Elasticity of substitution and growth: normalized CES in the diamond model

Transcription

1 Economic Theory 268 (Red.Nr. 785) (2003) Elasiciy of subsiuion and growh: normalized CES in he diamond model Kaz Miyagiwa 1, Chris Papageorgiou 2 1 Deparmen of Economics, Emory Universiy, Alana, GA 30322, USA 2 Deparmen of Economics, Louisiana Sae Universiy, Baon Rouge, LA , USA ( cpapa@lsu.edu) Received: Ocober 27, 2001; revised version: February 25, 2002 Summary. I is ofen assered ha he more subsiuable capial and labor are in he aggregae producion he more rapidly an economy grows. Recenly his has been formally confirmed wihin he Solow model by Klump and de La Grandville (2000). This paper demonsraes ha here exiss no such monoonic relaionship beween facor subsiuabiliy and growh in he Diamond overlapping-generaions model. In paricular, we prove ha, if capial and labor are relaively subsiuable, a counry wih a greaer elasiciy of subsiuion exhibis lower per capia oupu growh in ransi and in seady sae. Keywords and Phrases: CES, Diamond overlappinggeneraions model, Economic growh. JEL Classificaion Numbers: E13, E23, O40. 1 Inroducion The developmen of modern growh heory owes much o he use of Cobb-Douglas aggregae producion funcion. However, i is also rue ha he general conclusions of growh heory have been somewha limied because of he uniary elasiciy of subsiuion beween capial and labor implied by he Cobb-Douglas producion funcion. This observaion has led Pichford (1960) o use he consan-elasiciyof-subsiuion (CES) producion funcion in he neoclassical growh model and We hank Rainer Klump, Theodore Palivos and an anonymous referee for heir valuable commens. Correspondence o: C. Papageorgiou

2 2 K. Miyagiwa, C. Papageorgiou invesigae he effec on growh of he elasiciy of subsiuion iself. 1 Amonghis findings is he fac ha an increase in he elasiciy of subsiuion no only increases he curvaure of he isoquan for a given level of oupu bu also causes he isoquan o shif inwards by makinglabor and capial more efficien. The laer effec has he implicaion ha despie iniially havingidenical endowmens of capial and labor an economy wih a higher elasiciy of subsiuion begins is growh pah a a greaer level of per-capia income han an economy wih a smaller elasiciy of subsiuion. Recenly, Klump and de La Grandville (2000) (henceforh KG) have used a cerain normalizaion procedure o correc he above bias inheren in he use of CES producion funcion. Usingheir normalized CES producion funcion in he Solow growh model hey have found ha a counry exhibiing a higher elasiciy of subsiuion experiences greaer capial and oupu per worker boh in ransiion and in seady sae. The objecive of his paper is o exend his line of research wihin he Diamond (1965) overlapping-generaions model. We believe i is a worhwhile exension since he Diamond model has increasingly been used in recen years o sudy economic growh as an alernaive o Solow model. Our main conclusion is ha he Diamond model does no exhibi he monoonic relaionship beween he elasiciy of subsiuion and capial and oupu per worker eiher in ransiion or in seady sae. More specifically, we show ha, if he elasiciy of subsiuion is sufficienly high, a furher increase in he elasiciy of subsiuion lowers oupu and income per worker boh in ransiion and in seady sae. Our resul obains because unlike in he Solow model, agens save a fracion of wage income in he Diamond model. As shown by KG, a higher elasiciy implies an increase in oal income bu also a fall in labor share. In a model where agens save ou of wage income, his makes i ambiguous how savings will be affeced by an increase in he elasiciy of subsiuion. We show ha if he elasiciy of subsiuion exceeds one by enough higher elasiciies imply lower growh and a lower seady sae capial sock. 2 The normalized CES producion funcion in he Solow model The normalizaion procedure proposed by KG is as follows. Given he sandard inensive-form CES producion funcion f(k )=Aδk ρ +(1 δ)] 1 ρ, where k is he capial per worker a ime, choose arbirary baseline values for capial per worker (), oupu per worker (ȳ) and he marginal rae of subsiuion beween capial and labor defined by m =f() f ()]/f () (primes denoe derivaives). Then, use hose baseline values o solve for he normalized efficiency parameer A(σ) = ) 1/ρ, and he normalized disribuion parameer δ(σ) = 1 ρ 1 ρ + m as a funcion of σ = 1 1 ρ, he elasiciy of subsiuion. Subsiuinghese normalized ȳ ( 1 ρ + m + m 1 There are wo ypes of CES funcions: he homoheic CES pioneered by Arrow e al. (1961), and he nonhomoheic CES derived by Sao (1975) and Shimomura (1999). This paper considers only homoheic CES.

