Uzawa(1961) s Steady-State Theorem in Malthusian Model

Transcription

1 MPRA Munich Personal RePEc Archive Uzawa(1961) s Seady-Sae Theorem in Malhusian Model Defu Li and Jiuli Huang April 214 Online a hp://mpra.ub.uni-muenchen.de/55329/ MPRA Paper No , posed 16. April 214 4:4 UTC

2 Uzawa(1961) s Seady-Sae Theorem in Malhusian Model Defu Li School of Economics and managemen, Tongji Universiy Jiuli Huang TEDA College, Nankai Universiy April 214 Absrac: This paper proves ha here is a similar Uzawa (1961) seady-sae growh heorem in a Malhusian model: If ha model possesses seady-sae growh, hen echnical change mus be purely land-augmening and canno include labor augmenaion. Keywords: Malhusian Model, Neoclassical Growh Model, Uzawa s Seady-Sae Theorem JEL Classificaions: O33;O41 Corresponding Auhor: Defu Li jldf@ongji.edu.cn, Tel: , Fax: Posal Address: School of Economics and Managemen, Tongji Universiy, 1239 Siping Road, Shanghai 292, China. 1

3 Uzawa(1961) s Seady-Sae Theorem in Malhusian Model 1. Inroducion Uzawa s heorem (1961) says ha for a neoclassical growh model o exhibi seady-sae growh, he echnological progress mus be Harrod-neural (purely labor-augmening). This resul raises he quesion as o why echnological progress canno be, say, Hicks neural or Solow neural. Many have explicily asked his quesion (see Fellner, 1961; Kennedy, 1964; Samuelson, 1965; Drandakis and Phelps, 1966; Acemoglu, 23, 29; Barro and Sala-i-Marin, 24; Jones, 25; Jones and Scrimgeour, 28) wihou achieving a clear answer, leaving he issue o be a puzzle for he growh heory. However, he above lieraure only discussed he requiremens for he neoclassical growh model concerning he direcion of echnical change. Does a Malhusian model (Malhus, 1798; Ricardo, 1817) also require limiing he direcion of echnical change along a seady-sae growh pah? While here seems o be no lieraure abou his quesion, i is imporan no only for an in-deph undersanding of he Malhusian model iself, bu also for solving he quesion as o why he neoclassical growh model mus limi he echnical change o be Harrod-neural along a seady-sae growh pah. Specifically, by comparing he wo ypes of environmens we can find ou wheher he resricion on he direcion of echnical change in seady-sae growh is special o he neoclassical growh model, or is required in oher models oo. Kremer (1993) consrucs and empirically ess a model of long-run world populaion growh combining he idea ha high populaion spurs echnological change, as implied by many endogenous growh models, wih he Malhusian assumpion ha echnology limis populaion. Lucas (22) resaed he Malhusian model in a neoclassical framework and proved ha even wih echnological progress and capial accumulaion, susained growh of per-capia income canno be achieved in ha environmen. While hese papers discussed he effecs of echnological progress in a Malhusian world, hey did no ask wheher a Malhusian model requires limiing he direcion of echnical change o generae seady-sae growh. Irmen (24) poined ou he srucural similariies beween he Malhusian and he Solow (Solow, 1956) models, bu did no address he aforemenioned quesion eiher. Differen from he above lieraure, his paper focuses precisely on ha quesion. To his end, by using he same mehod as Schlich (26), his paper proves ha for a Malhusian model o exhibi seady-sae growh, echnical change mus be purely 2

