## Appendix A Elasticities of production in neoclassical equilibrium

Neoclassical equilibrium in a system that produces a single output Y from the factors K, L, X is characterized by the maximum of profit (Y - C) at fixed total factor cost C(K, L, X). The cost C is given by

where PK, PL, PX are the unit prices of capital K, labor L and a third factor X (which need not be specified, although it can be equated either to commercial energy, E, or to useful work U).

Neoclassical economics assumes that all combinations of factors that are consistent with fixed total cost C are accessible without any further constraints, that is, they are mutually substitutable. This implies that the profit maximum lies somewhere within the interior of accessible K, L, X space (that is, not on a boundary). According to the Lagrange multiplication rule, the necessary condition for a local extremum in K, L, X space is that, in equilibrium, for some real number l, the gradient of Y - 1C must vanish:

dYdYdY dK' dL 'dX

It follows from the equality of the individual vector components that the neoclassical condition for economic equilibrium is given by dY dK dY dL dY dX "

(The special case of zero profit, where all of the output is allocated to the factor owners, corresponds to l = 1). Now multiply the first of these equations by K/Y; the second by L/Y and the third by X/Y, and introduce the elasticities a for K, b for L and g for X, as follows:

Y dK

Then, in equilibrium,

Finally, given constant returns to scale (a + b + g = 1) we get

Y = Y • (a + b + g) = l • (PKK + PLL + PXX) = l • C (A.6)

Substituting Y = 1C in the equilibrium conditions for a, b, and g, one obtains:

g = ~cr which are the cost shares of the three factors.

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