3 Elasiciy of subsiuion and growh: normalized CES in he diamond model 3 K C A Leonief 0 45 o Cobb-Douglas Perfec Subsiues B L Figure 1. Illusraion of de La Grandville s normalized CES producion funcion parameers ino he iniial equaion yields he normalized CES producion funcion: 2 f σ (k )=A(σ) {δ(σ)k ρ +1 δ(σ)]} 1 ρ. (1) Figure 1 illusraes he resul of his normalizaion. Despie disparae values for σ (0, 1, + ), all he isoquans for he baseline level of oupu are shown o go hrough he common poin (poin A) defined by (given by ray OA) and m (given by line BAC). The normalizaion generaes a family of dynamical pahs in he Solow growh model ha depend only on he value of σ. Pahs of capial per worker for hree values of σ are shown in Figure 2. Figure 2 differs from Figure 1 in KG (2000, p. 284) because here he Solow model is recas in a discree-ime seingo faciliae comparison wih he Diamond model below. More specifically, he pahs shown in he figure are generaed by he dynamical equaion k +1 = 1+n f σ(k ), where is he exogenous saving rae ou of oupu per worker, n is he exogenous labor growh rae and where for simpliciy capial is assumed o depreciae fully a he end of each period. 3 Despie he ranslaion ino he discree-ime seing, he KG resul is eviden; a counry havinga greaer value of σ clearly has more capial per worker in ransiion 2 For exensive discussions on he normalized CES funcion see Klump and Preissler (2000, pp ). 3 In consrucingfigures 2, 3, 4, we se =0.5, n =0.01. Parameric examples of he dynamic relaionship beween y +1 and y are available upon reques.

4 4 K. Miyagiwa, C. Papageorgiou Figure 2. Transiional pahs of per capia capial for differen σ in he Solow model and in seady sae han a counry endowed wih a lower value of σ. I follows ha, he greaer he value of σ, he greaer is income per worker boh in ransiion and in seady sae. 3 The normalized CES producion funcion in he diamond model In he Diamond (1965) overlapping-generaions model a new generaion is born a he beginning of every period. Agens are idenical and live for wo periods. In he firs period each agen supplies a uni of labor inelasically and receives a compeiive wage w σ, = f σ (k ) k f σ(k )=1 δ(σ)]a(σ)] ρ f σ (k )] 1 ρ. To make he model consisen wih he Solow model, assume ha agens save a fixed proporion of he wage income o finance consumpion in he second period of heir lives. All savings are invesed as capial o be used in he nex period s producion; ha is k +1 = 1+n w σ(k )= 1+n 1 δ(σ)]a(σ)]ρ f σ (k )] 1 ρ h σ (k ), (2) where n is he exogenous labor growh rae and where capial depreciaes fully. 4 Equaion (2) deermines he dynamical pah of capial per worker. Then, he dynamical pah of oupu per worker is obained from (1). 4 Alernaively, we could assume ha agens have preferences over consumpion in he wo periods of heir lives given by U(c 1,c2 +1 )=(1 )lnc1 + ln c2 +1, where ci +j is period i consumpion

