Endogenous Growth Models

Transcription

1 nognou Gowh Mol Noclaical Mol - g affc long-un gowh an aniion; mol on' ay anyhing abou how g i min; w wan o fin faco ha min g o w can mak i nognou ha way w can figu ou how w migh chang g hough policy iau - Noclaical ognou chnical chang g g can' b chang oiiv populaion gowh n > n uho: Solow No opulaion Gowh n ; Scal ffc If n >, hn y / y unboun uho: Rom 99 - vaiy Sgon, nan, Dinopoulo qualiy 99 ghion & owi 99 Goman & lpman 995 Schumpian nognou chnical chang R&D; innional invmn oiiv opulaion Gowh n > ; No Scal ffc ognou g uho: Jon moifi Rom; lik noclaical; n up wih g n Sgom qualiy nognou g uho: oung 998 owi 998 Dinopoulo & Thompon 998 Tchnology - Fi Co - chnology involv fi co which la o non-convii conomi of cal; clu pfcly compiiv mak Suppo oupu povi by, lvl of chnology; amoun of labo C b h co of chnology lvl Un pfc compiion, fim fac conan pic o w C If labo mak i pfcly compiiv, wag qual h valu of maginal pouciviy of labo: w Subiu ha back ino h pofi funcion: C C < Rom mol focu on impfc compiion; conomic pofi a qui bfo R&D can ak plac Dign - chnology ak fom of ign; inucion on how o pouc a paicula pouc Non-Rival - if on pon conum h goo, anoh can alo conum i a h am im wihou iminihing h fi pon' bnfi cluabl - i i poibl o pvn oh fom uing h goo; chnology i ma cluabl by kping c.g., fomula fo Cok, ncypion cabl TV, pan/copyigh, c. Incniv - cluabiliy povi incniv fo mak o i bcau fim can chag fo h pouc o vic cluabl Non-cluabl Rival iva Taiional Goo ppl Mohamma' Cok Common Goo aking o Fih in Ocan Non-Rival Collciv Goo Tchnology ay-p-viw TV ublic Goo Naional Dfn ighhou of 4

2 Rom Mol implifi 3 Sco - conomy ha 3 co ha a vically la. omognou Goo - pouc un pfc compiion wih ana Cobb-Dougla poucion funcion. Inmia Goo - pouc capial goo un monopoly impfc compiion; u only capial goo an chnology, no labo alhough i can b a an w g h am ul... u kping i impl 3. Rach Sco - pfc compiion pvail; only u labo Fi abo - + ; pli min gowh a Baic - a wih man fo ; olv pofi maimizaion poblm; ha gna man fo ; olv monopoly pofi maimizaion poblm; ha pofi gna incniv in hi mak Final Oupu -, i ; w' looking a ponial chnologi infini i + ampl - coni only ign i.., : Conan i - ach ign gna h am lvl of pofi ach pio π; wih an infini im hoizon, h icoun pofi i π/ gal of whn h ign i icov; h opimal amoun of ach ign i h am: i hi i call ymmy No: w' g h am concluion wih a fini im hoizon a long a ach ign ha h am lifpan Rul gnal fom: Diconinuii - if a nw ign i icov, ump i.., i no coninuou; ha man w can' u calculu; o fi h poblm, w'll look a a coninuum of nw ign:,, i i... bcau of ymmy i I + Rlaiv ic - ; Balaw' aw - onc you min pic an know quanii in boh mak, n pic i known Numai - pic of goo o $ o all oh goo a pic laiv o h numai.g., $5, compu an $, ca; if compu i numai, pic of ca i [compu/ca]; w'll h final goo a h numai Dpciaion - K C wh C i aggga conumpion a im Capial - by finiion capial i numb of inmia goo u in poucion of final goo o K i i * i * K W can aggga bcau all inmia goo a am o oupu on magin; ul of hi quaion i ha if h i no nw chnology, w on' g mo capial Subiu ino h poucion funcion: K K, o w hav labo augmning * # of uni of ach ign chnological chang u lik Solow mol mpiical Daa - /3... oughly /3 inpu ino oupu i labo an /3 i capial; fuh bakown how labo qually pli bwn kill an unkill Rqui inmia goo an labo Rqui capial Rqui labo Gna man fo conomic π u o buy # of ign of 4

