Market Structure and Schumpeterian Growth

Transcription

1 ak Sucu and Schumpan Gowh Val E. ambson Dpamn of Economcs P.O. Box Bgham Young Unvsy Povo, UT phon: fax: mal: vl@byu.du Kk. Phllps Dpamn of Economcs P.O. Box Bgham Young Unvsy Povo, UT phon: fax: mal: kk_phllps@byu.du hs vson: 4824 plmnay & ncompl, do no c Kywods: gowh, mak sucu E cods:, O3, O4

2 Absac W psn a dsc-m vson of an ohws sandad Schumpan gowh modl. Unlk connuous-m modls, wh dsc m modls, lna poducon funcons fo pobabls mak no sns as hy mply pobabls ga han on fo suffcnly hgh npus. W show ha n dsc m s possbl fo mo han on fm o nnova smulanously. How h pofs a dvdd n h cas of hs s s ccal fo aggga bhavo. Boh a monopols and a goup of ponal Band compos valu an addonal un of R&D npu only o h xn ha wll succd wh all oh npus n h ndusy fal. Ponally collusv fms plac h sam valu, bu also valu h succss of h own npus, vn f anoh fm also succds. Ths has mplcaons fo h choc a modlng famwok. If s a pvaln, modls wh lna funcons fo R&D n connuous m, whl acabl, wll mss mpoan bhavo. W psn vdnc conssn wh s bng pvaln n many nduss. In gnal qulbum, wh all pcs a ndognously dmnd, h amoun of R&D undakn by a goup of fms ha ngag n Band compon whn h s a s lss han h amoun undakn f h s a sngl R&D fm. In gnal qulbum, h amoun of R&D undakn by a goup of fms ha colluds whn h s a s ga han h amoun undakn f h s a sngl R&D fm. 2

3 Inoducon Schumpan gowh ass fom h sach and dvlopmn R&D acvs of nnovaos pusung h monopoly ns ha accu o nw popay chnologs. Th s a lag and nsghful lau on Schumpan gowh, ncludng paps by Aghon and How 992, Gossman and Hlpman 99, and Sgsom, Anan and Dnopoulos 99. Howv, h Schumpan gowh lau has hus fa gnod h ffcs of pos-nnovaon mak sucu whn sval nnovaos can b succssful a onc, ha s, whn s a possbl. I s asy o undsand why: n h connuous m modls ha a ypcal n hs lau, h pobably of a s nfnsmal. Aguably, hs gnos an mpoan aspc of aly. R&D pojcs ak m, and ha m s naually dnfd wh h lngh of a dsc m pod. If h pod lngh s subsanal hn smulanous ha s, sam pod dscovs of smla nnovaons a lkly o b common. Whh h addonal complxy of modlng s s wohwhl dpnds on how mpoan hy a mpcally. If hy aly occu n h al wold, hn h s ll o b los n usng connuous m modls and h fac ha h soluons a mo acabl maks hm vy aacv. Howv, f s occu fqunly hn h acably may no b woh h cos. W us daa on makups and on h avag gowh as of Solow sduals fom Phllps 993 o xamn h mpoanc of s. Th daa a a h wo-dg SI lvl. Assum hs nduss gow n dsc jumps of sz, qual o h goss makups, wh consan pobabls,. W can solv fo h pobabls of succss n R&D by usng g. Tabl psns h valus of µ and fo h sxn nduss wh makups a ga han on and h mpld valus fo. Gvn, w can calcula h pobably of a gvn an nnovaon occus, assumng qually szd fms. Ths s A famous xampl of such a mgh b h smulanous noducon of VHS and Ba vdo cass codng chnologs. los o hom, ads of acadmc lau can pobably hnk of numous xampls of smla das bng publshd a abou h sam m.

