Nancy J. Wulwick
Hamiltonian calculus originated as the mathematical counterpart of the physics of energy in the mid-nineteenth century. Economists have recently adopted the Hamiltonian formalism to develop the theory of optimal growth. 1 To what extent, the essay asks, have economists remarked upon the formal analogy between dynamic optimization in economics and in energetics? To what extent has the fact that the discourses of energetics and economics both translate into a common mathematical language served to legitimize economics? Have economists good reason to think that the analogy is empirically justified? Has the Hamiltonian formalism given economists the power to think freshly about problems of economic growth? Do economic growth theorists attempt to use the formalism adopted from energetics in ways that conflict with restraints imposed by the formalism? Is the calculus overly constrictive? That is, has the calculus, in inducing economists to ignore what it cannot handle, too narrowly circumscribed the issues that economists can treat in dealing with growth?
To answer those questions, the essay introduces the Hamiltonian dynamic system as presented originally by Hamilton (1834) and in the form of modern control theory (Pontryagin et al. 1962). The essay then examines three applications of the Hamiltonian dynamic system to problems of economic growth. The applications appeared in papers by Samuelson and Solow (1956) and Cass (1965), two precursors of New Classical growth theory, and the New Classical economist P. M. Romer (1990). The history of the Hamiltonian in economics is retrospective in that the focus of interest of the essay is on the New Classical growth models; thus the essay emphasizes the economics that the New Classical economists extracted from the Samuelson-Solow and Cass models. The conclusion of the essay suggests that the New Classical economists have adopted a formalism that is inappropriate to help economists better answer the old question, “What are the engines of growth?” (Rebelo 1987:2).
In order to develop its argument, the essay contains more mathematical exposition than most essays in the history and the methodology of economics. Most readers who suffer from mathematics anxiety will be comforted to learn