The following approximation holds for any variable X, when X and do not greatly differ:.
(4) .
Consequently, equation (2) may be written as:.
(5) .
or as:.
(6).
A comparison with the income identity, namely, Vi k wLi + rJi , shows that w = ' or w = a = '. A similar relationship holds between the output elasticity of capital and capital's share. Moreover, (1- '-') equals zero, so that the data will always suggest the existence of constant returns to scale, whatever the true technological relationship. .
What is the implication of all this? Start from the accounting identity and undertake the approximation in the reverse order from that outlined above. The two procedures are formally equivalent. The consequence of this argument is that, for reasonably small variations of L and J and with w and r constant (the last two are not essential, as we shall see below), a Cobb-Douglas multiplicative power function will give a very good approximation to a linear function. Since the linear income identity exists for any underlying technology, we cannot be sure that all that the estimates are picking up is not simply the identity. The fact that a good fit to the Cobb-Douglas relationship is found implies nothing, per se, about such technological parameters as the elasticity of substitution. .
To see this consider Figure 1, which shows the accounting identity expressed as Vi/Li= w + rJi/Li and the Cobb-Douglas relationship as Vi/Li = A(Ji/Li)a. The observations must lie exactly on the accounting identity. We have assumed, for the moment, that w and r are constant. (If they show some variation, then the observations would be a scatter of points around the line where the slope and the intercept represent some average value of r and w.) The Cobb-Douglas approximation is given by the solid curved line, cd, which is tangent to the income identity, ab. If, however, we mistakenly statistically fit a Cobb-Douglas function to these data, we will find the best fit depicted by the dotted curved line, ef.