Biased Technological Change, Human Capital. and Factor Shares.

Transcription

1 Biased Technological Change, Human Capial and Facor Shares. Hernando Zulea 1 April 26, I am graeful o Venea Andonova and Andrés Zambrano for commens and suggesions.

2 Absrac We propose a one-good model where echnological change is facor saving and cosly. We consider a producion funcion wih wo reproducible facors: physical capial and human capial, and one no reproducible facor. The main predicions of he model are he following: (a) The elasiciy of oupu wih respec o he reproducible facors depends on he facor abundance of he economies. (b) The income share of reproducible facors increases wih he sage of developmen. (c) Depending on he iniial condiions, in some economies he producion funcion converges o AK, while in oher economies long-run growh is zero. (d) The share of human facors (raw labor and human capial) converges o a posiive number lower han one. Along he ransiion i may decrease, increase or remain consan. Journal of Economic Lieraure classi caion: 011, 031, 033. Keywords: endogenous growh, human capial, facor using and facor saving innovaions, facor income shares.

3 1 Inroducion The works by Cobb and Douglas (1928) and Kaldor (1961) creaed a paradigm for macroeconomics. The idea ha labor income share does no decrease or increase wih ime or wih he sage of developmen have had imporan implicaions in macroeconomics and growh heory. Considering an aggregae producion funcion, if facor income shares are consan and he price of each facor is deermined by is marginal produciviy hen he elasiciy of oupu wih respec o each facor is also consan. In oher words he consancy of facor shares implies ha he Cobb-Douglas is a good approximaion for he aggregae producion funcion. Subscribing o his paradigm, almos all of he lieraure on economic growh accouning assumes ha he elasiciy of oupu wih respec o capial (and labor) is consan (see Easerly and Levine, 2002; Young, 1994 and Solow, 1957, among ohers). However, economic growh models of biased innovaions predic a posiive correlaion beween capial abundance and capial income share 1. Indeed, if facor prices are deermined by marginal produciviy of facors hen labor saving innovaions reduce he income share of workers and increase he capial income share. In more general erms, he income share of no reproducible facors decreases wih he sage of developmen while he income share of reproducible facors increases. Therefore, he heoreical argumen described above implies ha he income share of capial should be posiively correlaed 1 Sudies of facor saving innovaions and economic growh have consanly grown in number. Some examples are Zeira (1998), Pereo and Seaer (2006) and Zulea (2007), among ohers. 1

4 wih he sage of developmen. 2 We propose a model of facor saving innovaions o explain he absence of a secular rend in labor shares. We consider a model wih wo reproducible facors: human capial (H) and physical capial (K) and one no reproducible facor (L). Assuming ha any echnology can be obained paying a cos, capial abundan economies have more incenives o adop capial inensive echnologies. In he same way, in counries where he capial inensiy of he echnology is higher agens have more incenives o save. This produces a viruous circle driving capial abundan economies o long run growh. We consider a one good economy and assume a se of Cobb-Douglas producion funcions of he form Y = AK H L 1. Thus we can rewrie he producion funcion de ning y as oupu per uni of no reproducible facor, y = Y L = Ak h L can be undersood as a combinaion of no reproducible facors, i.e., raw labor and land. Therefore an increase in () is physical capial-using (human capial-using), labor saving and land saving echnological change. The main resuls of he model are he following: (i) The elasiciy of oupu wih respec o reproducible facors depends on he facor abundance of he economies. (ii) The income share of reproducible facors increases wih he sage 2 There are wo oher heoreical reasons why he elasiciy of oupu wih respec o reproducible facors, namely, physical capial and human capial, should be posiively correlaed wih he sage of developmen: he Hecksher-Ohlin heory of Inernaional Trade and he heory of Inernaional Capial Flows. 2

