ElasticityofSubstitution and Growth: Normalized CES in the Diamond Model

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1 ElasiciyofSubsiuion and Growh: Normalized CES in he Diamond Model Kaz Miyagiwa Deparmen of Economics Louisiana Sae Universiy Baon Rouge, LA Chris Papageorgiou Deparmen of Economics Louisiana Sae Universiy Baon Rouge, LA February, 2001 Absrac Klump and de La Grandville (2000) used he \normalized" Consan Elasiciy of Subsiuion (CES) speci caion o prove ha he Solow growh model exhibis a posiive relaionship beween per capia oupu and he elasiciy of subsiuion boh in ransiion and in seady sae. This paper shows ha heir resul does no exend o he Diamond overlapping generaions model. In paricular, heir resul is reversed when capial and labor are relaively subsiuable; counries wih a higher elasiciy of subsiuion have lower per capia oupu and growh. JEL Classi caion Numbers: E13, E23, O40. Keywords: CES, Diamond Overlapping Generaions Model, Economic Growh.

2 1 Inroducion In arecenpaper, Klump and delagrandville (2000) uilized he \normalized"consanelasiciy of Subsiuion (CES) producion funcion in he Solow (1956) growh model and found ha a counry endowed wih a greaer elasiciy of subsiuion experiences greaer capial and oupu per worker boh in ransiion and in seady sae. The objecive of his paper is o examine wheher heir resul carries over o he Diamond (1965) overlapping-generaions model. Such examinaion is warraned because he Diamond model has increasingly been used in recen years o sudy economic growh as an alernaive ohesolowmodel. Our main ndingisha he Klump-de LaGrandville resul does no hold in he Diamond model; in paricular, heir resul is reversed ifhe elasiciy of subsiuion is su±cienly large. 2 The Normalized CES Producion Funcion in he Solow Model Oliver de La Grandville (1989) suggesed ha a meaningful examinaion of he properies ofdi eren members of he same family of CES producion funcions requires he following normalizaion. Given he sandard inensive-form CES producion funcion f(k ) = A[±k ½ + (1 ±)]1 ½, where k is he capial per worker a ime, choose arbirary baseline values for capial per worker ( ¹ k), oupu per worker (¹y) and he marginal rae of subsiuion beween capial and labor de ned by ¹m = [f( ¹ k) ¹ kf 0 ( ¹ k)]=f 0 ( ¹ k) (primes denoe derivaives wih respec o k). Then, use hose baseline values o solve for he normalized e±ciency parameer A(¾) = ¹y ³ ¹ k 1=½, 1 ½ +¹m ¹k+¹m and he normalized k disribuion parameer ±(¾) = ¹ 1 ½ 1 as a funcion of ¾ = ¹k 1 ½ +¹m 1 ½, he elasiciy of subsiuion. Subsiuing hese normalized parameers ino he iniial equaion yields he normalized CES producion funcion: 1 f¾(k) = A(¾)f±(¾)k ½ +[1 ±(¾)]g1 ½ : (1) Figure 1 illusraes he de La Grandville normalizaion. Despie disparae values for ¾, all he isoquans for a given iniial level of oupu (¹y) are shown o go hrough he common poin (poin A) de ned by ¹ k (given by ray OA) and ¹m (given by line BAC). As shown by Pichford (1960), an increase in ¾ wihou he normalizaion causes no only an increase in he curvaure of he 1 For exensive discussions on he normalized CES funcion see de La Grandville (1989, p.476), and Klump and Preissler (2000, pp.44-45). 1

