Proceeding backwards and up the saddle-path in the final Regime 1 (Fig 1), either (a) n& n will fall to 0 while n& M1 / n M1

Transcription

1 hemicl Appenix o Pogessive evices Fo convenience of efeence equion numbes in his hemicl Appenix follow fom hose in he Appenix in he icle. A. Poof of Poposiion Poceeing bckws n up he sle-ph in he finl egime Fig eihe n will fll o 0 while / n emins 0 o b / n becomes posiive while / / n is sill posiive. e consie fis he le possibiliy which implies h egime hs poceeing bckws given wy o egime in Appenix A.. e lso noe n impon peliminy fc. A he nsiion poin beween egimes n G & / G = v & / v / n = v & / v / n wih ll v & k / vk being coninuous will hve o be coninuous: shoul i jump isconinuously he sme will hve o be ue of / n n / n iniviully which implies isconinuous jump in since / n equls 0 n using equion A cn be seen o be coninuous he nsiion poin violing he lbou mke equilibium coniion 36 ll ohe ems in 36 cn be shown o be coninuous. Thus / n n / n will ech hve o be coninuous s well n so will Z & / Z implying h he slope of he sle-ph he nsiion poin is lso coninuous. The sme pplies ny ohe nsiion poin beween wo egimes. Fomlly hen bckw nsiion fom egime o will occu if he sle-ph cus he / n = 0 locus of egime poin which / n fom eihe egime which e equivlen poin which / n = 0 is posiive. I is esily shown h his equies h he sle-ph cu he / n = 0 locus nywhee o he igh of y fom he oigin given by A Z = θ ] ] G Fom equions A n 6 i my be seen h he hoizonl inecep of he / n = 0 locus of egime lies sicly beween he oigin n he hoizonl inecep of he G & / G = 0 locus of egime he sle-ph in egime mus lie o he igh of his le locus. Denoing he inesecion poin beween his le locus n he bove y by poin H i follows h he / n = 0 locus mus pss o he igh of poin H filing which he sle-ph will cu i o he lef of he bove y. I uns ou h in he negively-slope cse which equies < / 3 his cnno be sisfie while in he ohe cses epening on pmee vlues i cn. Afe sighfow mnipulions we obin he following necessy lowe boun on if nsiion fom o is o occu: A mx. /3 /{3 θ ] }] The secon gumen on he igh excees /3 when θ is is lowe boun given by he igh sie of 64 flls s θ ises n my fll below /3 when θ eches hypoheicl mximum of. In he egion whee he secon gumen oes no fll below /3 n θ e hus subsiues. A high implies h innovion in will pesis longe in he gowh pocess n high θ h

2 innovion in will commence elie n eihe of hese will conuce o hese innovion phses ovelpping wih ech ohe. e nex show h poceeing bckws ino egime fom he nsiion poin he sle-ph will emin negively-slope i cnno un ighws implying h poceeing fows now boh G n Z e flling no cn i un ownws implying h G n Z e ising fows. In he fome cse veicl line wn some vlue of G o he igh of he hypoheicl uning-poin woul inesec boh ms of he sle-ph once when G & / G is negive n gin lowe vlue of Z when G & / G is posiive: howeve fom A4 G & / G is n incesing funcion of Z so such scenio is no possible. A pecisely nlogous gumen cn be employe o exclue he le cse shoul he coefficien of G now in A5 be posiive o zeo. If he coefficien is negive h of Z cn be seen o be negive lso n he Z & / Z = 0 locus in his cse will be negively-slope n will hve negive hoizonl inecep. eing up he esuling phse-igm i is esily shown h if he sle-ph is iniilly negively-slope s i is he nsiion poin fom egime o i will going bckws lwys emin so n hus no un own. Poceeing bckws long he sle-ph in egime poin will be eche whee n = 0 since fom A n A4 he veicl inecep of he 0 locus lies / / n = below h of he G & / G = 0 locus. Fom A n A3 if θ θ ] G Z is posiive h poin / n will be posiive hee n convesely. Alhough we hve elie gue h θ θ is likely o be negive we lso gue h excees ω n foioi θ n moeove poceeing up he sle-ph G is eclining n Z is ising. Thus i ppes plusible o suppose h / n emins posiive n so he bckws cossove is fom egime o 3: he lenive cse is quie sighfow wih he bckws jecoy nsiing fom egime o egime 4 whence fom he esuling phse igm i hs o coninue bckws in he sme nohwes iecion n hen upon cossing he / n = 0 locus nsiion o egime 6 nlyze below n emin hee ll he wy ows he veicl xis if necessy. Agin i cn be shown h in egime 3 he bckws jecoy will minin nohwes movemen. I will hen s i conveges ows he veicl xis eihe emin ll he wy in egime 3 o nsiion o egime 6 in which / n is 0 n only / n is posiive n emin hee. Fom A9 i is no possible o hve pmee configuion such h he coefficien of G/ is negive n h of Z/ posiive. In ll ohe cses wih he bckws cossove fom egime o 3 occuing in nohwes iecion he esuling phse igms show h he jecoy cnno nsiion o ny ohe phse. In egime 6 ll hee loci G & / G = 0 / n = 0 n Z& / Z = 0 will inesec common poin in he posiive qun n will ll be negively-slope wih he fles he lgebiclly lges slope belonging o he G & / G = 0 locus followe by he Z & / Z = 0 n hen he / n = 0 loci. The bckw sle-ph jecoy will hus exen ll he wy ows he veicl xis. I shoul be menione h hee ppes o exis sligh

