Par I Growh Growh is a vas lieraure in macroeconomics, which seeks o explain some facs in he long erm behavior of economies. The curren secion is an inroducion o his subjec, and will be divided in hree secions. In he firs secion, we se forh he moivaion for he heory: he empirical regulariy which i seeks o explain. The second secion is abou exogenous growh models; ha is, models in which an exogenous change in he producion echnology resuls in income growh as a heoreical resul. Finally, he hird secion inroduces echnological change as a decision variable, and hence he growh rae becomes endogenously deermined. 1 Facs in long-run macroeconomic daa 1.1 Kaldor s sylized facs The firs five facs refer o he long run behavior of economic variables in a given economy, whereas he sixh one involves an iner-counry comparison. 1) The growh rae of oupu g y is consan over ime. 2) The capial-labor raio K L grows a a consan rae. 3) The capial-income raio K y is consan. 4) Capial and labor shares of income are consan. 5) Real raes of reurn are consan. 6) Growh raes vary across counries, persisenly. 1.2 Oher facs Besides hese classical facs, here also oher empirical regulariies which growh heory mus accoun for. These are: (i) Y L very dispersed across counries. (ii) The disribuion of Y L does no seem o spread ou (alhough he variance has increased somewha). (iii) Counries wih low incomes in 1960 did no on average show higher subsequen growh (his phenomenon is someimes referred o no absolue (β) convergence ). 1
(iv) There is, hough, condiional convergence : Wihin groups classified by 1960 human capial measures (such as schooling), 1960 savings raes, and oher indicaors, a higher iniial income y 0 (in 1960) was posiively correlaed wih a lower growh rae g y. This is sudied by performing he growh regression : g 1960 1990 y, i = α + β log y 0i + γ log edu 0i + ε i i = 1,..., n Then conrolling for he iniial level of educaion, he growh rae was negaively correlaed wih iniial income for he period 1960-1990: β < 0. Whereas if he regression is performed wihou conrolling for he level of educaion, he resul for he period is β = 0 - no absolue convergence, as menioned above. (v) (Foreign?) Trade volume seems o correlae posiively wih growh. (vi) Demographic growh ( feriliy ) is negaively correlaed wih income. (vii) Growh in facor inpus (capial K, labor L, land,...) does no suffice in explaining oupu growh. The idea of an explanaion of growh is due o Solow, who envisaged he mehod of growh accouning. Based on a neoclassical producion funcion y = z F (K, L,...) he variable z capures he idea of echnological change. If goods producion is performed using a consan reurns o scale echnology, operaed under perfec compeiion, hen (by an applicaion of he Euler Theorem) i is possible o esimae how much ou of oal producion growh is due o each producion facor, and how much o he echnological facor z. The empirical sudies have shown ha he conribuion of z (he Solow residual ) o oupu growh is very significan. (viii) Workers end o migrae ino high-income counries. 2
2 Exogenous growh In his secion we will sudy he basic framework o model oupu growh, by inroducing an exogenous change in he producion echnology, ha akes place over ime. Mahemaically, his is jus a simple modificaion of he sandard neoclassical growh model ha we have seen before (maybe we should say ha we had been sudying he sandard neoclassical no-growh model!). Two basic quesions arise, one on he echnique iself, and one on is reach. Firs, we may ask how complicaed i will be o analyze he model. The answer is quie reassuring o you: i will jus be a relaively easy ransformaion of maerial we have seen before. The second quesion is wha is he power of his model: Wha ypes of echnological change can be sudied wih hese ools? We will separae he issue of growh in wo componens. One is a echnological componen: is growh feasible wih he assumed producion echnology. The second one is he decision making aspec involved: will a cenral planner choose a growing pah? Which ypes of uiliy funcion allow wha we will call a balanced growh pah? Then his secion of he course is organized in hree secions. The firs and second ones address he echnological and decision making issues, respecively. In he hird one, we will sudy a ransformaion o he exogenous growh model ha will help us in he analysis. 