Inflation in Lake Wobegone

In Lake Wobegone there are 1+n firms. The first firm raises or lowers its price by a mean-zero random amount, e. The remaining n identical firms wait to see what the first firm has done, then raise their prices by B times the average expected inflation rate, or lower their prices by B times the average expected deflation rate.

In rational expectations equilibrium, the Lake Wobegone inflation rate p is determined by:

p = Bp[n/(n+1)] + e/(n+1)

Which gives us:

p = e/(n(1-B)+1)

If 0 < B < 1+(1/n) this equilibrium looks sensible. A positive random shock at the first firm causes positive inflation, but the effect gets smaller as n gets larger.

If B > 1+(1/n) this equilibrium looks strange. A positive random shock at the first firm causes negative inflation.

The first equilibrium is the standard equilibrium in a New Keynesian model, where the central bank follows the Howitt-Taylor principle, promising to raise the nominal interest rate by enough, if expected inflation rises above target, to make each firm want to raise its price by less than expected inflation.

That second equilibrium is the Neo-Fisherian equilibrium in a New Keynesian model, where the central bank holds the nominal interest rate fixed at the (real) natural rate of interest, so if expected inflation rises above target, each firm wants to raise its price by more than expected inflation.

Empirically, that second equilibrium doesn't look very plausible to me. When the government raises taxes on smokes, for example, we normally see a small temporary increase in inflation, not a decrease.

Comments

[Lord]

How about if the UK increased the VAT tax to reduce deficits? A transitory increase followed by an expected decrease due to reduced demand, depending on time scale?

[Nick Rowe]

Lord: fair point. The Canadian VAT (GST) was a cleaner example, since it was approximately revenue neutral.

[notsneaky]

But is this a comment on the relevance of the model or a policy prescription? Is it saying "the Central Bank should not target interest rates" or is it saying "look, if I tweak the usual assumption about your monetary rule, this model produces nonsense, hence it's not a good model"? Or both (which would need some kind of auxiliary argument)?

Also, the following came to mind while thinking of this - I'm not sure if it's a good analogy but let me set it out there. Think of the game of Tic-Tac-Toe. The Nash equilibrium of that game is a pair of sequences of moves by each player such that the outcome is a tie. Is this equilibrium "empirically plausible"? Based on experience, yes (although I always somehow loose when I play my kids). But suppose that initially, "early in the game", one player makes a mistake. Then the outcome flips completely and the other player will win for sure. But the possibility of that initial mistake early on dramatically changing the outcome does not NECESSARILY make the "it's a tie" equilibrium implausible.

[Nick Rowe]

notsneaky: I think an orthodox New Keynesian economist would read this post and say "Yep. This is a variant on what we've been saying all along. The RE equilibrium in a NK model is (approximately) correct if the central bank follows the Howitt-Taylor Principle, but the RE equilibrium makes no sense if the central bank pegs a nominal interest rate, so don't let central banks peg nominal interest rates, and don't listen to those Neo-Fisherians who tell you that pegging a higher nominal interest rate will raise the RE equilibrium inflation rate."

The difference is that orthodox NKs normally talk about the B > 1 RE equilibrium being unlearnable. Or do what Garcia-Schmidt & Woodford did in their latest paper, to show that the B > 1 RE equilibrium will be totally wrong if agents do "reflection" rather than pure RE.

In other words, this post was not intended as a critique of the NK model as such, or of NK macroeconomists. It was intended as a critique of a (mis)use of that model by Neo-Fisherians.

I think I like your Tic-Tac-Toe analogy. The policy implication is: if your objective is to choose a game so that your kids nearly always draw, then don't make them play Tic-Tac-Toe, even if the Nash Equilibrium is draw. Because they are unlikely to get anywhere close to the Nash equilibrium, given that real-world kids make mistakes. (The only way your analogy fails is that tic-Tac-Toe only has 3 possible outcomes: win, draw, lose, rather than a continuum, so we can't talk about "nearly drawing".)

