# The Phillips Curve, Again

December 4, 2017

The Phillips Curve is back. In saying so, I do not mean to imply that being “back” refers to a sudden reappearance of a stable empirical relationship between unemployment (or the output gap) and inflation. The Phillips Curve is back in the same way that conspiracy theories about the assassination of JFK are back after the recent release of government documents. In other words, the Phillips Curve is something that people desperately want to believe in, despite the lack of evidence.

The Phillips Curve is all the rage among central bankers. Since the Federal Reserve embarked on quantitative easing, they have been ensuring the public that QE would not be inflationary because of the slack in the economy. Until labor market conditions tighten, there would be little threat of inflation. Then, as the labor market tightened, the Federal Reserve warned that they might have to start raising interest rates to prevent these tightening conditions from creating inflation.

What is remarkable about this period is that the Federal Reserve has undershot its target rate of inflation throughout this entire period — and continues to do so today. So what does this tell us about the Phillips Curve and what can we learn about monetary policy?

If one looks at the data on unemployment and inflation (or even the output gap and inflation), you could more easily draw Orion the Hunter as you could a stable Phillips Curve. Fear not, sophisticated advocates of the Phillips Curve will say. This is simply the Lucas Critique at play here. If a Phillips Curve exists, and if the central bank tries to exploit it, then it will not be evident in the data. In fact, if you take a really basic 3-equation-version of the New Keynesian model, there is a New Keynesian Phillips Curve in the model. However, when you solve for the equilibrium conditions, you find that inflation is a function of demand shocks, technology shocks, and unexpected changes in interest rates. The output gap doesn’t appear in the solution. But fear not, this simply means that monetary policy is working properly. The Phillips Curve is apparently like the observer effect in quantum mechanics in that when we try to observe the Phillips Curve, we change the actual result (this is a joke, please do not leave comments about why I’ve misunderstood the observer effect).

However, I would like to submit that even this interpretation is problematic for thinking about monetary policy and defending the Phillips Curve. In the New Keynesian model, we get an equation that looks like this:

$pi_t = beta E_t pi_ + kappa y_t$

where $pi$ is the rate of inflation, $y_t$ is the output gap, and $kappa$ and $beta$ are parameters. This equation is an equilibrium condition of the model. Since it is an equilibrium condition, it always holds. This equilibrium condition can be derived by (1) having a monopolistically competitive firm solve a profit-maximization problem with a Rotemberg-esque quadratic adjustment cost associated with prices, (2) solving for a symmetric equilibrium, and (3) log-linearizing around the steady state. So this is an equilibrium condition for the aggregate economy. When you look at this equation, you would think that you can use this equation for some intuition about the evolution of inflation. To demonstrate how silly it would be to do so, let’s assume that people in the economy are sufficiently patient that we can re-write this equation as:

$pi_t = E_t pi_ + kappa y_t$

So you look at this equilibrium condition and you get a very New Keynesian interpretation of the world. It looks as though inflation is explained by changes in expected inflation and changes in the output gap. However, this interpretation is wrong. This equation is an equilibrium relationship. Thus, I could just as easily re-write this equation as

$y_t = frac (pi_t - E_t pi_)$

Hmm. Now we have something that looks like an expectations augmented Phillips Curve with the direction of causation moving in the opposite direction. Now, it looks as though unexpected changes in inflation are causing changes in the output gap.

So what is a central bank to do?

Actually, using this equation alone, we can’t say anything at all! This equation is just an equilibrium relationship. Without knowing anything else about the economy, this tells us nothing. We have one equilibrium equation with two unknowns. In addition, we have a rational expectation about inflation, which implies that the expectation is model-consistent. In order to know what a model consistent expectation is, we need to have a model from which we can form expectations. In other words, this equation tells us absolutely nothing in isolation from a bigger model.

For example, suppose that we are in a world with the gold standard. Let $p_t$ be the log of the price level. A reasonable assumption would be that $p_t$ follows a random walk:

$p_t = p_ + e_t$

or

$pi_t = e_t$

Combining this with our Phillips Curve would give us

$y_t = frac pi_t = frac e_t$

So output and inflation are driven by shocks to the price level. There is no exploitable relationship between inflation and the output gap, despite the fact that (a) regressing the output gap on inflation would yield a positive coefficient, and (b) the model features a New Keynesian Phillips Curve. This is important because the best evidence that we have when it comes to the Phillips Curve is from the gold standard era.

In addition, if the quantity theory holds, then the rate of inflation and the expected rate of inflation would be determined by the path of money supply. Output would then adjust to fit the equilibrium condition that looks like a Phillips Curve. This was the view of Fisher and Friedman, for example.

What all of this means is that even given the fact that the New Keynesian model features an equation that resembles the Phillips curve, this does not imply that there is some predictive power that comes from thinking about this equation in isolation. In addition, it certainly does not imply that changes in the output gap cause changes in the rate of inflation. There is no direction of causation implied by this one equilibrium condition.

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