Robert Waldmann and Simon Wren-Lewis, in discussing Paul Romer's history of macro 1977-1982, bring up the Phillips curve. I've also written about it on occasion.

I thought I'd have a look at the Phillips using the DSGE form of the information equilibrium model. Turns out it results in something really cool ... Here are the relevant equations from the DSGE form link:

\text{(1) } n_{t} = \sigma_{t} + \left( \frac{1}{\kappa} - 1 \right) (m_{t} - m_{t-1}) + n_{t-1}

$$

$$

\text{(2) } \pi_{t} = \left( \frac{1}{\kappa} - 1 \right) (m_{t} + m^{*}) + c_{m}

$$

$$

\text{(4) } \ell_{t} = n_{t} - \pi_{t} + c_{\ell}

$$

Here $n$ is nominal output, $m$ is base money (minus reserves), $\pi$ is the price level, and $\ell$ is the total employed. The symbol $\sigma$ represents 'nominal shocks'. They are the stochastic part of the model, and they're typically positive. They represent the difference between where $n$ is at time $t$ and where it should be based on the change in $m$ from time $t-1$ to time $t$ alone -- essentially equation (1). The rest of the symbols are constants, with $\kappa$ being the IT index (approximately constant over the short run).

I had hoped to show some kind of relationship between changes in the total employed ($\ell_{t} - \ell_{t-1}$) -- and thus changes in unemployment -- and inflation ($\pi_{t} - \pi_{t-1}$). But the math led me to something I didn't expect. With a bit of algebra, you can show that labor growth is given by:

\ell_{t} - \ell_{t - 1} = \sigma_{t}

$$

*regardless*of the information transfer index $\kappa$. Those nominal shocks I've talked about since this post? They are basically changes in the number of employed. That's why they're typically positive and typically around a few percent.

Effectively, employment growth is the part of nominal output (NGDP) that is left over after accounting for inflation. Thus there shouldn't be any relationship between inflation and unemployment -- i.e. the Phillips curve isn't real. This even applies to a version of the model consistent with expectations, since we could easily write:

E_{t} \ell_{t+1} - \ell_{t} = E_{t} \sigma_{t+1}

$$

That is to say expected changes in unemployment are the expected shocks to the economy

*after*accounting for inflation.
How does this look empirically? Pretty good ($\sigma$ in blue, $\ell$ in yellow):

Jason, I read all your posts but I rarely feel like I understand them. Intuitively, I love the idea of modeling our collective groping towards equilibrium as being like a gradual increase in entropy, as we all try to make sense of the partial information that each of us is able to absorb. I like it much better than modeling each economic participant as a hyper-rational lifetime-utility-maximizing agent with complete knowledge of everything, which strikes me (and most laypeople) as ... well, let's just say, optimistic.

ReplyDeleteBut I haven't been able to get to the point where I can actually follow the math in your posts. Which annoys me, because while I'm not as good at math as you are, I'm no slouch, I participated in the US math olympiad in high school, I got through differential equations and div-grad-curl etc at MIT. And what you're writing doesn't look complicated.

But I get lost. I don't know whether n_t is NGDP or log(NGDP). I don't see why year-over-year NGDP should be related to the size of the monetary base --- why would n_t be equal to a shock plus last year's NGDP plus a multiple of the change in the monetary base? I would have expected NGDP growth to be related to productivity growth, population growth, and price level growth. I am 100% convinced that you have good reasons for modeling things the way you are, but I'm just not following them.

Where I'd like to get to is the point where I can use the information transfer framework to model why NGDPLT is so much better than inflation targeting. The intuition is that when there's a negative shock to aggregate demand (or equivalently, a positive shock to demand for the medium of account), inflation targeting winds up with lower real output than NGDPLT, because with NGDPLT, the increase in the money supply allows nominal prices to adjust as needed until all potential output is sold, whereas with IT, all it takes is a supply constraint in one market to cause the central bank to hit its inflation target and incorrectly conclude that monetary policy is on target when in reality it's much too tight. Specifically, Kevin Erdmann has shown us (http://idiosyncraticwhisk.blogspot.com/2015/08/july-2015-inflation-and-our-very-low.html) that supply constraints in housing leads to shelter inflation which, in combination with downward nominal rigidity in other prices, leads to a persistent aggregate demand shortfall. Ironically, Fed tightening suppresses home construction that makes shelter inflation worse which cases the Fed to tighten, vicious cycle. What I'm trying to get to in this rambling is that the Fed's over-tight money in combination with downward nominal rigidity suppresses information transfer, whereas under NGDPLT, the price signal propagates much more quickly as some prices rise much faster than others. Basically, you'd see house prices going up even faster compared with the factors of production of houses, leading to more construction, rather than less as we get with inflation targeting.