3 voluion of Dign - pouciviy fin a numb of ign p pio of im p ach; hi valu i a conan an i pnn on h numb of ach an h numb of ign ha alay i Waning - w' aling wih coninuou im o inuiion in' cla ook a h ach; flow of ign p pio of im i Tha' qual o a pouciviy paam δ im h hou wok by h h ach im h numb of ign im h lngh of h im pio δ ggga Dign - δ δ, wh i chang in oal numb of ign ov im an i h oal numb of hou vo o ach Raliz ha δ... w can cancl ou h on ach i / Now ivi boh i by o g a of chnological chang: δ... no ha hi a i conan ov im an pn on oal hou of ach abo - full mploymn conain; oal amoun of labo i conan ov im no populaion gowh o all w can o i anf labo bwn ach of chnology an poucion of final goo: + Summay of Mol - 5 quaion: # of yp of inmia goo # of yp of compu # of inmia goo # of ach yp of compu K C C conumpion 3 K * 4 δ δ pouciviy paam 5 + Solving h Mol - w'll only focu on long-un ay-a quilibium; aniional ynamic fo hi mol a ha Baic Ia - w hav chnology u fo inmia goo u fo final goo; w'll olv hm backwa Final Goo - un pfc compiion; pofi a w i i i W ai wa h numai o ; ubiu qn o poblm bcom: FOC - ma, i i i i w i w i i i 3 of 4

4 To o oh FOC, wi obciv: [ i i i ] i w an aliz h ingal i maimiz whn h m in back i maimiz o h FOC i i i... hi i inv man fo i i ai Way - u h ymmy bfo aking FOC: ma, i w FOC - w an i i i Inmia Goo - ach goo ha am inv man... bcau of ymmy icu aly, w opp h i ubcip an only alk abou a h numb of ach ign w u h am numb of ach ign; inmia goo a pouc by monopoli ma Wag of Capial - in a i h wag co of uing capial bcau i' h oppouniy co valu of b alnaiv... coul u h inmia goo o inv i an g a un of FOC - [ ], o pic i a facion of which i conan in ay-a Ma ofi - >, o pofi i poiiv an a facion of oal al conan in ay-a Tchnology - "mak fo ign"; look a valu of h fim; l pic of a ign; h a way o olv fo : Sock Mak biag - a of un of bon mu qual h a of un of "ock" chnology Bon - un uing coninuou im Sock - un i ba on ivin + capial gain Divin - un fo puchaing chnology i monopoly pofi of inmia goo o ivi ha pofi by : Capial Gain - chang in pic of chnology ov pic pai: uing i ogh: Subiu: + + Cancl h : + Sock Mak biag quaion; alway hol mpiical Daa - wih S& 5; hol wihin.9 ov la ya 4 of 4

5 5 of 4 Dicoun ofi Sam - ba on pic of ign bing qual o pn valu of all fuu pofi; w'll u non-conan in a o ina of - w'll hav an ingal... hi i h way Rom i i τ ibniz Rul τ no funcion of o only m Diffnia w : + τ ook a fi m: an valua a man τ o Sinc, h fi m boil own o ook a con m: τ in' a funcion of o w can pull i ou of h ivaiv ' f f f ; in hi ca τ f U ibniz Rul again only m: ' f So igh m bcom: τ τ Now pull ou h : τ Combin m: + + am quaion Ky Diffnc - hi mho g o ock mak abiag quaion, bu can' go h oh way bcau of ingaion conan; baically w' hav c + which plain how w can hav "bubbl" in mak; ock mak abiag quaion ill hol bcau conan op ou whn w iffnia by, bu ock pic i acually high o low han Valu of Fim - olv ock mak abiag quaion fo : / ; ha' inananou pofi ivi by inananou in a minu gowh a of fim Say-Sa - boh mho gav u h ock mak abiag; un a-a: π