3 Figure 1: Illusraion of de La Grandville's normalized CES producion funcion K C A? = 0 0 B? =?? = 1 L isoquan for a given level of oupu; i also causes he isoquan o shif inward by making facors more e±cien. The de La Grandville normalizaion prevens such dispersions. The de La Grandville normalizaion generaes afamily ofdynamical pahs in he Solow growh model ha depend only on he value of ¾. Pahs of capial per worker for hree values of ¾ are shown in Figure 2. Figure 2 di ers from Figure 1 in Klump and de la Grandville (2000 p.284) because here he Solow model is recas in a discree-ime seing o faciliae comparison wih he Diamond model below. More speci cally, he pahs shown in he gure are generaed by he dynamical equaion k +1 = 1 +n f ¾(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. Despie he ranslaion ino he discree-ime seing, he Klump-de la Grandville resul is eviden; acounry havinga greaer value of ¾ clearly has morecapial per worker in ransiion and in seady sae han a counry endowed wih a lower value of ¾. I follows ha, he greaer he value of ¾, he greaer income per worker is boh in ransiion and in seady sae. 2

4 Figure 2: Transiional pahs of per capia capial for di eren ES in he Solow model 3 TheNormalizedCESProducionFuncioninheDiamondModel In he Diamond (1965) overlapping-generaions model anewgeneraion is born a hebeginningof every period. Agens are idenical and live for wo periods. In he rs period each agen supplies a uni of labor inelasically and receives a compeiive wage w ¾; = f ¾ (k ) k f 0 ¾ (k ) = [1 ±(¾)][A(¾)] ½ [f ¾ (k )] 1 ½ : Tomakehemodel consisenwih he Solowmodel, assumeha agens savea xed proporion of he wage income o nance 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 ¾(k ) = 1 +n [1 ±(¾)][A(¾)]½ [f ¾ (k )] 1 ½ h ¾ (k ); (2) where n is he exogenous labor growh rae and where capial depreciaes fully. 2 Equaions (2) 2 Alernaively, we could assume ha agens have preferences over consumpion in he wo periods of heir lives 3

5 deermines he dynamical pah of capial per worker. Then, he dynamical pah of oupu per worker is obained from (1). Seady saes fork (denoed by ) are soluions o he polynomial equaion k h ¾ (k) = 0: (3) If ¾ 1 (½ 2 [0; 1]), here always exiss one unique posiive seady sae for k, since lim k!0 h 0 ¾ (k) > 1 and lim (k) = 0. If ¾ < 1 (½ < 0), here are eiher zero or wo posiive and disinc k!+1 h0 ¾ seady-sae values for k ; depending on he value of he scale facor A(¾). 3 We now urn o our wo main ndings. (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 ½ ¹m ¹k, for any k > ¹k, (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 ½ ¹m k ¹, he higher he elasiciy of subsiuion, he lower he seady-sae level of capial and oupu per worker. Figure 3 illusraes he dynamical pahs of capial per worker in he Diamond model for ½ ¹m ¹ k, where we se ¹m = 1 and ¹ k = 5. As ¾ increases from 1:25 o 5 and o 1, he level of capial per worker falls boh in ransiion and in seady sae for any k > ¹ k; hereby reversing he Klump-de La Grandville resul. 4 given byu(c 1 ;c2 +1 ) = (1 )lnc1 +lnc2 +1, whereci +j is periodiconsumpion by he represenaive agen in period +j,j =0;1. The represenaive agen maximizesu(c 1 ;c2 +1 ) subjec o he consrain,c1 + c2 +1 R ¾;+1 w ¾;, where w¾; and R¾;+1 represen he reurns o labor and capial, respecively. Maximizaion yields he ransiion equaion,k+1 = 1+nw¾(k), which is equivalen o equaion (2). 3 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 seadysaes are disinc, anddependonwheher he iniial capial socklies above orbelow he locallyunsable equilibrium. The condiionsfor and characerizaionofmuliple equilibria in he Diamond (1965) model (see e.g. Azariadis1993, pp )remainuna ecedbyhe normalizaion. 4 Parameric examplesofhe dynamic relaionship beweeny+1 andy are available upon reques. 4