4 pod n h las column of Tabl fo vaous valus of. Th suls show ha s a lkly o mpcally mpoan fo mos nduss. Th mand of h pap xplos a dsc m, nfn-hozon modl ha s analogous o h connuous m modls ha domna h Schumpan gowh lau. Scons 2-4 spcvly dscb h h scos of h modl conomy: nnovaos, poducs, and consums. Poducs mploy labo n h poducon of a consumpon good wh h cun chnology. In ach pod nnovaos com no xsnc and mploy labo wh h goal of dscovng a labo-savng chnology and supplanng h cun poduc o poducs. Scon 5 psns paal qulbum analyss, assumng h ndusy s small nough o ak wags and ns as as gvn. uously, aggga R&D s h sam whn > and s sul n pof-dsspang Band bhavo as whn. Th dsbuon of R&D acoss > nnovaos s ndmna. Of cous, such ndmnacs a common n consan-uns-o-scal modls, bu ous s no such a modl. In dsc m, h pobably of succss canno b globally lna n npus. So h souc of h ndmnacy mus l lswh. W noduc a noon, calld consan uns o duplcaon, ha s npabl as havng nnovaos dcd how many ndpndn xpmns hy a gong o un smulanously dung h pod. In h Band cas, an addonal xpmn s of valu o an nnovao only f succds whn all oh xpmns fal. Ths s u whh h nnovao o s compos a conducng h oh xpmns; ndd s u whh o no h nnovao has compos. Snc h magnal valu of an xpmn dpnds only on h numb of xpmns and no on whch nnovaos a unnng hm, h oal qulbum numb of xpmns s ndpndn of h numb of nnovaos. Thus, n paal qulbum, h Band cas ylds h sam lvl of R&D as h monopoly cas. By conas, f s sul n collusv bhavo o, quvalnly, f a monopoly s andomly gand o on of h succssful sk-nual fms, hn h suls dff fom h Band cas. In h collusv cas an xpmn has valu whnv s succssful, so h aggga numb of xpmns s hgh han n h Band cas. In summay, f h numb of nnovaos xcds on, allowng colluson nducs hgh gowh. 2

5 Scon 6 xnds h analyss of Scon 5 wh a smpl gnal qulbum modl ha maks wags and ns as ndognous. If >, h al wag dpnds on whh h was a n h pvous pod bcaus of h ffc on mak sucu n h oupu mak. As a sul, h quvalnc of Band and monopolsc bhavo dos no cay ov fom h paal qulbum scnao. Scon 6 also consds h wlfa pops of h vaous mak sucus. As n h pvous lau s, fo xampl, Aghon and How 998 wlfa ffcs a ambguous. In all cass gowh may b h oo apd o oo slow, so may o may no b opmal o allow colluson o ncas h gowh a. Smulaons suggs ha, fo asonabl paams, h Band oucom xhbs nsuffcn gowh, suggsng ha allowng colluson n h vn of s may b wlfa-nhancng. Indd, gowh s subsanally ncasd owad bu no byond h opmum vn f h a only wo nnovaos. Th s an opmal numb of nnovaos such ha allowng colluson ylds gowh clos o h socally opmal lvl. Ths opmal numb of nnovaos s ah small. Fnally, h wlfa loss s asymmc n h sns ha h loss assocad wh allowng colluson and ovshoong h opmal numb of nnovaos by as much as naly a housand s sgnfcanly small han h loss assocad wh mposng Band compon. Scon 7 conans som concludng maks. 2 Innovaos onsd an ndusy ha uss labo o poduc a consumpon good. Th a counably nfnly many m pods ndxd by h posv ngs. Th pvalng chnology n pod s chaaczd by s consan oupu p wok, A >. A h bgnnng of ach pod nnovaos a bon. Th nnovaos mploy labo n hops of dscovng a nw chnology chaaczd by oupu p wok of A wh >. If an nnovao hs l woks, has a pobably l of dscovng h nw chnology. Succssful nnovaos f any cas sach and bgn poducon n pod. Unsuccssful nnovaos cas o xs. A consan uns o scal assumpon s wdly mposd n h Schumpan gowh lau. Spcfcally, s assumd ha s homognous of dg on ov 3

6 npu lvls sasfyng,. Of cous, consan uns canno hold globally bcaus s boundd abov by on. On h oh hand, h nuon bhnd consan uns ha popoonal ncass n npus should lad o a las a popoonal ncas n oupu s compllng. W concl hs logc wh s falu o hold globally whn appld o by akng h vw ha s msladng o hnk of as a ypcal poducon funcon. In ou vw, R&D undws xpmns whch may b succssful o unsuccssful. I s hn naual o assum ha an ncas n npus suls n a popoonal ncas n h numb of xpmns undwn. Th pops of can hn b dducd fom h pops of h sochasc oucoms of h xpmns. Th lngh of a m pod s naually dnfd wh h m qud o un an xpmn, and h nnovao s choc of labo dcas how many xpmns can un smulanously. A naual analog o consan uns o scal, whch w call consan uns o duplcaon, poss ha all xpmns hav h sam qud labo npu and h sam ndpndn pobabls of succss. φ b h amoun of labo qud o conduc on xpmn. Thn h numb of succsss n x xpmns qung labo npu of l xφ s bnomally dsbud. As s wll known, hs dsbuon s appoxmaly a Posson dsbuon f φ s small. Spcfcally, f an nnovao hs l uns of labo, h pobably ha has a las on succss and hnc dscovs h nw chnology s l l, wh >. Now consd h poblm facng ach of h nnovaos n ach pod. l j b h labo mployd by nnovao j n pod, l l l,..., l, l l j b l wh h j h ~ lmn movd, and l l l, l 2, Gvn possbly sochasc squncs of wag as w ~ w, w2,... and ns as ~, 2,..., a Nash ~ qulbum fo nnovaos s a squnc l l, l,... such ha, fo all nnovaos j n all m pods, l l ~ [ ν l, l w l] j agmax j, j l Fo any vaabl, say x, x~ wll dno a possbly sochasc squnc of valus of x bgnnng n pod. 4