6 Rach Wag - pouciviy p wok im valu of oupu: δ ouciviy Wok - pouciviy paam im numb of ign: δ Valu of Oupu - w u foun ha: Conum Bhavio - ui pouc i fo final goo, inmia goo, an chnology; la pic of h puzzl i conum i; aum uiliy i icoun a conan a : σ C ma.. Z Z + w C σ Conain - chang in a in fom a plu wag minu conumpion ynamic opimizaion... mayb n im C Soluion - C σ Slop of conumpion ay nohing abou lvl; no, if, lop i p conum will [vnually] conum mo in h fuu C C af C Summay of Soluion o Fa - w i i fom olving ma fom olving ma 3 fom olving ma 4 fom olving ma 5 π fom olving ock mak abiag qn 6 w R δ ach wag 7 C C σ fom conum bhavio Say-Sa Soluion - all 7 abov hol, bu w hav ay a numb ach ign inmia goo o w'll u * ; alo ha la o a ingl pic o * * abo Mak quilibium - wag in manufacuing of final goo i am a wag of ach o h' no movmn of wok fom on o h oh: w * δ * * Tak 5 an ubiu 4: [ * ] Subiu ino ha: * 6 of 4

7 lug ha ino h igh i of h wag quilibium quaion: [ * ] w * δ δ bunch of uff cancl ou: * 4 quaion & 4 Unknown - olv fo,, g, [] fom olving labo mak quilibium δ δ [] g δ olv whn alking abou voluion of ign [3] + fi labo [4] C g C σ fi pa i aumpion; con pa fom conum bhavio Solv [3] fo an plug i ino []: g δ Subiu fom [] ino hi: g δ δ δ gσ + Solv [4] fo an plug i ino hi: g δ δ δ Solv fo g: g σ + σ + Dinopoulo i lf vion in cla; I go con vion... hy' h am olicy - wha can w o o ag g? Subiiz R&D - δ g Conumpion Now - g... no u you can ag painc wih policy, bu w can look a ocii wih high aving a l conumpion an fin high g C K Gowh Ra - un ou vyhing gow a g... g C K... an fi o ln-iffnia ick giv g K K... fi o ln-iffnia ick giv g K oblm - in numao ugg lag yil lag g; ha woul ugg U.S. gowh i fa han ong Kong' i' no!; alo gow ponnially o g ha o gow ponnially i.., g... ha' no aliic ih 7 of 4

8 Schumpian Gowh - uing qualiy fo chnological pog ap - Sgom,.al. - R, 99 ghion & owi - inmia goo lik Rom; conomica, 99 Goman & lpman - coninuum of inui; 99 Dinopoulo - Ovviw; 993 Chaaciic - Dynamic Gnal quilibium Mol - can' u paial quilibium bcau mak gowing coul aw ouc away fom oh mak ouc Rplacmn - "caiv ucion"; pouc a plac by "b" pouc.g., ypwi, VS ap, cabuo Impfc Compiion - vlopmn of nw pouc qui a la mpoay monopoly o o R&D Uncainy - chaaciz all nw pouc vlopmn... only abou % uviv Sucu of Mol - "all h ifficul pa of conomic com ogh" On Goo - on inuy poucing a ingl conumpion goo Qualiy - qualiy of goo can b impov; all "vion" of final goo a pfc ubiu R&D Rac - uncainy Two civii - manufacuing of final oupu an R&D fo qualiy impovmn Full mploymn - fi labo foc Monopoly - fim ha win R&D ac noy monopoly; uaion of monopoly pn on n R&D ac ic imi - pic limi by pic of pviou goo an amoun of impovmn Sock Mak - u o financ R&D Uiliy - pnaiv conum ha inmpoal uiliy funcion: U ln[ z ] Dicoun a - Sub-uiliy - q q 3 z,,, q Dg of Impovmn - > i g of qualiy impovmn of pouc laiv o i immia pco Vion of Goo - q ; counably infini lvl of qualiy ouc Rplacmn Mchanim - ub-uiliy abov wok in conuncion wih picing o ffcivly limina pviou vion of h final goo ampl - uppo ach wok pouc uni of oupu an u wag of labo a numai; ha man wok uni $... MC C Now uppo conomy a wih only ; conum pn all hi mony on hi goo o man i, wh i aggga pniu, i pic of Suppo i icov; conum uiliy i + ; if an a a h am pic, conum buy all an no ; fim ha icov wan o iv pouc of ou of mak, bu alo wan o mak a much pofi a poibl o h chag h high pic h can fo ha ill ha conum chooing only... i.., wan o kp uiliy fom high han uiliy fom : ; if 3 8 of 4

