06 August 2021

Why it is illegal to use any differentiable production functions in economics

In the 18th and 19th centuries, economists spent a lot of time thinking about what determined the division of income between land, labour and capital. As the importance of agriculture diminished, the question transformed into one of determining how income was distributed between capital and labour. The problem was difficult and an answer elusive.

Then neoclassical economics resolved the problem in a nifty way. It drew up a production function showing how output was mathematically determined by physical inputs of capital and labour. When a partial derivative of the production function with respect to labour was taken one arrived at the marginal product of labour. A partial derivative of the production function with respect to capital gave the marginal return on capital. Clearly, the use of mathematics could resolve puzzles that verbal arguments could not.

On closer examination, however, the solution turns out to be not so nifty. A partial derivative of the production function with respect to labour means holding capital constant and determining the change in output due to a marginal change in labour alone. A partial derivative of the production function with respect to capital means holding labour constant and determining the change in output due to a marginal change in capital alone.

The problem of course is that a capitalist cannot produce an output without the aid of a worker. And a worker cannot produce an output unless he makes use of capital supplied by a capitalist. Else, the capitalist would not be a capitalist and the worker would not be a worker. It is not possible to produce an output by holding labour constant and varying only capital or by holding capital constant and varying only labour So, the partial derivative of the production function with respect to labour and the partial derivative with respect to capital do not exist. A production function like the Cobb-Douglas production function must therefore be ruled out because it has both a partial derivative with respect to labour and a partial derivative with respect to capital. Other production functions in common use in neoclassical economics must be rejected for the same reason.

Tens of well-known economics papers are therefore invalid. One thinks of such classics as Technical Change and the Aggregate Production Function by Robert Solow (1957) or Capital-Labor Substitution and Economic Efficiency by Kenneth Arrow, H.B. Chenery, B.S. Minhas and Robert Solow (1961). Real Business Cycle theory which makes use of similar production functions must also be disqualified for the same reason.

There is no denying that substitution between labour and capital can take place. A capitalist who employs 50 workers to manufacture shoes manually can set up an automated assembly line with five workers. But that substitution does not take place at the margin. Since it is not possible to disentangle the contributions of capital and labour in the final product, neoclassical economics is far from having found a solution to the question that exercised the minds of economists in the 18th and 19th centuries.

Both the capitalist and the worker can lay claim to the marginal product, which is a joint result of capital and labour. But the decision about how the rewards flowing from the marginal product are divided is made by the capitalist. He does not make it in a vacuum, though. If labour unions are strong or there is demand for the worker's labour from other capitalists the worker will get a greater share than otherwise. Similarly, if he has sufficient accumulated saving, he can hold out for a greater share whereas if he is asset-destitute he has to settle for whatever the capitalist gives him. The division depends on the balance of power between the capitalist and the worker. There are clear parallels between the marginal product of labour and marginal utility. In both cases, neoclassical economics invents a variable which has no basis in reality. Then it justifies this by constructing a function which it assumes to be differentiable, when, even by internal logic, there is no evidence of differentiability.

Looking back, it is astonishing how much of neoclassical economics is based on mathematical errors.

Category: Economics

Philip George
Understanding Keynes to go beyond him