7 ~ wh ν j, l j, l s h xpcd psn valu of h pof sam dscound o pod assocad wh noducng an nnovaon n pod. Th fs od condon assocad wh 2. assumng an no soluon s ~ l ν j, l j, l w 2.2 No ha ν j, dpnds on l j bcaus hs dmns how many compos h succssful nnovao can xpc o fac, ha s, how many oh succssful nnovaos ~ fom pod h wll b. No also ha ν j, dpnds on l bcaus hs dmns how long h succssful nnovao n pod wll njoy h lad n chnology. Th pops of ν j, wll b dvd blow. 3 Poducs In ach pod,, poducs ha s, h mos cnly succssful nnovaos njoy a sa-of-h-a chnology wh consan magnal cos w A. Poducs smulanously choos pcs o maxmz cun pofs. Thy fac compon fom ach oh as wll as fom on o mo of h pvous poducs ha hav h nx olds chnology wh magnal cos w A, wh >. If n pod h s a sngl poduc, o f h a sval poducs who can collud, hy chag h pvous poducs magnal cos. 3 Ths kps h pvous poducs ou of h mak and ans aggga pof of π w A P, wh s h quany dmandd. By conas, f h a sval poducs who a unabl o collud, hy chag h own magnal cos, P w A, and an aggga pof π n h sulng Band qulbum. Poducs n h aggga mploy N A uns of labo. Th pops of ν j,, h xpcd psn valu of h pof sam assocad wh noducng an nnovaon n pod, can now b dscbd. A succssful nnovao n pod has h mos ffcn chnology n pod wh pobably 3 Ths assums ha pofs a ncasng n pc blow w A. Such s h cas gvn h assumpons abou consums mposd blow. 5

8 γ. Thaf, h pobably ha s chnology mans h mos ffcn n pod τ condonal on havng bn h mos ffcn n pod τ - quals h pobably ha all nnovaos fal n pod τ -: γ τ l j, τ j j l j, τ Thus, condonal on bng succssful, an nnovao dscouns pofs n pod s back o pod by h faco s γ τ τ τ δ s whch ncopoas h pobably of suvval as wll as h ns a. Whn, f h sol nnovao n pod s succssful bcoms a monopols unl s suppland by a nw succssful nnovao. Th assocad valu of ν o b s dnod ν s s δ s wss s ν A 3. Whn >, R&D s somms occu. Suppos, gvn a, succssful nnovaos bcom Band compos haf. A succssful nnovao n pod njoys no unlss all oh nnovaos fal. Ths pobably s l Π j Σ jl. Thus b n hs cas, h valu ν j, o b dnod ν j, s 4 ν b Σ jl j, δ s wss s A 3.2 Fnally, suppos succssful nnovaos collud x pos and ha ach has an qual sha of h monopoly pof unl a nw chnology s dscovd. Thn h valu ν j, c o b dnod ν s j, c δ s wss j, α U U 2 s A # U ν 3.3 b v 4 Alhough can dff acoss fms, snc fms may nvs dffn amouns, h fm subscp s suppssd o smplfy noaon, and smlaly fo αj blow. 6