2.1 The echnology of exogenous growh 2.1.1 Feasibiliy of growh Given he assumpions on he producion echnology, on he one hand, and on he source of echnological progress, on he oher, we wan o analyze wheher he sandard neoclassical growh model is really consisen wih susained oupu growh. A leas from he poin of view of he producion side: is susainable oupu growh feasible? You probably guess he answer, bu le us go ino i in deail. The sandard case is ha of labor augmening echnological change (à la Solow). The resource consrain in he economy is: c + i = F (K, n }{{} ) = F (K, γ n }{{} ) hours efficiency unis where F represens a consan reurns o scale producion echnology, and γ > 1. The capial accumulaion law is K +1 = (1 δ) K + i Susained growh is possible, given he consan reurns o scale assumpion on F. Our objec of sudy is wha is called balanced growh: all economic variables grow a consan raes (ha could vary from one variable o anoher). In his case, his would imply ha for all, he value of each variable in he model is given by: y = y 0 gy c = c 0 gc K = K 0 gk i = i 0 gi n = n 0 gn balanced growh pah all variables grow a consan raes (ha could be differen) 3
This is he analogue of a seady sae, in a model wih growh. Our ask is o find he growh rae for each variable in a balanced growh pah, and check wheher such a pah is consisen. We begin by guessing one of he growh raes, as follows. From he capial accumulaion law K +1 = (1 δ) K + i if boh i and K are o grow a a consan rae, i mus be he case ha hey boh grow a he same rae. Tha is, g K = g i mus hold. By he same ype of reasoning, from he resource consrain we mus have ha g y = g c = g i. c + i = F (K, n ) = F (K, γ n ) y Nex, using he fac ha F represens a consan reurns o scale echnology (hence i is an homogenous of degree one funcion), we have ha ( ) F (K, γ n ) = γ K n F γ, 1 n Hence, since we have posulaed ha K and y grow a a consan rae, we mus have ha K γ n = 1 in addiion, since he ime endowmen is bounded, acual hours can no growh beyond a cerain upper limi (usually normalized o 1); hence g n = 1 mus hold. Bu his resuls in g K = γ, and all oher variables also grow a rae γ. Hence, i is possible o obain consan growh for all variables: A balanced growh pah is echnologically feasible. 2.1.2 The naure of echnological change From he analysis in he previous secion, i seems naural o ask wheher he assumpion ha he echnological change is labor augmening is relevan or no. Firs, wha oher kinds of echnological change can we hink of? Le us wrie he economy s resource consrain, wih all he possible ypes of echnical progress ha he lieraure alks abou: c + γ i i = γ z F ( γ K K, γ ) n n and we have: - γ i : Invesmen-specific echnological change. You could hink of his as he relaive price of capial goods showing a long erm decreasing rend, vis a vis consumpion goods. In fac his has been measured in he daa, and in he case of he US his facor accouns for 50% of growh (??? Is his correc?). - γ z: Neural (or Hicks-neural) echnological change. - γ K : Capial augmening echnological change. - γ n: Labor augmening echnological change. 4
The quesion is which ones of hese γ s (or which combinaions of hem) can be larger han 1 on a balanced growh pah. We can immediaely see ha if F is homogeneous of degree 1 (if producion echnology exhibis consan reurns o scale) hen he γ z is redundan, since in ha case we can rewrie: γ z F ( γ K K, γ ) [ ] n n = F (γ z γ K ) K, (γ z γ n ) n As for he admissible values of he oher γ s, we have he following resul. Theorem 1 For a balanced growh pah o hold, none of he shif facors γ (excep γ n ) can be larger han 1, unless F is a Cobb-Douglass funcion. Proof. In one of he direcions, he proof requires a parial differenial equaions argumen which we shall no develop. However, we will show ha if F is a Cobb-Douglass funcion hen any of he γ can be larger han 1, wihou invalidaing a balanced growh pah as a soluion. If F is a Cobb-Douglass funcion, he resource consrain reads: c + γ i i = ( γ K K ) α ( γ n n ) 1 α Noice ha we can redefine: o rewrie he producion funcion: γ n γ α 1 α K γ n ( γ K K ) α ( γ n n ) 1 α = K α ( γ n n ) 1 α In addiion, consider he capial accumulaion equaion: K +1 = (1 δ) K + i Dividing hrough by γ i, K +1 γ +1 i We can define γ i = (1 δ) K γ i K K γ i ĩ i γ i + i γ i and, replacing K in he producion funcion, we obain: ( c + ĩ = γ K γ i K ) α ( ) γ 1 α n n K +1 γ i = (1 δ) K + ĩ The model has been ransformed ino an equivalen sysem in which K +1, insead of K +1 is he objec of choice (more on his below). Noice ha since F is Cobb-Douglass, he γ s muliplying K can in fac be wrien as labor-augmening echnological drif facors. Performing he ransformaion, he rae of efficiency labor growh is: γ i can be γ n γ α 1 α K γ α 1 α i 5
