The Linex Model With Useful Work As A Third Factor
It is clear that the argument for introducing exergy E as a third factor in Equation 6.3 applies equally well to useful work U. We have therefore modified the scheme of Kümmel et al. by substituting useful work U for commercial energy (exergy) inputs E in their LINEX production function, bearing in mind that our underlying model economy must have at least two sectors because U is explicitly an intermediate product.11 The major justification for this formulation is the hope that all of the time dependence of 'technical progress' can be explained in terms of K, L and U. We also postulate that 8 = 0 and that a and b may be taken to be constants, independent of time, although we also consider the timedependent case.
The assumed marginal productivities are given by Equations 6.15 and 6.16. The constant returns to scale (Euler) condition, Equation 6.4 (also Equation 6.17), also holds. Partial integration and exponentiation yields the timeindependent linearexponential (LINEX) function analogous to Equation 6.18, except that U replaces E and A = 1:
We note that the above LINEX function satisfies the three socalled Inada conditions with respect to capital K, namely Y(0) = 0; Y'(0) = Y'(œ) = 0 (Inada 1963). Comparing Equation 6.18 with Equation 6.21, it is clear that the function g can be written which is a zeroth order homogeneous function of the variables, as required for constant returns to scale. In principle, a and b could still be functions of time.
It is interesting to note that by equating the two models for GDP, namely the CD function (Equation 6.1) and the LINEX function (Equation 6.18), one can obtain an expression for the A(t) multiplier in Equation 6.1, in terms of K, L and U, namely
It is evident that A(t) in this formulation is strongly dependent on U, and more weakly (and inversely) dependent on K and L. These variables are functions of time, of course, and U is the product of resource exergy input E times exergy conversion efficiency f as in Equations 6.19 and 6.20. In short, if the model (Equation 6.31) can be parametrized to fit the actual
GDP data reasonably well, A(t) can be explained approximately as a function of resource conversion efficiency. Numerical results and interpretations are discussed in Chapter 7.
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