9 wh αu s h pobably ha h s of succssful nnovaos s U, 2 - dnos h collcon of subss of nnovaos ha do no nclud j, and #U s h numb of succssful nnovaos n s U. No ha ν c j, can also b npd as h xpcd dscound payoff fom succssful nnovaon f a monopoly poson.g. a pan s gand andomly o on of h succssful nnovaos. 4 onsums In ach pod,, consums dvd h cun ncom bwn consumpon, and savngs n h fom of a full s of Aow-Dbu asss. Q dno h holdngs of ass n pod, puchasd n pod -. Wh ndpndn R&D fms ach h succdng o falng h a 2 dffn sas of nau. onsums cun ncom s compsd of h wag fo h on un of nlascally suppld labo, w, and h cun valu of h pvous pod s nvsmns, 2 Σ d Q. Takng h sochasc squnc of wag as, ns as, oupu pcs, and pofs as gvn, consums choos consumpon and nvsmns ach pod o maxmz h xpcd psn valu of lfm uly, Σ β U, subjc o h consan ha cun xpndus on consumpon and savngs mus qual cun ncom fo all and fo all alzaons of h sochasc pocss: P w 2 d Q 2 q Q wh h d dnos h payoff of on un of ass n pod and h q dnos s pc. Subjc o wll-known gulay condons, h consums poblm can b solvd cusvly usng h Bllman quaon: V Q, Ω maxu E{ β [ V Q, Ω ]} Q wh sasfs h budg consan, 2 { Q } Q and Ω s h nfomaon s usd o fom xpcaons n pod. Sandad dynamc pogammng chnqus yld h consums ncssay condons fo all and all alzaons: 7

10 q d U ' β E U ' ; 4. P P In ns of acably and compaably wh h s of h lau, w follow h common pacc of scng anon o h uly funcon U wh,. Th mak dscoun faco,, s h nmpoal pc of on un of consumpon omoow n ms of on un of consumpon oday. Puchasng on of ach avalabl Aow-Dbu asss gvs xacly on un of consumpon wh cany. W us hs fac, sum and manpula h I quaons n 4. and g P βe P Paal Equlbum onsd a paal qulbum modl of a sngl R&D ac wh cun wags a xognous o h ndusy. A succssful nnovaon by only on fm a m wll yld a sam of xpcd monopoly pofs, ν m, bgnnng n pod. Wh only on fm ngagd n R&D, quaon 2.2 ducs o l ν w m 5. Solvng hs fo l gvs h amoun of labo hd gvn h wag a, w, h m valu of fuu pofs, ν, and h as of R&D,. l w ln m v If mo han on fm ngags n R&D.. > and s sul n Band compon n h poduc mak, hn a fm cvs h sam of monopoly pofs f and only f s h only fm o nnova. In hs cas 2.2 bcoms l Σ l Σ l j j m j j [ ν ] w, wh s h pobably ha a las on oh fm nnovas. Ths ducs o ν m w 5.2 Σ l 8

11 wh Σ l s h aggga mploymn by all R&D fms. j j Supsngly, quaons 5. and 5.2 yld h sam aggga mploymn, snc l whn h s only on R&D fm. Thus, a monopolsc nnovao wll h h sam amoun of R&D as a goup of nnovaos ha xpc o ngag n Band compon f h s an R&D. Inuvly, all fms valu an addonal un of labo only o h xn ha wll succd wh all oh uns fal. Fo h monopolsc nnovao hs s bcaus owns any succss fom all oh uns anyway. Fo a mmb of a goup hs s bcaus any succss fom uns dos no own suls n zo pofs du o compon. 6 Smpl Gnal Equlbum A gnal qulbum s a ls of sochasc squncs ~ ~ ~ ~ l, N,, Q, ~, P ~, w ~, ~ π, q ~ such ha, fo all and all alzaons of h sochasc ~ ~ ~ pocss, l s Nash qulbum fo nnovaos, 2 and Q solv h consums poblm, 3 h full mploymn condons N a sasfd, 4 h cd mak clang condons, ha Q qual pofs n sa and sa fo all and, a ~ sasfd, and 5 h oupu pc squnc P and h pof squnc ~π a dvd as dscbd abov. As s gnally hough no unvsally 5 h cas n h Schumpan gowh lau, w sc anon o saonay qulba. onsumpon ss ov m as nw chnologs a dscovd. oupu n h fs pod b numa; ha s P. Th nau of qulbum dpnds on h assumpons mposd on h mak sucu. W agan consd h sucus: a sngl nnovao, 2 mulpl nnovaos ha dsspa pofs hough Band compon f mo han on s succssful, and 3 mulpl nnovaos ha succssfully collud f mo han on s succssful. Of cous, h fs sucu s a spcal cas of h of h la wo. W f o hs, spcvly, as h monopoly cas, h Band cas, and h collusv cas. 5 S Dssnbg and Nyssn