2.2 Choosing growh The nex issue o address is wheher an individual who inhabis an economy in which here is some sor of exogenous echnological progress, and in which he producion echnique is such ha susained growh is feasible, will choose a growing oupu pah or no. Iniially, Solow overlooked his issue by assuming ha capial accumulaion rule was deermined by he policy rule i = s y where he savings rae s [0, 1] was consan and exogenous. I is clear ha such a rule can be consisen wih a balanced growh pah. Then he underlying premise is ha he consumers preferences are such ha hey choose a growing pah for oupu. However, his is oo relevan an issue o be overlooked. Wha is he generaliy of his resul? Specifically, wha are he condiions on preferences for consan growh o obain? Clearly, he answer is ha no all ypes of preferences will work. We will resric aenion o he usual ime-separable preference relaions. Hence he problem faced by a cenral planner will be of he form: } max {i, c, K +1, n } { β u (c, n ) =0 s.. c + i = F (K, γ n ) K +1 = i + (1 δ) K (CP) K 0 given For his ype of preference relaions, we have he following resul: Theorem 2 Balanced growh is possible as a soluion o he cenral planner s problem (CP) if and only if (where ime endowmen is normalized o 1 as usual). u (c, n) = c1 σ v (1 n) 1 1 σ Proving he heorem is raher endeavored in one of he wo direcions of he double implicaion, because he proof involves parial differenial equaions. Also noice we say ha balanced growh is a possible soluion. The reason is ha iniial condiions also have an impac on he resuling oupu growh. The iniial sae has o be such ha he resuling model dynamics (ha may iniially involve non-consan growh) evenually lead he sysem o a balanced growh pah (consan growh). No any arbirary iniial condiions will saisfy his. Commens: 1. v (1 n) = A fis he heorem assumpions; hence non-valued leisure is consisen wih balanced growh pah. 2. Wha happens if we inroduce a sligh modificaions o u (c, n), and use a funcional form like u (c, n) = (c c)1 σ 1 1 σ? 6
c can be inerpreed as a minimum subsisence consumpion level. When c ges large wih respec o c, risk aversion decreases. Then for a low level of consumpion c, his uiliy funcion represenaion of preferences will no be consisen wih a balanced growh pah; bu, as c increases, he dynamics will end owards balanced growh. This could be an explanaion o observed growh behavior in he early sages of developmen of poor counries. 2.3 Transforming he model You should solve he exogenous growh model for a balanced growh pah. To do his, assume ha preferences are represened by he uiliy funcion c 1 σ v (1 n) 1 1 σ You should ake firs order condiions of he cenral planner s problem (CP) described above using his preference represenaion. Nex you should assume ha here is balanced growh, and show ha he implied sysem of equaions can be saisfied. Afer solving for he growh raes, he model can be ransformed ino a saionary one. We will do his for he case of labor-augmening echnology under he consan reurns o scale. The original problem is { } max β c1 σ v (1 n) 1 {i, c, K +1, n } =0 1 σ ( ) c + i = γ K n F γ, 1 n (GM) s.. K +1 = i + (1 δ) K K 0 given We know ha he balanced growh soluion o his Growh Model (GM) has all variables growing a rae γ, excep for labor. We define ransformed variables by dividing each original variable by is growh rae: ĉ = c γ î = i γ K = K γ and hus obain he ransformed model: { max β ĉ1 σ γ (1 σ) } v (1 n) 1 {î, ĉ, K +1, n } =0 1 σ s.. ( ) ) K (ĉ + î γ = γ γ n F γ, 1 n K +1 γ +1 = [î + (1 δ) K ] γ K 0 given 7
