Monday, August 15, 2011

A response to Nick Rowe: Fragility at the limit (amv)

I promised Nick Rowe to respond to his post on what he calls "fragility at the limit'. It is a follow-up on the debate on Kocherlakota's argument that it is possible to determine long-run inflation expectations by setting nominal yields according to the Fisher relation. Let me start with a disclaimer: I do not believe that central banking is or should be using nominal yields in the way Kocherlakota suggests. I am interested in the theoretical model underlying Kocherlakota's argument. I further believe there is a grain of truth in his statement, since I 'believe' in long-run neutrality. As I mentioned many times before, I side with Sumner & his allies on NGDP- or (at least) price-level targeting.

Nick introduces an interesting argument. If I got him right, he accepts that Kocherlakota's argument is a sound REE-prediction. He however argues that the model is 'fragile'. To show what he means, he partitions the set of agents by defining a fraction f of adaptive agents on [0,1] (and, accordingly, a fraction (1-f) of RE-agents). The same fraction partitions the bahavior of firms, that is, f gives the fraction of sluggish price setters, while (1-f) gives the set of instantaneous price setters.

He starts with f=1 and argues, correctly, that an 1% increase in nominal yields suggests a 1% increase in real yields and, thus, initiates a sluggish deflationary (or disinflationary) process. As f approaches zero, that is, as the fraction of RE-agents who adjust prices instantaneously increases, the same 1% increase in nominal yields results in ever faster deflationary processes. As f approaches zero, increasing nominal yields suggests "instant explosive deflation". At the limit, however, that is, for f=0, Kocherlakota's model applies, that is, a 1% increase in nominal yields suggest 1% higher inflation expectations.

Nick concludes: "Suppose that the predictions in the limit (as the assumptions approached the model's assumptions) were totally different from the predictions at the limit. That would be a model that is fragile in the limit. And that, to my mind, would be a totally useless model."

Now, in this post I do not want to talk about monetary policy. As I mentioned before (in some comment section), I dislike DeGrauwe-type models, because they suggest that RE is a restriction on individual behavior. I rather side with Arrow, who convincingly argues that RE as an individual assumption is prohibitively restrictive. RE is a general equilibrium notion; REE defines a class of GE models; REE-price systems reflect information as if agents had the super-human capabilities Arrow is referring to. What is needed, of course, is some kind of mechanism, like Sandroni's market selection process, that establishes convergence in case of agents with bounded cognitive and information-processing capabilities. In case of monetary macro models, other learning processes are relevant, like Howitt's, who argues that if QTM-logic applies, and if central banks can manipulate nominal interest rates only by adjusting monetary base in the inverse direction, then a Kocherlakota-type REE is unlearnable. Note, however, that this is about the REE's dynamic stability and not about Nick's concept of 'fragility at the limit'.

To show what I mean, let me take an example that is totally unrelated to monetary policy. Let me talk about the Sonnenschein-Mantel-Debreu results (SMD). The Debreu version (1974) claims that given any function, say g, from the unit price simplex to a k-dimensional commodity space satisfying continuity, homogeneity, Walras's Law, and a boundary condition, then for every compact subset of the price simplex there exists an exchange economy with k 'well-behaved' agents such that the aggregate excess demand of that economy is equal to g(p) on the compact subset. Thus, if structure is imposed by restrictions on individuals (those restrictions that suggest utility and profit maximization),  most of it is lost by aggregation (only continuity, homogeneity, and Walras' Law survive the adding-up of individual demands). Thus, sufficiency conditions for global uniqueness and global stability imposed on the system like gross substitutionality do not have microfoundations.

The SMD results are true for arbitrary dispersions of individual characteristics like preference and endowments. Imagine a convergent sequence of Arrow-Debreu exchange economies with a large but finite number of agents and commodities, decreasing in the dispersions of such primitives. States differently: for any sufficiently large n in the index set we have arbitrary dispersions of primitives; as n increases, dispersion decreases; in the limit, all agents have identical preferences and endowments. The last such economy in the sequence before we reach the limit is one with identical preferences and collinear endowments. As shown by Kirman and Koch (1986), the SMD results still apply. Up to this economy, there is no hope for global uniqueness and stability. But this is evidently not true for the limit economy, which in fact is a quasi-Crusoe economy. Global uniqueness and stability is trivial. Thus, as in Nick's framework, the sequence is discontinuous, that is, it jumps at the limit.

According to Nick's definition, we can say that the SMD results are 'fragile at the limit'. But his core statements that this implies 'uselessness' does not follow. It just tells us if we find an economy with strictly identical agents, we can hope for global uniqueness and stability. Otherwise, we have to make other restrictions than those on individuals (see Werner Hildenbrand on this). What this shows is that Nick's argument is not generically true as it is implied to be. An even though the SMD results tell us something about global stability of each economy in our sequence (or rather the lack thereof), Nick's notion of 'fragility at the limit' is about the continuity properties of a sequence of models (or rather about the lack thereof). As he insists, many economist do not pay enough attention to stability properties. I also believe that this is important. But as I tried to argue, his notion of 'fragility at the limit' is not concerned with dynamic stability issues.