12 6. Sngl Innovao Whn, h succssful nnovao always bcoms a monopols. Pos a saonay qulbum wh a consan ns a,, and a consan oupu pc, P. Thn, snc h monopoly pc always pvals, P w A fo all. Gvn an nnovaon n pod, consumpon wll b consan a unl fuh nnovaon occus and 3. ducs o s ν fo all. Now, by h dfnon of h oupu poducon funcon and h labo mak clang condon, A fo all. Subsung h pvous h laonshps no h nnovao s fs od condon 2.2 and manpulang ansfoms no [ ] 6. Equaon 4. n hs cas s β E{ In a saonay qulbum hs mpls β [ ] Ths and h dfnon of mply } 6.2 β[ ] Now 6. and 6.2 povd wo quaons n and. Th fom xhbs dcasng n whl h la xhbs ncasng n. Th unqu soluon wll xs wh > f β β. Th lf sd of hs condon flcs h xpcd pofably of sach whl h gh sd flcs h dg o whch fuu pofs a dscound. Thus, f h lklhood of succss flcd n and h gan fom

13 nnovaon flcd n a suffcnly hgh and h fuu s no dscound oo havly as flcd n β hn qulbum wll xhb posv nvsmn n sach. Ohws, n qulbum and only 6.2 s sasfd, yldng no gowh and an ns a qual o h a of m pfnc. Th qulbum valus of h oh vaabls can b dvd n a saghfowad mann. 6.2 ulpl Innovaos: Band Whn > R&D s wll somms occu. Whn hy do h sucu of h qulbum dpnds on poduc bhavo. Ths subscon assums ha mulpl poducs dsspa pofs hough Band compon. Poducon n hs cas s ga han ha wh only on poduc. Ths lads o cycls of monopoly and compon n h poducon sco whch do no xs n h oh mak aangmns w consd. Pos a saonay qulbum xhbng ns as, and wh h supscp dnos a vaabl s valu whn h s a monopoly poduc and a dnos s valu whn poducon s compv. As bfo, pos a consan oupu pc, P. Whn a pod- nnovao, say nnovao j, s h only succssful fm, njoys monopoly pofs unl s suppland. In hs cas h oupu pc sasfs P w A. If s no h only succssful fm, and s Band compos wll dsspa h pofs by chagng an oupu pc nomalzd o on of 6 w A P Now, snc consumpon wll b consan a fo as long as h succssful pod- nnovao s no suppland, and snc posv pofs only accu f nnovao j s h only succssful fm, 3.2 ducs o bk ν j, k k ;, fo all. By h dfnon of h oupu poducon funcon and h labo mak clang condon, 6 Snc oupu s numa, h ncasd compon s flcd n a hgh nomnal wag, ah han a low nomnal oupu pc.

14 k A ; k k, fo all. Subsung h pvous fou laonshps no h nnovao s fs od condon 2.2 and manpulang ylds [ ] Th a wo condons bcaus R&D mploymn and ns as a dffn whn cun poducs comp han s whn h s only on poduc. No ha 6.3 & 6.4 only dmn aggga R&D mploymn, h dsbuon of nvsmn acoss nnovaos s ndmna. Of cous, such ndmnacs a common n gnal qulbum modls wh consan uns o scal, bu hs s no such a modl. Inuvly, an nnovao s magnal xpmn s of valu f and only f succds whn all oh xpmns fal. Ths s u whh h nnovao o s compos a conducng h oh xpmns; ndd as llusad n scon 5, s u whh o no h nnovao has compos. W assum blow ha all R&D fms a qually szd: l, l fo all mj m cj c Th consum s Eul quaons 4.2 can b wn as wh wh β [ ] 6.5, β [ ] 6.6, Equaons povd fou quaons n,, &. Appndx shows ha hs s of quaons mpls > and >. Inuvly, whn poducon s compv, h al wag s hgh han whn h s a monopoly. Ths lads o a subsuon of labo away fom R&D and no poducon. 6.3 ulpl Innovaos: olluson Now suppos ha whn mulpl nnovaos a succssful hy collud and spl h monopoly pofs qually. Pos a saonay qulbum wh an ns a of, a 2