3 sly we biefly consie he cse in which he fis bckw nsiion is fom egime o 5 wih / n flling o 0 befoe / n becomes posiive if eve so h only is posiive. I is eily shown h in egime 5 ll hee loci G & / G = 0 Z & / n / Z = 0 n / n = 0 e negively slope n o no inesec in he posiive qun wih he / n = 0 locus being ouemos followe by he Z & / Z = 0 locus. Beween he G & / G = 0 n Z & / Z = 0 loci he bckws jecoy will hve nohwes movemen n cnno nsiion o ny ohe egion in his egime s phse igm. Poceeing bckws he jecoy my eihe emin in egime 5 ll he wy ows he veicl xis o nsiion o egime 3 in which / n lso becomes posiive. A necessy coniion fo he le is > / ] which ensues h he veicl inecep of he / n = 0 locus of egime 3 is bove h of he G & / G = 0 locus of egime 5 noing h he wo egimes e equivlen ny poin long he / n = 0 locus. A he sme ime shoul no be so high h he fis bckws nsiion ws fom egime o in which cse he peceing nlysis pplies. houl he sysem coss ino egime 3 he peceing nlysis hen pplies fom hen bckws. This complees he poof of he Poposiion. B. Fis-Oe Coniions of he ocil Plnne s Opimizion Poblem n ome Peliminy elionships A3 = Y / q n φ = 0 whee / / A4 Y = A n x n x A5 A6 A7 A8 A9 A30 x x x x = τ θ θ / q n = 0 φ = Y / x q n φ n = 0 = Y / x q n φ n = / x = 0 τθ q n φ n = 0 = / x τθ q n φ n = 0 = q n q n 0 ; = 0 if 0 φ > heoeicl possibiliy h inse of nsiing bckws fom egime 3 o 6 he jecoy nsis fom egime 3 o 5 in which / n = 0 n only / n > 0. A sufficien coniion o exclue his is > /. The likelihoo of his sligh heoeicl possibiliy is somewh gee if is close o o excees bu his is iself unlikely since i woul en o imply low she of lbou in Y. e hus ignoe his possibiliy lso becuse i oes no ffec he ges of Gowh pen ienifie in he ex. 3

4 A3 n q& = q n A3 q& = q A A34 q& = q = q n H n q n 0 ; = 0 if 0 φ > = q Y / n q q n φ x ] = q τθ n q q n φ x] = q Y τθ / n q / q n φ x x ] A35 φ 0; φ = 0 A36 im e q n = im e qn = im e qn = 0 s well s he equions of moion 3 wih eplce using he lbou-mke-cleing coniion. The foegoing coniions give ise o some useful peliminy egime-inepenen elionships no epenen on whehe ny given n & k / nk is posiive o 0: A37 Y / = q n φ = Y / n x = Y / nx = τ θ θ / = τθ / n x = τθ / nx / / / A38 Y = q n φ n n obine fom equions -3 A37 n he sme nomlizion of A = q / q / n q / q / n Y q / q / n s employe peviously A39 & = τθ qn A40 & = q n τθ] whee A4 = A4 & = q n Finlly A39 is nonline fis-oe iffeenil equion in q n lone which hs eihe he egenee soluion A43 q n = τθ / fo ll o in he non-egenee cse cn be solve o yiel A44 q n = τθ / ] C e / whee C is n biy consn. The le soluion oes no sisfy he nsvesliy coniion howeve n hence A43 hols iespecive of egime. C. Implicions of he Foegoing Coniions Equliy of q n n q n ove sicly posiive ime inevl implies s poine ou in he icle h hei especive es of chnge e equl ove his inevl which fom A39 n A40 implies h = Y is fixe in his inevl n given by 73 in he icle. Fom A38 fixiy of n hence Y n of q n = q n fixe by A43 implies h n hs lso o be fixe s ssee in he ex. 4

