Just because an agent has rational expectations does not mean he can solve the central planner's problem. That's why we need markets, to coordinate the plans and expectations of individual agents, each with their own local knowledge. Hayek , the socialist calculation debate, and all that. Only if the representative agent knew he was the representative agent would he be able to solve the aggregate version of the central planner's problem, just by introspection and rational expectations.

The representative agent cannot be assumed to know he is the representative agent. He observes the shocks that hit him, but does not observe the shocks that hit other agents. He cannot distinguish individual-specific from economy-wide shocks. (Just like Lucas '72.) (Update 5: as Adam P. notes in comments, the representative agent is really a composite agent, and may not be any individual agent. And even if he is an individual, it may be a different individual from one day to the next.)

If the central bank raises the nominal interest rate, what happens to actual and expected inflation depends on why people think the Bank did it.

This is for David Andofatto, in response to his recent post (and for anyone else who might be interested).

Suppose the economy is humming along nicely, with actual and expected inflation at the 2% target, and actual and expected output at potential output. And then all of a sudden the Bank raises the nominal interest rate by 1%, for no reason at all. What happens next?

Actual (and expected) inflation would need to rise by 1% to keep real interest rates the same and keep output at potential. But will it in fact rise by 1% and will output actually stay at potential?

We know the Bank did it for no reason at all, because I said so. But agents in the economy don't know that. How agents react will depend on why they think the Bank did it.

1. If agents think that the Bank had raised the inflation target from 2% to 3% (perhaps because the Bank announced it was doing that), they might expect 3% inflation going forward, so the real interest rate does not change, actual and expected output remain at potential, and actual inflation may rise to 3%. (Whether this is exactly what happens depends on the exact nature of the Phillips Curve, in particular on whether inflation is a jump variable, like in the Calvo model. If actual inflation is sticky/inertial, it can't happen.)

2. If they think that the Bank has kept the inflation target at 2%, the results will be very different. Let's explore that possibility now.

The representative agent knows there have been no shocks to his own consumption and investment plans, but he does not know he is the representative agent. He figures there must have been a positive shock to the consumption and investment plans of the representative agent, otherwise the Bank would not have raised the nominal interest rate to keep inflation at the 2% target. He thinks the Bank has raised the nominal interest rate exactly enough to offset the shock to the representative agent's consumption and investment plans. So the representative agent expects no change in inflation. He continues to expect 2% inflation. And he continues to expect output to remain at potential. He raises his price by 2%, but with the real interest rate now higher, he cuts his own current consumption and investment.

The representative agent is now surprised to discover he has sold less output than he expected to sell. But again, he does not know he is the representative agent, and may think this negative demand shock is specific to his own output. He raises his price by less than 2%, because he plans to lower his relative price, just like a monopolistically competitive firm faced with a negative demand shock.

If the central bank raises the nominal interest rate for no reason at all, but the representative agent thinks the Bank did it to keep inflation at the 2% target, it causes inflation to fall below 2%.

(We would get exactly the same result if we assumed the natural rate of interest fell, but the representative agent did not know it had fallen, because he thinks it's a shock specific to him, and the Bank kept the nominal interest rate the same (maybe because of the ZLB). The result would be a fall in inflation below target, and not the rise in actual and expected inflation that would be needed to keep output at potential.)

3. Now let's assume the representative agent knows the Bank raised the nominal interest rate for no reason at all. The representative agent's reaction will most likely be: "WTF? Let's fire the Governor!" (More charitably, he might assume the Governor's hand trembled temporarily.)

[A question I have ducked here is how much information the representative agent thinks the Bank will have about the representative agent. But since the representative agent does not know he is the representative agent, there is nothing contradictory about assuming the Bank knows more about the representative agent than the representative agent knows about the representative agent!]

It makes no sense for economists to talk about the effects of nominal interest rates on inflation (or anything else) without talking about how agents will interpret those changes in nominal interest rates. What is the Bank trying to do when it changes nominal interest rates? What does it mean? Is the Bank trying to change the inflation target? Or trying to defend the existing inflation target? Central banks know this, of course. That's why they have communications strategies, and announce monetary policy targets.

And it makes no sense for economists to use representative agent models of rational expectations where they assume the representative agent knows he is the representative agent. The representative agent does not know the representative agent's preferences, technology, and expectations. That does not contradict rational expectations.

We can imagine a world in which there is some central agent who acts like the conductor of an orchestra, or the coxswain of a racing eight, who helps coordinate individual agents' expectations and actions onto some mutually beneficial Schelling focal point. We can imagine such a central agent seeing a fall in the natural rate of interest, and the Bank unable to cut nominal interest rates, and calling out to all agents: "Everybody will expect higher inflation from now on, because that's the only way we are going to keep the Euler equation satisfied with output at potential!", But in the real world, the only central agent who can play that role is the central bank itself, by announcing a higher inflation target (or NGDP target, or whatever). It ain't going to happen by itself, rational expectations or not.

Update: Here's Noah Smith on David. And Steve Williamson's response to his critics.

By the way, here's my empirical evidence in support of my view:

For the last 20 years, the Bank of Canada has said it has been targeting 2% inflation. And the Bank of Canada has said it has been doing this by raising nominal interest rates whenever it thinks that inflation would be above 2% otherwise, and lowering nominal interest rates whenever it thinks that inflation would be below 2% otherwise. And actual inflation has averaged almost exactly 2% over the last 20 years.

If the Bank of Canada had got it the wrong way around, and had been turning the steering wheel the wrong way, this would be an amazing fluke. If the bus driver had been turning the steering wheel clockwise to make the bus turn left, the bus would almost certainly, after the first shock, be spinning round in ever-tighter circles, and would never get anywhere near its announced destination. Unless the Bank has been lying to everyone all along and had secretly been doing the exact opposite of what it says it has been doing.

Suppose, just suppose, that I ever did become convinced that the data on inflation supported Steve Williamson's theory. I would then conclude that everything I had ever learned in economics was totally wrong, and that the rate of inflation was in fact determined by a benevolent deity, or some secret cabal operating in the national interest, that ensured that the actual and expected rate of inflation always adjusted to ensure continuous full employment despite whatever stupid thing the Bank of Canada might do with nominal interest rates. I would throw out methodological individualism, and the idea that agents act in their own self-interest in setting prices and forming beliefs about prices. I would decide that the Functionalist Fallacy was not a fallacy after all.

Update 2: and here's Scott Sumner on the dangers of reasoning from a price (interest rate) change, and more on the empirical evidence.

Update 3: at the back of my mind there's some sort of game-theoretic Bayes-Nash equilibrium which I'm trying to sketch out. When one player (the agent) observes another player (the Bank) make a move, what does that move reveal about the other player's beliefs and preferences (target)? Assume agents' priors are that inflation will be at the Bank's announced target, to resolve the indeterminacy problem.

Update 4: of course, the problem at the back of all this debate is the accursed Neo-Wicksellian indeterminacy. If central banks did something sensible, like moving something with $ in the units rather than something with the units 1/years, we wouldn't be arguing about all this. The idea that central banks set nominal interest rates is a horrible social construction of reality.