15 consan oupu pc, P, and symmc nvsmn, l j j. Thn, snc h monopoly pc always pvals, P w A fo all. Snc consumpon wll b consan a fo as long as h succssful pod- nnovaos a no suppland, 3.3 ducs o α U # U c ν Fnally, A U 2 Subsung hs quaons no h nnovao s fs od condon 2.2 and manpulang ylds! 6.7!! Of cous, whn, hs cosponds o h monopoly cas. Th summd m s h xpcd nvs of h numb of an nnovao s succssful compos, and s dclnng n. Thus h sasfyng 6.7 s dclnng n. Th consum s Eul quaon s h sam as h monopoly cas 6.2 and s poducd h as β[ ] Equlbum valus of and solv 6.7 and If h s a soluon o 6.7 and 6.8 ha s, f β β hn h s a unqu saonay qulbum and xhbs a posv lvl of nvsmn; ohws, h unqu saonay qulbum xhbs no nvsmn n R&D. Appndx 2 shows ha s lag whn > han whn. Snc s h cas of a sngl nnovao w xamnd abov, follows ha mo aggga R&D labo s hd wh mulpl nnovaos f collusv poducon s xpcd. 7 As bfo, n qulbum ff β β. 3

16 Σ α j j j I s asy o vfy, gvn h pops of h bnomal dsbuon mply s dclnng n, ha an ncas n h numb of nnovaos dcass l : facd wh mo compos, ach nnovao nvss lss A socal plann s poblm 2 A A, A,,...} b h s of aanabl chnology lvls. A saonay { A socal plan s a funcon λ : A [,] ha assgns a lvl of nvsmn o ach lvl of chnology. Dfn h socal plann s poblm as choosng a saonay socal plan o maxmz h xpcd psn valu of uly, E β subjc o h consan ha λ A [,] fo all A A. W A, λ b h xpcd psn valu of mplmnng h plan λ f h cun chnology s A. No ha h choc of uly funcon mpls W A, λ' W A, λ f λ ' A λ A fo all A A. V A max W A, λ. Sandad connuy and compacnss agumns mply ha V A s wll dfnd f λ > β, ha s, f h maxmal gowh a sn oo lag lav o h a of m pfnc. I follows fom h pops of W and h opmaly of V ha V A V A. λ * b an opmal saonay plan. Thn, fo any A A, sandad cusv agumns mply V A * [ A λ A] β * * [ λ A V A λ A V ] A wh, call, l. Solvng fo A V ylds A V A β * λ A A λ [ ] * A λ * Snc V A s maxmal, λ * A mus maxmz fo all A. Dffnang V A wh spc o λ *, sng h sul qual o zo, and manpulang ylds β λ* A * λ A { [ ]} λ* A λ* A β Gaphcally, an ncas n shfs 6.4 and 6.5 o h lf n l spac, llusang h sul 4

17 No ha h lf sd of 6.9 s scly dcasng n λ * A, mplyng a unqu opmal lvl of nvsmn. No also ha h opmal lvl of nvsmn s ndpndn of A, ha s, h socal plann would opmz by choosng h sam valu fo n vy pod. Unfounaly, 6.9 canno b xplcly solvd fo h opmal fo puposs of compason wh h oucoms dscussd n pvous scons. Numcal smulaons a acabl howv. Th nx scon psns som xampls. 7 Som numcal smulaons I s wll known ha Schumpan gowh modls can xhb h nsuffcn, opmal, o xcssv gowh lav o h socal opmum. S, fo xampl, Aghon and How 998. Th nuon s compllng. On h on hand, succssful nnovaos as h n fuu ajcoy of h conomy, bu njoy pofs only unl suppland; hs suggss gowh as wll b oo slow. On h oh hand, succssful nnovaos dsoy h pofably of cun poducs, a socal loss ha nnovaos do no nnalz; hs suggss ha gowh as wll b oo hgh. Eh ffc may domna. An mplcaon of ou analyss s ha h wlfa pops of qulbum also dpnd on mak sucu. If h a a las wo nnovaos, and f gowh s nsuffcn n h Band cas, can b ncasd owad and vn byond h opmal a by allowng colluson. If gowh s oo hgh n h Band cas, allowng colluson wosns h poblm. To povd a fl fo h possbl magnuds of hs ffcs, som smulaons a pod n Tabl 2. Fo vaous paam valus, h abl compas h socal opmum, h monopoly oucom, Band oucoms fo vaous numbs of nnovaos, and h collusv oucom wh h sam numbs. In ach smulaon, mposng Band compon n h cas of s subsanally ducs gowh blow h socal opmum. Fuh ducs blow h lvl found n h monopoly cas. Gowh s sgnfcanly ncasd by allowng colluson vn f h a only wo nnovaos. In h las scon w llusa a cas wh colluson can vn lad o R&D lvls hgh han h socal opmum. 8 oncludng Rmaks 5