9 conum pn all hi mony on ih goo w hav ubiu h ino h uiliy icion an olv fo : an ; Sinc minimum pic of pouc of i MC wag w, hn h limi pic of i q q ; if w, conum a iniffn bwn an bu aum hy wich inananouly i.., alway choo high qualiy whn iniffn q q In Gnal - a wih uiliy: q q q q q ofi of Monopoli - aum on wok manufacu on uni of goo q inpnn of h lvl of qualiy i.., on ma wha lvl of qualiy i, only ak wok Dman fo q q if q w w ubiu w maginal co fo q if > w q Fim Obciv - ma q w q w w i.., pofi a popoional w o conumpion lik h Rom mol Soluion - o maimiz pofi, fim chag maimum pic o w Mol ival of Innovaion - oion oc - chaaciz by inniy µ "vlociy of innovaion" # of vn ha will ak plac ov any inval of lngh ~ oion µ µ g [ vn occu ]! Tim T you will hav o wai fo X o occu ~ ponnial F T [ vn occu bfo T ] cf q µ T T f T F' T µ µ pf pobabiliy ha vn will occu omim wihin h ho inval bwn T an T T + i appoimaly µ µ Inananou obabiliy - T lim µ µ µ T inananou pobabiliy ha vn on' occu i pc Tim Bwn Inval - / µ q µ Inpnn Fim - a inniy lvl; inananou pobabiliy i pobabiliy ha fim icov innovaion a am im i zo ggga abo - labo u fo R&D i ; fim ' labo i o µ + ; µ Diminihing Run - mau by <... w'll u o kp hing impl π q q q+ 9 of 4

10 Combin i all: inananou pobabiliy ha a la on fim will icov nw pouc a im i µ µ "Fai" Rach - aum fim ' laiv inananou pobabiliy of ucc qual i µ ha of R&D ouc: µ obabiliy of Succ - iniviual fim' inananou pobabiliy of icoving n high qualiy goo i µ Solv "fai" ach aumpion fo µ an ub ino hi pobabiliy: µ µ γ γ Sub fo µ i.., µ fom abov: µ Fim Doing R&D - no icoun aning of winn of R&D ac a V pc Dicoun aning - hav o muliply by pobabiliy of winning: V Co fo R&D - w pc Dicoun ofi - V γ w F ny - if w aum f ny ino R&D Rac, pc pofi will b zo γ γ V w V w o if w l, V w ; hi i h ock mak valu of h fim Financing R&D - fim iu ock: "If I win R&D ac by im, ockhol g monopoly pofi unil n R&D ac; if I on' win, ockhol g zo"; vy iky ock Typ of Fim - monopoli poucing q ; R&D fim looking fo q+ Rik F Run - ; un on ik f bon in im i Run fo Sock - of fim ha ha monopoly on cun goo; V γ V γ + V V V ivin + capial gain * fim uviv - valu of fim * fim iappa No: if hi i h am a h ock mak fomula u in Rom Mol V Tick - V V ofolio fficincy - pc un of cuiy of iing monopoli mu b qual o h ik f a of un: V γ γ + V V V γ γ ' cancl: + + V V lim : V + + V V γ γ of 4

