Sunday, August 8, 2010

A Conjecture on Equilibrium Selection

An interesting question is posed in the latest post by Rajiv Sethi on growth prospects for Sub-Saharan Africa. The core issue is whether, in a world characterized by multiple equilibria, it is historical inertia (transmitting local equilibria from one period to the next) or expectations (converging airlessly on new, possibly distant equilibria) that perform the selection. Sethi pulls out an old paper by Paul Krugman that zeroes in on the question.

The Krugman paper, as described by Sethi (I haven’t read the original) uses increasing returns in a two-good model to construct a two-equilibrium result, and the history-expectations dichotomy is resolved by the assumption that expectations will be decisive if the initial conditions are consistent with the attainment of either equilibrium.

My conjecture: this result, which is to say the assumption that convergent expectations can generate a leap from one local equilibrium to another provided either is feasible, depends on the number of potential equilibria and the complexity of the processes that generate them. Of course, if an economy’s structure is perfectly known to agents, complexity is not a problem, but it is safe to say that this knowledge is unattainable. (I’ve been trying, and it’s unattainable to me.) What we have, then, is a constraint on the role of expectations derived from the likelihood that they can converge on a multi-dimensional outcome complexly different from that inherited from the past.

Krugman’s model, if it is described properly, obscures this by proposing an extremely simple and transparent equilibrium selection choice. No doubt there are cases where such an assumption is warranted. But, as I argued long ago, the problem of multiple equilibria becomes far more general and complex when one incorporates interaction effects compared to increasing returns. (In technical terms, if you are looking at a matrix of send-order partial derivatives, increasing returns show up on the principle diagonal, while interaction effects are non-zero terms off the diagonal.) The factors that cause multiple equilibria entailing good x may be lodged in non-x sectors, interacting with x only through derived demands. Indeed, the processes may be so ramified that they resist any realistic strategy to disentangle them, such as would be required for a selection process based on expectations.

This is why, in the end, I think history plays by far the largest role in equilibrium selection. (It is also why I think that equilibrium methods are ill-suited to economic analysis.)

1 comment:

Anonymous said...

i wonder how much the model's assumption of just 2 possible equilibria feeds into the conclusion. in general, i think one can devise such simple models with just equilibria which can simulate any higher dimensional model with multiple equilibria, but just as working with say hexadecimal may be more realistic than working in binary, 'the devil' (reality) is in the details. but as krugman himself wrote (i think in nyrb) what he has been succesful at is geting tractable (publishable) models rather than interesting or relevant ones--except in the sense that perishing is not very interesting or that relevant (except possibly if you are a polar bear and can piss off a palin.)