6 Figure 3: Transiional pahs of per capia capial in he Diamond model when ½ ¹m ¹k Moreover, if ½ < ¹m ¹ k he relaionship beween he ¾ and he level of capial per worker is no unique because, as shown in Figure 4, he dynamical pahs of k for di eren values of ¾ cross each oher a some k > ¹ k. 5 Why do our resuls conras wih hose of Klump and de La Grandville? The Diamond model di ers from he Solow model in oneimporanrespec: individual savings comeouofwage income in he former and ou of oal (wage and renal) income in he laer. A useful way o demonsrae he di erence is o ered by Galor (1996). Suppose ha he fracion saved ou of wage income, w, di ers from he fracion saved ou of renal income, r, possibly because of di erences in preferences or endowmens among agens. Then he law of moion for capial per worker in he normalized CES producion funcion is k +1 = w f¾ (k ) f n ¾(k )k r + 1 +n f0 ¾(k )k : (4) 5 Inconsrucing Figure 4, we se ¹m=3, ¹ k =5 o keep he diagram from geing cluered. 5

7 Figure 4: Transiional pahs of per capia capial in he Diamond model when ½ < ¹m ¹k Since w = r = in he Solow model while w = and r = 0 in he Diamond model, he dynamical pah in he former conains he addiional erm, ou of renal income. 1+n f0 ¾ (k )k, ha represens savings For example, when ¾ = 1 (capial and labor are perfec subsiues), equaion (4) reduces o k +1 = ¹y ¹m (1 +n)( k ¹ + ¹m) + ¹y (1 +n)( k ¹ + ¹m) k ; in he Solow model. Thus, k +1 is a linear posiive funcion of k wih he verical inercep a ¹y¹m (1+n)( ¹ k+¹m) and he slope ¹y (1+n)( ¹. On he oher hand, in he Diamond model equaion (4) k+¹m) reduces o ¹m¹y k +1 = (1 +n)(¹k + ¹m) : ¹m¹y Thus, k +1 is a horizonal line a (1+n)( ¹ k+¹m).6 Then, as k grows from he common iniial value ¹k, he enire capial inensiy pah of he Solow model lies above he pah of he Diamond model. 6 The former line is depiced by he parameric curve¾¼1 in Figure 2, while he laer line is depiced by he parameric curve¾ =1 in Figure 3. 6

8 Now, o ge an inuiion of our resul di ereniae equaion (4) wih respec o ¾ o +1 = w (1 ¼ ¾(k ) f ¾ (k 1+ n {z (?) + r 1 ¾ (k ) ¼ + f ¾ (k ; (5) {z (+) where ¼ f0 ¾(k)k f¾(k) is he renal income share. The rs and he second erm on he RHS of equaion (5) show he change in wage and renal incomes, respecively, due o a change in ¾. The second expression is clearly posiive while he rs is generally ambiguous in sign. No maer wha, he second expression mus dominae he rs in he Solow model (where w = r = ) > 0 as shown by Klump and de La Grandville (2000). In he Diamond model (where w = and r = 0) he second expression in equaion (5) is absen. Wihin he rs expression, (1 ¼ ¾(k ) represens a posiive e ec of¾on wage income due o an increase in labor produciviy for a given wage income share (1 ¼ ). The second erm f ¾ (k represens a negaive e ec of ¾ on wage income due a decrease in he wage income share riggered by subsiuion of capial for labor. If he wage income share (1 ¼ ) is su±cienly small, hen he negaive e ec dominaes and our resul follows. Indeed ½ ¹m ¹ k implies (1 ¹¼) < 1=2 which is su±cien o obain our resul. 7 4 Conclusion In his paper we have shown ha he posiive relaionship beween he elasiciy of subsiuion and economic growh discovered recenly by Klump and de la Grandville does no carry over o he Diamond model. Thus, wheher he elasiciy of subsiuion has a posiive or negaive e ec on economic growh depends on our view of he world, ha is, on he paricular framework (Solow vs. Diamond) we believe as a beer represenaion of he world. Boh our work and ha of Klump and de la Grandville ake he elasiciy of subsiuion as exogenous. However, as poined ou by Hicks (1932), he aggregae elasiciy of subsiuion iself is likely o be inuenced by facors ha also a ec economic growh. Thus, endogenizing he elasiciy of subsiuion in he conex of a growh model seems like a naural nex sep in his line of research. 7 k Since ¹¼ = ¹ ¹k+¹m implies ha ¹m ¹ k 1 ¹¼ ¹¼ ; subsiuion ino½ ¹m ¹ k yields ¹¼ 1 1+½. Given ha½ 2 (0;1] under ½ ¹m ¹ k,½ ¹m ¹ k nally implies ¹¼ 1=2. 7