5 of developmen. (iii) Depending on he iniial condiions, in some economies he producion funcion converges o AK while in oher economies long-run growh is zero. (iv) The income share of reproducible facors converges o a posiive number lower han one and along he ransiion i may decrease, increase or remain consan. Three pieces of empirical evidence moivae our work: 1. In he eld of empirical economic growh, Durlauf and Johnson (1995) and Du y and Papageorgiou (2000) nd ha as economies grow heir echnologies become more inensive in reproducible facors, ha is, he elasiciy of oupu wih respec o reproducible facors is higher in rich economies. 2. Wih regard o he behavior of facor income shares, we know ha: (i) In developed counries he share of agriculure in oal oupu is usually smaller han i is in developing counries. By he same oken, he share of agriculure in oal oupu is reduced as economies grow. Since land is a major inpu in agriculure bu no in oher secors hese facs sugges ha land income share may decrease wih he sage of developmen. Consisenly, from 1870 o 1990 in he Unied Saes he share of land in Ne Naional Produc has been coninuously reduced (see Rhee, 1991 and Hansen and Presco, 2002). (ii) Over he pas 60 years, he US relaive supply of skilled work has increased rapidly. However, here has no been a downward rend for he reurns o college educaion. On he conrary, over his period, he college premium has increased (see Krueger, 3

6 1999; Krusell e.al., 2000 and Acemoglu 2002). Moreover, according o Goshalk (1997), real wages of unskilled labor signi canly fell during he pas few decades (iii) In addiion, i has been argued ha labor income share does no decrease or increase wih developmen (Gollin, 2002). However, he sandard measure of labor income share includes skilled and unskilled labor income share, ha is, i includes human capial. In he same way, he sandard measure of capial income share includes land income share. Therefore, i seems ha he income share of no reproducible facors (land and unskilled labor) has decreased, while he income share of reproducible facors has increased during he 20h cenury. 3. Blanchard (1998) noes ha since he early 80 s a dramaic decrease in labor income share has occurred in Europe (5 o 10 percenage poins of GDP) and suggess ha his decline could be explained by non-neural changes in echnology. There are several models of biased echnological change where labor share are relaively consan. 3 Boldrin and Levin (2002) build a model where he producion funcion is Leonief so, facor prices are deermined by opporuniy cos and no by marginal produciviy. Zulea and Young (2006) and Zeira (2006) consider models wih wo nal goods and assume ha labor saving innovaion can only be implemened in he capial inensive secor. In his seing, he e ec of labor saving innovaions is compensaed in he aggregae wih he in- 3 Kennedy (1964), Samuelson (1965) and Drandakis and Phelps (1966) are pioneers in his lieraure. 4

7 crease in he share of he labor inensive secor. 4 To he bes of our knowledge his is he rs model where physical capial using innovaions are accompanied wih human capial using innovaions in such a way ha he labor income share does no presen a clear decreasing rend. In hen nex secion we presen he model. Then we conclude in he las secion. 2 The Model 2.1 The cos of changing echnology We assume ha here are di eren qualiies of physical and human capial: Any ype of physical capial embodies a echnology (, :) Capial ypes ha embody more capial inensive echnologies are more cosly. In paricular, we assume ha for 1 uni of oupu devoed o build capial goods of ype ; he number of capial goods is given by K ; = 1 + [ln( )] + [ln( )] where is a measure for scale, and are echnological parameers such ha 1 and 1: We also assume ha each plan can only operae wih one echnology so, only one ype of capial is used a a ime and we can drop he subindex ; : For simpliciy, we choose no reproducible facors as a measure of scale, so if 4 While his paper examines he dynamics of labor share in response o echnical change, various heoreical and empirical research has explored he non-echnical deerminans of labor share, e.g., Gomme and Greenwood (1995) and Boldrin and Horvah (1995) (unemploymen insurance/labor conracs); Blanchard (1997), Blanchard and Wolfers (2000), Benolila and Sain-Paul (2003) and Kessing (2003) (labor adjusmen coss and bargaining power); Berola (1993) ( scal policy); and Ambler and Cardia (1998) (monopolisic compeiion). 5

