Abstract
This paper investigates an infinite-horizon problem in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak.
Similar content being viewed by others
References
Aghion, Ph. and Howitt, P., Endogeneous Growth Theory, Cambridge, MA: MIT Press, 1998.
Asheim, G., Ethical Preferences in the Presence of Resource Constraints, Nord. J. Political Economy, 1996, vol. 23, pp. 55–67.
Asheim, G., Justifying, Characterizing and Indicating Sustainability, Dordrecht: Springer Netherlands, 2007.
Bardi, M. and Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton — Jacobi — Bellman Equations, Boston, MA: Birkhäuser, 1997.
Barro, R. and Sala-i-Martin, X., Economic Growth, 2nd ed., Cambridge, MA: MIT Press, 2003.
Blanchard, O. J. and Fischer, S., Lectures on Macroeconomics, Cambridge, MA: MIT Press, 1989.
Caplin, A. and Leahy, J., The Recursive Approach to Time Inconsistency, J. Econ. Theory, 2006, vol. 131, pp. 134–156.
Carr, J., Applications of Centre Manifold Theory, Appl. Math. Sci., vol. 35, New York: Springer, 1981.
Cass, D., Optimal Growth in an Aggregative Model of Capital Accumulation, Rev. Econ. Stud., 1965, vol. 32, pp. 233–240.
Chichilnisky, G., An Axiomatic Approach to Sustainable Development, Soc. Choice Welfare, 1996, vol. 13, pp. 231–257.
Chichilnisky, G., What Is Sustainable Development?, Land Econ., 1997, vol. 4, pp. 467–491.
Ekeland, I. and Lazrak, A., Equilibrium Policies When Preferences Are Time-Inconsistent, http://arxiv.org/abs/math/0808.3790 (2006).
Ekeland, I. and Lazrak, A., Being Serious about Non-Commitment: Subgame Perfect Equilibrium in Continuous Time, http://arxiv.org/abs/math/0604264 (2006).
Ekeland, I. and Lazrak, A., The Golden Rule When Preferences Are Time-Inconsistent, Math. Financ. Econ., 2010, vol. 4, pp. 29–55.
Ekeland, I., From Frank Ramsey to René Thom: A Classical Problem in the Calculus of Variations Leading to an Implicit Differential Equation, Discrete Contin. Dyn. Syst., 2010, vol. 28, no. 3, pp. 1101–1119.
Ekeland, I., Karp, L., and Sumaila, R., Equilibrium Management of Fisheries with Overlapping Altruistic Generations: Working Paper, http://www.ceremade.dauphine.fr/~ekeland/Articles/Karp.pdf (2011).
Ekeland, I. and Témam, R., Convex Analysis and Variational Problems, Classics in Appl. Math., vol. 28, Philadelphia, PA: SIAM, 1987.
Shane, F., Loewenstein, G., and O’Donoghue, T., Time Discounting and Time Preference: A Critical Review, J. Econ. Lit., 2002, vol. 40, pp. 351–401.
Hartman, Ph., Ordinary Differential Equations, Classics in Appl. Math., vol. 38, Philadelphia, PA: SIAM, 2002.
Heal, G., Valuing the Future: Economic Theory and Sustainability, New York: Columbia Univ. Press, 1998.
Heal, G., Intertemporal Welfare Economics and the Environment, in Handbook of Environmental Economics: Vol. 3, K.-G. Mäler and J.R. Vincent (Eds.), Amsterdam: Elsevier, 2005.
Karp, L., Non-Constant Discounting in Continuous Time, J. Econ. Theory, 2007, vol. 132, pp. 577–568.
Karp, L. and Lee, I. H., Time-Consistent Policies, J. Econ. Theory, 2003, vol. 112, pp. 353–364.
Krusell, P. and Smith, A., Consumption-Savings Decisions with Quasi-Geometric Discounting, Econometrica, 2003, vol. 71, pp. 365–375.
Koopmans, T., On the Concept of Optimal Economic Growth, in The Economic Approach to Development Planning, Amsterdam: Elsevier, 1965.
Kuznetsov, Yu.A., Elements of Applied Bifurcation Theory, 3rd ed., Appl. Math. Sci., vol. 112, New York: Springer, 2004.
Harris, C. and Laibson, D., Dynamic Choices of Hyperbolic Consumers, Econometrica, 2001, vol. 69, pp. 935–957.
Li, Ch.-Zh. and Löfgren, K.-G., Renewable Resources and Economic Sustainability: A Dynamic Analysis with Heterogeneous Time Preferences, J. Environmental Econ. Management, 2000, vol. 40, pp. 236–250.
Phelps, E. and Pollak, R. A., On Second-Best National Saving and Game-Equilibrium Growth, Rev. Econ. Stud., 1968, vol. 35, pp. 185–199.
Phelps, E., The Indeterminacy of Game-Equilibrium Growth, in Altruism, Morality and Economic Theory, E. S. Phelps (Ed.), New York: Russell Sage Found., 1975, pp. 87–105.
Ramsey, F.P., A Mathematical Theory of Saving, Econ. J., 1928, vol. 38, pp. 543–559.
Romer, D., Advanced Macroeconomics, 4th ed., New York: McGraw-Hill, 2011.
Sumaila, U.R. and Walters, C. J., Intergenerational Discounting: A New Intuitive Approach, Ecological Econ., 2005, vol. 52, pp. 135–142.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Alain Chenciner on his 70th birthday
Rights and permissions
About this article
Cite this article
Ekeland, I., Long, Y. & Zhou, Q. A new class of problems in the calculus of variations. Regul. Chaot. Dyn. 18, 553–584 (2013). https://doi.org/10.1134/S1560354713060014
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354713060014