e., r = (1- ')V/K .
Consequently, partially differentiating the accounting identity, Vi = wLi + rJi, with respect to L gives gV/gL = w and it follows that (gV/gL)/(L/V) = a = wL/V. The argument stemming from the identity has not made any economic assumptions at all (e.g., it does not rely on the marginal productivity theory of factor pricing or the existence of perfectly competitive markets and optimising behaviour of firms). Consequently, the finding that the putative output elasticities equal the observed factor shares cannot be taken as a test that factors of production are paid their marginal products. This is a position, however, that was not accepted by Douglas (1976) himself. "A considerable body of independent work tends to corroborate the original Cobb-Douglas formula, but more important, the approximate coincidence of the estimated coefficients with the actual shares received also strengthens the competitive theory of distribution and disproves the Marxian." However, it is noticeable that, in his survey, Douglas fails to mention the Phelps Brown (1957) paper.
If the output elasticity of labour and the share of labour's total compensation are merely "two sides of the same penny", could it be that the Cobb-Douglas is simply an alternative way of expressing the income identity and, as such, has no implications for the underlying technology of the economy? This was the proposition that Simon and Levy (1963) proved some eight years later.
Following Simon and Levy (1963) and Intriligator (1978), the isomorphism between the Cobb-Douglas production function and the underlying accounting may be simply shown. The Cobb-Douglas, when estimated using cross-section (firm, industry or regional) data, is specified as:.
.
Vi = ALi 'Ji' (2).
where A is a constant. .
Equation (2) may be expressed as:.
(3).
where and are the values of some reference observations, such as those of the average firm or the base year.