An ambiguity in production functions

I had never given much thought to production functions, a standard way for economists to formalise thinking about production, until a rather frustrating discussion with commenter Martin on Scott Sumner's Money Illusion blog provoked me into thinking about them a bit more and I came to realise there is an ambiguity in production functions. Or, at least, an easy way to be misled about them.

The standard production function is the Cobb-Douglas production function:



Where

▪ Y = total production (the monetary value of all goods produced in a year)
▪ L = labor input
▪ K = capital input
▪ A = total factor productivity
α and β are the output elasticities of labor and capital, respectively: values that are constants determined by available technology.

The ambiguity is what do we mean by input? It is easy to think of the input as, for example, machinery (capital) and workers (labour). But that is not true. You can have all the machinery and workers you like, but if they are not actually doing anything, there is no production.

The inputs are the using of the machinery and the labour: the capital and labour services. Measuring total amounts of capital and labour available may tell us something about total production capacity, but they do not tell us anything about actual production. The function itself is about use, not capacity.

And the using of the capital and labour involves all sorts of extra complexities. How efficiently are they being used? How well allocated are they? What incentives to use them efficiently are operating? What information flows are operating? What risks are being generated? How are they being managed? Yes, total factor productivity (A) (aka the Solow residual) is part of the function but, as Wikipedia correctly says, the Solow residual “is a question rather than an answer”.

In running a firm, having the capital and labour is merely the first step. The difficult bits are in directing and managing their effective use. Which is then an iterative process, as capacity is accumulated or dispensed with according to what comes out of said use. Nor, as Yoram Barzel points out, can the boundaries of a firm be captured by a production function. Firms engage in all sorts of activities that are not merely applying labour to capital for some specific output.

One cannot produce something from nothing; but merely having something, merely having capital and labour, does not tell us what is produced, how well and to what value. Hence the failure of “just add capacity” (typically capital) analyses of, and policies for, economic growth.

Production functions are not independent of institutional frameworks, they are deeply and profoundly embedded in them. But if one is misled into thinking production functions are about capacity, rather than use of such capacity, it is easy to be misled about that.