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O_CHAPTER8 - Chapter 8 Other Major Tools for Optimal Growth...

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Chapter 8.Other Major Tools for Optimal GrowthTheory: ThePontryaginMaximum Principle and the Dorfmanian Complex dynamic systems, implying in particular inequality constraints have imposed to extend the classical results of the calculus of variations. This work has been carried after the Second World War by Richard Bellman in the U.S., in the form of dynamic programming, and by Lev Pontryagin and his associates in Russia. This latter contribution is also called °optimal control theory±; its central result is known as the Pontryagin maximum principle. The full-²edged maximum principle requires some very advanced mathe- matics and its proof extends over some 50 pages 1 . We will give here only the result in its simplest form, using the economic interpretation of this principle that was given by Robert Dorfman 2 (1969). The power of Robert Dorfman³s analysis will be obvious: not only does it permit to solve dynamic optimi- sation problems, but it will also enable us to obtain the classical results of the calculus of variations (the Euler-Lagrange equation and its extensions) through economic reasoning. But the reason why Dorfman³s contribution is so important is that it goes well beyond a very clever, intuitive, explanation of the Pontryagin maximum principle. Robert Dorfman introduced a new Hamiltonian, which has pro- found economic signi´cance. To honor Professor Dorfman³s memory, we call this ±modi´ed Hamiltonian±a Dorfmanian. In turn this Dorfmanian can be extended to obtain all high-order equations of the calculus of variations (the Euler-Poisson and the Ostrogradski equations). 1 The original contribution by Pontryagin and his associate is : L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes (translated by K.N. Trirogo/). New York, 1962. Very precise and complete results of optimal control theory can be found in K.J. Arrow, ±Applications of Control Theory to Economic Growth±, American Mathematical Society, Mathematics of the Decision Sciences, Part 2. Providence 1968, pp. 85-119; K.D. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy. Stanford University Institute for Mathematical Studies in the Social Sciences, 1968. 2 Professor Robert Dorfman (1917-2002) taught at Berkeley and at Harvard Univer- sity. His article °An Economic Interpretation of Optimal Control Theory± (1969) is a masterpiece. 1
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The reader will notice in this chapter and the coming ones that many tools are at its disposal to solve the simple variational problems of optimal economic growth: he can use the classical calculus of variations, the Pontrya- gin maximum principle and the Dorfmanian. He may then ask two questions: is one of the methods to be prefered? And if the choice between methods is completely open, would some be redundant? For simple problems like the ones we will deal with, there is indeed complete equivalency in the methods.
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