## Stability of General Equilibrium and Monetary Policy: Baby Steps

Over at Equitable Growth: The very sharp Nick Rowe has a useful piece today giving the baby-step intuition behind Schmidt and Woodford's argument that, no, expected and actual inflation do not as a rule decline one-for-one when a central bank lowers nominal interest rates:

: Understanding Schmidt and Woodford on Neo-Fisherianism: "Suppose you are really bad at algebra... can't solve...

...X = 0.5X. So you make a tentative first guess at the answer, say X=1, plug your guess into the right hand side, get X=0.5, which is your second guess, which you plug into the right hand side again, to get X=0.25, which is your third guess, and so on. Eventually your guesses converge to X=0... tatonnement (groping) towards the answer, just like the Walrasian auctioneer who solves the supply and demand equations in micro by raising prices if there's excess demand, and cutting prices if there's excess supply. But... if the equation is X = 2X... your guesses will diverge further and further away from the right answer.... READ MOAR

Now let's look at the Neo-Fisherian question. Let P be actual inflation, let Pe be expected inflation, Y be the output gap, i the nominal interest rate set by the central bank, and r the natural rate of interest.: P=Pe+aY... a>0 (Phillips Curve). Y=-c(i-Pe-r)... c>0(IS Curve).... P=(1+ac)Pe-ac(i-r). If the central bank holds i fixed... P=bPe+stuff... b>1. You can see the problem. If people try to solve for the rational expectations equilibrium using the tatonnement process, it won't converge.... The rational expectations equilibrium does exist, and is unique, but there is almost zero chance the agents in the model will solve for it by tatonnement.... But if, by sheer fluke, they did guess lucky first time, the solution is: P=Pe=i-r. Yep, it's the Neo-Fisherian result.... If we want to make 1>b, so the tatonnement does converge, the central bank needs to set the nominal interest rate as a function of actual (or expected) inflation. And it needs to ensure that the nominal interest rate increases more than one-for-one with actual (or expected) inflation. That's the Howitt/Taylor principle....

We have to interpret this model as saying that the rational expectations equilibrium is implausible if the central bank pegs the nominal interest rate.... Most people are not very good at algebra, the real world is a lot more complicated than any macro model, and people are all different. It's already stretching it to assume that people can solve the model by tatonnement in their heads, infinitely quickly, provided 1>b. Most of us mortals have to watch what happens, and revise our expectations in the light of experience...

Nick is right: a real model working in real time with real agents and real expectations is greatly, greatly superior either to a model that assumes a rational expectations equilibrium without inquiring into its stability, and greatly superior to a model that looks for a stable rational expectations under reasonable assumptions about expectation-revision tatonnement. But if you do not get there under reasonable assumptions about expectation-revision tatonnement, you will not get there under a a real model working in real time with real agents and real expectations either.

One of my interlocutors says that I should look at the so-called "neo-Fisherian" claim (which Irving Fisher would laugh it as revealing a truly awesome degree of stupidity) as performing not an economic knowledge-advancing but rather a community-sociological function. It allows most of the under briefed economists who didn't do their homework to understand the situation in 2008-9 to climb down from the position that their nuttier members like likes of Clifford Asness, Niall Ferguson, and Douglas Holtz-Eakin are still clinging to--that the burst of inflation is coming any day now. And it allows them to do so without admitting that Paul Krugman, Mike Woodford, Gauti Eggertsson, and the rest of the dirty f---ing hippie new Keynesians were right about the state of the economy in 2008-9. And that is why they are so averse to requiring sensible stability-based selection rules for classifying model equilibria.

He may be right.

Alternatively, they may just still not have done their homework.