Mathematical Systems Theory and Economics I / II pp 241-292 | Cite as

# Applications of Pontryagin’s Maximum Principle to Economics

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## Keywords

Capital Accumulation Capital Good Transversality Condition Balance Growth Balance Growth Path
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## Footnotes for Lecture I

- 1.Frank P. Ramsey, “A Mathematical Theory of Saving”, Economic Journal, Vol. 38,1928, pp. 543–59.CrossRefGoogle Scholar
- 2.Two such important papers are: P. A. Samuelson and R.M. Solow, “A Complete Capital Model Involving Heterogeneous Capital Goods”, Quarterly Journal of Economics, Vol. 70, November 1956, pp. 537–62, and T.C. Koopmans, “On the Concept of Optimal Economic Growth” in Semaine d’Etude sur Le Rôle de l’Analyse Econométrique dans la Formulation de Plans de Developpement, October 1963, Pontifical Academy of Sciences, Vatican City, 1965, pp. 225-87.CrossRefGoogle Scholar
- 3.For the seminal paper in descriptive one-sector growth theory, see R. M. Solow, “A Contribution to the Theory of Economic Growth”, Quarterly Journal of Economics, Vol. 70, February 1956, pp. 65–94.CrossRefGoogle Scholar
- 4.In this lecture, I will omit explicit indications of time dependence when such omission does not lead to confusion.Google Scholar
- 5.For references to the GR literature and its history, see E.S. Phelps, “Second Essay on the Golden Rule of Accumulation”, American Economic Review, Vol. 55, No. 4, September 1965, pp. 793–814.Google Scholar
- 6.See Phelps, op. cit.Google Scholar
- 7.Or consumption per citizen if workers are a given constant fraction of the total population.Google Scholar
- 8.L.S Pontryagin et al., The Mathematical Theory of Optimal Processes, New York and London: Interscience Publishers, 1962. See especially Chapter I, pp. 1–74.zbMATHGoogle Scholar
- 9.Pontryagin et al., op. cit., pp. 17-74. The correspondence between my notation and that of Pontryagin et al. follows: The planning problem treated in this lecture is a special case of the problem treated in my “Optimal Programs of Capital Accumulation for an Economy in Which There is Exogenous Technical Change”, in K. Shell (ed.), Essays on the Theory of Optimal Economic Growth, Cambridge, Mass, and London: M.I.T Press, 1967.Google Scholar
- 10.Pontryagin et al., op.cit., pp. 45-57.Google Scholar
- 11.Cf., e.g., L. S. Pontryagin, Ordinary Differential Equations, Reading Mass.: Addison-Wesley, 1962, especially pp. 159–67 and pp. 192-99.zbMATHGoogle Scholar

## References to the Literature; Applications of the Maximum Principle to the Theory of Optimal Economic Growth.

- 1.Bardhan, P.K., “Optimum Foreign Borrowing”, in K. Shell (ed.), Essays on the Theory of Optimal Economic Growth, Cambridge, Mass. and London: M.I.T. Press, 1967.Google Scholar
- 2.Bruno, M., “Optimal Accumulation in Discrete Capital Models”, in Essays on the Theory of Optimal Economic Growth, op. cit.Google Scholar
- 3.Cass, D., “Optimum Growth in an Aggregative Model of Capital Accumulation”, Review of Economic Studies, Vol. 32 (July 1965), pp. 233–240.CrossRefGoogle Scholar
- 4.Chase, E.S., “Leisure and Consumption” in Essays on the Theory of Optimal Economic Growth, op. cit.Google Scholar
- 5.Datta Chaudhuri, M., “Optimum Allocation of Investment and Transportation in a Two-Region Economy”, in Essays on the Theory of Optimal Economic Growth, op. cit.Google Scholar
- 6.Kurz, M., “Optimal Paths of Capital Accumulation under the Minimum Time Objective”, Econometrica, Vol. 33 (January 1965), pp. 42–66.MathSciNetzbMATHCrossRefGoogle Scholar
- 7.Nordhaus, W.D., “The Optimal Rate and Direction of Technical Change”, in Essays on the Theory of Optimal Economic Growth, op.cit.Google Scholar
- 8.Ryder, H.E., “Optimal Accumulation in an Open Economy of Moderate Size”, in Essays on the Theory of Optimal Economic Growth, op.cit.Google Scholar
- 9.Shell, K., “Toward a Theory of Inventive Activity and Capital Accumulation”, American Economic Review, Vol 56 (May 1966), pp. 62–68.Google Scholar
- 10.Shell, K., “Optimal Programs of Capital Accumulation for an Economy in Which There is Exogenous Technical Change”, in Essays on the Theory of Optimal Economic Growth, op. cit.Google Scholar
- 11.Shell, K., “A Model of Inventive Activity and Capital Accumulation”, in Essays on the Theory of Optimal Economic Growth, op. cit.Google Scholar
- 12.Sheshinski, E., “Optimal Accumulation with Disembodied Learning by Doing”, in Essays on the Theory of Optimal Economic Growth, op. cit.Google Scholar
- 13.Stoleru, L.G., “An Optimal Policy for Economic Growth”, Econometrica, Vol. 33 (April 1965), pp. 321–348.zbMATHCrossRefGoogle Scholar
- 14.Uzawa, H., “Optimum Technical Change in an Aggregative Model of Economic Growth”, International Economic Review, Vol. 6, No. 1 (January 1965), pp.18–31.zbMATHCrossRefGoogle Scholar

