The generalized Solow model with endogenous growth

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1 Page of 8 The generalized Solow model wih endogenous growh December 205 Alecos Papadopoulos PhD Candidae Deparmen of Economics Ahens Universi of Economics and Business papadopalex@aueb.gr This sprung ou of he Graduae Macro I class I decided o elaborae on a shor secion of he main exbook's chaper on Endogenous Growh models, in order o clarif he properies of he long-run equilibrium, he issue of ransiional dnamics in endogenous growh models, and how one can simulae hem. I. Endogenous Growh in he generalized Solow model "Solow" growh model implies ha he invesmen rae(s) are fixed."generalized" Solow model means ha boh phsical and human capials are presen. The model was inroduced b Mankiw e al. (992) in order o enhance he basic Solow model of growh so ha i fis beer he available daa. In heir version growh was no endogenous. Specificall he considered a model of he form a Y( ) A ( ) ( ) L( ) g a() h( ) e, h( ) ( ) L( ) 0 s Y, s Y where noaion should be obvious. g e is he exogenous echnical progress erm ha augmens he efficienc of labor, wih is iniial level normalized o uni, and we also

2 Page 2 of 8 assume ha populaion L () grows a consan rae n. The iniial values of echnical progress and labor are normalized o uni. When he producion funcions exhibis consan reurns o scale in,, L : g ( ) ( )( ) g e L L e Y() A( ) ( ) L( ) ( ) A( ) ( ) ( ) ( )( ) In fac his is he criical issue as o wheher he model will exhibi capabili for endogenous growh or no. If ou see an absrac producion funcion F,, L for which i is assumed ha F,, L F,, L facors, hen he per capia producion funcion, i.e. consan reurns o scale in all hree Y LF L, L, Y L f ( k, h) will exhibi diminishing reurns o scale in kh, and he model won' be capable for endogenous growh (so such an absrac formulaion is equivalen o assuming ). If exogenous echnical progress exiss, he per capia magniudes will grow bu onl because of i. We are ineresed in he "endogenous growh" case, for which we need o impose. ere, he erm g e disappears from he producion funcion. If ou wan o be ver pedanic, does no impl ha labor efficienc is fixed: we have ever righ o assume ha labor's "producive poenial" is increasing bu somehow, i does no find is wa ino he producion funcion, and so i does no conribue o oupu: in fac he case, g 0 could be used o model organizaional failure: firms emplo people ha increase heir individual efficienc, bu he producion funcion in place (he work processes, he organizaional srucure, he producion ssem in general) fails o le his increased efficienc produce resuls (leading hus o a declining oupu per efficienc uni and reasonabl o disappoined workers ha feel ha he are being wased). Bu in our model, when assuming, we also assume g 0, and he onl source of exogenous growh remaining is he populaion growh. So a he ver leas, in he balanced

3 Page 3 of 8 growh pah, oal magniudes will grow a rae n, while per capia variables will be consan. Bu we wan more han ha. Le's see how we can ge i. I.. Source of endogenous growh Under we have a ( ) h( ) ( ) L( ) and so Y( ) A( ) ( ) and in per capia erms ( ) ( ) ( ) Ak h [] wih k s ( ) n k( ), h s ( ) n h( ) [2] Noe ha here he producion funcion exhibis consan reurns o scale in, and correspondingl in kh,, and his is he reason ha endogenous growh in per capia erms can arise in his model. Le's see wh, exacl. To obain an expression for he growh rae of a variable (in coninuous ime), we ake is naural logarihm and hen is derivaive wih respec o ime: ln ( ) ln A ln k( ) ( )ln h( ) d ln ( ) k h ( ) [3] d k h The above relaion of growh raes does no impl ha he hree per capia variables (oupu, phsical capial, human capial), grow a he same rae. I jus ells us how he growh rae of per capia oupu depends on he growh raes of he oher wo.

