David Moyes says Jordan Pickford has been a better player than Dele Alli this season. This set me wondering about marginal productivity theory.

To see my point, think about how we'd test Moyes' claim. We could look at what the two teams did in games which Alli and Pickford missed. But there are too few of these to draw robust inferences, and doing so would be impossible for a player who hadn't missed any games*. Instead, we could compare how Sunderland would have done if Pickford were replaced with a next-best alternative to how S***s would have done with a next-best replacement for Alli. In effect, we're asking: what are their marginal products?

I suspect that Pickford's marginal product consists in converting heavy defeats into narrower ones: this still leaves Sunderland relegated, only with a lesser goal difference. Alli, by contrast, has converted draws or losses into wins. That's a bigger difference.

This, I think, highlights three problems with marginal productivity analysis.

The first is that marginal product is the result of a hypothetical question. For example, in considering my own marginal product, I ask: how would the IC do without me? That's a hypothetical to which we cannot give a precise answer. This is true of much of neoclassical economics. As Noah says:

Demand curves aren't actually directly observable. They're hypotheticals - "If the price were X, how much would you buy?"

In this, he's echoing Sraffa:

The marginal approach requires attention to be focused on change, for without change either in the scale of an industry or in the 'proportions of the factors of production' there can be neither marginal product nor marginal cost. In a system in which, day after day, production continued unchanged in those respects, the marginal product of a factor (or alternatively the marginal cost of a product) would not merely be hard to find - it just would not be there to be found. (Production of Commodities by Means of Commodities, pv)

Secondly, in football there's a high ratio of noise to signal: performances are due in part to luck. This is true not just in football. Consider two men of equal ability who become CFOs of start-ups. One start-up becomes massive, the other struggles. The man who joined the former will be many times richer than the latter. But that's more to do with fortune than with human capital or marginal product: firms don't usually grow big because they've got a slightly better CFO than the firm down the road. Anyone who's mind isn't befuddled by Randian nonsense will know that our lives and incomes are the product of luck. But we know that people are terrible at distinguishing luck from skill - in part because they suffer from the outcome bias.

Thirdly, imagine a top goalkeeper were playing for a better side than Sunderland. His performances would then make a difference to his team's points: a couple of great saves per game would convert losses to draws or wins, rather than 4-0 defeats into 2-0 ones. His marginal product would be higher.

This tells us that, in teams, an individual's marginal product is beyond his control: if Pickford had better colleagues, his marginal product would be higher. A similar problem arises in many large firms. As the late Herbert Scarf wrote:

If economists are to study economies of scale, and the division of labor in the large firm, the first step is to take our trusty derivatives, pack them up carefully in mothballs and put them away respectfully; they have served us well for many a year. But derivatives are prices, and in the presence of indivisibilities in production, prices simply don't do the jobs that they were meant to do. They do not detect optimality; they aren't useful in comparative statics; and they tell us very little about the organized complexity of the large firm.

For me, flaws such as these mean that marginal product theory doesn't make much sense as an explanation of wage levels. We should abandon it as a mental model in favour of bargaining (pdf) models. In these, matches between workers and jobs lead to surpluses, and the surplus is divided according to the balance of power.

In such models, human capital raises wages insofar as it generates surplus and gives its holders outside options which enhance their bargaining power. But human capital and "marginal product" aren't the whole story. All the things that affect bargaining power, such as technology and unions, also matter. Such models are consistent with the theory that inequality is due to the rise of superstar firms (pdf). They're also consistent with the fact that minimum wages don't destroy many jobs. And they help explain rising CEO pay better than marginal product theory.

What I'm appealing for here is for economists to abandon unscientific just-so stories and to think instead about the real world. In this world, wages are determined not by unobservable entities such as marginal product but by - among other things - power (pdf).

* S***s did well during Harry Kane's injury. Few would say this is evidence that Kane is a poor player, and not should they.