The neoclassical version of the Johansen-vintage (putty-clay) growth model

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1 Jon Vislie Sepemer 2 Lecure noes ECON 435 The neoclssicl version of he Johnsen-vinge (puy-cly) growh model The vrious elemens we hve considered during he lecures cn e colleced so s o give us he following se of equions, sed on neoclssicl ssumpions - n exogenous sving re (connecing curren income o he purchse of new equipmen ) nd exponenil growh in ol lour force. (The presen model is more in line wih he version proposed y Phelps.) We lso ssume one commodiy eing produced in his economy. This good is used o produce new cpil equipmen; nd unis of mesuremen hve een chosen so h one uni of he finl good is used o produce one uni of equipmen. The price of he finl good s well s cpil equipmen is se equl o one. Then we hve he following model, when we impose n niciped rel wge schedule, w ( ) gw( ), where g is inerpreed s he expeced re of chnge: () X ( ) x( ) d (2) N ( ) n( ) d (3) N ( ) Ne (4) x () f( n (), k (), ) (5) k() sx() (6) x ( ) w ( ) n ( ) r( ) r k k r e ( rg) r( ) r( ) g( ) r e n n r r g e (7) f e d f r (8) f e d e we () d f w () Equion () is supply funcion for ol oupu of he finl good, ville (equl o income), nd is mde up of oupu from ll plns of differen vinges of equipmen eing used. Equion (2) defines ol lour requiremen on ll When we lk ou vinge, we hve o e i creful when ime is coninuous. Equipmen of vinge is he moun of inslled equipmen during he shor inervl [, d ], k( ) d, which is comined wih he numer of mn-hours on his equipmen, s given y n( ) d, o produce n

2 2 plns eing opered. Relion (3) shows demogrphic dynmics; populion or lour force increses n exogenous re equl o % per uni of ime. (2) nd (3) consiue n equilirium condiion in he lour mrke, where demnd (2) is equl o supply of lour s given y (3). In (4) we hve he ex ne funcion or ll ville echniques or modes of producion ( choice-of-echnique funcion ), king ino ccoun h echnologicl progress is emodied; s given y he index in he producion funcion. New echniques of producion or know-how cn only e uilized y invesing in new equipmen produced. (We neglec he fc h physicl cpil equipmen normlly is compleed during some producion period.) In (5) we hve h sving is fixed frcion of ol income, nd his is uomiclly invesed in new equipmen. (This condiion is n equilirium condiion, s required sving equls invesmen.) In (6) we hve he scrpping condiion, defining he oldes pln in use. The ge of he oldes opering pln is, i.e. of he vinge,, h erns zero qusi-ren. A ls (7) nd (8) re firs-order condiions for n enrepreneur mximising he difference eween he presen discouned vlue of expeced fuure qusi-rens nd he cos of purchsing new equipmen ny, over he expeced life-spn of he pln, king ino ccoun h he producion echnique nd cpciy will e fixed s long s he pln is opered. The ojecive is: r( ) ( nk, ) ( (), (), ) ( ) () () Mx e f n k w n d k when ssuming h he enrepreneur cn perfecly predic he rue life-spn of he pln. We cn hen consider (7) s demnd for new equipmen, wheres (8) is he demnd for lour o e used long wih equipmen inslled. The model hs 8 equions nd 8 vriles, deermining XNxnkwr,,,,,,, for ny ; hence i seems o hve soluion. The prices w nd r will e equilirium prices, king such vlues so h we hve equilirium in he lour mrke nd h demnd for new equipmen is compile wih he sving (he frcion of he ol oupu h is invesed). In his model we should hve o ke ino ccoun h () nd lso h we hve o incorpore how expecions re formed. 2 The wge ph h he enrepreneur uses in his clculions will e sed upon wh he or she nicipes ou he fuure. In fc he expeced growh re g in he wge funcion is exogenous, nd he soluion will herefore depend on wh he oupu equl o x( ) d. Noe lso h is n endogenous vrile nd hence should e wrien s funcion of. 2 The role of price expecions in similr ype of model, however wih declining cpil efficiency over ime, is nlysed oh heoreiclly nd numericlly, in E. Biørn nd P. Frenger, 992, Expecions, Susiuion, nd Scrpping in Puy-Cly Model ; Journl of Economics (Zeischrif für Nionlökonomie) 56 (2), pp