4 Page 4 of 8 From he laws of moion of he wo capials we have k () k s( ) n k( ) s n k k() h () h s( ) n h( ) s n h h() Insering ino he expression of he per capia oupu growh rae, k h ( ) ( ) ( ) s n ( ) s ( ) n k h k( ) h( ) ( ) ( ) s ( ) s n [4] k( ) h( ) or alernaivel ( ), ( ) ( ), ( ) s f k h s f k h n [5] k h Equaion [5] connecs he per capia growh rae of oupu wih he marginal producs of he wo capials. We see ha wha we need for a consan oupu growh rae is ha he marginal producs of he wo capials are consan (no equal). We know ha he reason for zero long-run growh in per capia erms (or in per efficienc erms, if echnical progress is posiive), wih onl phsical capial presen, is he fac ha he marginal produc of capial diminishes as he level of capial accumulaes. Wha happens in he curren model is ha because we have wo capials, wo accumulaed facors, "he one can help he oher mainain a consan and posiive marginal produc", even hough boh heir levels increase, and hus leading (possibl) o posiive growh rae of per capia oupu.

5 Page 5 of 8 I.2 Economic models and he long run growh rae Our models should bear some resemblance wih he real world, no maer how absrac he are. The real world daa ells us ha a) per capia growh is observed b) i is relaivel sable, per econom, in mos economies c) raios of basic macroeconomic variables, like consumpion or capial over oupu, alhough no sable over ime, he do no appear o end o exremes (o "zero" or o "infini"). Our models should be able o replicae broadl hese empirical regulariies (and ohers of course, see "slized facs of growh"). The firs empirical regulari demands from us some source of per capia growh. We iniiall came up wih an exogenous source (echnical progress), and now we are examining endogenous sources. The second regulari, demands ha in he long-run he model leads o hese growh raes being evenuall consan. The hird one demands ha hese growh raes are equal, a leas in he long-run. In economics speak, we codif his b saing ha we need o obain a "balanced growh pah" in per capia erms, i.e. ha we need o have k h gb gb 0 [6] k h Noe ha he requiremen ha he common growh rae is posiive is a separae one. Mahemaicall he model does no exclude he case k h k h 0 ) Equal growh raes From k ( ) h k h

6 Page 6 of 8 he condiion for an equal growh rae for all hree variables is k h ( ) ( ) k s s n s n [7] k h k( ) h( ) h s Noe how he consan-reurns-o-scale proper is criical here). Also, remember Noe ha he above sas onl ha he growh raes will be equal, no ha he will also be consan hrough ime. And cerainl, his condiion does no impl ha his growh rae (of he per capia magniudes, i.e. over and above n) will also be posiive. 2) Consan (and equal) growh rae To obain a consan growh rae, i mus be he case ha i is expressed in erms of parameers ha are considered fixed. We have alread obained he expression (eq. [4]) ( ) ( ) s ( ) s n k( ) h( ) Wriing he per capia producion funcion explicil we have Ak( ) h( ) Ak ( ) h( ) s ( ) s n k( ) h ( ) k( ) ( ) ( ) k( ) ( ) n s A h s A h [8] Imposing he condiion for an equal growh rae we have

7 Page 7 of 8 ( ) g s A s s s A s s n B As s ( ) s s n B g As s n [9] and now we see ha he common growh rae will also be consan. Sill, he above do no impl ha he growh rae will be posiive. 3) Posiive (as well as consan and equal) growh rae The wa o obain a posiive growh rae (ha is also consan and common) for he per capia magniudes is simple (and unique): We assume ha he values of he parameers are such ha he lead o gb 0. This is wha "endogenous growh" models do: he allow for he possibili of an endogenous consan and equal growh rae, and hen, he parameers mus be such as o deliver i as posiive also. In conras, he sandard Solow model, does no allow for endogenous posiive growh rae. Isn' his fixing of parameers "o our liking" oall arbirar? No, because we are doing i in order o replicae real-world experience. We see ha he approach o solve and characerize endogenous growh models is differen. Assume ha ou aemped a "sandard" approach, meaning, "when ou see a differenial equaion, pu i equal o zero o see wha happens". So se k h 0, and proceed from here... ou will find ha in such a case, As s n g 0. Bu his onl gives B us he condiion on he parameers under which he per capia growh rae will be zero, because b assuming k h 0 we esseniall impose a priori exacl ha, so his is no a proof ha he model is such ha i leads o zero per capia growh rae. Moreover, he goal of

