EconoQuantum ISSN: Universidad de Guadalajara México

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1 EooQuatum ISSN: Uiversidad de Guadalajara Méxio Plata Pérez, Leobardo; Calderó, Eduardo A modified versio of Solow-Ramsey model usig Rihard's growth futio EooQuatum, vol. 6, úm. 1, 2009, pp Uiversidad de Guadalajara Zapopa, Jaliso, Méxio Dispoible e: Cómo itar el artíulo Número ompleto Más iformaió del artíulo Págia de la revista e redaly.org Sistema de Iformaió Cietífia Red de Revistas Cietífias de Améria Latia, el Caribe, España y Portugal Proyeto aadémio si fies de luro, desarrollado bajo la iiiativa de aeso abierto

2 A modified versio of Solow-Ramsey model usig Rihard s growth futio Leobardo Plata Pérez Eduardo Calderó 1 Abstrat: We ivestigate the osequees of itroduig Rihard s Growth futio as a produtio futio i Solow-Swa ad Ramsey models. Poverty traps appear i a atural maer. jel lassifiatio: O41, C61. Itrodutio I this paper, we modify two of the foudatioal models of eoomi growth theory: the Solow-Swa model ad the Ramsey model. We mae the same assumptios that appeared i the origial wors, exept the oe that refers to the eolassial produtio tehology. Istead, we itrodue a Rihard s growth futio. This hage permits us to model the growig proess with ireasig returs whe the use of apital per apita is low ad dereasig returs whe the produtio fator is used i large quatities. Rihard s futio has bee used to model growth proesses i biology. This futio geeralizes Logisti futio ad permits to hage the iflexio poit. Our approah is osistet with the miroeoomi oept the three steps of produtio, whih appears i well ow textboos lie Call & Holaha (1983) or Pidy & Rubifeld (1992). The existee of poverty traps is obtaied with tehologies begiig with ireasig returs. The steady-state preseted i Solow (1956) ad Swa (1956) models are obtaied as a speial ase of this approah. We preset four types of equilibrium i the ase of the Solow- Swa model. We itrodue Rihards s futio i the model of Ramsey (1928) too. The typial saddle poit appears as equilibrium i this ase. 1 Faultad de Eoomía de la Uiversidad Autóoma de Sa Luis Potosí. lplata@ uaslp.mx; jealdero@olmex.mx. We tha the fiaial support provided through the CONACYT Projet from the Mexia Govermet ad C09-FRC from UASLP.

3 66 Mesa 1: Eoomía iteraioal y desarrollo Vol. 6. Núm. 1 Geeralized Solow s model Let us assume the Solow model (Barro ad Sala-i-Marti, 2004, preset the stadard formalizatio of this framewor). We itrodue Rihard s produtio futio as follows. A (1) f () ( 1 + e β σ ) 1 λ All parameters are greater tha ero, f(0)>0 meas that there is some free produtio, A is the limit of f() whe goes to ifiity. The futio is always ireasig i, it is ovex whe if ad oave whe if. The poit if is the ifletio poit whih represets the hage from ireasig returs to dereasig returs. Istead of the lassial Iada oditios lim f () 0, lim f () we ow have that the first 0 oditio is the same but the seod beomes f (0)>0. The fudametal equatio is ow: (2) sa ( 1 + e β σ ) 1 λ ( δ + η) From equatio (2) we a obtai four ases of equilibrium. Case 1: Solow s model ( if 0) This ase appears whe the fudametal equatio has oly oe equilibrium poit. This a our whe the tehology always presets dereasig returs. The equilibrium is a stable poit. Oe example of this is obtaied i the ase of A 100; s 0.3; β l3; σ 2; λ 3; δ 0.1; η f() sf() 240

