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November  2021, 26(11): 5827-5848. doi: 10.3934/dcdsb.2021178

Population dynamics and economic development

1. 

Department of Economics, University of Foggia, Largo Papa Giovanni Paolo Ⅱ, 1, I–71121 Foggia (FG), Italy

2. 

Department of Law, University of Pisa, Via Collegio Ricci, 10, I–56126 Pisa (PI), Italy

3. 

Department of Law, University of Naples 'Federico Ⅱ', Via Mezzocannone 16, I–80134 Naples (NA), Italy

* Corresponding author: Luca Gori

Received  October 2020 Revised  May 2021 Published  November 2021 Early access  July 2021

Fund Project: The first author is supported by National Natural Science Foundation of China Grant 11801566 and the Fundamental Research Funds for the Central Universities of China grant 19CX02059A

This research develops a continuous-time optimal growth model that accounts for population dynamics resembling the historical pattern of the demographic transition. The Ramsey model then becomes able to generate multiple determinate or indeterminate stationary equilibria and explain the process of the transition from a state with high fertility and low income per capita to a state with low fertility and high income per capita. The article also investigates the emergence of damped or persistent cyclical dynamics.

Citation: Andrea Caravaggio, Luca Gori, Mauro Sodini. Population dynamics and economic development. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5827-5848. doi: 10.3934/dcdsb.2021178
References:
[1]

E. AccinelliJ. G Brida and S. London, Crecimiento económico y trampas de pobreza: ¿cuáles el papel del capital humano?, Inv. Econ., 66 (2007), 97-118.   Google Scholar

[2]

C. Azariadis and A. Drazen, Threshold externalities in economic development, Q. J. Econ., 105 (1990), 501-526.   Google Scholar

[3]

R. J. Barro and X. Sala-i-Martin, Economic Growth, 2$^{nd}$ edition. Cambridge (MA) US: MIT Press, 2003. Google Scholar

[4]

G. S. Becker, An Economic Analysis of Fertility, Demographic and economic change in developing countries. National Bureau Committee for Economic Research. Princeton (NJ) US: Princeton University Press, 1960. Google Scholar

[5]

J. Bhattacharya and S. Chakraborty, Contraception and the demographic transition, Econ. J., 127 (2017), 2263-2301.   Google Scholar

[6]

K. Blackburn and G. P. Cipriani, A model of longevity, fertility and growth, J. Econ. Dynam. Control, 26 (2002), 187-204.   Google Scholar

[7]

D. E. Bloom, D. Canning and J. Sevilla, The Demographic Dividend. A New Perspective on the Economic Consequences of Population Change, Santa Monica (CA) US: RAND Corporation, 2003. Google Scholar

[8]

D. E. BloomD. CanningG. Fink and J. E. Finlay, Fertility, female labor force participation, and the demographic dividend, J. Econ. Growth, 14 (2009), 79-101.   Google Scholar

[9]

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

[10]

A. Bucci and L. Guerrini, Transitional dynamics in the Solow-Swan growth model with AK technology and logistic population change, BE J. Macroecon., 9 (2009). Google Scholar

[11]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev. Econ. Stud., 32, 233–240. Google Scholar

[12]

E. Canton and L. Meijdam, Altruism and the macroeconomic effects of demographic changes, J. Popul. Econ., 10 (1965), 317-334.   Google Scholar

[13]

P. Diamond, National debt in a neoclassical growth model, Am. Econ. Rev., 55 (1965), 1126-1150.   Google Scholar

[14]

L. Fanti and L. Gori, Fertility-related pensions and cyclical instability, J. Popul. Econ., 26 (2013), 1209-1232.   Google Scholar

[15]

L. Fanti and P. Manfredi, The Solow's model with endogenous population: A neoclassical growth cycle model, J. Econ. Dev., 28 (2003), 103-115.   Google Scholar

[16] R. W. Fogel, The Escape from Hunger and Premature Death, Cambridge University Press, New York (NY) US, 2004.   Google Scholar
[17]

H. G. Gaddy, A decade of TFR declines suggests no relationship between development and sub-replacement fertility rebounds, Demogr. Res., 44 (2021), 125-142.   Google Scholar

[18]

D. Gollin, Getting income shares right, J. Polit. Econ., 110 (2002), 458-474.   Google Scholar

[19] O. Galor, Unified Growth Theory, Princeton University Press, Princeton (NJ) US, 2011.   Google Scholar
[20]

