July  2020, 7(3): 197-208. doi: 10.3934/jdg.2020014

The Solow-Swan model with endogenous population growth

1. 

Departamento de Métodos Cuantitativos - Facultad de Ciencias Económicas y de Administración, Universidad de la República (Uruguay), Eduardo Acevedo 1139, Montevideo, CP 11200, Uruguay

2. 

Instituto de Economía - Facultad de Ciencias Económicas y de Administración, Universidad de la República (Uruguay), Lauro Müller 1921, Montevideo, CP 11200, Uruguay

Received  February 2020 Revised  April 2020 Published  July 2020

Fund Project: Our research was supported by CSIC-UDELAR (project "Grupo de investigación en Dinámica Económica"; ID 881928, Initiation to Research Program - 2017 "Crecimiento económico y dinámica de la población: teoría y análisis empírico"; ID 406)

This paper presents a reformulation of the classical Solow-Swan growth model where a dynamic of the endogenous population is incorporated. In our model, the population growth rate continually depends on per capita consumption. We find that – as in the classic Solow-Swan model – there is a steady state for the capital-labour ratio, which is always lower than that deduced from the original model with zero population growth rate, but it is not necessarily unique. Under certain conditions, there is an odd amount, and only the smallest and the largest are locally stable. Finally, a study of comparative static of stationary states is performed by varying the total factor productivity, and the results are compared with those of the original model. It is found that the effects of exogenous variables on endogenous variables differ from the original model.

Citation: Gaston Cayssials, Santiago Picasso. The Solow-Swan model with endogenous population growth. Journal of Dynamics and Games, 2020, 7 (3) : 197-208. doi: 10.3934/jdg.2020014
References:
[1]

E. Accinelli and J. G. Brida, Population growth and the Solow-Swan model, International Journal of Ecological Economics and Statistics, 8 (2007), 54-63. 

[2]

E. Accinelli and J. G. Brida, The Ramsey model with logistic population growth, Economics Bulletin, 3 (2007), 1-8. 

[3]

E. Accinelli and J. G. Brida, The dynamics of the Ramsey economic growth model with the von Bertalanffy population growth law, Applied Mathematical Sciences, 1 (2007), 109-118. 

[4]

N. S. Bay, On the attraction of positive equilibrium point in Solow economic discrete model with Richards population growth, Journal of Applied Mathematics and Bioinformatics, 2 (2012), 177.

[5]

G. Becker, An economic analysis of fertlity, in Demographic and Economic Change in Developed Countries, Universites-Natonal Bureau Commitee for Economic Research.

[6]

G. S. Becker and R. J. Barro, A reformulation of the economic theory of fertility, The Quarterly Journal of Economics, 103 (1988), 1-25.  doi: 10.3386/w1793.

[7]

G. BeckerK. Murphy and R. Tamura, Human capital, fertility and economic growth, Journal of Politcal Economy, 98 (1990), 12-37. 

[8]

G. Becker and G. Lewis, On the interaction between the quantity and quality of the children, Journal of Politcal Economy, 81 (1973), 279-288.  doi: 10.1086/260166.

[9]

J. Benhabib and K. Nishimura, Endogenous fuctuations in the Barro-Becker theory of fertility, Demographic Change and Economic Development, (1989), 29–41.

[10]

J. G. Brida and E. L. Maldonado, Closed form solutions to a generalization of the Solow growth model, Applied Mathematical Sciences, 1 (2010), 1991-2000. 

[11]

D. Cai, An economic growth model with endogenous carrying capacity and demographic transition, Mathematical and Computer Modelling, 55 (2012), 432-441.  doi: 10.1016/j.mcm.2011.08.022.

[12]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Review of Economic Studies, 32 (1965), 233-240.  doi: 10.2307/2295827.

[13]

L. C. Corchón, A Malthus-Swan-Solow model of economic growth, Journal of Dynamics & Games, 3 (2016), 225-230.  doi: 10.3934/jdg.2016012.

[14]

M. Ferrara and L. Guerrini, The Ramsey model with logistic population growth and benthamite felicity function revisited, Wseas transactions on mathematics, 3 (2009), 97-106.  doi: 10.1017/s0013091500002637.

[15]

M. Ferrara and L. Guerrini, Hopf bifurcation in a modified Ramsey model with delay, Far East Journal of Mathematical Sciences, 68 (2012), 219-225. 

[16]

O. Galor and D. N. Weil, Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond, American Economic Review, 90 (2000), 806-828.  doi: 10.1257/aer.90.4.806.

[17]

O. Galor, From stagnation to growth: Unified growth theory, Handbook of Economic Growth, (2005), 171–293. doi: 10.2139/ssrn.651526.

[18]

T. C. Koopmans, On the concept of optimal economic growth, in The Econometric Approach to Development Planning, North-Holland, Amsterdam, 1965.

[19]

N. G. MankiwD. Romer and D. N. Weil, A contribution to the empirics of economic growth, The Quarterly Journal of Economics, 107 (1992), 407-437. 

[20]

F. P. Ramsey, A Mathematical Theory of Saving, Economic Journal, 38 (1928), 543-59.  doi: 10.2307/2224098.

[21]

P. M. Romer, Endogenous growth and technical change, Journal of Political Economy, 98 (1990), 71-102. 

[22]

R. Solow, A contribution to the theory of economic growth, The Quarterly Journal of Economics, 70 (1956), 65-94.  doi: 10.2307/1884513.