3 3 enrepreneurs nicipe ou fuure prices nd fuure re of ineres, s well. 3 (Here we ssume h r is independen of.) There is no explici depreciion or decy of cpil equipmen s long s he pln or equipmen sill erns posiive qusi-ren. Hence s long s he equipmen is opering, i is inc in he sme wy s i ws when i ws inslled. (We hve sudden deh due o osolescence, nd no due o he sudden deh coming from fixed echnicl lifeime of he equipmen.) There is no disemodied echnicl progress h cn enefi producer who is suck wih his or her ps choice. In his model, wih emodied echnicl progress, he role of sving is imporn in he sense h invesmen will urn up s new nd modern equipmen, which will replce older, less effecive equipmen. When older equipmen is scrpped, relively more lour will e relesed per uni equipmen eing scrpped. (Noe h he pln level, wih rel wge expeced o increse, mrginl produciviy of lour hired nd comined wih new equipmen, will exceed he curren wge, nd herefore lso exceed he verge produciviy of lour used on equipmen eing scrpped.) We now sk he following quesions: Will he model exhii lnced equilirium chrcer in he sense of prilly lnced growh ph, s oh ol oupu nd vinge oupu grow he sme re, so s o keep () sisfied for ll? Do we hve similr requiremen s o lour force nd ol lour requiremen for vinges in use; i.e. so s o hve Ne n( ) d for ny? If yes, wh is he implied growh re for he oupu? The firs quesion will e nswered when we ssume h he ex ne funcion is of he Co-Dougls ype; x () f( n (), k (), ) An () k () e wih, where he erm e shows he efficiency of he cpil equipmen, long wih he pplied lour, he de when new equipmen is inslled. (This erm cpures he level of know-how he de when new equipmen is inslled; his re of emodied echnologicl growh is exogenous nd given y % per uni of ime.) The working hypohesis is he: Cn we find growh re G so h () is sisfied for ll ; i.e., wih X ( ) X() e G nd x ( ) x() e G? 3 An erly conriuion discussing he role of expecions in he vinge model is Kemp nd Thãn, On clss of Growh Models, Economeric, Vol. 34, No. 2 (April; 966), pp See lso E.Phelps Susiuion, Fixed Proporions, Growh nd Disriuion, Inernionl Economic Review, Vol. 4 (Sep., 963), pp

4 4 If his hypohesis is correc, nd when we suppose o e fixed, we hve: G G G x() G G( ) x() G () () () () G G G X e x e d x e e e X e x() G If he iniil vlues re reled o ech oher so h X() e G, hen we hve lnced growh in new cpciy nd in ggrege oupu, so s o hve () sisfied for ll. We furhermore hve: N () Ne. For he required lour o e used long wih new equipmen inslled o grow he sme re s ol lour force; n () ne, he following mus hold, when is fixed: n n () N Ne n( ) d n e d e e e n Hence; if N e, hen oh N nd n will exhii lnced growh s well, nd grow re % per uni of ime. G G From (5) we hen hve k() sx() sx() e ke, elling us h new equipmen will grow he re G % per uni of ime, s well. Using hese condiions in he incremen funcion, we ge direcly soluion for he growh re G: G G ( G ) x() An() k() e x() e A n e k e e An k e ( G ) G ( G ) x() e e e G G G The growh re h will suppor equilirium for ny is given y G. Remrk: If we hve he sndrd neoclssicl model wih no disincion eween ex ne nd ex pos; where X() AN() K() e, K () sx() nd N ( ) N () e, nd, we ge: K () san ( () e ) K () e, where he rend erm is disemodied echnicl progress-componen differen from he emodied pr ove; he re of echnicl progress needs o e inerpreed differenly; lso he prmeers (,) mus e

5 5 inerpreed differenly! We cn wrie he differenil equion s K () san [ K ()] e. This is Bernoulli differenil equion h cn e solved ( ) using he following formul you re no expeced o solve such differenil equion; consider i s n exmple: Le ( ) ( ) Z : K ( ) K K Z K K Z e Z ( ) ( ) e, sano ( ) ( ) Hence we ge Z () e so s o ge: K Z, or s( ) AN ( ) K () e, wih s( ) AN s( ) AN K() K. Hence s( ) AN ( ) K ( ) K ( e ). Insering his ino he mcro producion ( ) s AN ( ) funcion we ge: X ( ) ANe K ( e ) e. X () We cn show h. (Equliy if, hen X () ( ) ( ) ( ) [ ] ( ) s AN s AN [ ] X() A N e e AN e ( ) s AN AN e ) The sympoic growh re will e reched erlier in he neoclssicl world s compred o he puy-cly world, ecuse in puy-cly world we sr ou wih cpil equipmen h cnno e lered momenrily hisory prevens us from doping he mos efficien cpil equipmen he sme ime. (Also he inerpreions of he prmeers (,) should differ; in he neoclssicl model hese re reled o ol inpu use, wheres in he puy-cly model hey re reled o he incremens.) Le us go ck o he puy-cly world. The remining discussion is i enive, nd shows h we cn mke some inferences: The mrginl pln used is deermined from he scrpping condiion: x x e x x w () e [ e ] e n ne n n G ( ) ( ) () () ( G)( ) () ( G) ( G) ( ) ( )