O. Galor and D. N. Weil, The gender gap, fertility, and growth, Am. Econ. Rev., 86 (1996), 374-387.   Google Scholar

[21]

O. Galor and D. N. Weil, Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyondm, Am. Econ. Rev., 90 (2000), 806-828.   Google Scholar

[22]

L. Gori and M. Sodini, Endogenous labour supply, endogenous lifetime and economic development, Struct. Change Econ. Dynam., 52 (2020), 238-259.   Google Scholar

[23]

L. Gori and M. Sodini, A contribution to the theory of fertility and economic development, Macroecon. Dynam., 25 (2021), 753-775.   Google Scholar

[24]

L. Gori, P. Manfredi and M. Sodini, A Parsimonious Model of Longevity, Fertility, HIV Transmission and Development, Macroecon. Dynam., Forthcoming, 2019. doi: 10.1017/S1365100519000609.  Google Scholar

[25]

L. GoriE. LupiP. Manfredi and M. Sodini, A contribution to the theory of economic development and the demographic transition: Fertility reversal under the HIV epidemic, J. Demogr. Econ., 86 (2020), 125-155.   Google Scholar

[26]

L. Guerrini, The Solow–Swan model with a bounded population growth rate, J. Math. Econ., 42 (2006), 14-21.  doi: 10.1016/j.jmateco.2005.05.001.  Google Scholar

[27]

C. I. Jones, The Shape of Production Functions And the Direction of Technical Change, NBER Working, 2004. Google Scholar

[28]

S. Kalemli-Ozcan, Does the mortality decline promote economic growth?, J. Econ. Growth, 7 (2002), 411-439.   Google Scholar

[29]

S. Kalemli-Ozcan, AIDS, "reversal" of the demographic transition and economic development: Evidence from Africa, J. Popul. Econ., 25 (2012), 871-897.   Google Scholar

[30]

S. Kalemli-Ozcan and B. Turan, HIV and fertility revisited, J. Dev. Econ., 96 (2011), 61-65.   Google Scholar

[31]

T. C. Koopmans, On the Concept of Optimal Economic Growth, Koopmans TC, ed. The Econometric Approach to Development Planning, Amsterdam: North-Holland, 1965. Google Scholar

[32]

A. B. Krueger, Measuring labor's share, Am. Econ. Rev., 89 (1999), 45-51.   Google Scholar

[33]

P. Krugman, History versus expectations, Q. J. Econ., 106 (1991), 651-667.   Google Scholar

[34]

N. P. Lagerlöf, The Galor–Weil model revisited: A quantitative exercise, Rev. Econ. Dynam., 9 (2006), 116–142. Google Scholar

[35]

H. Leibenstein, Economic Backwardness and Economic Growth, New York (NY) US: Wiley, 1957. Google Scholar

[36]

P. LorentzenJ. McMillan and R. Wacziarg, Death and development, J. Econ. Growth., 13 (2008), 81-124.   Google Scholar

[37]

M. Livi-Bacci, A Concise History of World Population, 6$^{th}$ edition, Malden (MA) US: Wiley-Blackwell, 2017. Google Scholar

[38]

K. Matsuyama, Increasing returns, industrialization, and indeterminacy of equilibrium, Q. J. Econ., 106 (1991), 617-650.   Google Scholar

[39]

S. Marsiglio and D. L. Torre, Population dynamics and utilitarian criteria in the Lucas–Uzawa model, Econ. Model., 29 (2012), 1197-1204.   Google Scholar

[40]

T. Palivos, Endogenous fertility, multiple growth paths, and economic convergence, J. Econ. Dynam. Control, 19 (1995), 1489-1510.   Google Scholar

[41]

K. Prettner and H. Strulik, It's a sin. Contraceptive use, religious beliefs, and long-run economic development, Rev. Dev. Econ., 21 (2017), 543-566.   Google Scholar

[42]

F. P. Ramsey, A mathematical theory of saving, Econ. J., 38 (1928), 543-559.   Google Scholar

[43]

M. Roser, H. Ritchie and E. Ortiz-Ospina, World Population Growth, WorldInData.org., 2 (2013), Retrieved from: https://ourworldindata.org/world-population-growth. Google Scholar

[44]

I. Seidl and C. A. Tisdell, Carrying capacity reconsidered: From Malthus' population theory to cultural carrying capacity, Ecol. Econ., 31 (1999), 395-408.   Google Scholar