Noice ha we can wrie β ĉ1 σ γ (1 σ) v (1 n) 1 = β ĉ1 σ v (1 n) 1 1 σ 1 σ where =0 =0 β = β γ (1 σ) + =0 β 1 γ (1 σ) 1 σ Then we can simplify he γ s o ge: { max {î, ĉ, K +1, n } s.. ĉ + î = n F β ĉ1 σ v (1 n) 1 + =0 1 σ =0 ( ) K, 1 n K +1 γ = î + (1 δ) K β 1 γ (1 σ) } 1 σ (TM) K 0 given And we are back o he sandard neoclassical growh model ha we have been dealing wih before. The only differences are ha here is a γ facor in he capial accumulaion equaion, and he discoun facor is modified. We need o check he condiions for his problem o be well defined. This requires ha β γ 1 σ < 1. Recall ha γ > 1, and he usual assumpion is 0 < β < 1. Then: 1. If σ > 1, γ 1 σ < 1 so β γ 1 σ < 1 holds. 2. If 0 < σ < 1, hen for some parameer values of γ and β, we may run ino a poorly defined problem. 3. If σ > 2, hen we can afford o have β > 1! Nex we address he issue of he sysem behavior. If leisure is no valued and he producion echnology ( ) K f(k) F L, 1 + (1 δ) K L saisfies he Inada condiions: - f(0) = 0 - f ( ) > 0 - f ( ) < 0 - lim k f ( ) = 0 - lim k 0 f ( ) = Then global convergence o seady sae obains for he ransformed model (TM): lim = ĉ ĉ lim î = î lim k = k 8
Bu, consrucion of ĉ, î, and k, his is equivalen o ascerain ha he original variables c, i, and k grow a rae γ asympoically. Therefore wih he saed assumpions on preferences and on echnology, he model converges o a balanced growh pah, in which all variables grow a rae γ. This rae is exogenously deermined; i is a parameer in he model. Tha is he reason why i is called exogenous growh model. 9
3 Endogenous growh The exogenous growh framework analyzed before has a serious shorfall: growh is no ruly a resul in such model. I is an assumpion. However, we have reasons (daa) o suspec ha growh mus be a raher more complex phenomenon han his long erm produciviy shif γ, ha we have reaed as somehow inrinsic o economic aciviy. In paricular, raes of oupu growh have been very differen across counries for long periods; rying o explain his fac as merely he resul of differen γ s is no a very insighful approach. We would prefer our model o produce γ as a resul; hus, we look a endogenous growh models. Bu wha if he counries ha show smaller growh raes are sill in ransiion, and ransiion is slow? Could his be a plausible explanaion of he persisen difference in growh? Wha does our model ell us abou his? A leas locally, he rae of convergence can be found from log y log y = λ (log y log y) where λ is he eigenvalue smaller han 1 in absolue value found when linearizing he dynamics of he growh model (around he seady sae). Recall i was he roo o a second degree polynomial. The closes λ is o 1 (in absolue value), he slowes he convergence. Noice ha his equaion can be rewrien o yield he growh regression: log y log y = (1 λ) log y + (1 λ) log y + α where (1 λ) is he β parameer in he growh regressions, log y shows up as log y 0 ; (1 λ) is he γ, and log y he residual z; and finally α (usually called γ 0 ) is he inercep ha shows up whenever a echnological change drif is added. In calibraions wih reasonable uiliy and producion funcions, λ ends o become small in absolue value - hence, no large enough o explain he difference in growh raes of Korea and Chad. In general, he less curvaure he reurn funcion shows, he fases he convergence. The exreme special cases are: 1. u linear λ = 0 - immediae convergence 2. f linear λ = 1 - no convergence The more curvaure in u, he less willing consumers are o see heir consumpion paern vary over ime - and growh is a (persisen) variaion. On he oher hand, he more curvaure in f, he higher he marginal reurn on capial when he accumulaed sock is small; hence he more willing consumers are o pu up wih variaion in heir consumpion sream, since he reward is higher. 3.1 The AK model Le us recall he usual assumpions on he producion echnology in he neoclassical growh model. We had ha F was consan reurns o scale, bu also ha he per capial producion funcion f verified: - f(0) = 0 - f ( ) > 0 - f ( ) < 0 - lim x 0 f ( ) = - lim x f ( ) = 0 10
Wih he resuling shape of he global dynamics (wih a regular uiliy funcion): Long run growh is no feasible. Noice ha whenever he capial sock k exceeds he level k, hen nex period s capial will decrease: k < k. In order o allow long run growh, we need he inroduce a leas some change o he producion funcion: We mus dispose of he assumpion ha lim f ( ) = 0. f should no x cross he 45 o line. Then lim f ( ) > 0 seems necessary for coninuous growh o obain. In addiion, if we have ha x lim f ( ) = 1 (ha is, he producion funcion is asympoically parallel o he 45 o line), hen exponenial x growh is no feasible - only arihmeic growh is. Then we mus have lim f ( ) > 1 for a growh rae o be x susainable hrough ime. The simples way of having his, is assuming he producion echnology o