Abstract A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N-person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.

Abstract：
A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N-person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.

Berkovitz, L.D. Optimal control theory. Springer-Verlag, New York, 1974

[3]

Björk, T., Murgoci, A. A general theory of Markovian time inconsistent stochasitc control problem. working paper (September 17, 2010). Available at SSRN: http://ssrn.com/abstract=1694759

[4]

Böhm-Bawerk, E.V. The positive theory of capital. Books for Libraries Press, Freeport, New York, 1891

[5]

Ekeland, I., Lazrak, A. Being serious about non-commitment: subgame perfect equilibrium in continuous time. Available online: http://arxiv.org/abs/math/0604264

[6]

Ekeland, I., Privu, T. Investment and consumption without commitment. Math. Finan. Econ., 2: 57-86 (2008)

Grenadier, S.R., Wang, N. Investment under uncertainty and time-inconsistent preferences. Journal of Financial Economics, 84: 2-39 (2007)

[9]

Herings, P.J., Rohde, K.I.M. Time-inconsistent preferences in a general equilibriub model. Econom. Theory, 29: 591-619 (2006)

[10]

Hume, D. A Treatise of Human Nature. First Edition, 1739; Reprint, Oxford Univ. Press, New York, 1978

[11]

Jevons, W.S. Theory of Political Economy. Mcmillan, London, 1871

[12]

Krusell, P., Smith, A.A.Jr. Consumption and saving decisions with quasi-geometric discounting. Econometrica, 71: 366-375 (2003)

[13]

Laibson, D. Golden eggs and hyperbolic discounting. Quarterly J. Econ., 112: 443-477 (1997)

[14]

Malthus, A. An essay on the principle of population, 1826; The Works of Thomas Robert Malthus, Vols. 2-3, Edited by E. A. Wrigley and D. Souden, W. Pickering, London, 1986

[15]

Marin-Solano, J., Navas, J. Non-constant discounting in finite horizon: the free terminal time case. J. Economic Dynamics and Control, 33: 666-675 (2009)

[16]

Marshall, A. Principles of Economics. 1st ed., 1890; 8th ed., Macmillan, London, 1920

[17]

Miller, M., Salmon, M. Dynamic games and the time inconsistency of optimal policy in open economics. The Economic Journal, 95: 124-137 (1985)

[18]

Palacios-Huerta, I. Time-inconsistent preferences in Adam Smith and Davis Hume. History of Political Economy, 35: 241-268 (2003)

[19]

Pareto, V. Manuel d’′economie politique. Girard and Brieve, Paris, 1909

[20]

Peleg, B., Yaari, M.E. On the existence of a consistent course of action when tastes are changing. Review of Economic Studies, 40: 391-401 (1973)