[45]

A. Sen, The concept of development, Chenery, H. and T. N. Srinivasan, eds., Handbook of Development Economics, North-Holland, Amsterdam, 1 (1988), 9–26. Google Scholar

[46]

L. Spataro and L. Fanti, The optimal level of debt in an OLG model with endogenous fertility, Ger. Econ. Rev., 12 (2011), 351-369.  doi: 10.1111/j.1468-0475.2011.00534.x.  Google Scholar

[47]

E. Spolaore and R. Wacziarg, How deep are the roots of economic development?, Journal of Economic Literature, 51 (2013), 325-369.   Google Scholar

[48]

R. M. Solow, A contribution to the theory of economic growth, Q. J. Econ., 70 (1956), 65-94.   Google Scholar

[49]

K. Tabata, Inverted U-shaped fertility dynamics, the poverty trap and growth, Econ. Lett., 81 (2003), 241-248.  doi: 10.1016/S0165-1765(03)00188-5.  Google Scholar

[50]

F. Wirl, Stability and limit cycles in one-dimensional dynamic optimisations of competitive agents with a market externality, J. Evol. Econ., 7 (1997), 73-89.   Google Scholar

[51]

A. Yakita, Human capital accumulation, fertility and economic development, J. Econ., 99 (2010), 97-116.   Google Scholar

show all references

References:
[1]

E. AccinelliJ. G Brida and S. London, Crecimiento económico y trampas de pobreza: ¿cuáles el papel del capital humano?, Inv. Econ., 66 (2007), 97-118.   Google Scholar

[2]

C. Azariadis and A. Drazen, Threshold externalities in economic development, Q. J. Econ., 105 (1990), 501-526.   Google Scholar

[3]

R. J. Barro and X. Sala-i-Martin, Economic Growth, 2$^{nd}$ edition. Cambridge (MA) US: MIT Press, 2003. Google Scholar

[4]

G. S. Becker, An Economic Analysis of Fertility, Demographic and economic change in developing countries. National Bureau Committee for Economic Research. Princeton (NJ) US: Princeton University Press, 1960. Google Scholar

[5]

J. Bhattacharya and S. Chakraborty, Contraception and the demographic transition, Econ. J., 127 (2017), 2263-2301.   Google Scholar

[6]

K. Blackburn and G. P. Cipriani, A model of longevity, fertility and growth, J. Econ. Dynam. Control, 26 (2002), 187-204.   Google Scholar

[7]

D. E. Bloom, D. Canning and J. Sevilla, The Demographic Dividend. A New Perspective on the Economic Consequences of Population Change, Santa Monica (CA) US: RAND Corporation, 2003. Google Scholar

[8]

D. E. BloomD. CanningG. Fink and J. E. Finlay, Fertility, female labor force participation, and the demographic dividend, J. Econ. Growth, 14 (2009), 79-101.   Google Scholar

[9]

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

[10]

A. Bucci and L. Guerrini, Transitional dynamics in the Solow-Swan growth model with AK technology and logistic population change, BE J. Macroecon., 9 (2009). Google Scholar

[11]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev. Econ. Stud., 32, 233–240. Google Scholar

[12]

E. Canton and L. Meijdam, Altruism and the macroeconomic effects of demographic changes, J. Popul. Econ., 10 (1965), 317-334.   Google Scholar

[13]

P. Diamond, National debt in a neoclassical growth model, Am. Econ. Rev., 55 (1965), 1126-1150.   Google Scholar

[14]

L. Fanti and L. Gori, Fertility-related pensions and cyclical instability, J. Popul. Econ., 26 (2013), 1209-1232.   Google Scholar

[15]

L. Fanti and P. Manfredi, The Solow's model with endogenous population: A neoclassical growth cycle model, J. Econ. Dev., 28 (2003), 103-115.   Google Scholar

[16] R. W. Fogel, The Escape from Hunger and Premature Death, Cambridge University Press, New York (NY) US, 2004.   Google Scholar
[17]

H. G. Gaddy, A decade of TFR declines suggests no relationship between development and sub-replacement fertility rebounds, Demogr. Res., 44 (2021), 125-142.   Google Scholar

[18]

D. Gollin, Getting income shares right, J. Polit. Econ., 110 (2002), 458-474.   Google Scholar

[19] O. Galor, Unified Growth Theory, Princeton University Press, Princeton (NJ) US, 2011.   Google Scholar
[20]