8 L i is he amoun of no reproducible facors used by he rm i hen for 1 uni of oupu devoed o build capial goods of ype ; he sock of capial is given by 5, K i = 1 + L i [ln( i )] + L i [ln( i )] Following Barro and Sala-i-Marin (1995) we also assume ha physical and human capial are produced wih he same echnology. 6 Therefore, he oal amoun of asses of he rm a i can be wrien as a i = K i + H i L i [ln( )] L i [ln( )] (1) De ning a K;i = K i L i [ln( i )] and a H;i = H i L i [ln( i )] he oupu produced by a rm i using K unis of capial of ype i ; i can be wrien as Y i = A(a K + [ln( i )] L i ) i (a H + [ln( i )] L i ) i L 1 i i i (2) In oher words, we assume ha he producion echnology in each period is chosen from a ime invarian se of echnologies where a more superior echnology also enails larger coss on acquiring capial. We are aware of he fac ha new hings are invened each year and many of he echnologies we use nowadays were recenly creaed. However, he aim of he paper is no o ex- 5 This funcion is chosen because of is racabiliy and he main resuls of he model do no depend on such an assumpion. See Zeira (2005) or Pereo and Seaer (2006) for di eren coss funcions. 6 This simplifying assumpion does no a ec he resuls of he model. 6

9 plain he process of creaion of new echnologies bu o explain (i) why some socieies adop capial inensive echniques of producion while ohers use labor inensive echnologies and (ii) wha are he e ecs of hese di erences in erms of funcional disribuion of income and economic growh. 7 Markes are compeiive so rms choose he echnology in order o maximize oupu, max i; i Y i s: i 0 and i 0: As a resul, in he inerior soluion, he echnology is given by (complee derivaion in he Appendix 4.1), i = K i L i ln 1 + Ki L i ln K i L i K i L i and i = H i L i ln 1 + Hi L i ln H i L i H i L i Noe ha, holding he res consan, any increase in he size of he rm a ecs K i ; H i and L i in he same proporions, so he equilibrium levels of and are independen of he size of he rm. If all rms use he same echnology and face he same marke prices hen for any pair of rms i and j; Ki L i K L = Ki L j = K L ; where is capial per uni of no reproducible facors. Similarly, for any pair of rms i and j; Hi L i facors. = Hj L j = H L ; where H L is human capial per uni of no reproducible 7 An exension of he model where echnologies are creaed wih a probabiliy p(s) such ha 0 1 and p 0 (s) > 0 would deliver he same predicions of he model we are presening. 7

10 Therefore, he equilibrium echnology (common for every rm) is given by, k ln k = max 0 ; k ln k + 1 and = max 0 ; h ln h h ln h + 1 Noe ha under his seing he decenralized soluion is no opimal. Indeed, if he producion funcion is Cobb-Douglas and markes are compeiive hen he wage is given by w = (1 )Ak : Therefore, if make zero pro s, hen he ineres rae mus be lower han he marginal produciviy of asses because rms mus pay for he echnology. This fac may have ineresing implicaions for opimal axaion. However, we wan o focus on he e ecs ha facor saving innovaions have on he behavior of facors shares and he simples way o presen his idea is solving he planner problem. 2.2 The dynamic problem. For he dynamic problem we assume consan populaion, homogenous agens, in nie horizon and logarihmic uiliy funcion. To obain he dynamic resricions of he problem we di ereniae equaion 1, _a K + _a H = _a = _ k + _ h + _ + _ 8