be represened by a funcion of he form: f(k) = A k wih A > 1. More generally, for any depreciaion rae δ, we have ha he reurn on capial is (1 δ) k + f(k) = (1 δ) k + A k = (1 δ + A) k à k so he requiremen in fac is A > δ for exponenial growh o be feasible. The nex quesion is wheher he consumer will choose growh, and if so, how fas. We will answer his quesion assuming a CES uiliy funcion (needed for balanced growh), wih non-valued leisure. The planner s problem hen is: { } U = max β c1 σ {c, k +1 } =0 =0 1 σ s.. c + k +1 = A k where σ > 0. The Euler Equaion is c σ = β c σ +1 A The marginal reurn on savings is R = A, a consan. This seems like a price. Before, we he (gross) reurn on savings was R = f k + 1 δ 11
For example, in he Cobb-Douglass case i was α k 1 α which ends o 0 if k goes o infiniy. Back o he Euler Equaion, we have ha he growh rae of consumpion mus saisfy: c +1 c = (β A) 1 σ The savings rae is a funcion of (all) he parameers in uiliy and producion - he reurn funcion. Noice ha his implies ha he growh rae is consan as from = 0. There are no ransiional dynamics in his model; he economy is in he balanced growh pah from he sar. There will be long run growh provided ha (β A) 1 σ > 1 (PC I) This does no quie sele he problem, hough: an issue remains o be addressed. If he parameer values saisfy he condiion for growh: is uiliy sill bounded? Are he ransversaliy condiions me? We mus evaluae he opimal pah using he uiliy funcion: So he necessary condiion is ha U = [ β =0 [ ] ] (β A) 1 1 σ σ c1 σ 0 1 σ [ ] β (β α) 1 1 σ σ < 1 (PC II) The wo Parameer Condiions (PC I) and (PC II) mus simulaneously hold for a balanced growh pah o obain. Remark 3 (Disorionary axes and growh) Noice ha he compeiive allocaion in his problem equals he cenral planner s (Why?). Now suppose ha we have he governmen levy a disorionary ax on (per capia) capial income and use he proceeds o finance a lump-sum ransfer. Then he consumer s decenralized problem has he following budge consrain: c + k +1 = (1 τ k ) R k + τ while he governmen s budge consrain requires ha τ k R k = τ This problem is a lile more endeavored o solve due o he presence of he lump-sum ransfers τ. Nowihsanding his, you should know ha τ k (he disorionary ax on capial income) will affec he long run growh rae. Remark 4 (Explanaory power) Is he model realisic? Le us address his issue by saing he assumpions and he resuls obained: Assumpions The A K producion funcion could be inerpreed as a special case of he Cobb-Douglass funcion wih α = 1 - hen labor is no producive. However, his conradics acual daa, ha shows ha labor is a hugely significan componen of facor inpu. Clearly, in pracise labor is imporan. Bu his is no capured by he assumed producion echnology. We could imagine a model where labor becomes unproducive; for example assuming ha F (K, n ) = A K α n 1 α hen if lim α = 1, we have asympoic lineariy in capial. Bu his is unrealisic. 12
Resuls The growh has become a funcion of underlying parameers in he economy, affecing preferences and producion. Could hen he dispersion in cross-counry growh raes be explained by differences in hese parameers? Counry i s Euler Equaion (wih a disorionary ax on capial income) would be: ( ) c+1 = [ β i A i (1 τ i k)] 1 σ i c i Bu he problem wih he AK model is ha, if parameers are calibraed o mimic he daa s dispersion in growh raes, he simulaion resuls in oo much divergence in oupu level. The dispersion in 1960-1990 growh raes would resul in a difference in oupu levels wider han he acual. Remark 5 (Transiional dynamics) The AK model implies no ransiional dynamics. However, we end o see ransiional dynamics in he daa (recall he condiional convergence resul in growh regressions). 3.2 Romer s exernaliy model The inellecual preceden o his model is Arrow s learning by doing paper in he 1960s. The basic idea is ha here are exernaliies o capial accumulaion, so ha individual savers do no realize he full reurn on heir invesmen. Each individual firm operaes he following producion funcion: F ( K, L, K ) = A K α L 1 α K ρ where K is he capial operaed by he firm, and K is he aggregae capial sock in he economy. We assume ha ρ = 1 α; so ha in fac a cenral planner faces an AK-model decision problem. Noice ha if we assumed ha α + ρ > 1, hen balanced growh pah would no be possible. The compeiive equilibrium will involve a wage rae equal o: w = (1 