycl, Th Amrca Ecoomc Rvw, 996, 86 (3): Loury, G. C., Markt Structur ad Iovato, Th Quartrly Joural of Ecoomcs, 976, 93 (3): Mullga, J. G., Llars, E., Markt Sgmtato ad th Dffuso of Qualty- Ehacg Iovatos: Th Cas of Dowhll Skg, Th Rvw of Ecoomcs ad Statstcs, 003, 85 (3): Rgaum, J. F., O th Dffuso of Nw Tchology: A Gam Thortc Approach, Th Rvw of Ecoomc Studs, 98a, 48 (3): Rgaum, J. F., Markt Structur ad th Dffuso of Nw Tchology, Th Bll Joural of Ecoomcs, 98b, (): Romaa Joural of Ecoomc Forcastg /006