Book contents
- Frontmatter
- Contents
- Introduction
- PART I POSITIVE GROWTH THEORY
- PART II OPTIMAL GROWTH THEORY
- 7 Optimal growth theory: an introduction to the calculus of variations
- 8 Other major tools for optimal growth theory: the Pontryagin maximum principle and the Dorfmanian
- 9 First applications to optimal growth
- 10 Optimal growth and the optimal savings rate
- PART III A UNIFIED APPROACH
- In conclusion: on the convergence of ideas and values through civilizations
- Further reading, data on growth and references
- Index
8 - Other major tools for optimal growth theory: the Pontryagin maximum principle and the Dorfmanian
Published online by Cambridge University Press: 01 February 2010
- Frontmatter
- Contents
- Introduction
- PART I POSITIVE GROWTH THEORY
- PART II OPTIMAL GROWTH THEORY
- 7 Optimal growth theory: an introduction to the calculus of variations
- 8 Other major tools for optimal growth theory: the Pontryagin maximum principle and the Dorfmanian
- 9 First applications to optimal growth
- 10 Optimal growth and the optimal savings rate
- PART III A UNIFIED APPROACH
- In conclusion: on the convergence of ideas and values through civilizations
- Further reading, data on growth and references
- Index
Summary
Complex dynamic systems, in particular those with inequality constraints, have made it necessary to extend the classical calculus of variations. This work has been carried after the Second World War by Richard Bellman in the United States, 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-fledged maximum principle requires some very advanced mathematics and its proof extends over some 50 pages. We will indicate here only the result in its simplest form, using the beautiful economic interpretation of this principle that was given by Robert Dorfman (1969). The power of Robert Dorfman's analysis will be obvious: not only does it permit us to solve dynamic optimization 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 Dorfman's 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 profound economic significance. To honour Professor Dorfman's memory, we call this “modified 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).
- Type
- Chapter
- Information
- Economic GrowthA Unified Approach, pp. 199 - 209Publisher: Cambridge University PressPrint publication year: 2009