O. Galor and D. N. Weil, The gender gap, fertility, and growth, Am. Econ. Rev., 86 (1996), 374-387.   Google Scholar

[21]

O. Galor and D. N. Weil, Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyondm, Am. Econ. Rev., 90 (2000), 806-828.   Google Scholar

[22]

L. Gori and M. Sodini, Endogenous labour supply, endogenous lifetime and economic development, Struct. Change Econ. Dynam., 52 (2020), 238-259.   Google Scholar

[23]

L. Gori and M. Sodini, A contribution to the theory of fertility and economic development, Macroecon. Dynam., 25 (2021), 753-775.   Google Scholar

[24]

L. Gori, P. Manfredi and M. Sodini, A Parsimonious Model of Longevity, Fertility, HIV Transmission and Development, Macroecon. Dynam., Forthcoming, 2019. doi: 10.1017/S1365100519000609.  Google Scholar

[25]

L. GoriE. LupiP. Manfredi and M. Sodini, A contribution to the theory of economic development and the demographic transition: Fertility reversal under the HIV epidemic, J. Demogr. Econ., 86 (2020), 125-155.   Google Scholar

[26]

L. Guerrini, The Solow–Swan model with a bounded population growth rate, J. Math. Econ., 42 (2006), 14-21.  doi: 10.1016/j.jmateco.2005.05.001.  Google Scholar

[27]

C. I. Jones, The Shape of Production Functions And the Direction of Technical Change, NBER Working, 2004. Google Scholar

[28]

S. Kalemli-Ozcan, Does the mortality decline promote economic growth?, J. Econ. Growth, 7 (2002), 411-439.   Google Scholar

[29]

S. Kalemli-Ozcan, AIDS, "reversal" of the demographic transition and economic development: Evidence from Africa, J. Popul. Econ., 25 (2012), 871-897.   Google Scholar

[30]

S. Kalemli-Ozcan and B. Turan, HIV and fertility revisited, J. Dev. Econ., 96 (2011), 61-65.   Google Scholar

[31]

T. C. Koopmans, On the Concept of Optimal Economic Growth, Koopmans TC, ed. The Econometric Approach to Development Planning, Amsterdam: North-Holland, 1965. Google Scholar

[32]

A. B. Krueger, Measuring labor's share, Am. Econ. Rev., 89 (1999), 45-51.   Google Scholar

[33]

P. Krugman, History versus expectations, Q. J. Econ., 106 (1991), 651-667.   Google Scholar

[34]

N. P. Lagerlöf, The Galor–Weil model revisited: A quantitative exercise, Rev. Econ. Dynam., 9 (2006), 116–142. Google Scholar

[35]

H. Leibenstein, Economic Backwardness and Economic Growth, New York (NY) US: Wiley, 1957. Google Scholar

[36]

P. LorentzenJ. McMillan and R. Wacziarg, Death and development, J. Econ. Growth., 13 (2008), 81-124.   Google Scholar

[37]

M. Livi-Bacci, A Concise History of World Population, 6$^{th}$ edition, Malden (MA) US: Wiley-Blackwell, 2017. Google Scholar

[38]

K. Matsuyama, Increasing returns, industrialization, and indeterminacy of equilibrium, Q. J. Econ., 106 (1991), 617-650.   Google Scholar

[39]

S. Marsiglio and D. L. Torre, Population dynamics and utilitarian criteria in the Lucas–Uzawa model, Econ. Model., 29 (2012), 1197-1204.   Google Scholar

[40]

T. Palivos, Endogenous fertility, multiple growth paths, and economic convergence, J. Econ. Dynam. Control, 19 (1995), 1489-1510.   Google Scholar

[41]

K. Prettner and H. Strulik, It's a sin. Contraceptive use, religious beliefs, and long-run economic development, Rev. Dev. Econ., 21 (2017), 543-566.   Google Scholar

[42]

F. P. Ramsey, A mathematical theory of saving, Econ. J., 38 (1928), 543-559.   Google Scholar

[43]

M. Roser, H. Ritchie and E. Ortiz-Ospina, World Population Growth, WorldInData.org., 2 (2013), Retrieved from: https://ourworldindata.org/world-population-growth. Google Scholar

[44]

I. Seidl and C. A. Tisdell, Carrying capacity reconsidered: From Malthus' population theory to cultural carrying capacity, Ecol. Econ., 31 (1999), 395-408.   Google Scholar