4 land-augmening and canno include labor augmenaion. 2. The Malhusian Model Consider an economy wih a neoclassical producion funcion F. In paricular, his funcion relaes, a any poin in ime, he quaniy produced, Y(), o labor inpu L() and land inpu T() and is characerized by consan reurns o scale in hese inpus. Due o echnological progress, i shifs over ime, and we wrie: Y( ) F[ T( ), L( ), ] (1) wih F( T, L, ) F( T, L, ),for all 4 (, L,, ) R T (2) Land inpu, T, grows exponenially a rae : T( ) e T, (3) If =, hen he land is invarian. Bu even hough >, he key resul of Malhusian model will sill be valid. The labor inpu, L, change over ime according o he Malhusian assumpion ha populaion growh depends on he level of income per capia. The higher ha level is, he higher is he birh rae and he lower he moraliy rae, implying a higher rae of populaion growh. Le n () denoe he oal populaion growh rae, and b () and d () he birh and moraliy rae, respecively. Le per-capia income be given by ) Y( ) / L( ). Then he populaion growh funcion is defined as n( ) b[ )] d[ )], b ( )/ ), d( )/ ) (4) From equaion (4), i is obvious ha n( ) b[ )] d[ )] ) ) ) (5) 3 Seady-sae Theorem in he Malhusian Model: If he sysem (1)-(4) possesses a soluion where Y() and L() are all nonnegaive and grow a consan raes, g and n, respecively, hen ( ) F[ T( ), L( ), ] G[ e g T( ), L( )] (6) According o his heorem, exponenial growh requires echnological progress o be purely land-augmening, wih a rae of progress of g-τ. 3

5 Proof : By assumpion we have g Y( ) Ye (7) From equaion (4) and (8), we can obain Taking ime derivaives yields n L( ) Le (8) n b( y) d( y) b[ ye ] d[ ye ] (9) by e dy e (1) which implies ( b d )( g n) (11) According o he Malhusian assumpion: b, d so ha b d. Therefore, we mus have Define g n (12) G( T, L) G( T, L,) (13) As Y G L, ), Y ( T g n Y e, L L e, T T e, and G is linear homogeneous, we can wrie Y Y e g G[ T e ( g ), L e ] (14) As g=n, his proves he heorem. 4 Conclusion This paper proves ha here is a seady-sae growh heorem in a Malhusian model which is analogous o Uzawa s in a neoclassical environmen. In paricular, for a Malhusian model o exhibi seady-sae growh, echnical change mus be purely land-augmening and canno include labor-augmenaion. The resul shows ha he resricion on he direcion of echnical change in seady-sae growh is required no only for he neoclassical growh model bu also for oher models. References: 1. Acemoglu, Daron, 23, Labor- and Capial-Augmening Technical Change, Journal of European Economic Associaion, Vol.1 (1), pp

6 2. Acemoglu, Daron, 29, Inroducion o Modern Economic Growh, Princeon Universiy Press. 3. Barro, Rober and Xavier Sala-i-Marin, 24, Economic Growh. MIT Press, Cambridge, MA. 4. Drandakis, E. M., and Edmund S. Phelps, 1966, A Model of Induced Invenion, Growh, and Disribuion, Economic Journal, Vol. 76 (34), pp Fellner, William, 1961, Two Proposiions in he Theory of Induced Innovaions, Economic Journal, Vol. 71(282), pp Irmen, Andreas, 24, Malhus and Solow a noe on closed form soluions. Economics Bullein, Vol.1, No. 6pp Jones, Charles I., 25, The Shape of Producion Funcions and he Direcion of Technical Change. Quarerly Journal of Economics 2: Jones, Charles I., and Dean Scrimgeour, 28, A New Proof of Uzawa s Seady-Sae Growh Theorem, Review of Economics and Saisics, Vol. 9(1), pp Kennedy, Charles M., 1964, Induced Bias in Innovaion and he Theory of Disribuion, Economic Journal, Vol. 74 (295), pp Kremer, Michael,1993, Populaion Growh and Technological Change: One Million b.c. o 199. Quarerly Journal of Economics 18: Lucas, Rober E., 22, The Indusrial Revoluion: Pas and Fuure. In Lecures in Economic Growh. Cambridge, Mass.: Harvard Universiy Press. 12. Malhus, Thomas R.,1798, An Essay on he Principle of Populaion. London: W. Pickering. 13. Ricardo, David,1817, On he Principles of Poliical Economy and Taxaion. Cambridge: Cambridge Universiy Press, Samuelson, Paul A., 1965, A Theory of Induced Innovaion along Kennedy-Weisäcker Lines, Review of Economics and Saisics, Vol. 47(4), pp Schlich, Ekkehar, 26, A Varian of Uzawa s Theorem, Economics Bullein, Vol. 5 (6), pp Solow, Rober M.,1956, A Conribuion o he Theory of Economic Growh, Quarerly Journal of Economics, Vol. 7, February, pp Uzawa, Hirofumi, 1961, Neural Invenions and he Sabiliy of Growh Equilibrium, Review of Economic Sudies, Vol. 28, February, pp