[notsneaky]

I think I know the difference between an "orthodox" New Keynesian economist and a "non-orthodox" New Keynesian economist (and I'm assuming "non-orthdodox" here does not mean "heterodox" bur rather something like Neo-New-Keynesian") but if you could clarify as to what you think is the difference that'd be great (mostly because sometimes I need validation that I'm not crazy)

[Nick Rowe]

notsneaky: an orthodox (sensible) New Keynesian says "A 1% increase in the inflation target causes the nominal interest rate set by the central bank to (eventually) increase by 1%". A Neo-Fisherian would use the same model but reverse causality, and say "If the central bank increases the nominal interest rate by 1% that will (eventually) cause the inflation rate to rise by 1%."

[Bob]

On smokes, the tax incidence is 99% on the smoker so perhaps a special case.
http://markwadsworth.blogspot.co.uk/2011/04/price-elasticity-explains-who-bears-tax.html?m=0

[Bob]

Identical firms, not one of which says "let's keep our prices the same and run an overtime shift"??

[Nick Rowe]

Bob: They have to pay extra for overtime, so marginal cost rises. So no.

[Bob]

A capitalist system without any fixed capital or fixed costs?
Interesting stuff.

[Nick Rowe]

Bob: you can add fixed costs in if you like. It makes no difference to the conclusion. (In fact, I was implicitly assuming those fixed costs do exist, because I am implicitly assuming monopolistic competition, not perfect competition.) It's the shape of the *marginal* cost curve that matters for pricing, not the *average total cost* curve.

Plus, we are doing macro not micro, so we are talking about *all* firms increasing output, not just *one individual* firm. So even if every individual firm has a horizontal MC curve, if *all firms* expand output and employment together they will be bidding up wages in the competition to hire labour away from each other, so each firm will find its MC curve will *shift up*, when *all firms* expand together. Avoid the fallacy of composition.

So lose the snark, and start learning some basic micro and macro economics.

[Bob]

Lol
In Nick Rowe's crazy world of 'monopolistic competition' all racing cars from all those different teams cross the finish line at *precisely* the same time. Nobody can work hard. Nobody can work smarter. Nobody makes a mistake. There is no learning on the job. There is no spare capacity. No time. No administered prices. No improved methods.

In other words no actual competition and therefore no capitalism.

And of course by inference no export licence to the real world.

[Nick Rowe]

Bob: "In Nick Rowe's crazy world of 'monopolistic competition' all racing cars from all those different teams cross the finish line at *precisely* the same time."

Nope. Re-read my post. One firm is different from all the rest. I show that the presence of just that one difference matters (if B > 1). Now convince me that it matters, for the point I want to make here, if it's more than one firm.

Or go away and troll someone else. Because it's not very original to point out that models are simplifications of reality. We all know that. Read Borges

[Bob]

Look, we're not talking about *all* firms expanding output. We're talking about *some* firms expanding output and *some* firms trying to push prices. The firms with the output expansion will stop the price push, forcing them to increase output and productivity as well (but by then they are on the back foot and will shrink instead).
Do you agree with that?

[Nick Rowe]

Bob: have a look at this old post

If you rig the assumptions just right, you can get the model to yield the predictions that you like. (You need the MC curve in the second diagram to lie on top of the MR curve). But the assumptions you need to rig to get that result are not very plausible. (Roughly speaking, you would need a very elastic labour supply curve, and real-world labour supply curves don't seem to be very elastic.)

But this discussion is way off topic for this post, which is about the direction of causation in the correlation between nominal interest rates and inflation.

[Bob]

"If you rig the assumptions just right, you can get the model to yield the predictions that you like."
I'm sure you can.
What!?
That's not how you're supposed to do things. This is the old "if the theory contradicts the facts, the facts are wrong."
IMV most of econ is BS.

[Nick Rowe]

Bob: Oh Christ. That was precisely my point!

"If you [Bob] rig the assumptions just right, you [Bob] can get the model to yield the predictions that you [Bob] like." But those assumptions would not be plausible.

Forget it.