From J* To U*: What My Conjecture Is About

I have little doubt that my previous article on J* -- a definition that I invented -- was confusing to most of my readers. As I wrote, I reverted to a mathematical style of writing. It is likely that inventing a concept and proving it does not exist is a pastime that would mainly be of interest to mathematicians (and philosophers). However, I have a real-world target in mind: NAIRU. All we need to do generalise the theorem procedure, and we can prove that a similar concept -- U* -- does not exist in the current institutional structure. We can then use that information to annihilate any definition of NAIRU that ends up being equivalent to U*.

Why not take on NAIRU directly, a reader might ask? This is because economists are not mathematicians. They use any number of different concepts, and assume that they are the same thing. It is a waste of time trying to prove the incoherence of each of these concepts; we just prove that U* cannot exist, and we can then just prove the equivalence of any particular definition of NAIRU to U* as needed.

Obviously,. that seems to be a rather grandiose assertion. I could easily be wrong. The most obvious hurdle is that there could be a flaw in my J* non-existence proof. I have thrown it out there, and I am waiting for it to be shot down. A more intelligent approach would have been to approach people privately and get their opinion, but hey, I decided to roll the dice.

The next hurdle is generalisation. I have added some comments below (which I added to the original article as well). When I first did the proof, I had very little idea how to generalise it; I just needed to pin down a proof for a single case. However, if we start pulling theoretical tricks out of the nonlinear robust control toolkit, the job might be easily done. If we want to generalise the result, things could get pretty ugly from a mathematical point of view. For example, norms of series will not converge. I dealt with this in my doctoral thesis in an inelegant fashion; mathematicians may have a better way to approach it.

If my guess is correct, finding a counter-example model is much tougher than it would appear. It is not enough to pick a model that violates one of the many assumptions I imposed: the model cannot be approximated by any such system in steady state. Is such a model going to be anything other than pathological?

Furthermore, lack of a steady state might not be enough. If we move into the frequency domain, we can just pin down the zero frequency behaviour. If this works, we can hit any system with oscillatory or even chaotic behaviour.

Trivial Definitions of NAIRU

The problem with mainstream economic analysis is the tendency to interchange variables that are the result of statistical methods and theoretical concepts. It is obvious that if we define NAIRU to be the result of some statistical procedure, it exists. The problem is that such a definition is trivial: it has no predictive power.

Imagine that I define "NAIRU" to be the 12-month moving average of the unemployment rate. We know that if we take any reasonable definition of steady state, the observed unemployment rate has to converge to my definition of NAIRU. However, seems clear that this is a trivial observation: this estimate of NAIRU has essentially no predictive power for other steady states.

I imagine that the economists producing NAIRU estimates would be very unhappy with the suggestion that their fancy models are equivalent to taking the 12-month moving average of the unemployment rate. However, I see no evidence that their estimates have any more predictive power than that definition.

What is not Implied by this Conjecture

My arguments here are theoretical. I have little doubt that they could be misinterpreted by those who do not understand economic theory. I assume that my readers would not make these particular arguments, but one could conceive that someone somewhere might make comments that are equivalent to the following statements.

• Non-existence of U* implies no trade-off between unemployment and inflation. This conjecture does not imply that we would not see a trade-off between unemployment and inflation. All the conjecture states is that the trade-off has to include the effect of fiscal policy. We can adjust the unemployment rate (within limits) by changing fiscal policy, without affecting the steady state inflation rate.
• Why don't policymakers drive the unemployment rate to zero? That question presupposes that policymakers care about the fate of the unemployed; I see little evidence of such care in the post-1990s world. Secondly, there is no mechanism -- other than the Job Guarantee -- to force the "involuntary" unemployment rate to zero. (If a Job Guarantee were implemented, there would still be residual unemployment from people on jobless benefits, or conducting a job search. Since they could take a Job Guarantee job, such unemployment might be viewed as voluntary. Whatever.) Traditional demand-management techniques used in the 1960s failed, for reasons outlined by Hyman Minsky (and presumably others) at the time. Within the simplistic class of models studied, the "fiscal coherence" assumption does imply a non-zero lower bound for the observed J or unemployment rate does exist: we have a theoretical limit for how low tax rates can be an still preserve price stability. In the real world, we have clearly never flirted with that theoretical lower bound.

Remarks on Generalisation

Remark. We do not need to assume that economic models lie in the class of models described here. It may only be enough to specify that they can be approximated by such a model in steady state. This is obviously a much wider class of model behaviour. We can formalise the notion of approximating a system by applying the definitions used in robust control theory. (My doctoral thesis is the only reference I can offer off-hand that covers the issues I see.)

Remark. The steady state assumption is needed to allow us to apply algebra in the proof. If we were willing to delve into more advanced mathematics, it seems possible that we could specify conditions in frequency domain terms, and just look at the zero frequency ("DC") component of signals. Since most time series are expected to be bounded away from zero for infinite periods of time, there would be obvious convergence issues for the Fourier transform. We would need to do our analysis on finite intervals, and then take limits. This would be tedious, but the generalisation may result in a more elegant proof.

(c) Brian Romanchuk 2017