The Pontryagin maximum principle and problems of optimal economic growth. (English) Zbl 1215.49001

Aseev, S. M.; Kryazhimskiĭ, A. V.

Proc. Steklov Inst. Math. 257, 1-255 (2007); translation from Tr. Mat. Inst. Steklova 257, 5-271 (2007); ISBN: 978-5-02-035603-0.

This extensive paper is divided in two chapters, the first containing the mathematics underpinning the economic models (mainly a version of the maximum principle for finite dimensional control systems) and various economic applications. The second chapter is devoted to a single model (see below).
Chapter 1. The (finite-dimensional differential autonomous) system is
\[ x'(t) = f(x(t), u(t)) \quad (t \geq 0), \qquad x(0) = x_0, \tag{1} \]
where the measurable control \(u(t)\) belongs to a control set \(U\). The infinite time horizon functional to be maximized is
\[ J(x, u) = \int_0^\infty e^{- \rho t} g(x(t), u(t))\,dt. \tag{2} \]
The assumptions on (1) are standard: differentiability of \(f(x, u)\) in \(x\) and continuity in \((x, u)\), a growth condition insuring existence of solutions for all \(t \geq 0\) and convexity conditions implying (together with additional requirements on \(g(x, u))\) the existence of optimal controls (the case where solutions may blow up in finite time is also briefly covered). The setting in most treatments of control problems for (1) is a finite interval \([0, T];\) the present infinite time horizon situation (which the authors consider essential in the treatment of economic systems) introduces complications illustrated with the following example. If we make a “change of time” \(\tau(t) = 1 - e^{-\rho t}\iff t(\tau) = - \log (1 - \tau)/ \rho\) the time interval becomes \(0 \leq \tau \leq 1.\) The functional retains its regularity,
\[ J(x, u) = {1 \over \rho}\int_0^1 g(x(\tau), u(\tau)) d\tau, \]
but the system loses it: it becomes
\[ x'(\tau) = {1 \over \rho (1 - \tau)}f(x(\tau), u(\tau)), \]
whose right hand side blows up at \(\tau = 1,\) the terminal time of the control interval.
After various examples in Section 1, the mathematics begins in Section 2 with an existence proof for optimal controls (based on a finite horizon approximation). Then the validity of Pontryagin’s maximum principle for (1)–(2) is established (giving, as usual, necessary conditions for optimality of a pair \((x(t), u(t))\) in terms of the adjoint variable \(p(t))\) together with transversality conditions at infinity. In Section 5 the results are applied to the problem of optimal investment in basic production assets of an abstract enterprise where the coordinates of the vector \(p(t)\) are interpreted as marginal or shadow prices. In the rest of the chapter various additions to the theory are given. Among these: an approximation of the control problem (1)–(2) by a sequence of finite horizon problems, proved first for an affine system
\[ x'(t) = f_0(x(t)) + \sum_{j=1}^m f_j(x(t)) u^j(t) \]
and then for the general system, and sufficient optimality conditions based on the maximum principle. This chapter also includes further examples and economical applications such as models of economic growth.
Chapter 2. The mathematical machinery developed in the first chapter is applied to the study of two interconnected macroeconomic systems (e.g., countries). In the second system (the technological follower) the R&D (research and development) sector is largely oriented to absorption of part of the knowledge produced by the R&D sector of the first system (the technological leader). The object is to study the optimal behavior of the technological follower as a subsystem.
The paper contains an ample bibliography of 83 items and (especially at the end of Chapter 1) a detailed comparison of the results with earlier results in the literature.

Reviewer: Hector O. Fattorini (Los Angeles)

MSC:

49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49K15	Optimality conditions for problems involving ordinary differential equations
91B62	Economic growth models
91B74	Economic models of real-world systems (e.g., electricity markets, etc.)
91B32	Resource and cost allocation (including fair division, apportionment, etc.)
91B02	Fundamental topics (basic mathematics, methodology; applicable to economics in general)

Keywords:

Pontryagin’s maximum principle; infinite horizon; adjoint vector; models of economic growth; optimal investments; production assets; marginal prices; shadow prices; technological leader; technological follower

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