11 Therefore, he resricions can be expressed in he following way: _k = u Ak h _h = a Ak h _ = b ( ) _ = z ( ) c c Ak Ak h h c c (3) (4) (5) (6) where u is he share of savings devoed o increase he sock of physical capial, a is he share of savings devoed o increase human capial, b is he share of savings devoed o human capial using echnological changes; z is he share of savings devoed o physical capial using echnologies changes and u + a + b + z = 1: The planner problem is he sandard one: maximize he presen discouned uiliy of he represenaive agen subjec o he resricions, Z Max log c e d s:: (3), (4), (5), (6) 0 0 (7) 9

12 and he ransversaliy condiions are e lim k = 0!1 c lim!1 e c h = 0 where 0 and 0 are he iniial echnologies. From he rs order condiions i follows ha for he inerior soluion, _c = Ak 1 c h = Ak h 1 (8) and k ln k = max 0 ; k ln k + 1 h ln h = max 0 ; h ln h + 1 (9) (10) From equaion 8 i follows ha, in he inerior soluion, k h = (11) Therefore, he producion funcion can be rewrien as A( ) k + he opimal consumpion growh rae is and _c c = ( ) 1 ( ) Ak + 1 (12) 10

13 Equaion 12 is he consumpion growh rae of any model where physical and human capial are produced wih he same echnology (see Barro and Salai-Marin, 1995). However, in our seing and are variables no parameers, and grow wih he economy. For his reason he consumpion growh rae may no decrease as he economy accumulaes asses. Indeed, he growh rae of consumpion is increasing whenever k 1 (he proof is presened in he Appendix 4.3). From equaions 9, 10 and 11, in he inerior soluion he variables h, and can be expressed as funcions of k h = h(k ); = (k ); = (k ) where h 0 (k ) 0; 0 (k ) 0 and 0 (k ) 0: Therefore, we can de ne k min as follows: De niion 1 k min is he capial per uni of no reproducible facors such ha he opimal consumpion growh rae is zero, ha is, ((k min )) 1 (kmin) ((k min )) (kmin) k (kmin)+(kmin) 1 = A : Now, given ha he growh rae of consumpion is increasing whenever k 1 and h 0 (k ) 0; 0 (k ) 0 and 0 (k ) 0; from de niion 1 i follows ha: 1. If k > k min hen he opimal consumpion growh rae is posiive. 2. If k < k min hen he opimal consumpion growh rae is negaive. Therefore, for economies where reproducible facors are relaively scarce i can be opimal o consume he enire oupu. On he oher hand, if he share 11

14 of reproducible facors is high hen consumpion growh rae is posiive. As we show below, wo candidaes for opimal pah may arise: one wih a small amoun of reproducible facors in he long-run and anoher wih an in nie amoun of reproducible facors. 2.3 Long run There are wo basic ypes of long-run equilibria ha he model can suppor. Firs, here is a neoclassical seady-sae where = 0 ; = 0 and _c c = _ k k = _h h = 0: The second one is a Balanced Growh Pah (BGP) where + = 1 and _c c = k _ k = h _ h = 0: Seady Sae Consider an iniial echnology 0 ; 0 such ha ( 0 ) 1 0 ( 0 ) 0 < A and ( 0) 0 ( 0 ) 1 0 < A In his case, he seady sae physical and human capial per uni of no reproducible facors are k =! 1 ( 0 ) 1 0 ( 0 ) 1 0 A 0 0 h = (0 ) 0 ( 0 ) 1 0 A Noe ha k < 1 and h < 1. Therefore, from equaions 9 and 10 i follows ha 12

15 if k k and h h hen = 0 and = 0 : Therefore, economies ha are abundan in no reproduclible facors are likely o converge o a seady sae Balanced Growh Pah From equaions 9, 10 and 12 i follows ha, _c lim = ( ) 1 ( ) c Ak + 1 (13) k!1 Form equaions 9 and 13 i follows ha: 1. If + 1 and ( ) 1 ( ) A < 2 hen he problem has a nie value soluion (proof in he Appendix 4.4) 2. If + 1 and ( ) 1 ( ) > A hen capial abundan economies, ha is economies where k > k min, converge o a BGP (proof in he Appendix 4.5). From resuls 1 and 2 i follows ha if + = 1 hen he dynamic problem can be solved and capial abundan economies converge o a Balanced Growh Pah while poor economies converge o a seady sae. Now, in he Balanced Growh Pah he growh rae of human and physical capial mus be he same, _k h k = _ h ; so u Ak + 1 c k! = a 1 Ak + 1! c k 13