5 b Anlogously fom A30-A3 n A39-A4 i is esily seen h if ny wo of / n / n n / n e sicly posiive ove sicly posiive ime inevl equiing h he es of chnge of he coesponing q k n k be equl hen n hence Y cnno chnge uing h inevl. D. Fuhe Anlysis of he ocilly Opiml Ph e fis suy egime P5 in which only / n cn be posiive. eing n N n n / / enoe q n n n n especively n using A40 n ohe equions egime P5 is chceize by: A45 & / / / = { N nˆ n τθ } A46 / n = / / τ N nˆ n n fuhe equion in & / A47 below wih n fixe n n vying: i shoul be noe h he & i / i iffeenil equions i = e ll egime-inepenen bu which n i vibles in hose equions e fixe n which chnging e egime-epenen. ih n fixe n ˆ o be eemine subsequenly A45 n A46 genee phseigm in n - spce. Boh & / = 0 n / n = 0 loci e posiively slope n we obin Fig below. ince s will be seen P5 will going fow be succeee by P he sey-se vlue of is simply he invese of q n s given in A43. I is esily shown h sic posiiveness of he sey-se vlue of fn. 5 of min ex implies h he wo loci in Fig inesec in he posiive qun ˆ sy n h < ˆ. Thus in P5 hee exiss unique noh-es jecoy commencing n 0 no wn n leing o poin A in Fig n when A is ine hee will be n insnneous swich o P eniling no chnge in o = q n - bu n insnneous swich in lbou evoe o &D fom innovion in se o se. A ising uing P5 implies flling q n unil i flls o equliy wih he unchnging q n which poin he swich occus: noe lso h he sey-se poin A is no he inesecion of he wo loci unlike in cusomy nlyses. The nlysis is of couse coniionl on he vlue of n ˆ : highe vlue of his woul cuse he loci o shif lefws excep he oigin bu he finl vlue of emins while he finl vlue of n will be lowe. e nex exmine egime P6 in which only / n 0. Using A4 n ohe equions n leing enoe q n - we now hve / / A47 & / = N n n0 A48 / n = / / τ N n n0 n A45 bove wih n now fixe n n vying. ih n fixe in his egime n 0 we hve phse-igm in n spce. e equie h he pouciviy em fo se be lge enough h / < filing which his egime cnno fom 5

6 p of he opiml soluion 3 : he esuling phse-igm is vey simil o Fig wih n n eplce by n n n wih he / n = 0 locus cuing he & / = 0 locus fom bove bu wih he le locus ising coninully wihou finie sympoe. e lso noe h if poceeing fows now P5 is o succee P6 his will hve o occu poin which he ising ph cus he ising fom bove fe which ises bove q n flls below q n n boh convege o hei especive sey-se vlues excly he sme finie ime he ime which he swich o P occus. 4 Heuisiclly i is helpful o visulize hese evelopmens by using hoizonl n veicl coss-secions of hee-imensionl phse-igm Fig 3 below. In nlyzing his we noe fis h ou sysem is sequenilly block-ecusive : in egime P5 A45 n A46 fom self-conine ynmic sysem in n spce while he behviou of epens on h of n n ; in P6 A47 n A48 e self-conine n influence he behviou of. In he hoizonl n plne in Fig 3 wn fo P5 we hve o voi clue wn jus he & / = 0 locus s well s he jecoy BA commencing fom n 0 n ening he sey-se poin A. In he veicl plne we hve fo he sme egime wn coss-secion of he conemponeous jecoy CD which hs o ech he sey-se poin D pecisely he sme ime s poin A below is ine he & 0 locus fo P5 is no wn. Noice h poin D lies bove he 45 o line since / = >. e now come o he ciicl poin. Poceeing bckws fom he sey-se poins A n D long he especive coss-secions o he s poin of egime P5 which is whee n = n0 5 wh ssunce is hee h he s poin poin C will lie on he 45 o line s i mus s expline elie he cossove ino P5 occus excly when =? To ssue his n ˆ hs o be chosen ccoingly which seves o pin own he soluion fo his vible enoe n. Thus he peceing egime P6 will hve o pevil unil n ises o his vlue n he finl sey-se vlue of n is hen eemine by his n he sey-se vlues of Y n. I follows h n is funcion of he iniil coniion n 0 : he sey-se vlues of n n n e hus no inepenen of his picul iniil coniion of he moel! e lso noe h s n ˆ is vie he phs BA n CD will chnge excep fo hei ening vlues n n he equiemen h poin B lie somewhee long he n = n 0 line so s o chieve equliy of he s vlues of n of P5 when n ˆ = n n b he vey beginning of he opiml gowh pocess when P6 commences he vlues of he co-se vibles q n q will s usul hve o be 3 n coul hen only ise if flls n woul hen coninue o fll even fe n ceses ising n e h evenully violes he nsvesliy coniion. 4 The sey-se vlue = / which & / = 0 mus hus excee which cn be shown o equie h no be oo f below ohewise he sysem woul nsiion iecly fom P6 o P n he finl vlue of woul be iffeen: i mus howeve be below ohewise P6 woul no be p of he opiml ph which woul simply nsiion fom P5 o P. 5 ince n is consn uing P6 which s poine ou below is he fis phse of he opiml gowh pocess is vlue n 0 he s of P5 is he sme s is vlue he commencemen of he enie opiml gowh pocess. 6