11 Solv fo V: V V γ + V mpiical Suppo - cun ach how ha hi quaion wok fo h S& 5; on' m o wok fo iniviual fim bcau i ifficul o ima abo - wok mak uni of oupu; oupu i inc w' auming w w numb of wok in manufacuing i / Confuion ov - abo u fo R&D i - hi i h oiginal finiion Inniy of R&D oion poc i µ - w aum γ o inniy i Inananou pobabiliy ha nw icovy i ma uing inval i µ - w aum γ o inananou pobabiliy i Fo a fim, hi pobabiliy i µ Rik of faul i pobabiliy of nw vlopmn in which w u ai wa if γ γ Conum Maimizaion - ma U ln[ z ] olv in W Soluion w Summay of Mol - 6 quaion: q w q w w monopoly pofi; w w w maimiz monopoly pofi; w q γ 3 V w f ny o R&D o zo pofi coniion; w an γ 4 V ock mak pofolio fficincy coniion; γ V γ + V 5 N + full mploymn 6 conum uiliy maimizaion Say-Sa Soluion - all 6 quaion hol all h im, bu in ay a w know ha 6 o ha man Fom 3 w hav V o ha man V fom 4 V + of 4

12 - Sub an : V + Solv fo : +... lin in, pac i.., conumpion v. invmn wih conan lop 5 giv anoh lin in, pac ffc of Siz of conomy - N ~ an ~ ; mo populaion impli mo R&D invmn o gowh inca; hi i poblm wih mol o w'll mov h cal ffc la Conum ainc - ~ an ~ ; mo impain high icouning fo fuu conumpion man conum mo, bu inv l Taniional Dynamic - baically o h am hing w i fo ay a, bu w can' aum 6 Fom 3 w hav V o ha man V Sub ino 4: V + ~ Solv fo pofi: + Tha qual : + N Solv fo in a: ~ Sub ino 6: ~ N ha Diagam - look a ; o min icion of movmn, hol conan an u + > ; + >... ha man inca abov an ca blow i Th a wo abl poin on h gaph, bu nih i a faibl quilibium ll Conumpion - all conumpion an no invmn i no aional; fim ha incniv o o R&D an hav monopoly pow fov... no aliic ll Invmn - h oh abl poin on h gaph on' maimiz uiliy bcau h' no conumpion Say-Sa - h ay-a poin i an quilibium, bu i i unabl; aniion a ump in an aliic bcau i chon by conum an by fim an boh choic a ma inananouly Gowh Ra - of uiliy: U ln[ z ]... ally u n o focu on inananou uiliy q [ z ] ln[ ] q ln ln ln + N ~ N ~ ~ N ~ N + lop i - / N + lop i - N N ~ N N N of 4

13 lnz Subiu : ln[ z ] q ln + ln ln w F, pc valu of inananou uiliy qln + ln ln q i numb of innovaion o a i ump incmnally, b i govn by a oion poc o q pc numb of innovaion fom now unil im i F Gowh Ra ln # wok in R&D gowh a of uiliy F, lnz Wlfa nalyi - ub F, ino U ln[ z ] o compa ocially opimal lvl of invmn an conumpion o h lvl min by mak quilibium ma, U Solv ingal fi: U [ ln + ln ln ].. N + [ ln + ln ln ] [ ln ] + [ ln ] [ ln] a wo ingal a ay bcau [ ln ] ln ln ln ln [ ln ] Fi ingal n ingaion by pa: b a b b ] a X X X... l X an a ln an ln a conan w : X ln ln ln + ln ln ln ln ln u i all ogh: U + ln ln + ' an / [ ] ln ln Ral conumpion pniu Invmn ln Now i' a aic poblm: ma U ln ln +.. N +, Thi i a claic conum opimizaion poblm ana wll, bhav iniffnc cuv an a lina bug conain; l oluion ocial opimum b ˆ, ˆ ll fuu gowh fom invmn Dicoun 3 of 4

14 Dioion - on' hav nough infomaion o giv a gnal amn abou h ocial wlfa, bu h gaph how ha i' poibl o hav oo much invmn oo lil conumpion o oo lil invmn oo much conumpion; uually h la N No nough invmn ubiiz - + ~ Ê U 3 ~ ˆ N U U Too much invmn a N - + Ê ~ U 3 U U ˆ ~ N 4 of 4