I have never thought of the macroeconomic production function as a rigorously justifiable concept. … It is either an illuminating parable, or else a mere device for handling data, to be used so long as it gives good empirical results, and to be abandoned as soon as it doesn’t, or as soon as something else better comes along.
– Robert Solow
When speaking about production and output, economists generally refer to ‘factors of production;’ things are inputted into the production process to produce something else. Most of the time, they use the two factors ‘capital’ and ‘labour.’ They are a firm’s presumed inputs in theories of the firm and supply curves, where a firm takes their values as inputs and, after some mathematical manipulation, produces a certain amount of output. They are also used in a macroeconomic model known as a ‘production function,’ which does something similar for the entire economy. There are various different production functions that use different maths, and include other variables such as technology or productivity – the most famous one is known as Cobb-Douglas.
The problem with this form of estimation is that it has long been known to be logically questionable. Anyone who has taken a science class past a basic level will know that checking your units – that they are consistent and balance out on both sides of the equation – is emphasised repeatedly. But this seems to be thrown out of the window in the basic analysis of production functions and firm behaviour.
The analysis of production takes two physical inputs – most likely capital and labour. Generally, the inputs are also assumed to be clay-like; available in infinitely small quantities. The inputs are combined (as far as I can see, this means flung together inside a black box) and produce a physical output of some other good, which is of course also infinitely divisible and clay-like. Labour is measured in terms of hours of work; capital in terms of money. This is where the problems start.
The Cambridge Capital Controversies revealed many problems with using a monetary value to measure capital equipment, certainly within a theory of distribution. However, there is another, far more simple and perhaps more fundamental objection: by definition, we are supposed to be measuring physical units of input. This means it is simply not coherent to measure in terms of cost. If we were to opt for measuring in terms of cost as a rule, then what would be the justification for not lumping labour in with capital, and just having a single input, perhaps labelled ‘stuff’? The answer is the justification for not doing the same with capital.
If we decide to use physical inputs, it seems there are ways around the problem. Instead of labelling one input ‘capital,’ we could consider a certain type of capital good – say, shovels with which to equip some ditch-digging labourers. It is fair to assume these are roughly the same and so we can add them up. However, this method lays bare problems that the blanket term ‘capital’ previously obscured.
First, we clearly need more than just people and shovels to dig a ditch. We might need wheelbarrows, land, a skip, sustenance for the labourers, transport for labourers, perhaps a supervisor – in fact, there is potentially an incredibly large amount of factors of production, something I’ve noted before. It becomes computationally difficult or even impossible to include everything that contributes to production, and some factors will simply be immeasurable.
Second, it is clear that these objects are not perfectly divisible. In the examples of ‘capital’ and ‘labour,’ we could divide both money and labour time into infinitely small units. But once we allow for production being ‘lumpy,’ functions are no longer smooth and differentiable, and as such marginal productivities simply do not make sense.* Furthermore, this belies the idea of an elasticity of substitution – the rate at which you can substitute one input for the other – since taking away a ‘lump’ will simply make output fall to zero (this is also something I’ve touched on before).
Economists will likely have various rebuttals to this style of thinking. The most used will be that Cobb-Douglas and various theories of the firm make good, testable predictions. But actually their predictions leave a lot to be desired – firms do not behave how economists predict, and the Cobb-Douglas production function has poor empirical results (economists generally refer to the initial estimations made by the creators of the model, but things have changed since then).
The other defense will be similar but not quite the same: it is just a simplification, used to illuminate a particular aspect of a problem. Well, the fact is that making counterfactual assumptions about the nature of a system does not illuminate anything; it simply tells us about a different universe. Furthermore, simplifications cannot be internally consistent. Even within the logic of ‘labour’ and ‘capital,’ it has been shown repeatedly that the conditions under which either of them can be aggregated are incredibly stringent. Similar arguments apply to other aggregate parameters used by economists, such as aggregate measures of technology or productivity.
Simple macroeconomic production functions smack of trying to turn macro into ‘applied microeconomics.‘ But it has repeatedly been shown that aggregation problems will always be present, and that it is best to study emergent phenomena rather than try extrapolate microeconomic parameters until they have no real meaning. At the other end, microeconomic production is just an attempt to reduce everything to ‘rigorously’ derived smoothly differentiable intersecting lines, rather than simply accepting empirical realities about firms and micro behaviour, and opening up the firm to see what happens inside instead of treating it as a black box.
Overall, it seems the whole idea of production functions and factors of production as anything other than vague, qualitative concepts is something of a dead end.
*I similarly expect that, once we allow that preferences may be lumpy, utility functions are no longer smooth. But lumpy preferences is something for another time.