9 Appendix Proof of Theorem 1 Rewrie equaion (2) as k +1 = = 1 +n [f ¾(k ) k f¾ 0(k )] 1 +n [f ¾(k )(1 ¼ )]; where ¼ f0 ¾ (k )k f ¾ (k ) is he renal income share. Di ereniaing wih respec o ¾ +1 = (1 ¼ 1+ n ¾(k ) f ¾ (k : = ¾ 1 1 h 2 ½ y 2 ¼ ln ¹¼ + (1 ¼ ¼ )ln 1 ¹¼ i 1 ¼ = ¾ 1 (1 ¼ 2 )¼ ln k ¹k and de La Grandville (2000, pp ) +1 = ½ (1 ¼ )y 1+ n ¾ 2 ½ 2 = (1 ¼ )y 1+ n ¾ 2 ½ 2 ¼ ln ¹¼ +(1 ¼ )ln 1 ¹¼ ¼ 1 ¼ ¼ ln ¹¼ + (1 ¼ )ln 1 ¹¼ +½ 2 ¼ ln k ¼ 1 ¼ ¹k k where ¹¼ = ¹ ¹k+¹m. From Ã! ¹¼ ¹k ½ y = ; ¼ k ¹y µ 1 ¹¼ y ½ = ; 1 ¼ ¹y we obain k ¹k = ¹¼ ¼ + y ¾ 2(1 ¼ )¼ ln k ¾ ¹k from Klump ; (A1) 1 ¼ 1 ¹¼. Subsiuing his ino he las erm in he brackes of equaion (A1) +1 = (1 ¼ )y 1 +n ¾ 2 ½ 2 ¼ ln ¹¼ +(1 ¼ ¼ )ln 1 ¹¼ ¹¼ ½¼ 1 ¼ lnµ ¼ Since he logarihmic funcion is sricly concave, we have ha 1 ¼ : 1 ¹¼ ln ¼ ¹¼ < ¼ ¹¼ 1 ) ln ¼ ¹¼ > 1 ¼ ¹¼ ) ln ¹¼ ¼ > 1 ¼ ¹¼ ; (A2) ln 1 ¼ 1 ¹¼ < 1 ¼ 1 ¹¼ 1 ) ln 1 ¼ 1 ¹¼ > 1 1 ¼ 1 ¹¼ ) ln > 1 1 ¼ 1 ¹¼ 1 ¼ 1 ¹¼ ; (A3) µ ¹¼ 1 ¼ ln < ¹¼ µ 1 ¼ ¹¼ ¼ 1 ¹¼ ¼ 1 ¹¼ 1 ) ln 1 ¼ > 1 ¹¼ 1 ¼ ¼ 1 ¹¼ ¼ 1 ¹¼ : (A4) 8

