The question of whether Gareth Bale is worth €100m raises questions about the validity of marginal productivity theory.

Eamonn Butler writes:

In free markets, people pay us for the goods and services we produce because they value those products. So market rewards do depend, very directly, on the value that we deliver to other members of our society.

This ignores a crucial distinction - between actual, observable value and expected value. Bale's fee, and most of his salary, is determined by Madrid's expectation of his value. But it's quite possible that this will exceed Bale's actual value to Madrid.

So how can Eamonn say that "rewards depend...on the value that we deliver"? One possibility is that he's talking about averages over the long-term. This is reasonable for many professions. If a company were to hire, say, electricians at £200,000 a year it would go bust because electricians don't create that much value. The excess supply of electricians would then bid their wages down to levels compatible with their value.

But this reasoning cannot apply in our case. When Sp*rs fans sang "there's only one Gareth Bale" they were - uniquely in their miserable lives - correct. It makes no sense to speak of the average fee of all Bales; Madrid are paying a fortune for him precisely because he's unique*. It might be that, averaged over all possible futures, Bale will be worth £30m a year - his fee and salary amortized over his six year contract. But this is unobservable and hence unscientific.

A more promising interpretation of Eamonn's comment is that, averaged across all footballers, wages equal value. Sure, some are overpriced and overpaid, but others - by the standards of the average footballer - are underpaid.

This claim, though, runs into an obvious problem. In aggregate, football clubs lose millions of pounds. Which is surely evidence that footballers, on average, are paid more than their average revenue product.

The objection to this is that revenue product is only part of value. We must add owners' non-pecuniary valuations - such as the gratification of their ego or the escaping of Russian politics.

But if we regard value in this sense, then Eamonn's claim becomes an untestable tautology. Whenever we see any worker or product changing hands at fancy prices, we can attribute it to some kind of "value." Sure, value's subjective. But it might be so subjective as to be unmeasurable. In fact, it's quite possible that the much-maligned labour theory of value is more scientific (pdf) than marginal productivity theory - insofar is it is both testable and, better still, compatible (pdf) with the evidence (pdf).

Here, though, we must draw a distinction between the validity of marginal productivity theory and the legitimacy of free market transactions**. It's quite coherent to argue that marginal productivity theory is silly, and yet defend market exchanges on Nozickean grounds that what consenting adults do should be their own business. Markets don't need to be value-maximizing to be legitimate.

* Bale is, in a sense a (near) monopoly, and monopolies charge a price above marginal product.

** I shall avoid the question of whether Madrid is operating in a free market.

Another thing: There are many complications here. One is that, in football, value is to a large degree zero-sum. Bale could justify his fee and salary by winning Madrid trophies. But if he does this, he's reducing the value of Barcelona players. Another is that, in team sports, an individual's marginal product depends upon complementarities with his team-mates; if these are favourable, his marginal product might seem big, and if their not it won't. This raises all sorts of issues.