## Footnotes to Lecture II

- 1.Meade, J.E., A Neoclassical Theory of Economic Growth, New York: Oxford University Press, 1961Google Scholar
- 2.Uzawa, H., “On a Two-Sector Model of Economic Growth, II”, Review of Economic Studies, Vol. 30 (1963), pp. 105–118.CrossRefGoogle Scholar
- 3.In the terminology of Meade, the factors of production are assumed to be perfectly malleable.Google Scholar
- 4.A very similar problem was treated by H. Uzawa in “Optimal Growth in a Two-Sector Model of Capital Accumulation”, Review of Economic Studies, Vol. 31 (1964), 1–24. In an unpublished paper entitled “A Skeptical Note on Professor Uzawa’s Two-Sector Model”, Professor Wahidul Haque of the University of Toronto points out some difficulties in the Uzawa analysis. The difficulties are avoided in this lecture by using the techniques developed for a more general purpose in my paper “Optimal Programs of Capital Accumulation for an Economy in Which There is Exogenous Technical Change”, in K. Shell (ed.), Essays on the Theory of Optimal Economic Growth, Cambridge, Mass, and London: M.I.T. Press, 1967.CrossRefGoogle Scholar
- 5.As long as k
_{1}(ω) ≠ k_{2}(ω) so that along the production possibility frontier (∂^{2}c/∂z^{2}) < O.Google Scholar - 6.Cf.L.S. Pontryagin, Ordinary Differential Equations, pp. 246-254.Google Scholar
- 7.For the case k
_{1}> k_{2}, we have that (∂p/∂ω) < O. Further aid in the construction of Figure II.3 can be found in my “Optimal Programs of Capital Accumulation for an Economy in Which There is Exogenous Technical Change”, op. cit.Google Scholar - 8.Help in deriving the full analysis can be found in D. Cass, “Optimum Growth in an Aggregative Model of Capital Accumulation”, Review of Economic Studies, Vol. 32 (July 1965), pp. 233–240.CrossRefGoogle Scholar

## Footnotes for Lecture III

- 1.L.S. Pontryagin et al., op. cit., pp. 189-191.Google Scholar
- 2.L.S. Pontryagin et al., op.cit., p. 81.Google Scholar
- 3.Cf., e.g., D. Cass, “Optimum Growth in an Aggregative Model of Capital Accumulation” Review of Economic Studies, Vol.32 (July 1965), pp.233–240.CrossRefGoogle Scholar
- 4.This conjecture is related to a conjecture made by Kenneth J. Arrow in private correspondence.Google Scholar
- 5.Actually Radner’s lecture considered the case of a denumerably infinite number of commodities. He showed that present value although representable by a linear function need not be representable by an inner product of prices and quantities.Google Scholar

