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 full²edged 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 EulerLagrange 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 highorder equations of the calculus of variations (the
EulerPoisson 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. 85119; 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 (19172002) 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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 Fall '08
 OLIVIERDELAGRANDVILLE
 Derivative, Calculus of variations, Professor Robert Dorfman, Dorfmanian

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