This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Footnotes for Lecture I
Frank P. Ramsey, “A Mathematical Theory of Saving”, Economic Journal, Vol. 38,1928, pp. 543–59.
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.
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.
In this lecture, I will omit explicit indications of time dependence when such omission does not lead to confusion.
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.
See Phelps, op. cit.
Or consumption per citizen if workers are a given constant fraction of the total population.
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.
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.
Pontryagin et al., op.cit., pp. 45-57.
Cf., e.g., L. S. Pontryagin, Ordinary Differential Equations, Reading Mass.: Addison-Wesley, 1962, especially pp. 159–67 and pp. 192-99.
References to the Literature; Applications of the Maximum Principle to the Theory of Optimal Economic Growth.
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.
Bruno, M., “Optimal Accumulation in Discrete Capital Models”, in Essays on the Theory of Optimal Economic Growth, op. cit.
Cass, D., “Optimum Growth in an Aggregative Model of Capital Accumulation”, Review of Economic Studies, Vol. 32 (July 1965), pp. 233–240.
Chase, E.S., “Leisure and Consumption” in Essays on the Theory of Optimal Economic Growth, op. cit.
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.
Kurz, M., “Optimal Paths of Capital Accumulation under the Minimum Time Objective”, Econometrica, Vol. 33 (January 1965), pp. 42–66.
Nordhaus, W.D., “The Optimal Rate and Direction of Technical Change”, in Essays on the Theory of Optimal Economic Growth, op.cit.
Ryder, H.E., “Optimal Accumulation in an Open Economy of Moderate Size”, in Essays on the Theory of Optimal Economic Growth, op.cit.
Shell, K., “Toward a Theory of Inventive Activity and Capital Accumulation”, American Economic Review, Vol 56 (May 1966), pp. 62–68.
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.
Shell, K., “A Model of Inventive Activity and Capital Accumulation”, in Essays on the Theory of Optimal Economic Growth, op. cit.
Sheshinski, E., “Optimal Accumulation with Disembodied Learning by Doing”, in Essays on the Theory of Optimal Economic Growth, op. cit.
Stoleru, L.G., “An Optimal Policy for Economic Growth”, Econometrica, Vol. 33 (April 1965), pp. 321–348.
Uzawa, H., “Optimum Technical Change in an Aggregative Model of Economic Growth”, International Economic Review, Vol. 6, No. 1 (January 1965), pp.18–31.
Footnotes to Lecture II
Meade, J.E., A Neoclassical Theory of Economic Growth, New York: Oxford University Press, 1961
Uzawa, H., “On a Two-Sector Model of Economic Growth, II”, Review of Economic Studies, Vol. 30 (1963), pp. 105–118.
In the terminology of Meade, the factors of production are assumed to be perfectly malleable.
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.
As long as k1 (ω) ≠ k2(ω) so that along the production possibility frontier (∂2c/∂z2) < O.
Cf.L.S. Pontryagin, Ordinary Differential Equations, pp. 246-254.
For the case k1> k2, 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.
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.
Footnotes for Lecture III
L.S. Pontryagin et al., op. cit., pp. 189-191.
L.S. Pontryagin et al., op.cit., p. 81.
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.
This conjecture is related to a conjecture made by Kenneth J. Arrow in private correspondence.
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.
Footnotes to Lecture IV
K. Shell and J.E. Stiglitz, “The Allocation of Investment in a Dynamic Economy”, Quarterly Journal of Economics, Vol. 81 (November 1967).
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.
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.
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.
Also cf. K. Shell, “Toward a Theory of Inventive Activity and Capital Accumulation” American Economic Pveview, Vol.56 (May 1966) pp. 62–68.
Or equivalently, a three-sector model in which the capital intensities in all sectors are identical.
Notice that p. has unit (C/Kj) j = 1, 2. Thus (P2/P1) has unit (K1/K2) which is the slope of the production possibility frontier in (Z1, Z2) space. By the one-sector assumption, the absolute value of the slope of the PPF is unity.
To see that the dimensions of the terms are consistent, write down the units of each as follows
Observe that in this case, at the point where p1 = p2, 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, p2 > p1, 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.
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 p1 = p2 = 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 k1/k2, the economy eventually reaches OA. The behavior of the real system (i.e., k1, k2) is independent of the particular prices chosen, provided that we remain in Regime I. Hence, the values of k1 and k2 along the path which goes from the initial value of (k1, k2) to a point on OA are determined at every point of time, and consequently, f1 and f2 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 P2(t*) = 1 where t* is the time at which f1 = f2, has a unique backward solution.
If the economy begins in A. in Regime II, it either moves into A2, from which point the story is familiar, or p2 = p1, before the economy gets to OG in which case it switches to Regime I. In Regime I and in A1, the economy cannot switch to Regime II and must proceed toward the origin O. The behavior in A3, and A4 is symmetrical to that in A1 and A2. Observe from Figure IV.2 that an economy in A5 or A6 ultimately must proceed to A1, A2, A3, or A4.
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, PDVi(O) ≠ PDVj(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.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1969 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shell, K. (1969). Applications of Pontryagin’s Maximum Principle to Economics. In: Kuhn, H.W., Szegö, G.P. (eds) Mathematical Systems Theory and Economics I / II. Lecture Notes in Operations Research and Mathematical Economics, vol 11/12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46196-5_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-46196-5_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-04635-6
Online ISBN: 978-3-642-46196-5
eBook Packages: Springer Book Archive