Monday, November 2, 2015

Musical chairs and Taylor rules

Scott Sumner recently reiterated his musical chairs model. John Handley has the best take on that post. However, I thought I'd try to analyze the musical chairs model -- here it is ...
The Musical Chairs model: 
1.  In the short run, employment fluctuations are driven by variations in the NGDP/Wage ratio.
2.  Monetary policy drives NGDP, by influencing the supply and demand for base money.
3.  Nominal wages are sticky in the short run, and hence NGDP shocks cause variations in employment in the same direction.
4.  In the long run, wages are flexible and adjust to changes in NGDP. Unemployment returns to the natural rate (currently about 5% in the US.)
So let's look at 1 and 3: (1) "In the short run, employment fluctuations are driven by variations in the NGDP/Wage ratio." (3) "Nominal wages are sticky in the short run, and hence NGDP shocks cause variations in employment in the same direction." This comes down to: In the short run, employment fluctuations are driven by variations in NGDP.

If prices are sticky along with wages, then the price level is slowly varying in the short run and therefore changes in RGDP are proportional to changes in NGDP and we can say: In the short run, employment fluctuations are driven by variations in RGDP. This is just Okun's law. So Sumner's 1 and 3 are just Okun's law.

Next we'll tackle number 4. Let's split 4 into two pieces. First piece first:
4a. In the long run, wages are flexible and adjust to changes in NGDP.
Since the short run changes in RGDP are already accounted for in the changes in employment (Sumner's 1 and 3), the primary change in long run NGDP must come from changes in the price level. This means 4a is really just the statement: Wages are a price.

Now for the second piece:
4b. Unemployment returns to the natural rate (currently about 5% in the US.)
This is just an assumption that a unique equilibrium exists -- that the fluctuations in 1 and 3 are around some level.

Sumner's 2 is a bit vague as presented here, but in previous posts (I link to here) