In ' we have a measure of the percentage change in net product that goes with a 1 per cent change in the intake of labour, when the intake of capital is held constant; but when we try to trace such changes by comparing one industry with another, and the net products of the two industries approximately satisfy, Vi = wLi + rJi, the difference between them will always approximate to the compensation at the wage rate w of the difference in labour intake. The Cobb-Douglas ' and the share of earnings in income will be only two sides of the same penny. (Phelps Brown, 1957, p.557).
V, w, L, r, and J are output (value added at constant prices), the average wage rate, the labour input, the average observed rate of return and the constant price value of the capital stock. (We use V and J to refer to the value measures; Q and K are used below to denote the physical measures of output and capital.) The subscript i denotes the ith firm or industry.
The argument therefore seems to be this. The output elasticity of the Cobb-Douglas production function is defined as (gV/gL)(L/V) and given the assumptions of the neoclassical theory of factor pricing, the marginal prodiuct of labour equals the wage rate, gV/gL = w. Given these assumptions, it also follows that a = wL/V, the share of labour in value added. .
However, as Phelps Brown pointed out in the quotation cited above, there is also an accounting identity that defines the measure of value added for all units of observation, whether they be the firm, the 1, 2, 3 or 4 digit SIC, or the whole economy. This is given by:.
Vi = wLi + rJi (1).
It should be emphasised that there are no behavioural assumptions underlying this equation, in that it is compatible with any degree of competition, increasing or decereasing returns to scale and the existence or not of a well-baehaved underlying production function. J is the constant price measure of the value of the capital stock (normally calculated by the perpetual inventory method) and r is the observed rate of profit normally calculated as the product of the share of profits in value added (1- ') and the output-capital ratio, i.