7 chosen o ensue h he esuling n phs ive he bove-inice P5 s vlue he sme ime. Foml vliion of he foegoing heuisic gumen uns ou o be highly complice ffi n in ecion D below we lineize he moel in he P5 phse n povie pecise lgebic soluion fo n clely showing is epenence on n 0. Finlly using A45 n A47 ny swich-poin beween P5 n P6 which = we hve & / & / = { ] τθ }. The squebckee em is posiive fn. 4 bove n is eclining s Y gows. Thus i is possible fo he enie em in bces o be posiive low vlue of Y implying swich fom se o n o be negive nohe hypoheicl swich poin highe vlue of Y bu i is no possible fo hee o hen be le swich o se gin. Given > howeve n coninuiy of he opiml i jecoies innovion in se n no se mus occu immeiely pio o he finl swich o P. Thus s ssee he fow sequence of sges is inee fom P6 o P5 o P. E. ineizion of egime P5 of he ocil Plnne s Poblem Teminl vlues n of egime P5 hve been povie elie n n n hence n e o be eemine. eing he supescip enoe cul minus sey-se vlue n x he column veco n ' pimes enoe nspose lineizion of A47 wih N n n0 eplce by N nˆ n A45 n A46 oun he sey se yiels in mix noion: A49 x& = Ax b whee he elemens of he mix A e A50 = = A5 = / A5 3 = / n A53 = 0 A54 = τθ] A55 3 = ] / n A56 3 = 0 A57 3 = n τ ] A58 = whee A59 = τ ] n b is he column veco 00 n '. Fom he hi ow of A49 we hus noe h is no 0 he sey-se vlue of x = 000' bu hee efes o he lef-hn eivive of n : s expline elie hee occus n insnneous swich o P he sey-se poin so h he igh-hn eivive of n is inee 0. A49 is block-ecusive sysem wih & n no epenen on : s such one eigenvlue of he sysem is simply = n he ohe wo e he eigenvlues of he lowe igh x sub-mix of A. I consiebly simplifies he lgeb if we now suppose 7

8 8 h he eeminn of his sub-mix 3 3 is 0 so h hese wo eigenvlues e equl: given h ou im is in p o simply illuse he exisence of Iniil Coniion Depenence in he moel n h lge numbe of pmees ppes in his eeminn so h seing i o 0 oes no ppe o enil picully sevee economic esicions such n lgebic simplificion ppes ccepble. ih epee oos he soluion of he n block kes he fom fo convenience he s ime of egime P5 is se 0: A60 = X b e 3 3 A6 n = ] X b b e whee n X e o be eemine. e my hen solve fo : A6 = b e N e / X 3 3 wih N o be eemine. Ou sysem hus hs 5 unknowns n X N n ˆ sy he eminl ime which he swich fom P5 o P occus. e lso hve 5 equions nmely ech of n hs o equl 0 ˆ = no hs o equl = 0 n 0 n n hs o equl he given iniil vlue 0 n. Thus n epens on 0 n s we will now explicily show. Afe some exceeingly eious eivions elie s well we finlly obin he following soluions. which is negive is coniionl on n he soluion of n implici equion: A63 ] ] ] ] z e τ = ] ] ] τ whee z = / n 0 n : inee n will ppe in his fom in ll ou soluion equions. e lso hve: A64 ˆ = ] ] τ ] z ] ] A65 X = ] ] τθ z ] ] ] ] τ τ A66 N = { e ˆ ] ] τθ θ τ } ˆ e