16 and u a = Using he Firs Order Condiions we nd ha lim (a + u ) = 1 (proof is in he Appendix 4.2) so, k!1 lim b = lim z = 0; so k!1 h!1 lim u = and lim a = k!1 h!1 Therefore, in he Balanced Growh Pah, c c = lim k k!1k = c c = lim h h!1h = (14) (15) Noe ha equaions 14 and 15 imply ha lim c k!1 h+k = : Therefore in he BGP he raio consumpion asses is equal o he discoun rae as derived in several endogenous growh models Transiion We already showed ha depending on he parameers ; ; A and here can be long run growh. Similarly, depending on he iniial condiions some economies can be rapped in a seady sae. In his secion assume + = 1 and characerize he behavior of some of he main variables along he ransiion and idenify he condiions under which an economy converges o a seady sae 14

17 and he condiions under which i converges o a Balanced Growh Pah. Using equaions 3 and 12 we can characerize he behavior of he consumpion capial raio c k, _c c k _ = ( ) 1 ( k ) Ak + 1 u Ak + 1 c k (16) From equaion 16 i follows ha _ k k _k k : If his is he case hen c k > u + A Bu noe ha A 1 ( ) 1 ( ) accumulaes capial, so u + A 1 _c c. To see why suppose ha _c 1 ( ) 1 ( ) u c > so c k > c k : decreases and u grows as he economy decreases as he economy ( ) 1 ( ) u grows. Therefore, if _c c > _ k k hen _c c k _ k grows wih ime and he economy canno converge o a BGP. Moreover, if which is no feasible. Proposiion 2 De ne ~ k = _c c > _ k k k 1 min ( 0) 1 0 ( 0 ) 0 hen c k k min hen he economy converges opimally o a seady sae. converges o in niy , if k0 < ~ k and k 0 < Proof. Suppose no, ha is, here exiss a k 0 such ha k 0 < k; ~ k 0 < k min and he economy presens long-run growh. 1. In order o have capial accumulaion or echnological change consumpion mus saisfy c 0 < Y 0 : 2. In he inerior soluion, he consumpion-capial raio decreases as he sock of capial grows and converges o as capial goes o in niy. Moreover as long as k < k m he growh rae of consumpion is negaive. Therefore, for any such ha k k m i mus be rue ha c < c 0 : 15

18 3. Since he consumpion-capial raio decreases wih ime and converges o in he long-run hen in he opimal pah c > k for any < 1: From 2 and 3 i follows ha given k 0 ; if here is an opimal pah wih longrun growh hen c 0 > k min: From 1, 2 and 3 i follows ha oupu a period zero mus be higher han k min; namely, ( 0 ) 1 0 ( 0 ) 0 Ak > k min so k 0 >! k min ( 0 ) 1 0 ( 0 ) 0 which conradics k 0 < ~ k Proposiion 3 If k 0 > k m hen he economy presens long-run growh. The proof is sraighforward. When k > k m he marginal produciviy of savings is higher han he discoun rae. Therefore, savings are used o increase K; H; and and he consumpion growh rae is posiive. 2.4 The Behavior of Facor Shares In he previous secions we describe he main resuls of he model and characerize he long run equilibrium. We nd ha he share of reproducible facors depends on he relaive abundance of he facors. In his secion we explore he possible equilibrium pahs of he labor income share. As we saed before, he sandard measure of labor income includes raw labor income and human capial income and he sandard measure of capial income includes physical 16