α) A K α L α K 1 α Le us assume ha leisure is no valued and normalize he labor endowmen L 1 in every. Assume ha here is a measure 1 of represenaive firms, so ha he equilibrium wage mus saisfy w = (1 α) A K So noice ha in his model, wage increases whenever here is growh, and he wage as a fracion of oal oupu is subsanial. The renal rae, meanwhile, is given by: R = α A The consumer s decenralized Euler Equaion will be, assuming CES uiliy: c +1 c = [β (R +1 + 1 δ)] 1 σ So, subsiuing for he renal rae, we can see ha he rae of change in consumpion is given by: g CE c = [β (α A + 1 δ)] 1 σ On he oher hand, i is immediae ha, since a planner faces an AK model, his chosen growh rae would be: g CP c = [β (A + 1 δ)] 1 σ Then gc CP > gc CE ; a compeiive equilibrium implemens a lower han opimal growh rae, consisenly wih he presence of exernaliies o capial accumulaion. 13
Remark 6 (Pros and cons of his approach) The following advanages and disadvanages of his model can be highlighed: + Overcomes he labor irrelevan shorfall of he AK model. There is lile evidence in suppor of a significan exernaliy o capial accumulaion. Noice ha if we agreed for example ha α = 1/3, hen he exernaliy effec would be immense. The model leads o a large divergence in oupu levels, jus as he AK. 3.3 Lucas human capial accumulaion model In he Lucas model, plain labor in he producion funcion is replaced by human capial. This can be accumulaed, so he echnology does no run ino decreasing marginal reurns. For example, in he Cobb- Douglass case, we have: F (K, H) = A K α H 1 α There are wo disinc capial accumulaion equaions: H +1 = ( 1 δ H) H + I H K +1 = ( 1 δ K) K + I K And he resource consrain in he economy reads: c + I H + I K = A K α H 1 α Noice ha, in fac, here are wo asses: H and K. Bu here is no uncerainy; hence one is redundan. The reurn on boh asses mus be equal. Unlike he previous model, in he curren seup a compeiive equilibrium does implemen he cenral planner s soluion (why can we say so?). The firs order condiions in he laer read: c : β c σ = λ [ ] K +1 : λ = λ +1 1 δ K + F K (K +1, H +1 ) [ ] H +1 : λ = λ +1 1 δ H + F H (K +1, H +1 ) Which lead o wo equivalen insances of he Euler Equaion: c +1 c = c +1 c = ( [ ( )]) 1 β 1 δ K K+1 σ + F K, 1 H +1 ( [ ( )]) 1 β 1 δ H K+1 σ + F H, 1 H +1 (EE-K) (EE-H) Noice ha if he raio K +1 H +1 remains consan hrough ime, his delivers balanced growh. Le us denoe x K H and we have ha: 1 δ K + F K (x, 1) = 1 δ H + F H (x, 1) Bu hen equilibrium in he asse marke requires ha x = x be consan for all ; and x will depend on δ H, δ K, and parameers of he producion funcion F. 14
Example 7 Assume ha δ H = δ K, and F (K, H) = A K α H 1 α. Then (EE-K) = (EE-H) requires ha: So α A x α 1 = (1 α) A x α x = α 1 α = K H From = 1 onwards, K = x H - i is like having he capial sock on he one side, and he sae variable on he oher. Then A K α H 1 α = A (x H ) α H 1 α = à H =  K where à A xα, and  A x1 α. In any case, his reduces o an AK model! Remark 8 (Pros and cons of his approach) We can highligh he following advanages and disadvanages of his model: + Labor is reaed seriously, and no resoring o ricks like exernaliies. The law of moion of human capial is oo mechanic-like : ( H +1 = 1 δ H) H + I H Arguably, knowledge migh be bounded above a some poin. On he oher hand, his issue could be couner-argued by saying ha H should be inerpreed as general formaion (such as on-he-job raining, eceera), and no narrowly as schooling. This model implies divergence of oupu levels; i is an AK model in essence, as well. 3.4 Romer s qualiaive echnological change 3.4.1 The model Based on he Cobb-Douglass producion funcion F (K, L) = A K α L 1 α, his model seeks o make A endogenous. One possible way of modeling his would be simply o make firms chose he inpus knowing ha his will affec A. However, if A is increasing in K and L, his would lead o increasing reurns, since A (λ K, λ L) (λ K) α (λ L) 1 α > λ A K α L 1 α An alernaive approach would have A being he resul of an exernal effec of firm s decisions. Bu he problem wih his approach is ha we wan A o be somebody s choice; hence an exernaliy will no work. The way ou of his A dilemma is o lif he assumpion of perfec compeiion in he economy. 15