Boy I hope that came out coherent.

Anyway, I guess I'm hoping for a pointer to what I'd need to read to really understand this post... and also hoping for your reaction to my idea that NGDPLT beats inflation targeting in an aggregate demand shortfall because NGDPLT improves the information transfer coefficient, i.e., reduces the time needed for price signals to affect market behavior, whereas downward nominal rigidity impedes information flow under inflation targeting.

Thanks,

-Ken

Hi Ken,

DeleteThanks for continuing to read along. The variables in this post are log-linearized variables (as they frequently appear in DSGE models). Therefore the model for the interest rate:

R = (k1 N/M)^k2

becomes

log(R0+dR) = k2 log(N0 + dN) - k2 log(M0+dM) + k2 log k1

log(R0)+dR/R0 = k2 log(N0) + k2 dN/N0 - k2 log(M0) - k2 dM/M0 + k2 log k1

Now set dR/R0 = r, etc so that:

log(R0)+ r = k2 (log(N0) + n - log(M0) - m) + k2 log k1

define c = k2 log k1 - log R0 and m* = log N0 - log M0 so that:

r = k2 (n - m + m*) + c

This helps you be able to read off what happens, ceteris paribus. A relative increase in M leads to a relative fall in the interest rate R (liquidity effect) and a relative increase in N leads to a relative rise in the interest rate (income effect).

Here are some lecture notes about log-linearization

http://www3.nd.edu/~esims1/log_linearization_sp12.pdf

Regarding the specifics of some of the equations, the idea is that when the IT index (kappa) is about 1/2 (and sigma = 0), you have the quantity theory of money (QTM). That's what equations (1) and (2) say. Increasing base money leads to a higher price level and higher nominal output. In fact, P ~ M and N ~ M² so that nominal growth g is equal to real growth r plus inflation i (g = r + i), but also real output R = N/P = M so r = i. When the QTM is true, you have NGDP growth being equal parts real growth and inflation (10% nominal growth is 5% real and 5% inflation).

However, the ideal QTM doesn't always work (kappa > 1/2) and there are those shocks (sigma), so real life is a bit more complex.

So what is going on with productivity, population and technology? The above post seems to say that sigma covers much of the population growth (inasmuch as the total employed roughly follows population). However the model is a purely macroeconomic model -- it basically assumes everything is random at the micro level. I like to think of it as the interactions of agents being so complex it looks random.

One way to view the the model is as an equaltion of state for a generalized ideal gas. Output is like energy (n R T), money is like volume (V) and the price level is like pressure (P): P V = n R T. This model doesn't tell us about how the atoms are moving, only how changes in the macro observables affect other macro observables assuming ignorance of how atoms work.

continued ...

DeleteIf kappa = 1/2, we actually have a purely QTM economy. A price level target is equivalent to an NGDP target -- the difference is between targeting M or M². Also, any shocks (sigma) which could include e.g. government spending or population growth can be offset if the central bank is targeting M or M². This is Scott Sumner's monetary offset (and in general, everything he says is true when kappa ~ 1/2).

However, as kappa approaches 1, we end up with N ~ exp(sigma t) and P ~ constant. Monetary policy becomes irrelevant and you control N via sigma (e.g. immigration and government spending).

In between, 1/2 < kappa < 1 (where we are), you could have a policy mix. However using monetary policy in that case brings you to kappa = 1 faster. That's because kappa ~ log M/log N. And log x grows more slowly with x as x increases (basically, the derivative of log x ~ 1/x). If kappa < 1, then N > M, so for a roughly equivalent increase you get faster changes in log M than log N. Thus most economies will drift towards kappa ~ 1 ... Japan is in the lead with the EU not far behind.

The way out of that? Generally: hyperinflation (printing money to fund government, helicopter drops, wars). It will reset kappa to a lower value closer to 1/2 (if you ever return to normal). This happened to most of the industrialized world after WWII.

That's the 30,000ft view of the information transfer model. I'm actually working on a paper (some links to the draft material appears on the sidebar -- as well as some other posts) that I'm going to put up as a draft soon ... hopefully before the end of August.

Here's a good place to start

http://informationtransfereconomics.blogspot.com/2015/02/information-equilibrium-paper-draft.html

Thank you very much for the detailed reply. This was so helpful. I read Eric Sim's paper and your first response now it's clear to me what log-linearization means. I'm out of time to go through the second response now, but I'll come back to it.

DeleteThanks again.

-Ken

If Eq. (1) has (1/k) instead of (1/k - 1), then result becomes:

ReplyDeletelt - lt-1 = sigmat + (mt - mt-1)