18 Ths pap has agud ha pos-nnovaon mak sucu mas n dscm Schumpan gowh modls. Havng a sngl nnovao ylds a socally subopmal lvl of R&D and a gowh a ha s oo low. Whn h pobably of smulanous dscovs s non-nglgbl, havng mo han on nnovao lows aggga gowh as f pofs a dsspad by Band compon n h vn of a. By conas, havng mulpl nnovaos can ncas gowh f hy a allowd o collud n h vn of a. Th ncasd gowh a, whch coms a h cos of addonal R&D xpndus, may o may no b wlfa mpovng, bu smulaons suggs ha allowng colluson may g wong by lss han pohbng colluson dos. Th suls of R&D s a usually no dncal pans. W hav modld nnovaon as a dscovy ha lows h cos of poducng goods. I s jus as asy o np nnovaon as an ncas n qualy of goods poducd whl cos mans consan. Whn s occu n qualy mpovmns h sul wll mos lkly b goods ha a mpfc subsus. In hs cas, h monopoly ns would no b complly dsspad and ou suls fom h colluson cas would apply. Evn f s sul n dncal goods, howv, s possbl ha h collusv cas s sll h mos lvan f pans a gand o only on fm. Fo xampl, f smulanous dscovs a awadd o h fs fm n ln a h pan offc, o by som andom pocss, hn h xpcd wad fom a wll b non-zo and h collusv cas appls. 6

19 7 Appndx ulpl Innovaos: Band W hav fou quaons, n fou unknowns, &,, W can w 6.4 as: [ ] A A ; Rcall 6.3: ] [ Hnc, mus b ha A. W can us 6.5 and 6.6 o g: and w A as: A Rcall h dfnons, k k, k k k ; k, Suppos. I s asy o s ha hs mpls A<. Nx, suppos <. Ths mpls > and >. I s possbl o w A abov as: D D A wh ]} [ { > D, and ]} [ { > D Snc <, w hav A<<.

20 Hnc mus b ha >. In hs cas, < and <, so D and D abov a boh ngav. Ths, n un, mpls ha >. 8

21 9 Appndx ulpl Innovaos: olluson Equaon 6.7 s:!!!!!! Spaang ou h fs m n h sum gvs:!!! A ] [!!! > A Wh A and hs ducs o 6. fom h sngl nnovao cas Fo a gvn lvl of, s hgh n h cas wh > han whn. Snc s dcasng n fo 6.7 and ncasng n fo 6.8, movng fom o > wll sul n an ncas n qulbum. As llusad blow:

22 Tabl Pobabls of Ts Infd fom an Tchnology Gowh & akups SI cod dscpon µ s 2 s 5 s s 2 Food 2.6% %.5%.83% 2.6% 2.26% 22 Txl mll poducs 3.76% % 24.3% 35.% 38.29% 4.99% 23 Appal & oh xls 2.22% %.53% 7.63% 9.53% 2.2% 24 umb & wood poducs 2.3%. 2.29% 5.98% 9.34%.43%.39% 25 Funu & fxus.47%. 4.68% 3.97% 6.25% 6.99% 7.65% 26 Pap 2.%.3 7.%.82% 2.89% 3.24% 3.55% 28 hmcals 3.55% %.42%.67%.76%.83% 29 Polum 2.69% % 4.95% 7.76% 8.67% 9.49% 3 Rubb & Plascs.74% % 8.56% 3.25% 4.73% 6.5% 32 Son, lay & Glass.49% % 2.3% 3.64% 4.8% 4.48% 33 Pmay als.27%.3 2.%.53%.85%.95%.5% 34 Fabcad als.53% % 6.4% 9.58%.69%.68% 35 achny 3.% % 22.46% 32.88% 35.95% 38.55% 36 Elccal & Elconc 3.82% % 4.54% 7.4% 7.98% 8.73% 37 Tanspoaon Equpmn 2.33%.5 5.5% 4.2% 6.63% 7.4% 8.% 38 Insumns 2.25% % 2.38% 3.9% 32.98% 35.44% Avag of All 2.28% % 7.69%.59% 2.8% 3.84% 2

