I read Steve Williamson's new post. I could try and poke holes in his modelling. But it would be pointless, like a high-school debate. The disagreement is much more fundamental. I hope this is more constructive.

Inflation doesn't just happen. In a New Keynesian model, individual firms choose how much to raise their prices, and the inflation rate is the average of their individual choices.

Let's simplify massively. It's a one period game, so we can forget all about dynamics. All firms are identical. At the beginning of the period, all firms start out with exactly the same price. The natural (real) rate of interest r is then revealed. The central bank then announces a nominal interest rate R. Each firm then chooses how much to raise or lower its price. Firm i chooses the firm i inflation rate P(i). The inflation rate that maximises firm i's profit is given by P*(i) = P - b(R-P-r) where P is the average of all the firms' inflation rates, and b is some strictly positive parameter. A firm's losses, relative to its profit-maximising choice, are (P(i)-P*(i))2.

That's roughly the decision facing firms in a New Keynesian model, if the central bank holds the nominal interest rate fixed, and ignores the Howitt/Taylor Principle that says it should raise the nominal interest rate more than one-to-one in response to inflation.

By construction, this game has a unique Nash equilibrium P(i) = P = R-r. By construction, that unique Nash equilibrium is Neo-Fisherian. If the central bank raises the nominal interest rate, the equilibrium inflation rate rises one-for-one.

That equilibrium sucks. I wouldn't trust it one inch.

To see why I think that Neo-Fisherian equilibrium sucks, let's suppose the government imposes a price ceiling and a price floor. But let us suppose that ceiling and floor are never a binding constraint on the Neo-Fisherian equilibrium inflation rate.

Non-binding price ceilings and floors shouldn't make any difference, right? But in this case they do make a difference.

We now have three Nash Equilibria. The first is the original Neo-Fisherian equilibrium. The second is where all firms raise their prices to the ceiling, and each firm wants to raise its price some more, but isn't allowed to. The third is where all firms cut their prices to the floor, and each firm wants to cut its price some more, but isn't allowed to. The Neo-Fisherian equilibrium is only unique if the strategy space is unbounded. The Neo-Fisherian equilibrium is an artifact of building a model with an unbounded strategy space

It doesn't have to be a legislated price ceiling and floor. There might be some limit on how quickly some firms can print new menus or catalogues to raise or lower prices. Or it might be that if firms raise or lower prices too quickly the monetary system collapses. A complete monetary system collapse is an equilibrium too.

If the central bank instead announces a rule like R = r + (1+c)P, we get a unique equilibrium, with or without the non-binding price ceiling and floor.

Update: Narayana Kocherlakota asks what happens when we add a ZLB constraint to the model, so R >= 0%.

Let me first modify the central bank's rule (if it obeys the Howitt/Taylor Principle) to R = max{0, r + P* + (1+c)(P-P*)}, where P* is the central bank's inflation target.

If P* > -r we get two equilibria: the first where inflation equals the target; the second is where inflation hits the price floor, firms want to cut prices even more but can't, and the central bank wants to cut the nominal interest rate below 0% but can't.

If P* < -r we get only one equilibrium where inflation hits the price floor and the ZLB is binding.

By the way: yes, the New Keynesian model is wrong. Because it ignores the stock of money. When central banks hit the ZLB they did not let the stock of money fall without limit. They increased it. Which is why the monetary system did not implode.