# The Economic Growth Engine

## How Energy and Work Drive Material Prosperity

## Robert U. Ayres and Benjamin Warr

# Appendix A: Elasticities of production in neoclassical equilibrium

## Robert U. Ayres and Benjamin Warr

## Extract

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 Ϫ C) at ﬁxed total factor cost C(K, L, X). The cost C is given by C(K, L, X ) ϭ PK · K ϩ PL · L ϩ PX · X (A.1) 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 Ϫ lC must vanish: = 1 Y 2 lC 2 5 =Y 2 l=C 5 c 'Y 'Y 'Y , , d 2 l # 3 PK,PL,PX 4 5 1 0,0,0 2 'K 'L 'X (A.2) It follows from the equality of the individual vector components that the neoclassical condition for economic equilibrium is given by 'Y 5 l # PK 'K 'Y 5 l # PL 'L 'Y 5 l # PX 'X...

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