But monetary disequilibrium is a theoretical construct. We don't observe it directly. So we need an empirical proxy for monetary disequilibrium, so we can tell the central bank to minimise that empirical proxy.

If all prices were perfectly flexible, monetary policy wouldn't matter much. Monetary policy matters because not all prices are perfectly flexible, which means that bad monetary policy causes monetary disequilibrium, which is what happens when prices want to change but don't change. Recessions and booms are examples of monetary disequilibrium.

Deviations of actual unemployment from full employment might look like one possible proxy. Trouble is, "full employment" (or the "natural rate of unemployment", or "potential output" etc.) is itself a theoretical construct. It's not something we can observe. So telling the central bank to target "full employment", so as to minimise deviations of actual unemployment from full employment, didn't work very well.

So we switched to telling central banks to target a fixed rate of inflation instead. We told them to minimise deviations of actual inflation from target.

Is "deviation of inflation from (a fixed) target" a good empirical proxy for monetary disequilibrium?

Well, it's empirical all right, because we can measure inflation (more or less), and the data arrives with a fairly short lag (around one month). And 2% (or some such target) is simply a number, not a theoretical construct like "full employment".

But is "deviation of inflation from 2%" a good proxy for monetary disequilibrium?

On the face of it, it looks like a very bad proxy.

We are trying to measure something that happens because some prices don't change, so we look at the prices that do change? How the hell is that supposed to work? Well, perhaps surprisingly, there is one case where deviations of inflation from target would be a perfect proxy for monetary disequilibrium. That's the case where the firms that do change prices are exactly like the firms that don't change prices, except for that one difference.

And that one very special case is exactly the case assumed by the Calvo Phillips curve. Calvo's fairy flies around at random, touching firms with her wand, giving them permission to change prices. And precisely because she flies around at random, and there is a very large number of firms, the sample of firms that do change their prices are exactly the same as the remaining population of firms that don't change their prices. So if we see some firms cutting prices, we know that the remaining firms want to cut their prices too, but cannot, so there must be a monetary disequilibrium at those other firms, and it must be a recessionary monetary disequilibrium.

The randomness of the flight of the Calvo fairy is precisely the assumption that makes it easy to do the math. And it's precisely the assumption that makes inflation targeting the optimal monetary policy for minimising monetary disequilibrium. But if you think about it for a bit, it's not a very plausible assumption.

For example, suppose that all firms are identical, but some firms change their prices once a month and other firms can change their prices once a year. Assume a 0% inflation target, just for simplicity. If we see negative inflation in January, and 0% inflation in February, does that mean we have monetary disequilibrium in January, but no monetary disequilibrium in February? Of course it doesn't. In February, the firms that can only change prices once a year still want to cut their prices, but cannot. The economy is still in recessionary monetary disequilibrium, even though inflation is back at the 0% target.

If that were the only difference between firms, then price level path targeting would be the best way to minimise monetary disequilibrium. The central bank should aim to bring February's price level back up to where it was in December.

But presumably there are reasons why some firms change prices more frequently than others, and the two sets of firms will not be exactly the same in all other respects.

And even if all firms did change prices at the same frequency, on average, it is very unlikely that the firms that do change prices in any particular period are exactly the same as the firms that don't change prices in that period. There must be a reason why some firms did change prices and other firms didn't. Maybe there was a real shock, that caused relative demand to shift away from some firms, towards other firms. And if there is some threshold, like a small menu cost, so a firm won't change its price unless it wants to change it a lot, then if the distribution of relative demand changes is skewed, the prices that do change won't even tell us the correct sign of the monetary disequilibrium. All we are observing is one tail of the distribution of monetary disequilibrium.

Thinking about good monetary policy as minimising some empirical proxy for monetary disequilibrium does not tell us what the best proxy is. But it might help us think more clearly about the question. And it does help us understand why inflation targeting might not work. And it does tell us that using mathematically tractable assumptions, like the random flight of the Calvo fairy, could badly bias our search for the best monetary policy.