D n

The Hamiltonian function for a dynamical system is now defined as

and the canonical variable p is interpreted as the momentum of a particle. Note that if V is derived from a conservative force field, energy is conserved, so T + V is a constant and the time derivative of H must vanish. Hamilton's equations can then be expressed in a neat canonical form:

dX dt dp dt

The assumption of a single homogeneous capital-cum-consumption good is obviously problematic. However, Samuelson and Solow showed how, in principle, to generalize the Ramsey model to the case of heterogeneous capital goods, and even 'more realistic utility functionals not having the independently additive utilities of a simple integral' (Samuelson and Solow 1956). They concluded, among other things, that

Over extended periods of time an economic society can, in a perfectly straightforward way, reconstruct the composition of its diverse capital goods so that there may remain great heuristic value in the simpler J. B. Clark-Ramsey models of abstract capital substance. (Ibid., pp. 537-8)

In short, the more complicated models are solvable, in principle, though hardly in practice.

The basic Ramsey scheme with homogeneous capital can also be generalized to several variables and their time derivatives. For instance, we could include a variable representing an exergy resource stock R and the rate of change (that is, extraction) of that resource, which would be the current exergy supply E, which also happens to be the negative time derivative of R, R. (See for example Ayres 1988b). Integrals of this sort can be solved by well-known methods.

However, apart from mathematical tractability, there is really no reason to suppose that economic growth follows a consumption-maximizing path in equilibrium. Is maximizing aggregate consumption truly identical with maximizing utility? Are there no other growth drivers to be reckoned with? In fact, there are very strong reasons to suppose that the economy worships several gods, in different ways at different times. As pointed out earlier in this book, and emphasized by many economists, from Schumpeter to Kuznets, Schmookler, Abramovitz, Kaldor and Nelson, economic growth is not an equilibrium process.

But, as we have also noted, the standard model, in which energy plays no role or a minimal one, contradicts economic intuition, not to mention common sense. Indeed, economic history suggests that increasing natural resource (exergy) flows at ever-lower costs are a major fact of history. The declining costs of mechanical or electrical power (physical work per unit of time) in relation to the rising wages of labor have induced ever-increasing substitution of machines (mostly consuming fossil fuels) for human labor, as indicated in Figure 5.2. We think this long-term substitution has been the most important driver of economic growth since the industrial revolution.

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