Returning to the issue of production functions, the 'chain' requirement (a) is not satisfied by a product of single sector production functions. This is because the constant returns requirement must apply to the whole chain. The output of the first (extractive) sector is only one of the inputs to the downstream sectors. Additional inputs of labor and capital are also needed to add value. The expression (16.1) above is consistent with the chain requirement (a) because fE can be interpreted as the physical output of the extraction and
primary processing sector, while g expresses the combined value-added by any number of subsequent downstream sectors.
According to requirement (b) the product fgE must satisfy the Euler condition: it must be a homogeneous first order function of the variables, K, L and E. Since the term E is already first order, the Euler condition holds only if the product fg is homogeneous and of zeroth order in the same three production factors, K, L and E. The condition is satisfied by any function of simple ratios of the variables. (On the other hand we do not want fg to be an explicit function of time t. If t is not an independent variable the production function corresponds to an endogenous growth theory.)
Having calculated f we can now calculate g, from historical GDP data (in constant dollars). The result, GDP in 1992 dollars per unit of 'finished exergy' shows marked peaks during periods of upheaval such as wars and the Depression. The general trend declined during the first half of the century, but increased almost exponentially from 1960. The function g can be interpreted as a rough measure of the 'dematerialization' of the economy, in the very broad sense (counting fuels as materials). However, a more defensible measure of the material intensity of the economy is the ratio of exergy embodied in materials to the total exergy input to the economy. This ratio, although on a completely different scale, follows essentially the same pattern, although the impact of wartime
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