In he model o be presened, A will represen variey in producion inpus. The larger A, he wider he range of available producion (inermediae) goods. Specifically, in his economy capial and consumpion goods are produced according o he funcion A y = L β 0 x 1 β (i) di Where i is he ype of inermediae goods, and x (i) is he amoun of good i used in producion a dae. Therefore, here is a measure A of differen inermediae goods. You may noice ha he producion funcion exhibis consan reurns o scale. The inermediae goods x (i) are produced wih capial goods using a linear echnology: A 0 η x (i) di = K Tha is, η unis of capial are required o produce 1 uni of inermediae good of ype i, for all i. The law of moion and resource consrain in his economy are he usual: K +1 = (1 δ) K + I c + I = y We will assume ha an amoun L 1 of labor is supplied o he goods producion secor a ime. In addiion, we assume ha A grows a rae γ: A +1 = γ A Q: Given his growh in A, is long run oupu growh feasible? A: A key issue o answer his quesion is o deermine he allocaion of capial among he differen ypes of inermediae goods. Noice ha his decision is of a saic naure: he choice a has no (dynamic) consequences on he fuure periods sae. So he producion maximizing problem is o: max x(i) { L β 1 A x 1 β 0 } (i) di s.. A 0 η x (i) di = K Since he objecive funcion is concave, he opimal choice has x (i) = x for all i. This oucome can be inerpreed as a preference owards variey - as much variey as possible is chosen. Replacing he opimal soluion in he consrain: A 0 η x di = K A x η = K ((I)) And maximized producion is: A y = L β x 1 β di 0 = L β A x 1 β ((II)) 16
Using (I) in (II), ( ) 1 β y = L β 1 A K η A = Lβ 1 η 1 β Aβ K 1 β Clearly A β grows if A grows a rae γ. Then he producion funcion is linear in he growing erms. Therefore he answer o our quesion is Yes: A balanced growh pah is feasible; wih K, y and A growing a rae γ. So he nex issue is how o deermine γ, since we are dealing wih an endogenous growh model. We will make he following assumpion on he moion equaion for A : A +1 = A + L 2 δ A Where L 2 denoes labor effor in research and developmen, and L 2 δ is he number of new blueprins ha are developed a ime, as a consequence of his R&D. This moion equaion resembles a learning by doing effec. Exercise: Assume ha leisure is no valued, and oal ime endowmen is normalized o 1. Then he amoun of labor effor allocaed o he producion and o he R&D secors mus saisfy he consrain: L 1 + L 2 = 1 Assume ha here is an individual who consumes he consumpion goods c. Solve he planning problem o obain γ. 3.4.2 The decenralized problem We will work wih he decenralized problem. We assume ha here is perfec compeiion in he final oupu indusry. Then a firm in ha indusry solves a ime : { A } A max L β 1 x 1 β (i) di w L 1 q (i) x (i) di x (i), L 1 0 0 Noice ha he firm s is a saic problem. w and q (i) are aken as given. Equilibrium in he final goods marke hen requires ha hese are: A w = β L 1 β 1 x 1 β (i) di 0 q (i) = (1 β) L β 1 x β (i) (( )) As for he inermediae goods indusry, insead of perfec, we will assume ha here is monopolisic compeiion. There is only one firm per ype i (a paen holder). Each paen holder akes he demand funcion for is produc as given. Noice ha ( ) is jus he inverse of his demand funcion. All oher relevan prices are also aken as given. In paricular, he renal rae R, paid for he capial ha is rened o consumers. Then he owner of paen i solves: { π (i) = max q (i) x (i) R K} i K i s.. x (i) η = K i 17
or equivalenly, using ( ) and he consrain, { π (i) = max K i (1 β) L β 1 The firs order condiions for his problem are: (1 β) 2 L β 1 η1 β K i β ( ) K i 1 β R K i η π (i) > 0 is admissible! The firm owns a paen, and obains a ren from i. However, his paen is no cos free. I is produced by R&D firms, who sell hem o inermediae goods producers. Le p P denoe he price of a paen a ime. = R } Then ideas producers solve: { } max p P (A +1 A ) w L 2 A +1, L 2 s.. A +1 = A + L 2 δ A And we will assume ha here is free enry in he ideas indusry: hence, here mus be zero profis from engaging in research and developmen. Noice ha here is an exernaliy (someimes called sanding on he shoulders of gians ). The reason is ha he decision involving he change in A, A +1 A, affecs fuure producion via he erm δ A +j in he equaion of moion for A. Bu his effec is no realized by he firm who chooses he change in A. Hence his is a second reason why he planner s and he decenralized problems will have differen soluions (he firs one was he monopoly power of paen holders). The zero profi condiion in he ideas indusry requires ha he price p P order condiion δ A = w where w is as deermined in he marke for final goods. p P be deermined from he firs Once his is solved, if p C denoes he dae-0 price of consumpion (final) goods a, hen we mus have p P p C = s=+1 π s (i) p C s As a resul, nobody makes profis in equilibrium. The invenors of paens appropriae he exraordinary rens ha inermediae goods producers are going o obain from purchasing he righs on he invenion. 