5 Elasiciy of subsiuion and growh: normalized CES in he diamond model 5 Seady saes for k (denoed by ) are soluions o he polynomial equaion k h σ (k) =0. (3) If σ 1 (ρ 0, 1]), here always exiss one unique posiive seady sae for k, since limh k 0 σ(k) > 1 and lim k + h σ(k) =0.Ifσ<1 (ρ <0), here are eiher zero or wo posiive and disinc seady-sae values for k, dependingon he value of he scale facor A(σ). 5 We now urn o our wo main findings. (All proofs are in he Appendix.) Theorem 1 Suppose ha a counry is represened by he one-secor Diamond model wih a normalized CES aggregae producion funcion. If ρ m, for any k >, (A)he higher he elasiciy of subsiuion he lower he level of capial and oupu per worker a any sage of he ransiion pah, and (B)he higher he elasiciy of subsiuion he lower he growh raes of capial and oupu per worker along he ransiional pah. Theorem 2 Suppose ha a sable seady sae exiss in he one-secor Diamond model wih a normalized CES aggregae producion funcion. If ρ m, he higher he elasiciy of subsiuion, he lower he seady-sae level of capial and oupu per worker. Noe ha ρ m implies σ m > 1. Figure 3 illusraes he dynamical pahs of capial per worker in he Diamond model for ρ m, where we se m =1and =5.Asσ increases from 1.25 o 5 and o, he level of capial per worker falls boh in ransiion and in seady sae for any k >. 6 Moreover, if ρ< m he relaionship beween σ and he level of capial per worker is no unique because, as shown in Figure 4, he dynamical pahs of k for differen values of σ cross each oher a some k >. 7 Why do our resuls conras wih hose from he Solow model? The Diamond model differs from he Solow model in one imporan respec: individual savings by he represenaive agen in period + j, j =0, 1. The represenaive agen maximizes U(c 1,c2 +1 ) subjec o he consrain, c 1 + c2 +1 R σ,+1 and capial, respecively. Maximizaion yields he ransiion equaion, k +1 = w σ,, where w σ, and R σ,+1 represen he reurns o labor wσ(k), which is 1+n equivalen o equaion (2). 5 When here are wo posiive seady saes, he larger of he wo is locally asympoically sable. In his case, he rivial seady sae (k =0) is also locally asympoically sable. The domains of aracion of he wo sable seady saes are disinc, and depend on wheher he iniial capial sock lies above or below he locally unsable equilibrium. The condiions for and characerizaion of muliple equilibria in he Diamond (1965) model (see e.g. Azariadis, 1993, pp ) remain unaffeced by he normalizaion. 6 Duffy and Papageorgiou (2000) use panel daa echniques o esimae a cross-counry aggregae CES producion specificaion. For heir enire sample of 82 counries hey find ha he elasiciy of subsiuion beween capial and (skilled) labor is significanly greaer han uniy. 7 In consrucingfigure 4, we once again se m =1, =5.

6 6 K. Miyagiwa, C. Papageorgiou Figure 3. Transiional pahs of per capia capial in he Diamond model when ρ m Figure 4. Transiional pahs of per capia capial in he Diamond model when ρ< m