## Footnotes to Lecture IV

- 1.K. Shell and J.E. Stiglitz, “The Allocation of Investment in a Dynamic Economy”, Quarterly Journal of Economics, Vol. 81 (November 1967).Google Scholar
- 2.F.H. Hahn, “Equilibrium Dynamics with Heterogeneous Capital Goods”, Quarterly Journal of Economics, Vol 80 (November 1966), pp. 633–646; “On the Stability of Growth Equilibrium”, Memorandum, Institute of Economics, University of Oslo, April 19, 1966.zbMATHCrossRefGoogle Scholar
- Also see P.A. Samuelson, “Indeterminacy of Development in a Heterogeneous-Capital Model with Constant Saving Propensity”, in K. Shell (ed.), Essays on the Theory of Optimal Economic Growth, Cambridge, Mass, and London: M.I.T. Press, 1967.Google Scholar
- 3.Cf., e.g., P.A. Samuelson and R.M. Solow, “A Complete Capital Model Involving Heterogeneous Capital Goods”, Quarterly Journal of Economics, Vol.70 (November 1956), pp. 537–562.CrossRefGoogle Scholar
- Also cf. K. Shell, “Toward a Theory of Inventive Activity and Capital Accumulation” American Economic Pveview, Vol.56 (May 1966) pp. 62–68.Google Scholar
- 4.Or equivalently, a three-sector model in which the capital intensities in all sectors are identical.Google Scholar
- 5.Notice that p. has unit (C/K
_{j}) j = 1, 2. Thus (P_{2}/P_{1}) has unit (K_{1}/K_{2}) which is the slope of the production possibility frontier in (Z_{1}, Z_{2}) space. By the one-sector assumption, the absolute value of the slope of the PPF is unity.Google Scholar - 6.To see that the dimensions of the terms are consistent, write down the units of each as followsGoogle Scholar
- 7.Observe that in this case, at the point where p
_{1}= p_{2}, the system lacks uniqueness of momentary equilibrium. But this nonuniqueness lasts only for a moment. The amount of capital accumulation which occurs during that moment is infinitesimal, regardless of the value which a takes on in that moment. The next moment, p_{2}> p_{1}, and the economy’s path is unaffected by what happens in the moment of nonuniqueness of equilibrium. Hence, although the economy lacks uniqueness of momentary equilibrium, it is not causally indeterminate.Google Scholar - 8.It can be shown that for any given initial endowment there exists one and only one initial price which will get the economy to the OA ray at exactly the same moment that p
_{1}= p_{2}= 1. First, we observe that if the economy remains in Regime I, it must, in finite time, cross OA, since. The right-hand side, in the region below OA, is clearly bounded away from zero, and hence in finite time, no matter what the initial value of k_{1}/k_{2}, the economy eventually reaches OA. The behavior of the real system (i.e., k_{1}, k_{2}) is independent of the particular prices chosen, provided that we remain in Regime I. Hence, the values of k_{1}and k_{2}along the path which goes from the initial value of (k_{1}, k_{2}) to a point on OA are determined at every point of time, and consequently, f_{1}and f_{2}are determined as functions of time alone. Since the Cobb-Douglas production function is analytic, the price differential equation satisfies the Lipschitz condition and hence the price differential equation, with the terminal condition P_{2}(t*) = 1 where t* is the time at which f_{1}= f_{2}, has a unique backward solution.Google Scholar - 9.If the economy begins in A. in Regime II, it either moves into A
_{2}, from which point the story is familiar, or p_{2}= p_{1}, before the economy gets to OG in which case it switches to Regime I. In Regime I and in A_{1}, the economy cannot switch to Regime II and must proceed toward the origin O. The behavior in A_{3}, and A_{4}is symmetrical to that in A_{1}and A_{2}. Observe from Figure IV.2 that an economy in A_{5}or A_{6}ultimately must proceed to A_{1}, A_{2}, A_{3}, or A_{4}.Google Scholar - 10.It turns out that if \( \begin{array}{*{20}{c}} {\lim } \\ {{\text{t}} \to \infty } \end{array}{{\text{p}}_{\text{1}}}({\text{t)}}{{\text{e}}^{{\text{ - R(t)}}}} \ne {\text{O then }}\begin{array}{*{20}{c}} {\lim } \\ {{\text{t}} \to \infty } \end{array}{{\text{p}}_{\text{j}}}({\text{t)e}}{{\text{ }}^{{\text{ - R(t)}}}} = O,({\text{i}} \ne {\text{j)}}{\text{.}} \). Therefore if a capitalist may not resell capital, PDV
_{i}(O) ≠ PDV_{j}(O). But this is a strange restriction. Robert Hall in his M.I.T. Ph.D. dissertation, “Essays on the Theory of Wealth”, 1967, classifies economies in which for some i \( \begin{array}{*{20}{c}} {\lim } \\ {{\text{t}} \to \infty } \end{array}{{\text{p}}_{\text{1}}}{{\text{e}}^{{\text{ - R(t)}}}} \ne {\text{O}} \) as “speculative boom” economies. This definition suggests that there is something basically “unsound” about such economies. However, we already know from Professor Radner’s lectures and from my Lecture III, that there exist efficient competitive equilibrium economies for which the present value linear functional does not have an integral representation.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1969