Optimal Savings and the Value of Population

Proceedings of The National Academy of Sciences, Vol. 104, No. 47, pp.18421-18426, November 20, 2007

36 Pages Posted: 15 Jan 2008 Last revised: 12 Feb 2008

See all articles by Kenneth J. Arrow

Kenneth J. Arrow

Stanford University - Department of Economics

Alain Bensoussan

University of Texas at Dallas - Naveen Jindal School of Management; Hong Kong Polytechnic University - Faculty of Business; Ajou University

Qi Feng

Purdue University - Krannert School of Management

Suresh Sethi

University of Texas at Dallas - Naveen Jindal School of Management

Date Written: April 2, 2007

Abstract

We study a model of economic growth in which an exogenously changing population enters in the objective function under total utilitarianism and into the state dynamics as the labor input to the production function. We consider an arbitrary population growth until it reaches a critical level (resp. saturation level) at which point it starts growing exponentially (resp. it stops growing altogether). This requires population as well as capital as state variables. By letting the population variable serve as the surrogate of time, we are still able to depict the optimal path and its convergence to the long-run equilibrium on a two-dimensional phase diagram. The phase diagram consists of a transient curve that reaches the classical curve associated with a positive exponential growth at the time the population reaches the critical level. In the case of an asymptotic population saturation, we expect the transient curve to approach the equilibrium as the population approaches its saturation level. Finally, we characterize the approaches to the classical curve and to the equilibrium.

Keywords: economic growth, optimal control, phase diagram, dynamic programming, optimal savings, genuine savings

JEL Classification: O41, C61

Suggested Citation

Arrow, Kenneth J. and Bensoussan, Alain and Feng, Qi and Sethi, Suresh, Optimal Savings and the Value of Population (April 2, 2007). Proceedings of The National Academy of Sciences, Vol. 104, No. 47, pp.18421-18426, November 20, 2007 . Available at SSRN: https://ssrn.com/abstract=1083144

Kenneth J. Arrow

Stanford University - Department of Economics ( email )

Landau Economics Building
579 Serra Mall
Stanford, CA 94305-6072
United States

Alain Bensoussan

University of Texas at Dallas - Naveen Jindal School of Management ( email )

800 West Campbell Rd
SM 30
Richardson, TX 75080-3021
United States
9728836117 (Phone)

HOME PAGE: http://www.utdallas.edu/~axb046100/

Hong Kong Polytechnic University - Faculty of Business

Dept SEEM
Systems Engr * Engr Mgmt
Hong Kong, Hong Kong
China

Ajou University ( email )

Ajou
France

Qi Feng

Purdue University - Krannert School of Management ( email )

1310 Krannert Building
West Lafayette, IN 47907-1310
United States

Suresh Sethi (Contact Author)

University of Texas at Dallas - Naveen Jindal School of Management ( email )

800 W. Campbell Road, SM30
Richardson, TX 75080-3021
United States

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