6 6 As is consn in equilirium he erm wihin rckes mus e consn, s well. The rel wge will hen hve o increse ( G )% per uni of ime. Hence; he rel wge will now grow re G. This re of growh should perhps e niciped y invesors; les mong hose enrepreneurs hving rionl expecions. Wh ou he ge of he oldes pln in use, nd wh ou he equilirium re of ineres? For ny se of prmeers, is consn, u will chnge for se of new prmeers. The vriles re ffeced y he expecions he invesors hve ou he wge increse; s given y w () gw (). Le us collec he remining equions in our growh model. We hve: (7)' x () r fk k r () e In his condiion we know h oh equipmen nd new producive cpciy will grow he sme re; hence he rio eween x nd k is fully deermined y he x() rio of heir iniil vlues;. k x() Ank n r Then we hve, on using : A (*) r k k k e A ls, from (8) we hve, when insering for he equilirium wge previling ; w(): ( rg) ( rg) x( ) r e r e x() n r r n() r g e r g e n (8)' f w ( ) [ e ] e When using he scrpping condiion nd h G, we ge: ( G) ( G) x() r e x() r e e [ e ] e e (**) n r g e n r g e ( rg) ( rg) ( G) ( G) ( G) r r The condiions (*) nd (**) give us wo condiions o deermine r nd, oh s funcions of he expeced re of wge increse g. Consider now wo ypes of expecions formion:

7 7 Zero foresigh or nïve (sic) expecions in he sense h he invesors sysemiclly mke miskes; g Perfec foresigh or rionl expecions in he sense h he invesors cn look hrough he mechnism of he model nd predic correcly he equilirium re of increse in rel wge; i.e., g If he invesors form nïve or sic expecions (nd never lern), hey ssume h wge will sy consn, wihg. Insed of king ino ccoun h lour will e more expensive in he fuure nd hen fce lower fuure qusi-rens hn expeced, he enrepreneurs will choose oo lour-inensive echnology. When experiencing h rel wge in fc is incresing, hey will hve o scrp he equipmen erlier hn plnned ecuse qusi-rens re exhused sooner. From (**) we hve, wih g : ln ln ln, s ln. e e If he enrepreneurs hve nïve expecions, hey will choose oo lour-inensive echnique for he pln. From (**) we hen see h he higher is he produciviy increse of newly inslled equipmen, s given y, he lower is, h is, he lower is he ge of he oldes equipmen in use he cpil equipmen is scrpped more frequenly. Here high mens high emodied echnicl progress h cn only e ken dvnge of y inslling new equipmen. (Noe h does no depend on he sving re.) If high urnover of equipmen due o high growh in echnicl know-how (hrough high re of emodied echnicl progress), he shorer durion will inslled cpil equipmen hve. If he invesors form rionl expecions or hve perfec foresigh s o he rue re of increse in rel wge, hey nicipe correcly h g. Given his expecions, he enrepreneurs will choose cpil-inensive echnique, comined wih lour so h he mrginl produciviy of lour is ove he curren wge re. This mkes, ce.pr., he expeced life-spn of he pln inslled longer. Less lour hired will prolong he period over which he firm cn ern posiive qusi-rens. (The higher is he expeced re of increse in rel wge, he more cpil-inensive echnique will e chosen.) Noe lso h if, hen he correcly niciped re of wge increse, g s well. And s we noed ove, for he cse when one correcly nicipes h g, we hve, s. If here is no emodied echnicl progress, he rionlly expeced wge will e consn, moiving he enrepreneur

8 8 o choose pln wih very long durion nd much longer hn wh is implied y he echnicl lifeime. Hence, in his cse, scrpping is echnicl decision, nd he echnicl lifeime will hen se limi on he pln s lifespn. In noher very ineresing ricle from 967, on Some Prolems of Pricing nd Opiml Choice of Fcor Proporions in Dynmic Seing, Economic, Vol. 34 (My), pp. 3-52, Leif Johnsen derived se of efficiency condiions wihin similr, u more disggreged seing hn he mcro-version ove. One prolem in his dynmic seing, wih fixed fcor proporions ex pos, is wheher dynmic opimliy cn e relised in sysem wih decenrlised decisions, when decisions re highly influenced y expecions ou fuure prices. As he sid: For prcicl purposes one mus herefore conclude h i is necessry o hve consisen nd correc expecions ou fuure prices in order h decenrlized decisions wih profi mximizion shll led o fulfilmen of he opimliy requiremens. He ws no oo opimisic s o wheher free mrke would cope wih h issue he suggesed some mrke-dminisror or plnning ody which should pulish prediced prices; hence his srong ineres in so-clled indicive plnning. Leif Johnsen ws hed of modern dynmic heories h rely hevily on fuures mrkes nd rionl expecions. Bu s we know ody, for hese mrkes o funcion well, some very srong ssumpions hve o e imposed, s oh unceriny nd prive informion will oscure he role of rnsmiing informion hrough he price sysem.