3.4.3 Balanced growh pah Nex we solve for a (symmeric) balanced growh pah. We assume ha all variables grow a (he same, and) consan raes: K +1 = γ K A +1 = γ A c +1 = γ c L 1 = L 1 L 2 = L 2 w +1 = γ w 18
Wih respec o he inermediae goods x (i), we already know ha an equal amoun of each ype of hem is produced each period: x (i) = x. In addiion, we have ha his amoun mus saisfy: A η x = K Since boh A and K (are assumed o) grow a rae γ, hen x mus remain consan for his equaion o hold for every. Hence, x = x = K A η Then he remaining variables in he model mus remain consan as well: I is up o you o solve his problem: R = R π (i) = π p P = p P q (i) = q Exercise: Wrie down a sysem of n equaions and n unknowns deermining γ, L 1, L 2, eceera. Afer ha, compare o he planner s growh rae γ which you have already found. Which one is higher? 19
4 Concluding Remarks 4.1 Dealing wih convergence One of he key elemens o es he explanaory power of boh he exogenous and he endogenous growh models is heir implicaions respecing convergence of growh raes among differen counries. Recall ha: Exogenous growh vs Endogenous growh A K α L 1 α does no lead o divergence A K leads o divergence in relaive income levels Is i possible o solve he riddle hrough appropriae calibraion? Using α = 1/3, he exogenous growh framework leads o oo fas convergence. A brillian soluion is o se α = 2/3. The closer o 1 α is se, he closer he exogenous growh model looks like he AK model. Bu we are no so free o play around wih α. This parameer can be measured from he daa: α = K F K y = K R y A possible soluion o his problem is o inroduce a mysery capial, so ha he producion funcion looks like: y = A K α L β S 1 α β Or, alernaively inroduce human capial as a hird producion facor, besides physical capial and labor: y = A K α L β H 1 α β 4.2 Dealing wih reurns on invesmen We will explore he argumen developed by Lucas [] o sudy he implicaions of he growh model in crosscounry differences in raes of reurn on invesmen. This will allow us o sudy how acual daa can be used o es implicaions of heoreical models. There is a significan assumpion made by Lucas: Suppose ha i was possible o expor US producion echnology (or know how ) o oher counries. Then he producion funcion, boh domesically and abroad, would be y = A K α L 1 α wih a differen level of K and L in each counry, bu he same A, α, and capial depreciaion level δ. Then imagine a less developed counry whose annual (per capia) oupu is a sevenh of he US oupu: y LDC y US = 1 7 Using per capia variables (L US = L LDC = 1), he marginal reurn on capial invesmen in he US is calculaed as: R US = α A K α 1 US δ and he parameers α and δ ake values of 1/3 and.1, respecively. 20
The ne reurn on capial in he US can be esimaed o be 6.5% per annum, so he gross rae is: R US = 1.065 Manipulaing he Cobb-Douglass expression a lile, α A K α 1 US = α A Kα US K US = α yus K US Now, wha is he reurn on capial in he less developed counry? R LDC = α yldc K LCD δ We have ha So 7 = y US y LDC = A Kα US A K α LDC = ( KUS K LDC ) α y LDC = 7 1 y US K LDC 7 1 α K US y LDC = 7 1 α α yus K LDC K US and, using α = 1/3, Then We know from he daa ha Therefore, y LDC K LDC = 7 2 yus K US 1.065 = 1 3 yus K US.1 y US K US =.495 y LDC K LDC = 49 yus = 49.495 K US = 24.255 Which implies ha he (gross) rae of reurn on capial in he less developed counry should be: R LDC = 1 24.255.1 = 7.985 3 This is saying ha if he US producion echniques could be exacly replicaed in less developed counries, he ne reurn on invesmen would be 698.5%. This resul is sriking since if his is he case, hen capial should be massively moving ou of he US and ino less developed counries. Of course, his riddle migh disappear if only we le A LDC < A US. 21
5 References Lucas, Rober E. Jr., Why doesn capial flow from rich counries o poor counries?, 22