19 capial income and land income. In order o describe he behavior of he sandard measures of facor shares we need o rewrie he producion funcion in he following way Y = AK H N l 1 where N is land, l is raw labor, is land income share, 1 is raw labor income share and N l 1 = L 1 : Therefore, he share of human facors is 1 and he share of non human facors is + : From secion 2 we obain he growh raes of and : Di ereniaing equaions 9 and 10 we nd _ 1 + ln k _k = ( ) ln k k (17) _ = ( ) 1 + ln h _h h ln h + 1 h (18) Therefore, as long as he economy accumulaes asses he share of physical and human capial in he producion funcion ( and ) grows. This implies ha along he ransiion he economy underakes echnological changes ha are human capial using, physical capial using, raw labor saving and land saving. Under hese condiions he share of human facors (1 ) remains consan if labor saving innovaions are always human capial using and land saving innovaions are always physical capial using, ha is, _ = _: In real life, innovaions can be physical capial using and raw labor saving or human capial using and land saving. Indeed, human facors shares move up and down (see Benolila and Sain Paul, 2003 or Young, 2005) However, in 17

20 he long run echnologies are likely o become more inensive in reproducible facors. In our model, as long as > 0 and > 0 in he long run boh labor income share and capial income share converge o a posiive number. The behavior of hese shares along he ransiion depends on he parameers and and on he e ecs ha innovaions have on he share of land : In any case, his model is consisen wih he empirical evidence regarding he behavior of facor shares. 3 Conclusions We presen a one good model of economic growh wih wo reproducible facors where echnological change is facor saving and facor shares are deermined by echnology. Assuming ha echnologies can be changed paying a cos we nd ha agens in capial abundan economies are more likely o adop capial inensive echnologies han agens in poor economies. As a resul, he elasiciy of oupu wih respec o reproducible facors depends on he relaive abundance facors. We also show ha capial abundance simulaes innovaions ha save no reproducible facors and ha savings are higher in economies where he echnology is more capial inensive For his reason, rich economies may achieve long-run growh while poor economies may converge o a seady sae. Since facor prices are given by marginal produciviy, as economies grow, he income share of he reproducible facor grows while he income share of no 18

21 reproducible facors decreases. This predicion is consisen wih he old resul of consan labor income share. Indeed, human capial accumulaion simulaes human capial-using innovaions and increases human capial income share. The increase in human capial income share can counerweigh he reducion in raw labor income share in such a way ha oal labor income share (including remuneraion for human capial) remain consan. The same logic can be applied o land and physical capial. References [1] Ambler, S. and Cardia, E., The Cyclical Behavior of Wages and Pro s under Imperfec Compeiion. Canadian Journal of Economics 31 (1), [2] Barro, R. J. and Sala-i-Marin, X., Economic Growh. McGraw-Hill Advanced Series in Economics. [3] Benolila, S. and Sain-Paul, G., Explaining Movemens in he Labor Share. Conribuions o Macroeconomics 3 (1), [4] Berola, G., Facor Shares and Savings in Endogenous Growh. The American Economic Review 83 (5), [5] Blanchard, O. J., The Medium Run. Brookings Papers on Economic Aciviy 2,

22 [6] Blanchard, O. J. and Wolfers, J., The Role of Shocks and Insiuions in he Rise of European Unemploymen: The Aggregae Evidence. Economic Journal 110 (462), [7] Boldrin, M. and Horvah, M., Labor Conracs and Business Cycles. Journal of Poliical Economy 103 (5), [8] Boldrin, M. and Levine, D. K., 2002 Facor Saving Innovaion. Journal of Economic Theory 105 (1), [9] Cobb, C. and Douglas, P.H., A heory of Producion. American Economic Review 18, [10] Du y, J.. and Papageorgiou, C., A Cross-Counry Empirical Invesigaion of he Aggregae Producion Funcion Speci caion. Journal of Economic Growh 5 (1), [11] Durlauf, S. and Johnson, P., Muliple Regimes and Cross-Counry Growh Behavior. Journal of Applied Economerics 10 (4), [12] Drandakis, E. M. and Phelps, E. S., 1966 A Model of Induced Invenion, Growh and Disribuion." Economic Journal 76, [13] Easerly, W. and Levine, R., 2001 I s No Facor Accumulaion: Sylized Facs and Growh Models. World Bank Economic Review 15 (2), [14] Gollin, D., Geing Income Shares Righ. Journal of Poliical Economy 110 (2),