[45]

A. Sen, The concept of development, Chenery, H. and T. N. Srinivasan, eds., Handbook of Development Economics, North-Holland, Amsterdam, 1 (1988), 9–26. Google Scholar

[46]

L. Spataro and L. Fanti, The optimal level of debt in an OLG model with endogenous fertility, Ger. Econ. Rev., 12 (2011), 351-369.  doi: 10.1111/j.1468-0475.2011.00534.x.  Google Scholar

[47]

E. Spolaore and R. Wacziarg, How deep are the roots of economic development?, Journal of Economic Literature, 51 (2013), 325-369.   Google Scholar

[48]

R. M. Solow, A contribution to the theory of economic growth, Q. J. Econ., 70 (1956), 65-94.   Google Scholar

[49]

K. Tabata, Inverted U-shaped fertility dynamics, the poverty trap and growth, Econ. Lett., 81 (2003), 241-248.  doi: 10.1016/S0165-1765(03)00188-5.  Google Scholar

[50]

F. Wirl, Stability and limit cycles in one-dimensional dynamic optimisations of competitive agents with a market externality, J. Evol. Econ., 7 (1997), 73-89.   Google Scholar

[51]

A. Yakita, Human capital accumulation, fertility and economic development, J. Econ., 99 (2010), 97-116.   Google Scholar

Figure 1.  A portion of the stable manifold on which the trajectories converging to the steady state $ P^{\ast } $ occur. Parameter set: $ \alpha = 0.18 $, $ \beta = 0.84 $, $ \delta = 0.13 $, $ \varepsilon = 0.4 $, $ \rho = 0.3 $, $ \sigma = 1.25 $, $ a = 40.5 $, $ b = 0.5 $. Stationary state equilibrium: $ P^{\ast }: = (k^{\ast },c^{\ast },L^{\ast }) = (0.257,0.749,82.315) $
Figure 2.  Existence of multiple equilibria depending on $ \beta $
Figure 3.  Global indeterminacy. Starting from the same initial conditions $ k_{0} = 0.455 $ and $ L_{0} = 104.843 $, an infinity of initial choices on $ c_{0} $ lead to the indeterminate stationary state equilibrium $ P_{2}^{\ast } $, while there exists a unique choice ($ c_{0}^{1} = 0.833 $) leading to the saddle $ P_{1}^{\ast } $. Parameter set: $ \alpha = 0.18 $, $ \beta = 0.84 $, $ \delta = 0.13 $, $ \varepsilon = 0.4 $, $ \rho = 0.3 $, $ \sigma = 1.25 $, $ a = 40.5 $, $ b = 0.5 $ and $ x = 0.7 $. Stationary state equilibria: $ P_{1}^{\ast }: = (k_{1}^{\ast },c_{1}^{\ast },L_{1}^{\ast }) = (0.594,0.833,80.677) $ and $ P_{2}^{\ast }: = (k_{2}^{\ast },c_{2}^{\ast },L_{2}^{\ast }) = (4.372,2.925,79.676) $
Figure 4.  Local indeterminacy scenario in terms of capital (Panel A) and population size (Panel B)
Figure 5.  Three-dimensional phase portrait in the space $ (k,c,L) $ showing: (i) the trajectory converging to $ P_{1}^{\ast } $ starting from the initial condition $ (k_{0},c_{0}^{1},L_{0}) = (0.346,0.565,3.178) $, and (ii) a trajectory converging to a limit cycle $ \Gamma $ around $ P_{2}^{\ast } $ starting from the initial condition $ (k_{0},c_{0}^{2},L_{0}) = (0.346,0.452,3.178) $. Economic and demographic variables permanently fluctuate around $ P_{2}^{\ast } $. Parameter set: $ \alpha = 0.54 $, $ \beta = 1 $, $ \gamma = 1.5672 $, $ \delta = 0.7 $, $ \varepsilon = 0.4 $, $ \rho = 0.3 $, $ \sigma = 7.04 $, $ a = 0.37 $, $ b = 0.2 $ and $ x = 1.1 $. Stationary state equilibria: $ P_{1}^{\ast }: = (k_{1}^{\ast },c_{1}^{\ast },L_{1}^{\ast }) = (0.221,0.288,3.305) $ and $ P_{2}^{\ast }: = (k_{2}^{\ast },c_{2}^{\ast },L_{2}^{\ast }) = (0.673,0.336,3.03) $
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