## Exergy In The Cobbdouglas Model Of Growth

The simplest mathematical form that satisfies the constant returns to scale (Euler) condition and integrability conditions (Appendix A) is the single-sector, two-factor Cobb-Douglas production function:

where the constant returns condition implies that a + b = 1 (6.2)

It is traditional (as noted in Chapter 5) to interpret the marginal productivities a and b (elasticities of output) as factor payments shares for capital and labor in the national accounts. This is convenient because the national accounts are actually constructed in terms of payments to labor (wages, salaries) and payments to capital (interest, dividends, royalties). This makes such an interpretation seem natural.

It seems natural in this spirit to add a third factor such as exergy E, as follows:

where A(t) is the 'Solow residual', that is, the growth component that is not explained by either capital accumulation or increased labor supply. The constant returns condition implies that a + b + g = 1 (6.4)

The factor payments shares interpretation is not valid, however, when a third factor is introduced. As already explained, this is because segregating 'payments to exergy' amounts to considering exergy production as a separate sector, or sectors. Payments to 'exergy' are really payments to farmers, lumber companies, coal mines or oil and gas producers, mostly for labor and capital. These firms taken as a group constitute a sector or sectors. As a fraction of all payments (GDP), payments to this sector are comparatively small, that is, only 4 percent to 5 percent for most OECD countries. This implies - according to the standard neoclassical (single sector) interpretation noted in the last chapter - that the marginal productivity of resource inputs must be correspondingly small, too small to account for consumer price changes or GDP growth changes (for example, Denison 1979). The income allocation theorem (Appendix A), which is based on a single sector, single 'composite' product model, does not hold for a multi-sector, multi-product model.

The growth equation is the total time derivative of the production function, dY _ (a 'K b ''L g 'E 1 'A dt VK dt L dt E dt A dt

The last term reflects the possibility that some part of the growth cannot be explained in terms of K, L, E and is therefore a function of time alone.

We can now define the four output elasticities a, b, g and S, where S can be thought of as the marginal productivity S of 'technical progress' as follows, assuming constant returns to scale:

dIn Y KdY

g = ——- = 1 — a — b (constant returns) (6.8) d ln E

where a, b and g are all functions of K, L and E. The integrability conditions are not trivial. Mathematically, they require that the second-order mixed derivatives of the production function Y with respect to all factors K, L, E must be equal. In words, these conditions imply that the integrals along any two paths between two points in factor space are equal. It is quite conceivable that this condition might not hold. If it does not hold, integrals along different paths between the same two points would depend on the path. The economic interpretation of such a situation might be a regime change, such as the breakdown of the centrally planned Soviet economy in 1989 and its replacement by free-market capitalism. The integrability condition requires that a a a K— + L— + E— = 0 (6.10)

The most general solutions to these three equations are:

The simplest (trivial) solutions are constants, namely: a = a0, ß = ß0 and g = 1 - a - ß. We consider other solutions of the above equations later. For the single-sector two-factor case, we then obtain the original Cobb-Douglas function where a0 + ß0 = 1 (g = 0) and the usual choices for a0 and ß0 are 0.3 and 0.7, corresponding to the time-averaged cost shares for capital and labor, respectively, in the national accounts.

Figures 6.3a and 6.3b graph the key factors of production, for the US and Japan, over the period 1900-2004. Figure 6.4 (for the US) shows clearly that the C-D function with resource inputs E as a third independent variable, but retaining the constant returns condition and with an exponent (corresponding to marginal productivity) proportional to the share of payments to resource inputs in the national accounts, does not explain historical US growth over the long run. Similar results could easily be shown for Japan and other industrialized countries.7

Reverting to the standard Solow model, and its accompanying assumptions, A(t) can be fitted independently to the unexplained residual that was once called 'technological progress' or, more recently, total factor productivity (TFP). We have done this, as shown in Figure 6.5. The 'best fit' for the technical progress function over the whole period 1900-98 (shown in the graph) is A(t) = exp[0.05&? (t -1900)] where t is the year. In other words, throughout the 20th century, growth attributable to exogenous technical progress or TFP in the US has averaged 3.9 percent per annum. However, there have been significant deviations from the average growth rate in certain periods, for example, below trend in the 1930s and above trend in the early postwar decades.

It is important to recognize that the third factor E is not truly independent of the other two. This means that not all combinations of the three factors are actually possible. In particular, capital and resource flows are strongly - and obviously - synergistic, hence correlated. Indeed, capital - except for residential housing and money - can be defined for our purposes as the collection of all energy-conversion machines and information-processing equipment plus structures to contain and move them. Thus capital goods are activated by energy (exergy) flows, while exergy has no economic function in the absence of capital goods.

The Cobb-Douglas function assumes constant marginal productivities over the entire century from 1900-98. This is also unrealistic. The essential result that holds true in general is the following: including resource (exergy) inputs in the model as a third factor of production cannot explain long-term growth, but the imputed marginal productivity of resource inputs is much greater than the factor-payments share (for example, Kümmel et al. 1985, 2000; McKibben and Wilcoxen 1994, 1995; Bagnoli et al. 1996). We will arrive at a similar conclusion subsequently by a different route, in Chapter 7. Figure 6.3a GDP and factors of production (USA, 1900-2005) Figure 6.3b GDP and factors of production (Japan, 1900-2005) Figure 6.4 US GDP, 1900-2000 (actual versus three-factor Cobb-Douglas function, L (0.70), K (0.26), E (0.04)) Year

Figure 6.5 Technological progress function and Solow residual (USA, 1900-2005)

Year

Figure 6.5 Technological progress function and Solow residual (USA, 1900-2005) ## Guide to Alternative Fuels

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