The Cobb-Douglas function discussed above is the simplest solution of the growth and integrability conditions. However, the C-D function has serious weaknesses. The major weakness from our perspective is the built-in assumption that marginal productivities and elasticities of all factors are constant over the whole century. That assumption would be inconsistent with technological change.
Another approach (first demonstrated by Kümmel) is to choose the next-simplest non-trivial solutions of the growth equation and integrability equations (Kümmel 1980; Kümmel et al. 1985). This was done by selecting plausible mathematical expressions for the output elasticities a, ß and g based on asymptotic boundary conditions. To satisfy the Euler condition, these must be homogeneous zeroth order functions of the independent variables. Since the elasticities are partial logarithmic derivatives of the output Y (by definition), one can perform the appropriate partial integrations to obtain the corresponding production function, except for a constant term.
The first of Kümmel's proposed solutions can be thought of as a form of the law of diminishing returns (to capital). It is an asymptotic boundary condition conveying the notion that even in a hypothetical capital-i ntensive future state, in which all products are produced by machines, some irreducible need for labor L and exergy E will remain, namely:
Kümmel's second equation reflects the continuing substitution of labor by capital and exergy as capital intensity (automation) increases:
The assumption of constant returns to scale implies that, at every moment in time, g = 1 - a - b (6.17)
which is the constant returns condition. Partial integration of the growth equation yields the so-called LINEX (linear-exponential) function:
Y = AEexp a1'2(2 2 ('"IT)) + 31'261'>(E 2 ')
The functions (of time) a(t) and b(t) have been characterized by Kümmel as 'capital efficiency' and 'energy demand' respectively. It turns out that the multiplier A can be set equal to unity.
Not surprisingly, with time-dependent parameters a(t) and b(t), the GDP fits can be extremely good. On the other hand, neither a(t) nor b(t) has a straightforward economic interpretation. Hence, such a model is not ideal for forecasting. What is interesting, however, is the resulting calculated time-dependent productivities, which show a significant increase in exergy productivity and a decline in labor productivity, over time.8
We now propose a true two-sector model with a third factor consisting of 'useful work' (denoted U) performed by the economy, as a whole. By definition, the product of resource (exergy) inputs E times conversion efficiency f is equal to useful work performed U. There are two ways to measure E, one of which includes biomass (agricultural and forest products) plus non-fuel minerals, while the other version is limited to commercial fuels and other commercial energy sources.9 Having adopted the convention of an aggregate production function of the variables K, L and E, and a multi-sector 'process chain' approximation, we can write:
Evidently f1 is the conversion efficiency of the resource (exergy) inflow E into the first level intermediate product /1; this occurs in the first (extractive) sector. In the second sector, I1 is converted with efficiency f2 into the second intermediate product I2, and so on. The term g is just the ratio of output Y to the last intermediate product. Equation 6.19 is still an identity. It becomes a model only when we specify the intermediate products and functional forms.
As a first approximation, it is now convenient to assume that the economy is a two-stage system with a single intermediate product, denoted U. (To those skeptics who correctly point out that a two-stage approximation is much too simple for realism, we note that most of economic growth theory to date postulates a single-stage, single-sector, composite product model.) Then we have, to a first approximation:
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