10 Assume ha ¾ > 1 (½ 2 (0;1]) and k > ¹ k: Muliplying boh sides of he nal inequaliies in (A2), (A3) and (A4) by ¼ ; (1 ¼ ) and ½¼, respecively, yields he following inequaliy: ¼ ln ¹¼ +(1 ¼ ¼ )ln 1 ¹¼ ¹¼ 1 ¼ ½¼ 1 ¼ lnµ ¼ 1 ¹¼ µ > ¼ 1 ¼ µ +(1 ¼ ) 1 1 ¼ µ + ½¼ 1 ¹¼ 1 ¼ ¹¼ 1 ¹¼ ¼ 1 ¹¼ = ¼ ¹¼ (¹¼ ¼ ) + 1 ¼ 1 ¹¼ (¼ ¹¼) + ½ 1 ¹¼ (¼ ¹¼) µ 1 ¼ = (¼ ¹¼) 1 ¹¼ + ½ 1 ¹¼ ¼ ¹¼ µ µ ¼ ¹¼ = 1 ¼ + ½ ¼ (1 ¹¼) 1 ¹¼ ¹¼ µ µ ¼ ¹¼ = 1+ ½ ¼ 1 ¹¼ ¹¼ µ ¼ ¹¼ " Ã! # ¹k ½ + ¹m k = 1+ ½ ¹ k 1 ½ 1 ¹¼ ¹k k ½ ¹ k 1 ½ ; (A5) + ¹m where he las equaliy comes from he fac ha ¼ = k½ ¹ k 1 ½ k ½ ¹ k 1 ½ +¹m " Ã! ¹ k + ¹m k ½ Á(k ) 1 +½ ¹k 1 ½ # ¹k k ½ ¹ k 1 ½ ; + ¹m. The funcion is monoonically decreasing wih he horizonal asympoe a ½ ¹m ¹k : Therefore, if ½ ¹m ¹k, Á(k ) 0. Then since ¼ ¹¼ 1 ¹¼ 0, he las expression in (A5) is non-negaive. < 0. To prove ha oupu per worker is a decreasing funcion of he ¾ when ½ ¹m ¹k and k > ¹k; : We have jus shown +1 < 0 for ½ ¹m ¹ k and k > ¹ k. Given +1 is posiive for all k +1 > 0, < 0. This complees he proof of Theorem 1A. To prove Theorem 1B, de ne 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. Di ereniaion y = 1 +1 < 0; 8 ½ ¹m ¹ k ; and k > ¹ k; = 1 +1 < 0; 8 ½ ¹m ¹ k ; and k > ¹ k: This complees he proof. 9

11 Proof of Theorem 2 A seady sae, k = k +1 = k and herefore equaion (2) reduces o he polynomial equaion (3). Di ereniaing k wih respec o ¾ ¾ = 1+n (w ¾) n ; (B1) (w k )0 where (w ¾ )0 = h (1 ¼ i and (w k ) 0 = h i (1 ¼ )(f ) By Theorem 2, (w ¾ ) 0 < 0 for ½ ¹m ¹ k and k > k, so he numeraor is negaive. To show ha he denominaor is posiive, solve he de niion of ¼ = (f ) 0 k f and he seady-sae polynomial equaion k = 1+n [f (f ) 0 k ] simulaneously for f and (f ) 0 o obain f = (f ) 0 = (1+ n) (1 +n) Subsiuing equaions (B2), = ½¼ (1 ¼ ) k ¼ k 1 ¼ ; (B2) ¼ 1 ¼ : (B3) ino (w k )0 gives (w k )0 = (1 ¼ (1 +n) (1+ n) ½¼ ) (1 ¼ ) 1 ¼ 1 ¼ k (1+ n) = (1 ½)¼ : Then 1+n (w k )0 = (1 ½)¼ and hence 1 1+n (w k )0 = 1 (1 ½)¼ > 0. < 0. To prove ha he seady-sae oupu per worker is a decreasing funcion of he ¾ when ½ ¹m ¹ k and k > ¹ k; once again : > 0 for all k > 0; < 0 for ½ ¹m ¹ k and k > ¹ k as shown < 0 as desired. k 10

12 References Azariadis, C., Ineremporal Macroeconomics. Cambridge, MA: Blackwell Publishers, Diamond, P.A., \Naional Deb in a Neoclassical Growh Model," American Economic Review, 55: , delagrandville, O., \In Ques ofhesluskydiamond,"american Economic Review, 79: , Galor, O., \Convergence? Inferences from Theoreical Models," Economic Journal, 106: , Hicks, J.R., The Theory of Wages, London: MacMillan, Klump, R. and de La Grandville, O., \Economic Growh and Elasiciy of Subsiuion: Two Theorems and Some Suggesions," American Economic Review, 90: , Klump, R. and Preissler, H., \CES Producion Funcions and Economic Growh," Scandinavian Journal of Economics, 102:41-56, Pichford, J.D, \Growh and he Elasiciy of Subsiuion," Economic Record, 36: , Solow, R.M., \A Conribuion o he Theory of Economic Growh," Quarerly Journal of Economics, 70:65-94,