Job market time is coming up, which means I’m sure I’ll be posting more about some of the top job market papers. This short paper on arbitrage is not Glen’s JMP – more on that in a few days, as it is right up my alley and also the best JMP I’ve read this year – but interesting nonetheless. If you’re not familiar with Weyl, you may remember him as the guy who finished his PhD in one year at Princeton a few years back before signing on as a Harvard Fellow.
This paper concerns whether arbitrage is useful. The standard explanation is simple: arbitrage, by creating markets where one may not already exist, allows more efficient allocation of risk, and to the extent that such efficient allocation is useful in determining which projects are socially useful to pursue, allows more efficient production across the economy as a whole: a useful function rather than the biblical “den of thieves”.
But what if consumers disagree, subjectively, about the payoff of an asset? One may believe it is distributed N(-r,1) and another N(r,1), when ex-post, we will learn the true distribution is N(0,1). Assume that the two consumers cannot trade, and both hold zero units of the asset. If an arbitrageur comes along, the good will be relatively cheaper in the first market, and so the arbitrageur will buy cheap there and sell dear in the second market. Since risk-averse individuals would not want any of the asset if they knew the asset was N(0,1), this arbitrage-induced trade will lower ex-post “objective” utility. If the model is broadened to allow that asset prices also help effectively allocate capital to the right projects, and hence the role of arbitragers in collecting information is incorporated into the model, it is still easy to construct examples where arbitrage lowers utility, ex-post.
The point is well-taken, and Weyl strikes me as having a Samuelsonian writing style that skips on fluff and rather goes straight to the most salient aspects of a theoretical model. That said, I have two worries. First, I don’t really know what “objective utility” is. If consumers are subjective expected utility maximizers, then nothing in this model suggests that their ex-ante maximizing selves are made worse off by arbitrageurs. Indeed, as discussed on this site, there is an interesting line of research about how social planners should deal with the problem of “contradictory” subjective beliefs; I think of this as the “Problem of the Duel” reflecting Gilboa’s example of two sharpshooters in a potential duel who (impossibly) both believe themselves to be the quicker draw. The second problem concerns Aumann-style information flow. Why don’t I update my beliefs in this model when I see the existence of the arbitrageur willing to buy or sell the asset? I don’t even think you need Myerson-Satterthwaite to generate a no trade situation here. A simple model of asking an (honest) arbitrageur if he is still willing to buy, then updating, then asking again, ad infinitum, should cause convergence of beliefs as in Geanakoplos. If the arbitrageur is not honest, then some sort of mechanism problem needs to be written out. In any case, even a single and unavoidable information update would change subjective beliefs and therefore change the welfare calculations. In any case, the basic point is that there are massive methodological issues with “objective” social welfare which are not easy to paper over; credit to Weyl, at least, for acknowledging this, and pointing out that even if you reject the existence of “objective” social welfare, the fact that his model leads risk to be allocated to people other than the most risk loving is itself troubling in many contexts beyond consumer welfare.
http://www.people.fas.harvard.edu/~weyl/Second_Draft_Arbitrage.pdf
A Fine Theorem 