## Tuesday, September 29, 2015

### The Phillips curve and the information transfer index

Brad DeLong wrote about the Phillips curve yesterday, which inspired me to take on the Phillips curve again. I've long tried to understand it with the information transfer model. In my most recent attempt I noted that with constant information transfer index $\kappa$, there is no direct relationship between employment and inflation [1].

However, that leaves out changing $\kappa$ (I sometimes use the inverse and label it $k$). The model (see the draft paper) uses the form

$$\frac{1}{k} = \kappa = \frac{\log M/(\gamma M_{0})}{\log N/(\gamma M_{0})}$$

which means that a shock to $N =$ NGDP (a recession) is a shock to $\kappa$. That is how NGDP changes can impact the price level. The question is: what is the magnitude of the impact on inflation?

I solved for the coefficient of the leading order term in $\delta N$ taking $N \rightarrow N + \delta N$, deriving the coefficient $\alpha$ such that $\delta \pi \simeq \alpha \delta N/N$. Here is $\alpha$ vs time:

We can see that $\alpha$ is positive (and approximately 0.05) through the 60s and 70s, falling to approximately zero by the year 2000. That is to say a negative shock to NGDP reduces inflation during the 60s and 70s -- an NGDP shock of 5% should reduce inflation by 0.25 percentage points.

Now according to the link [1] above, NGDP shocks are roughly equal to labor shocks, so

$$\delta \pi \simeq \alpha \frac{\delta N}{N} \simeq \alpha \frac{\delta L}{L}$$

That means a negative shock to labor (a rise in unemployment) should result in a lower inflation rate. That is the traditional Phillips curve. It basically goes away after the 1970s -- interestingly coinciding with the adoption of the expectation-augmented version.

1. Is alpha related to kappa?

1. In the explicit form it depends on kappa and its derivative. But it's kinda messy as a formula...

2. Interesting! To me, this is one of the most important things that something like ITM could bring to economics- improved policy advice with regard to employment and the economy. Judging from the alpha graph above, I would imaging that with kappa ~ 1 (information trap) that inflation is relatively unresponsive to the inflation rate.

However, in a previous post, you estimated the "natural" rate of unemployment in a different way, and seemed to come out with a different take. Can you explain the apparent difference?

http://informationtransfereconomics.blogspot.com/2013/11/the-labour-supply-part-2.html

3. oops- inflation is unresponsive to the unemployment rate in the information trap.

4. This comment has been removed by the author.

5. In that link, the "natural rate of unemployment" is effectively a constant -- it is proportional to $\kappa_{L}/\kappa_{U}$. It doesn't depend on inflation.

In the above post, I didn't estimate the natural rate -- these are labor fluctuations. However, the formula $\delta L/L \sim \delta N/N$ is derived from the markets in that post.

6. Jason, as far as you know, is this the only explanation for the disappearing Philip's curve? In my extremely limited experience, I've seen lots of criticism of the concept (of the Philip's curve), but I've never seen anybody attempt an explanation for why things would have changed over the decades like this.

7. The Lucas critique version (though I don't know who said this specifically) is that when the Phillips curve started to be used for policy, it melted away due to agents adapting.

The expectations-augmented version is that people have come to expect low and stable inflation in the future starting in after the 80s when the central bank became credible with Volcker.