9 A67 τθ { τθ ] τ ] z] =. τθ ] τθ { } ]}. A63 n A67 joinly eemine n z fe which he emining unknowns e eemine. e hus suy hese wo equions nex. Inspecion of A63 shows h if z = 0 hs n explici soluion h is simply given by he igh sie of A63 he em in e equls his vlue of : he soluion is unique since he lef sie is sicly incesing in while he igh is consn. uccessive iffeeniion of A63 wih espec o z lso shows h is ecesing convex funcion of z n he sme is hus ue of he igh sie of A67 enoe H. The lef-sie of A67 H is ecesing line funcion of z n wih some lgebic effo i cn be shown h he veicl inecep z = 0 of H sicly excees h of H given h > : in he hypoheicl cse h hese e equl H will be ngen o H z = 0 n fo z > 0 will lie bove i. Fo > hee is hus unique soluion fo z enoe z < 6. To conclue z n n 0 will joinly eemine n n n n will joinly eemine n. e ssume h he iniil vlue of n is less hn n so h he fis phse of he opiml ph is inee P6: filing his he sysem will simply nsi fom P5 o P n he finl vlue of n will be eemine esiully fom n he iniil n. F. Policy fo he Decenlize Economy e h k k = enoe he subsiies o puchses of inpus fom goup k who hus now py h k p k fo he especive composie inpus n b k he subsiies o employes of lbou in &D civiies who hus fce wge coss of b k w. These shoul be insee in he elevn equions elie n in egime we finlly ive A68 / n = /{ θ / hb θ / hb ] G / h b ] Z } whee / / A69 Z = τ w / h / / / n. A70 / n = /{ θ / h b ] G } / / / / h n θ h θ / hb θ h / h / h / hb ] Z z will gully incese o some vlue less hn uniy: z = 6 As flls below implies n is infinie which is no possible since ˆ woul sill be finie hen. 9

10 To in he socil opimum i is necessy fo he vlues of n & k / nk k = fom hese equions o coincie wih he coesponing vlues in egime P 7 / n is of couse 0 in P. A fis equiemen fo his is h h k = k sn esul which pplies lso in ohe egimes. Also fom inspecion of he elevn equions one migh conjecue h b τ vn = τ b vn shoul coespon o q n we now hve h w = b vn = b vn in egime which jusifies he peceing bckee equliy: we le veify h his conjecue is coec. Nex we logihmiclly iffeenie he equion τ b vn = q n wih espec o noing h in P q n = q n is consn use A68 n is equie equliy wih he socilly opiml / n n lso noe h in 55 he secon igh-hn em shoul be ivie by h : we hen en up wih iffeenil equion in b lone whose only sble soluion is consn one A7 b = /{ θ θ ] ] } The ems in bces bove excluing he fis em up o / which is posiive fn. 5 of min ex so h b 0 : he opiml subsiy o &D in is sicly posiive. ubsiuing b ino A70 n equing pive n socilly opiml / n s we eive he opiml b b which is no subsiy shoul be given o &D in se which is of couse consisen wih he zeo vlue of / n in his egime. Nex we consie he ecenlize implemenion of P5 hough policy inevenion in egime 5. ince only n 0 in hese egimes only he equliies vn = / w = b τ mus hol n iffeeniing he le equliy noe h his equliy implies iffeenibiliy of b ove ime wih espec o n subsiuing fom elie equions we ive A7 b & / / / b = τ N nˆ n τθ / / ˆ N n n ] b ogehe wih egime P5 s iffeenil equions in n n. ince b is coninuous i mus convege o by he en of P5: lso he fis hee expessions on he igh of A7 e joinly posiive since hey equl /. If is close o so h he movemen of / N nˆ n is govene by he ising movemen of b & / b will hve o be posiive houghou which equies fom A7 h b cnno be below by oo much o begin wih since ohewise b & / b will become incesingly negive ove ime. If is lge i is conceivble h b will iniilly fll n hen ise lhough i ppes unlikely on economic gouns. e conclue h he opiml b is ime-vying unlike in he Gossmn- Helpmn moel n gully moves o fom below lhough conceivbly nonmonooniclly. 7 Fo his pupose i is useful o wok wih he oiginl / n n / n equions of P which e eive by solving A38 iffeenie wih espec o n hen fom 66 se equl o 0 n wih φ = 0 A39 A40 n he equion n / n = / = / / { θ θ ]/ } / θ ] fo / n / n q& / q n q& / q. 0

11 sly pecisely nlogous chceizion pplies in espec of he evoluion of b in he peceing phse in which ecenlize implemenion of P6 is equie. Thus eflecing he opiml sequence of phses of sucul chnge &D subsiizion evolves fom subsiizing inpu se o se o se while inemeie-inpu puchse is opimlly subsiize houghou.

12 & = 0 0 n = A 0 Figue D C? 0 45 o B A Figue 3 & = 0