23 [15] Gomme, P. and Greenwood, J., On he Cyclical Allocaion of Risk. Journal of Economic Dynamics and Conrol 19, [16] Goschalk, P., 1997, Inequaliy, Income Growh and Mobiliy: The Basic Facs, Journal of Economic Perspecives 11(2), [17] Hansen, G. and Presco, E., Malhus o Solow. American Economic Review 92 (4), [18] Heckscher, E. F., The E ec of Foreign Trade on he Disribuion of Income. Ekonomisk Tidskrif, [19] Kennedy, C., Induced Bias in Innovaion and he Theory of Disribuion. Economic Journal 74, pp [20] Krueger, A.,1999. Measuring Labor s Share American Economic Review 89, (2), [21] Krusell, P. Ohanian, L. Ríos-Rull, J. V. Violane, G., Capial-Skill Complemenariy and Inequaliy: A Macroeconomic Analysis. Economerica 68 (5), [22] Ohlin, B., Inerregional and Inernaional Trade. Cambridge: Harvard Universiy Press. [23] Pereo, P. and Seaer, J., 2006 Augmenaion or Eliminaion? Working Paper hp://ideas.repec.org/p/deg/conpap/c011_060.hml [24] Rhee, C., Dynamic Ine ciency in an Economy wih Land. Review of Economic Sudies 58 (4),

24 [25] Samuelson, P. A., A Theory of Induced Innovaion on Kennedy-von Weisacker Lines. Review of Economics and Saisics 47 (4), [26] Solow, R., Technical Change and he Aggregae Producion Funcion. Review of Economics and Saisics 39, [27] Young, A., The Tyranny of Numbers: Confroning he Saisical Realiies of he Eas Asian Growh Experience. Quarerly Journal of Economics 110: [28] Young, A. T., One of he Things We Know ha Ain So: Why US Labor s Share is no Relaively Sable. Working Paper, January 2005, hp://ssrn.com/absrac= [29] Zeira, J., Workers, Machines and Economic Growh. Quarerly Journal of Economics 113 (4), [30] Zeira, J., Machines as Engines of Growh. CEPR 2005, Discussion Papers [31] Zulea, H., 2006 "Facor Saving Innovaions and Facor Income Shares. Working Paper, hp:// [32] Zulea, H. and Young, A., Labor s Shares Aggregae and Indusry: Accouning for Boh in a Model of Developmen wih Induced Innovaion. Working Paper, hp://papers.ssrn.com/absrac=

25 4 Appendix 4.1 Choosing Facor Shares max i; i Y i s.. i 0 and i 0 i i = i A(K i ) i 1 (H i ) 1 i L i i i + A(K i ) i (H i ) 1 i L i i i ln k i + = 0 i i A(K i ) i (H i ) i 1 L 1 i i i + A(K i ) i (H i ) 1 i L i i i ln h i + = 0 i Therefore, he equilibrium echnology (common for every rm) is given by, k ln k = max 0 ; k ln k + 1 and = max 0 ; h ln h h ln h Savings and asses Di ereniaing equaions 9, 10 and 11 _ = 1 + ln k (k ln k + 1) _ 2 k (19) _ = h ln h (h ln h + 1) _ 2 h (20) _k k _ h h = (21) Combining equaions 19, 20 and 21 wih 3, 4, 5 and 6 23