8 Page 8 of 8 his model is o characerize he case where per capia magniudes grow, so i is of no real ineres o characerize he case k h 0. I.3 A noe on semanics and conceps One could argue ha he per capia growh here is no "reall" endogenous, since he savings/invesmen parameers are fixed, and do no come from an opimizaion framework. This appears a valid poin bu i is no: "endogenous" as opposed o "exogenous" means here "emerging from he producion acivi iself". And in he generalized Solow model, he endogenous growh ma come abou due o his invesmen acivi. In he exogenous growh models, growh comes from populaion growh and/or echnical progress ha exis ouside he resource consrain of he econom: he are like free gifs for which he econom need no pa or sacrifice anhing, need no devoe an resources o enjo he benefis from hem. There exis models where he growh rae of populaion and/or echnical progress become endogenous, deermined hrough an opimizing framework, and cruciall, in he conex of rade-offs dicaed b he scarci of resources: his is wha makes somehing endogenous, no wheher i is varing or fixed (and in general, separae in our minds he disincion "endogenous/exogenous" from he disincion "fixed/varing"). From anoher angle we someimes call "endogenous" wha can be decided upon. Indeed, bu in economic models decisions are aken under resource consrains. I.4 Phase diagrams and ransiional dnamics. In man models of endogenous growh, we have no ransiional dnamics. Specificall, in models wih wo pes of capial and an ineremporal opimizaion framework, we ge wo varians: if we can ransform one pe of capial o he oher, hen we have no ransiional dnamics: we fix a he beginning he raio of he wo capials a is opimal level b making wha ransformaion is necessar, and we sa here forever. If a srucural shif occurs (sa a parameer change), we do i again, and we sa here forever. In he second varian, invesmen is irreversible, i.e. we canno ransform one capial o he oher a all. In his case, here are ransiional dnamics: he opimal hing o do is o le

9 Page 9 of 8 he capial ha i is relaivel higher han opimal depreciae (zero invesmen in i) for as long as i is needed o reach he opimal capials raio (see Barro and Sala-i-Marin 2004, ch 5.. and 5..2) In he generalized Solow model wih endogenous growh, here is no opimizing framework: invesmen raes are fixed. This creaes he following issue: if he iniial socks of he wo capials do no saisf he raio required for long-run equilibrium (eq. [7]), will he model neverheless converge o he sead-sae? Isn' i possible, wih fixed and sricl invesmen raes, for i o diverge? We will show ha he sead sae of he model is globall sable, i.e. even if we sar wih an unbalanced capials raio, and even if we coninue invesing in boh capials, we will end up a he sead-sae. We will show his hrough he assumpion of a srucural shif in one of he invesmen raes: if such a hing happens, he econom (currenl assumed o be on he sead-sae given he previous se for parameer values), is now characerized b a capials raio ha i is no equal o he one implied b he new se of values. Sill, i will converge o he new one. This is equivalen o saring from a siuaion where he iniial value of he capials raio is no equal o s s. We need o consider issues of sabili of equilibrium, and relaed characerizaions of he balanced growh pah. Sabili relaes o fixed poins, and o have a fixed poin, one needs variables ha become consan in he long run. An obvious approach is o use growh raes (of per capia magniudes), and no levels of variables. Anoher is o define raios ha will be consan in he balanced growh pah, like he capials raio. In fac, we op for a combinaion: we will use he growh rae of per capia oupu, and he capials raio z k h. The evoluion of he second hrough ime will also ell us abou he growh raes of he wo capial socks. From eq. [8] we have saz ( ) saz n [0] and so

10 Page 0 of 8 ( ) s Az z ( ) s Az z 2 ( ) Az s z s z [] Also ( ) ( ) z z k h z s n s n z s Az s Az k( ) h( ) z Az s z s [2] Insering [2] ino [] we can wrie Az s z s Az s z s ( ) 2 ( ) Az sz s [3] Manipulaing eq. [0] saz ( ) saz n ( ) n Az s z s Az s z s s n s Az Az s z s Insering his in [3] we ge he ssem of differenial equaions

11 Page of 8 ( ) n s Az z Az s z s 2 [4] The zero change loci are 0 s Az n z 0 z s s [5] Noe ha ouside is zero-change locus, he oupu growh rae is everwhere declining. So we have he following phase diagram:

12 Page 2 of 8 Wha we learn from he above phase diagram is ha he oupu growh rae can approach is sead-sae value onl from above. So in an adjusmen process, if is below he new sead sae value, we expec o see i jump up, overshooing is long-erm value, and hen o decline. We urn now o consider srucural shifs in he form of an increase in he invesmen raes. I.4. An increase in he rae of invesmen in phsical capial s An increase in s moves he z 0 locus o he righ, bu leaves he oher one unaffeced, since i does no appear explicil in he relaed equaion [5]. The phase diagram in his case is The oupu growh rae jumps from poin E o poin D and hen sars o decline owards E. During he ransiion, he raio kh increases which means ha he phsical capial grows a a higher rae han he human capial.

