## Introducing Useful Work U

where f is the overall technical efficiency of conversion of 'raw' exergy inputs E into useful work output U. Note that E and U are measured in the same (energy) units, whence the ratio f = U/E is a dimension-less number. It can be interpreted as the efficiency of conversion of raw materials taken from nature into useful work (including finished materials).

To summarize: while discarding most of the neoclassical equilibrium and optimality assumptions as unnecessary, we retain the assumption that a production function of three factors (variables) is definable and meaning-ful.10 We also retain (notwithstanding some reservations) the assumption of constant returns to scale, meaning that the production function must be a homogeneous function of the first order (Euler condition). Hence, the term g on the right-hand side of Equation 6.21 can be interpreted as an aggregate production function provided it is homogeneous of order zero with arguments labor L, capital K, and useful work U.

The calculation of E and U and the calculation of the efficiency factor f are major computational undertakings in themselves, since much of the underlying data is not published, as such, in official government statistics. The time series for useful work U must be constructed from other time series and information about the history of technology. Details of these calculations, for the US, were presented in Chapter 4.

As already noted, the new variable U is an intermediate product, meaning that it is an output generated by one sector and utilized by another sector (or sectors) within the economy. A single-sector model is not adequate for the same reason already explained: at least two sectors are necessary. The first sector produces the intermediate product U from inputs of capital K*, labor L* and some fraction of the useful work U* (the exergy inputs to useful work can be regarded as free gifts of nature). It follows that the capital K*, labor L* and U* needed to produce the aggregate useful work output U should therefore be subtracted from the total inputs of K and L in the production function, to avoid double counting. In principle, as inputs to the first sector, one should calculate K*, L* and U* and subtract them from the totals K, L and U, respectively. Let

On the other hand, the second sector Y2 produces all 'downstream' goods and services (that is, GDP) from inputs of capital K - K*, labor L - L* and useful work U - U*.

However, it seems reasonable to postulate, as a first approximation, that capital, labor and useful work are used in the same proportions in the production of useful work U as they are in the economy as a whole. In fact, we assume that the mathematical form of the production functions Y1, Y2 and Y are identical in form, except for a constant multiplier. This being so, it follows that

whence we can write

It follows that

and therefore

Actually the above logic is not only applicable to the simple Cobb-Douglas case. It also applies to any production function that is homogeneous and of order unity, including the so-called LINEX function discussed next. To be sure, it is possible that the 'mix' of labor, capital and useful work inputs to the primary sector is slightly different than the mix of inputs applicable to the secondary (or other) sectors. For instance, the primary extraction and conversion sector may be slightly more capital-intensive and less labor-intensive than the downstream sector(s). However, adjusting for such small differences is a second-order correction.

Conceptually, the cost of producing useful work can be equated with the monetary value of the capital and labor consumed in the extractive and primary processing sector, plus the amount of useful work consumed within that sector. However, there are no quantitative data for any of these factors. Among the components of useful work, only electric power has a market price. This is undoubtedly a limitation on our model, although hopefully not a critical flaw.

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