7 Elasiciy of subsiuion and growh: normalized CES in he diamond model 7 come ou of wage income in he former and ou of oal (wage and renal) income in he laer. A useful way o demonsrae he difference is offered by Galor (1996). Suppose ha he fracion saved ou of wage income, w, differs from he fracion saved ou of renal income, r, possibly because of differences in preferences or endowmens amongagens. Then he law of moion for capial per worker in he normalized CES producion funcion is k +1 = w 1+n f σ(k ) f σ(k )k ]+ r 1+n f σ(k )k. (4) Since w = r = in he Solow model while w = and r =0in he Diamond model, he dynamical pah in he former conains he addiional erm, 1+n f σ(k )k, ha represens savings ou of renal income. Now, o ge an inuiion of our resul differeniae equaion (4) wih respec o σ o obain: = w ( ) f σ(k ) f σ (k ) π ] (5) 1+n }{{} w (?) + r f σ (k ) π + f σ (k ) π ], 1+n }{{} rk (+) where π f σ (k)k f σ(k ) is he renal income share. The firs (second) expression on he righ shows how wage (renal) income is affeced by a change in σ. As shown by KG, a higher elasiciy of subsiuion implies an increase in income and renal shares. Tha makes he effec on renal income clearly posiive since capial benefis from an increase in boh oupu and renal share. Bu he effec on wage income is ambiguous as labor share falls. In he Solow model where oal savings maer, however, hose income disribuion effecs cancel ou, leadingo k+1 > 0 as shown by KG. In conras, in he Diamond model where agens save a fracion of wage income, he second expression in (6) is absen. Wihin he firs expression, ( ) fσ(k) measures how wage income increases when oal oupu is increased, and his magniude clearly depends on labor share, ( ). The second erm, f σ (k ) π represens a fall in labor share riggered by an increase in subsiuabiliy of capial for labor. If labor share ( ) is sufficienly small, he second erm dominaes he firs, and our resul follows. Indeed ρ m implies () < 1/2 which is sufficien o obain our resul. 8 4 Conclusion In his paper we have shown ha he posiive relaionship beween he elasiciy of subsiuion and economic growh discovered recenly by KG (2000) does no 8 Since = + m implies ha m 1 ; subsiuion ino ρ m yields 1. Given ha 1+ρ ρ (0, 1] under ρ m, ρ m finally implies 1/2.

8 8 K. Miyagiwa, C. Papageorgiou carry over o he Diamond model. Thus, wheher he elasiciy of subsiuion has a posiive or negaive effec on economic growh depends on our view of he world, ha is, on he paricular framework (Solow vs. Diamond) we believe o be a beer represenaion of he world. Appendix Proof of Theorem 1. Rewrie equaion (2) as k +1 = 1+n f σ(k ) k f σ(k )] = 1+n f σ(k )(1 π )], where π f σ (k)k f σ(k ) is he renal income share. Differeniaingwih respec o σ yields = ( ) f σ(k ) f σ (k ) π ]. 1+n ] Subsiuing fσ(k) = 1 1 σ 2 ρ y 2 π ln π +(1 π )ln 1 1 π and π = 1 σ (1 2 π )π ln k from Klump and de La Grandville (2000, pp ) yields = { (1 π )y 1+n σ 2 ρ 2 π ln +(1 π )ln ] π + y σ 2 ()π ln k } where = = 1+n + m. From we obain k equaion (A1) gives = 1 π π 1 = 1+n ( )y σ 2 ρ 2 π ln π +(1 π )ln + ρ 2 π ln k ( ) ρ y =, π k ȳ = ( ) ρ y, ȳ ], (A.1). Subsiuinghis ino he las erm in he brackes of ( )y σ 2 ρ 2 π ln π +(1 π )ln ρπ ln ( )]. π

9 Elasiciy of subsiuion and growh: normalized CES in he diamond model 9 Since he logarihmic funcion is sricly concave, we have ha ln π < π 1 ln π > (A.2) ln > π, ln < 1 ln > 1 (A.3) ln > 1, ( ) ln < ( ) π π 1 ln (A.4) π > π. Assume ha σ>1 (ρ (0, 1]) and k >. Muliplyingboh sides of he final inequaliies in (A2), (A3) and (A4) by π, ( ) and ρπ, respecively, yields he followinginequaliy: π ln π +(1 π )ln ( >π ) +(1 π ) ( ρπ ln ( 1 (π )+ ρ = π ( π )+ (π ) ( 1 π =(π ) + ρ π ) ( )( π = + ρ π ) () ( ) π ( = 1+ρ π ) ( ) ( π + m = 1+ρ ) π ) ( + ρπ π ) k ρ ] 1 ρ k ρ, (A.5) 1 ρ + m kρ 1 ρ k ρ 1 ρ + m where he las equaliy comes from he fac ha π =. The funcion ( ) + m k ρ ] 1 ρ φ(k ) 1+ρ k ρ, 1 ρ + m is monoonically decreasingwih he horizonal asympoe a ρ m. Therefore, if ρ m, φ(k ) 0. Then since π 1 0 for k >, he las expression in (A5) is non-negaive. Consequenly, k+1 < 0. To prove ha oupu per worker is a decreasingfuncion of he σ when ρ m and k >, rewrie y+1 as y +1 = y +1. )