23 Tabl 2 Rsuls fom Smulaons β onopoly Band olluson Opmum no. of R&D fms R&D labo 3.% 2.99% 2.98% 2.98% 2.97% 3.56% 4.5% 4.26% 4.46% 2.58% % opmal 23.86% 23.76% 23.68% 23.65% 23.63% 28.3% 32.2% 33.83% 35.48%.% xpcd uly % wlfa loss -8.98% -9.2% -9.5% -9.6% -9.7% -7.6% -6.53% -6.2% -5.72%.% avg. gowh a 2.64% 2.63% 2.63% 2.62% 2.62% 2.95% 3.8% 3.28% 3.36% 4.78% β onopoly Band olluson Opmum no. of R&D fms R&D labo 2.79% 2.78% 2.77% 2.77% 2.77% 3.3% 3.75% 3.94% 4.3% 9.44% % opmal 29.6% 29.47% 29.37% 29.34% 29.3% 35.6% 39.78% 4.73% 43.7%.% xpcd uly % wlfa loss -2.98% -2.99% -3.% -3.% -3.% -2.42% -2.% -.84% -.69%.% avg gowh a 2.5% 2.5% 2.5% 2.5% 2.5% 2.8% 3.4% 3.3% 3.22% 4.53% β onopoly Band olluson Opmum no. of R&D fms R&D labo 6.89% 6.83% 6.73% 6.7% 6.67% 9.42% 2.5% 4.2% 6.3% 22.4% % opmal 3.2% 3.86% 3.4% 3.27% 3.4% 42.53% 56.49% 64.4% 73.62%.% xpcd uly % wlfa loss -53.5% -54.% % -55.4% % % -9.75% -2.84% -6.78%.% avg gowh a 6.43% 6.34% 6.24% 6.2% 6.9% 8.% 9.2% 9.43% 9.66% 9.92% 2

24 β onopoly Band olluson Opmum no. of R&D fms R&D labo 3.42% 3.39% 3.38% 3.37% 3.37% 3.8% 4.9% 4.2% 4.3% 8.67% % opmal 8.3% 8.7% 8.9% 8.7% 8.5% 2.35% 2.89% 22.48% 23.4%.% xpcd uly % wlfa loss -.27% -.3% -.33% -.34% -.35% -9.66% -9.2% -9.4% -8.89%.% avg gowh a.74%.73%.72%.72%.72%.89% 2.% 2.4% 2.8% 4.52% β onopoly Band olluson Opmum no. of R&D fms R&D labo 3.7% 3.3% 3.2% 3.% 3.% 3.65% 4.7% 4.38% 4.6% 4.48% % opmal 2.9% 2.95% 2.83% 2.8% 2.77% 25.23% 28.77% 3.27% 3.79%.% xpcd uly % wlfa loss % -65.7% % -65.3% % -6.3% % -56.5% -55.%.% avg gowh a 2.68% 2.66% 2.65% 2.64% 2.64% 2.99% 3.24% 3.33% 3.42% 4.87% β onopoly Band olluson Opmum no. of R&D fms R&D labo 2.52% 2.32%.56%.37%.2% 9.6% 29.86% 37.3% 48.7% 26.4% % opmal 47.9% 47.5% 44.22% 43.48% 42.89% 73.3% 4.23% 42.73% 83.89%.% xpcd uly % wlfa loss -5.7% -6.96% -8.% -8.4% -8.62% -.56% -.3% -.6% -.53%.% avg gowh a 95.63% 94.28% 93.36% 93.3% 92.95% 99.7% 99.94% 99.99%.% 99.85% 22

25 Rfncs [] Aghon, P. and P. How 992, A modl of gowh hough cav dsucon, Economca, 6, pp [2] Aghon, P. and P. How 998, Endognous Gowh Thoy, IT Pss. [3] Dssnb,. and. Nyssn 998, A smpl modl of Schumpan gowh wh complx dynamcs, ounal of Economc Dynamcs and onol, 22, pp [4] Gossman, G. and E. Hlpman 99, Qualy adds n h Thoy of Gowh, Rvw of Economc Suds, 63, pp [5] Hall, R. 988, "Th Rlaon bwn Pc and agnal os n U.S. Indusy." ounal of Polcal Economy, 96, pp [6] Phllps, K. 993, Qualy adds, Gowh and R&D: An Assssmn fom U.S. Indusy, ang-rochs onfnc Ss on Publc Polcy, 38, pp [7] Sgsom, P., T..A. Anan, and E. Dnopoulos 99, A Schumpan odl of h Poduc f ycl, Amcan Economc Rvw, 8, pp