26 u k z = 1 + ln k (1 ) (k ln k + 1) 2 u (22) b = 1 + ln h (1 ) h ln h + 1 a (23) a = z b (1 ) (1 h ) (24) Combining equaions 22 and 9 z = lim z = 0: k!1 1+k ln k (1 ) Similarly, combining equaions 23 and 10 b = Therefore, lim h!1 b = 0: 1+ln k (k ln k +1) 2 u : Therefore, 1+h ln h (1 ) Therefore, from equaion 24 i follows ha in he long run u k Finally, h k = : so 1+ln h (h ln h +1) 2 a : = a h. lim u = and lim a = k!1 h!1 4.3 The growh rae of consumpion increases as he amoun of reproducible facors grow. De ne r = ( ) 1 ( ) Ak + 1 log r = (1 ) log () + log() + log(a) + ( + 1) log k 24

27 combining wih equaion 11 and rearranging, log r = log () + log(h) + log(a) + ( 1) log k Di ereniaing, _r r = _ + h _ h + _ k log(h) + _ log k (1 ) _ k Combining wih equaion 17 and rearranging, _r r = (1 + ln k)k 1 k ln k ln k k 1 (1 ) _ k + h _ h + _ log(h ) Combining wih equaion 11 and rearanging, _r r = 1 + ln k (1 + ln k) ln k (1 ) _ k k + _ h h + _ log(h ) Therefore, if k > 1 and _ k k > 0 hen _r r > 0: 4.4 If + 1 and ( ) 1 ( ) A < 2 hen he problem has a nie-value soluion. The maximized Hamilonian is given by H 0 (a ; ) = u(c )e + (Y c ) : Therefore, we have o prove ha lim!1 u(c )e = 0 To simplify noaion we drop he index *. Noe ha lim!1 u(c ) = 1 and lim!1 e = 0; so in order o nd he limi we 25

28 di ereniae he expression u(c )e : U 0 (c ) U(c ) _c Recall ha he log uiliy funcion is a special case of he more general funcion CRRA, c1 U 0 (c ) U(c ) _c = 1 1 _c c c 1!1 : Indeed, lim 1 = log c 8, so Now, we use he log uiliy funcion, so = 1 and U 0 (c ) U(c ) _c = _c c : From equaion 8 _c c = ( ) 1 ( ) Ak + 1 U 0 (c ) U(c ) _c = ( ) 1 ( ) Ak and if + 1 and ( ) 1 ( ) A < 2 hen U 0 (c ) lim k!1 U(c ) _c ( ) 1 ( ) A 2 Therefore, if ( ) 1 ( ) A < 2 hen lim!1 U 0 (c ) U(c ) _c lim u(c )e = 0.!1 < 0 and 4.5 If + 1 and ( ) 1 ( ) > A abundan economies converge o a BGP. hen capial De ne he funcion f(k) = A ((k)) 1 (k) ((k)) (k) k (k)+(k) 1 1. If k > 1 hen f(k) is sricly increasing in k: 2. If + 1 hen lim k!1 f(k) = ( ) 1 ( ) A : Therefore, if 8 To show ha he uiliy funcion converges o he logarihmic funcion as! 1 we make use of L Hospial s rule. As! 1, boh he numeraor and denominaor of he funcion approach zero. We di ereniae boh he numeraor and he denominaor wih respec o and hen ake he limi of he derivaives raio as! 1. 26

29 ( ) 1 ( ) > A hen lim f(k) > 0: k!1 From 1 and 2, here exiss a ~ k such ha f(k) > 0 for any k > ~ k. Suppose no, hen for any k here exiss a nie number M such ha M > k and f(m) 0: f(k) is sricly increasing in k; so f(m) > f(k): Therefore, for any k; f(k) 0 and sup f(k) 0: 0: Finally, lim f(k) = sup f(k); so lim k!1 k!1 f(k) 0 which conradics lim f(k) > k!1 27