10 10 K. Miyagiwa, C. Papageorgiou We have jus shown ha k+1 < 0 for ρ m and k >. Given ha y+1 is posiive for all k +1 > 0, hen y+1 < 0. This complees he proof of Theorem 1A. To prove Theorem 1B, define he growh rae of capial per worker by g k = k +1 k 1 and he growh rae of oupu per worker by g y = y+1 y 1. Differeniaion yields g k = 1 k < 0, ρ m, and k >, y = 1 y +1 y < 0, ρ m, and k >. This complees he proof. Proof of Theorem 2. A seady sae, k = k +1 = k and herefore equaion (2) reduces o he polynomial equaion (3). Differeniaing k wih respec o σ yields k σ = 1+n (w σ) 1, (B1) 1+n (w k ) ] where (wσ) = (1 π ) f f π and (wk ) = (1 π )(f ) f π k ]. By Theorem 1, (wσ) < 0 for ρ m and k >k, so he numeraor is negaive. To show ha he denominaor is posiive, solve he definiion of π = (f ) k f and he seady-sae polynomial equaion k = 1+n f (f ) k ] simulaneously for f and (f ) o obain f = (1 + n) k 1 π, (B2) (f ) (1 + n) = 1 π. (B3) Subsiuingequaions (B2), (B3) and π k = ρπ (1 π ) k ino (w k gives ) (wk) =(1 π (1 + n) π (1 + n) k ρπ (1 π ) ) 1 π 1 π k (1 + n) = (1 ρ)π. Then 1+n (w k =(1 ρ)π and hence 1 ) 1+n (w k =1 (1 ρ)π > 0. ) Therefore k < 0. To prove ha he seady-sae oupu per worker is a decreasingfuncion of he σ when ρ m and k >, once again rewrie y as y = y k k. Given ha y k > 0 for all k > 0, and k < 0 for ρ m and k > as shown above, y < 0 as desired. π

11 Elasiciy of subsiuion and growh: normalized CES in he diamond model 11 References Arrow, K. J., Chenery, H. B., Minhas, B. S., Solow, R. M.: Capial-labor subsiuion and economic efficiency. Review of Economics and Saisics 43, (1961) Azariadis, C.: Ineremporal macroeconomics. Cambridge, MA: Blackwell Publishers 1993 Diamond, P. A.: Naional deb in a neoclassical growh model. American Economic Review 55, (1965) Duffy, J., Papageorgiou, C.: A cross-counry empirical invesigaion of he aggregae producion funcion specificaion. Journal of Economic Growh 5, ( ) Galor, O.: Convergence? Inferences from heoreical models. Economic Journal 106, (1996) Hicks, J. R.: The heory of wages. London: MacMillan 1932 Klump, R., de La Grandville, O.: Economic growh and elasiciy of subsiuion: Two heorems and some suggesions. American Economic Review 90, (2000) Klump, R., Preissler, H.: CES producion funcions and economic growh. Scandinavian Journal of Economics 102, (2000) Pichford, J. D.: Growh and he elasiciy of subsiuion. Economic Record 36, (1960) Sao, R.: The mos general class of CES funcions. Economerica 43, (1975) Shimomura, K.: A simple proof of he Sao proposiion on non-homoheic CES funcions. Economic Theory 14, (1999) Solow, R. M.: A conribuion o he heory of economic growh. Quarerly Journal of Economics 70, (1956)