1. Maddison subdivides capital into several categories (machines, structures, etc.) with different average lifetimes. The method is obviously subject to criticism, but to date nobody seems to have come up with an improvement that is workable.

2. The most consistent and persistent skeptic over the years has probably been Franklin Fisher (Fisher 1965, 1968, 1969a, 1969b, 1987, 1993; Felipe and Fisher 2003).

3. The language here is suggestive of an energy (or information) theory of value. Unfortunately, perhaps, the term 'value-added' is so thoroughly established in economics that it cannot reasonably be avoided. In any case, we are not espousing the discredited energy theory of value. For a more thorough discussion of the economy as a self-organized system of concentrating 'useful information', see Ayres (1994a, chapter 8).

4. It is worthwhile pointing out that Robert Solow's (1956) criticism of the Harrod-Domar model was to note that the so-called 'razor's edge' property of that model (which called for a very precise and impracticable matching of capital investment to labor-force growth) was a consequence of the assumption of fixed coefficients. Solow (and Swan) subsequently offered a theory that characterized technological advance as a shift in the production function (Solow 1957; Swan 1956).

5. Friedman actually said 'truly important and significant hypotheses will be found to have "assumptions" that are widely inaccurate, descriptive representations of reality, and in general the more significant the theory, the more unrealistic the assumptions, in this sense'. He went on to say 'To be important, therefore, a hypothesis must be prescrip-tively false in its assumptions'. His remarks have been generally interpreted to mean that the validity (that is, non-falsification) of a theory depends only on its predictive ability, not on the realism of its assumptions (van den Bergh et al. 2000). It should be noted that Friedman's remarks were intended to defend the unrealistic assumptions of neoclassical microeconomics against critics.

6. The theory of 'distance functions' that has recently emerged explicitly recognizes this fact (Faere and Grosskopf 1994; Faere 1988; Faere and Grosskopf 1993; Faere et al. 1994).

7. Retaining the constant returns condition but relaxing the (one-sector) assumption that productivity equals payments share in the national accounts enables a crude statistical fit, using OLS regression, with E as a third variable, and no time-dependent multiplier. (As it happens, this procedure is spurious, because the underlying distribution of residuals is not Gaussian, as it should be for OLS regressions to be valid.) In this case, the regression yields a negative value (—0.76) for the exponent of labor (L), a positive value (0.56) for the exponent of capital (K) and a positive value (+1.20) for the exponent of exergy E. In the case of Japan, the OLS 'best fit' exponents, with exergy as a third variable, are all positive and in the range [0-1]. But the fit itself is rather poor after 1980. In both countries, the fit is considerably better with U as the third factor. (Again, the OLS regression is spurious.)

8. Kümmel and colleagues have obtained extremely close fits for three countries using the LINEX function with energy (exergy) as the third variable, and fitting the functions a(t) and b(t) by a logistic function or a Taylor expansion, resulting in a five-parameter model. Fits have been obtained for the US and the Federal Republic of Germany (total economy, 1960-98), and for Japan (industrial output) over the period 1965-95. In all three cases, the R2 value is 0.999 and the Durbin-Watson coefficient is quite good

(DW= 1.46 for the US, 1.64 for Germany and 1.71 for Japan). The German results are remarkable, since they refer only to West Germany before 1990 and the merger with the so-called German Democratic Republic (GDR) in that year (Lindenberger et al. 2007).

9. Both versions of each variable, r and u have been tested statistically (see Ayres and Warr 2003). Both versions are defined and measured in terms of the thermodynamic measure already introduced. The more inclusive definition of resource inputs consistently provides a significantly better fit to the GDP data, regardless of choice of production function. We have done the OLS fits both with and without the constraint of constant returns. Without constant returns, the sum of the three calculated output elasticities turns out to be of the order of 1.3, which is implausibly high.

10. We do not assume that firms must operate on, or move along, the 'frontier' (by substitution among factors) as they would have to do if they were price-taking profit-maximizers operating at the least-cost point with perfect information in a perfectly competitive market. On the contrary, we regard the 'frontier' as the (fuzzy) locus of points in K-L-E space such that firms operating inside at a given time are uncompetitive and likely to decline, whereas firms outside the frontier are more likely to survive and grow. However, success or failure in an evolutionary model is not instantaneous, and a firm operating inside the frontier may be able to restructure or innovate to improve its competitive situation. This view is theoretically inconsistent with constant returns, atomistic competition, differentiability and various other assumptions underlying the notion of the production function (Sylos Labini 1995). For our purposes, we rely on the fact that there seems to be an empirical phenomenon that is consistent with the notion of aggregate capital.

11. The three-factor version of the Cobb-Douglas and LINEX models are already implicitly two-sector models since, in practice, the cost of exergy input E is not defined in terms of payments to 'nature' but rather to extractive industries that own natural resources, namely coal-mining, oil and gas drilling and hydro-electricity.

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