13 Page 3 of 8 Moreover, we have he following as regards he growh raes of he wo capials: k ( ) ( ) k s n k s k k k( ) k( ) ( ) ( ) ( ) ( ) k s k h k s h k [6] k( ) k( ) Analogousl we ge k h ( ) ( ) h s n h s k h h k h h( ) h( ) ( ) () h s k h [7] k () Since k h during he ransiion, we also ge k 0 meaning ha he growh rae of phsical capial is above is new long-run value and falls, while h 0 meaning ha he growh rae of human capial is below he new sead-sae value and increases. I.4.2 An increase in he rae of invesmen in human capial s An increase in s will shif he z 0 locus o he lef, and i will also increase he slope of he 0 locus. The new fixed poin is above he previous one. The phase diagram in his case becomes

14 Page 4 of 8 The oupu growh rae jumps from poin E o poin D and hen sars o decline owards E. During he ransiion, he raio kh decreases which means ha he phsical capial grows a a lower rae han he human capial. This also means ha k 0 so he growh rae of phsical capial is below he sead sae value and increases, while also h 0, meaning ha he growh rae of human capial is above he sead sae value and decreases.

15 Page 5 of 8 II. Discree version and simulaion of he model The discree version of he model we examine is Y A s Y ( ), s Y ( ) In per capia erms, hese become Ak h ( ) s k k h n n s h n n The growh rae of per capia oupu is defined as Ak h k h Ak h k h From he laws of moion of he wo capials, we have k s k n k h s h n h

16 Page 6 of 8 So s s n k h s A k h s A k h n s Az s Az [8] n For he variable z we have z k h k s k s h s h s k h z z saz s Az [9] Finall we have h k k s A k z n h s A h z n [20] [2] These las four equaions will be used in a Dnare scrip o simulae he model.

17 Page 7 of 8 II. Dnare scrip for model simulaion The following Dnare scrip simulaes changes in he invesmen raes, expressing he model in erms of he growh raes and of he capials raio. As is, i calculaes a large increase in s from 0.25 o I also provides a single plo wih he evoluion of he hree growh raes in he same diagram. I is insrucive o also simulae a decrease in he savings rae. % Generalized Solow Model in discree ime wih endogenous growh % % % % Two permanen shocks are inroduced in order o sud % he dnamic behavior of he model % %. A shock o he invesmen rae in human capial z % 2. A shock o he invesmen rae in phsical capial x % The model is expressed in erms of growh raes and he capials raio, % so ha i has a sead-sae. var g z gk gh; % g is per capia growh rae z =k/h he capials raio. ec varexo xk xh; parameers A alpha dela n sk sh; A=; alpha=0.333; dela=0.03; n=0.0; sk=0.25; sh=0.05; model; g=(/(+n))*(((sk+xk)*a*(z(-)^(alpha-))+- dela)^(alpha))*(((sh+xh)*a*(z(-)^(alpha)) + - dela)^(-alpha)) -; z= ((sk+xk)*a*(z(-)^alpha)+(-dela)*z(-))/((sh+xh)*a*(z(-)^alpha)+(- dela)); gk = (/(+n))*((sk+xk)*a*(z(-)^(alpha-))+-dela) - ; gh = (/(+n))*((sh+xh)*a*(z(-)^(alpha)) + - dela) - ; end; inival; g=0.045; z=5; gk = 0.045; gh= 0.045;

18 Page 8 of 8 xk =0; xh =0; end; sead; endval; g=0.045; z=5; gk = 0.045; gh = 0.045; xk = 0.5; xh = 0; end; sead; check; simul(periods=500); % Ploing %subplo(2,,); plo(z(:50,)); ile('raio of Phsical o uman Capial'); %subplo(2,,2); plo(g(:30,),'displaname',''); hold all plo(gk(:30,),'displaname','k'); hold all plo (gh(:30,),'displaname','h'); ile('growh raes'); References Barro RJ and Sala-i-Marin X (2004). Economic growh (2nd ed). MIT Press. Mankiw NG, Romer D and Weil DN (992). A Conribuion o he Empirics of Economic Growh, The Quarerl Journal of Economics, 07(2): pp