Infinite-horizon optimal control problems arise naturally in economics when dealing with dynamical models of optimal allocation of resources. Often, these problems are related to the study of economic growth processes. Therefore, they are called optimal economic growth problems. The goal functional to be maximized in optimal economic growth problems has a special form. It is defined by an improper integral that contains an exponential discounting factor. An economic model that leads to an optimal control problem of this type was first studied by F. Ramsey in the 1920s. Ramsey’s model had a strong impact on the formation of the modern economic growth theory, and became a prototype for a variety of further models. In the 1960s-1970s, the development of this field in economics was given great impetus by the discovery of the famous Pontryagin maximum principle.

Typically, optimal economic growth problems assume that the growth process is endless, which gives rise to specific mathematical features of the Pontryagin maximum principle. The most characteristic feature is that the so-called adjoint variables (also treated as “shadow prices”) may exhibit “pathological” behavior in the long run. This fact prevents application of standard versions of the Pontryagin maximum principle (associated with processes of finite durations) and appeals to developing new modifications of the Pontryagin technique, which would pay special attention to the above-mentioned “pathology”.

The aim of the course is to provide an introduction in optimal control theory with a focus on problems arising in growth theory. Recently developed “finite-horizon approximation” approach to deriving a modified maximum principle targeted specifically to infinite-horizon problems of optimal control is presented. The attention is focused on the characterization of the behavior of Pontryagin’s adjoint variables and Hamiltonian in a neighborhood of infinity. The developed methodology is illustrated by few meaningful examples from growth theory.

Prerequisites

The students (both mathematicians and economists) should have basic university knowledge in analysis and differential equations. Any knowledge in optimal control theory is not expected.

Literature

1. S.M. Aseev, A.V. Kryazhimskiy, The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons, SIAM Journal on Control and Optimization, 43, No. 3 (2004), 1094-1119.
2. S.M. Aseev, A.V. Kryazhimskiy, The Pontryagin maximum principle and optimal economic growth problems, Proceedings of the Steklov Institute of Mathematics, 257 (2007), 5-255.
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7. F.P. Ramsey, A mathematical theory of saving, Econ. J., 38 (1928) , 543—559.
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9. M.L. Weitzman, Income, Wealth, and the Maximum Principle, Cambridge, MA, Harvard Univ. Press, 2003.