[23]

T. W. Swan, Economic Growth and Capital Accumulation, Economic Record, 32 (1956), 334-361.  doi: 10.1111/j.1475-4932.1956.tb00434.x.

[24]

United Nations, Department of Economic and Social Affairs, Population Division, World Population Prospects: The 2015 Revision, Key Findings and Advance Tables, Working Paper No. ESA/P/WP.241, 2015.

show all references

References:
[1]

E. Accinelli and J. G. Brida, Population growth and the Solow-Swan model, International Journal of Ecological Economics and Statistics, 8 (2007), 54-63. 

[2]

E. Accinelli and J. G. Brida, The Ramsey model with logistic population growth, Economics Bulletin, 3 (2007), 1-8. 

[3]

E. Accinelli and J. G. Brida, The dynamics of the Ramsey economic growth model with the von Bertalanffy population growth law, Applied Mathematical Sciences, 1 (2007), 109-118. 

[4]

N. S. Bay, On the attraction of positive equilibrium point in Solow economic discrete model with Richards population growth, Journal of Applied Mathematics and Bioinformatics, 2 (2012), 177.

[5]

G. Becker, An economic analysis of fertlity, in Demographic and Economic Change in Developed Countries, Universites-Natonal Bureau Commitee for Economic Research.

[6]

G. S. Becker and R. J. Barro, A reformulation of the economic theory of fertility, The Quarterly Journal of Economics, 103 (1988), 1-25.  doi: 10.3386/w1793.

[7]

G. BeckerK. Murphy and R. Tamura, Human capital, fertility and economic growth, Journal of Politcal Economy, 98 (1990), 12-37. 

[8]

G. Becker and G. Lewis, On the interaction between the quantity and quality of the children, Journal of Politcal Economy, 81 (1973), 279-288.  doi: 10.1086/260166.

[9]

J. Benhabib and K. Nishimura, Endogenous fuctuations in the Barro-Becker theory of fertility, Demographic Change and Economic Development, (1989), 29–41.

[10]

J. G. Brida and E. L. Maldonado, Closed form solutions to a generalization of the Solow growth model, Applied Mathematical Sciences, 1 (2010), 1991-2000. 

[11]

D. Cai, An economic growth model with endogenous carrying capacity and demographic transition, Mathematical and Computer Modelling, 55 (2012), 432-441.  doi: 10.1016/j.mcm.2011.08.022.

[12]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Review of Economic Studies, 32 (1965), 233-240.  doi: 10.2307/2295827.

[13]

L. C. Corchón, A Malthus-Swan-Solow model of economic growth, Journal of Dynamics & Games, 3 (2016), 225-230.  doi: 10.3934/jdg.2016012.

[14]

M. Ferrara and L. Guerrini, The Ramsey model with logistic population growth and benthamite felicity function revisited, Wseas transactions on mathematics, 3 (2009), 97-106.  doi: 10.1017/s0013091500002637.

[15]

M. Ferrara and L. Guerrini, Hopf bifurcation in a modified Ramsey model with delay, Far East Journal of Mathematical Sciences, 68 (2012), 219-225. 

[16]

O. Galor and D. N. Weil, Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond, American Economic Review, 90 (2000), 806-828.  doi: 10.1257/aer.90.4.806.

[17]

O. Galor, From stagnation to growth: Unified growth theory, Handbook of Economic Growth, (2005), 171–293. doi: 10.2139/ssrn.651526.

[18]

T. C. Koopmans, On the concept of optimal economic growth, in The Econometric Approach to Development Planning, North-Holland, Amsterdam, 1965.

[19]

N. G. MankiwD. Romer and D. N. Weil, A contribution to the empirics of economic growth, The Quarterly Journal of Economics, 107 (1992), 407-437. 

[20]

F. P. Ramsey, A Mathematical Theory of Saving, Economic Journal, 38 (1928), 543-59.  doi: 10.2307/2224098.

[21]

P. M. Romer, Endogenous growth and technical change, Journal of Political Economy, 98 (1990), 71-102. 

[22]

R. Solow, A contribution to the theory of economic growth, The Quarterly Journal of Economics, 70 (1956), 65-94.  doi: 10.2307/1884513.

[23]

T. W. Swan, Economic Growth and Capital Accumulation, Economic Record, 32 (1956), 334-361.  doi: 10.1111/j.1475-4932.1956.tb00434.x.

[24]

United Nations, Department of Economic and Social Affairs, Population Division, World Population Prospects: The 2015 Revision, Key Findings and Advance Tables, Working Paper No. ESA/P/WP.241, 2015.

Figure 1.  Evolution of the world population. Period 1750-2100

Source: Max Roser and Esteban Ortiz-Ospina (2019) – "Future population growth". Published online at OurWorldInData.org. Retrieved from: https://ourworldindata.org/world-population-growth [Online Resource].

Figure 2.  World population growth rate. Period 1950-2100

Source: United State Census Bureau (September 2018) and United Nations (2019).

Figure 3.  Speed of convergence
Figure 4.  Even amount of equilibrias
Figure 5.  Possible dynamic solution
Figure 6.  Multiplex equilibria
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