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 & 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.   Google Scholar

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

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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.   Google Scholar

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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. Google Scholar

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

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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.  Google Scholar

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

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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.  Google Scholar

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

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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.   Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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. Google Scholar

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.   Google Scholar

[2]

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

[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.   Google Scholar

[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. Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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. Google Scholar

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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