4 A modified versio of Solow-Ramsey model Case 2: Equilibrium with flutuatios i the eoomi growth rate ( if > 0, good equilibrium) The ifletio poit is greater tha zero, eoomi growth rate presets flutuatios before reahig the uique stable steady state, whih is obtaied at high level of produtio. Oe example is obtaied with the followig values of parameters: A 100; s 0.3; β 25; σ 2; λ 5; δ 0.1; η f() sf() * 240 Case 3: Poverty traps ( if > 0, multiple equilibriums) The ifletio poit is greater tha ero but the eoomy presets several equilibriums. It is easy to he that uder our assumptios there are oly three equilibrium poits. Two of them are loally stable ad the oe i the middle is loally ustable. The first equilibrium that is greater tha zero is a poverty trap. Oly ireasig the output ould allow to leave the poverty trap. I this ase, the followig values a give this result, A 150; s 0.3; β 25; σ 0.1; λ 10; δ 0.1; η 0.05 f() 0.15 sf() 41.3q

5 68 Mesa 1: Eoomía iteraioal y desarrollo Vol. 6. Núm. 1 Case 4: Third world i the third world: ( if > 0, bad equilibrium) This ase appears whe we have oly oe equilibrium but δ+η is very large ompared to margial returs of f(). The eoomy reahes a stable state but produtio level is very low. I this ase, the depreiatio rate is destroyig produtio. f() 0.15 sf() * Geeralized Ramsey s model The model of Ramsey (1928) provides the miroeoomi foudatios to moder eoomi growth theory. As it is well ow, the osumer problem is as follows 1 θ Max U(0) e (ρ η)t t 1 1 θ dt s.t. & f () (δ + η) Usig Rihard s futio, it is easy to he that the eessary oditios produe the followig dyamial system & A ( 1 + e β σ ) 1 λ Cosumptio growth rate is therefore, (δ + η) γ & 1 ( θ A 1 + eβ σ ) 1+λ λ σ e β σ δ ρ λ

6 A modified versio of Solow-Ramsey model There are three solutios. I the first two ases, we have the typial saddle poit equilibrium. I the first ase (ase I), we have the Ramsey model beause the margial produtivity of apital is dereasig for all amout of. I the seod ase (ase II), the margial produtivity of apital is ireasig for small quatities of ad deeasig for large values of. Case I: A 100; β l3; σ 2; λ 3; δ 0.1; η 0.025; ρ Case II: A 100; β 25; σ 2; λ 5; δ 0.1; η 0.025; ρ I the last ase, as i ase III, the margial produtivity of apital is ireasig for small values of ad dereasig for large, but ow we fid that there are two steady states. The first equilibrium, as show i the graph below, is a ustable equilibrium. The seod oe is a saddle poit. The stable trajetory is show i the graph for the followig parameters. 2 2 The other trajetories are disarded for violatig the optimality oditios.

7 70 Mesa 1: Eoomía iteraioal y desarrollo Vol. 6. Núm. 1 Case III: A 150; β 25; σ 0.1; λ 10; δ 0.20; η 0.05; ρ Coludig remar The ilusio of heterogeeous agets i this framewor will be importat for future researh. Relatio betwee iome distributio ad growth rates a be explored i this model. Referees Barro, Robert J., Sala-i-Marti Xavier (2004). Eoomi Growth, New Yor; M Graw Hill. Call, Steve ad William Holaha (1983). Mireoeoomía 2ª Ediió, Editorial Iterameriaa. Pidy, Robert ad Daiel Rubifeld (1992). Miroeoomis 2d. Editio, New Yor: MMilla. Ramsey, Fra (1928). A Mathematial Theory of Savig, Eoomi Joural, Deember 1928 No. 38, pp Solow, Robert (1956). A otributio to the theory of Eoomi Growth, Quarterly Joural of Eoomis, Vol. 70, pp Swa, T. W. (1956). Eoomi Growth ad Capital Aumulatio, Eoomi Reord 32 (November) pp

Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2

Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2 Beroulli Numbers Beroulli umbers are amed after the great Swiss mathematiia Jaob Beroulli5-705 who used these umbers i the power-sum problem. The power-sum problem is to fid a formula for the sum of the