The Economic Growth Engine
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The Economic Growth Engine

How Energy and Work Drive Material Prosperity

Robert U. Ayres and Benjamin Warr

The historic link between output (GDP) growth and employment has weakened. Since there is no quantitively verifiable economic theory to explain past growth, this unique book explores the fundamental relationship between thermodynamics (physical work) and economics.
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Appendix A: Elasticities of production in neoclassical equilibrium

Robert U. Ayres and Benjamin Warr


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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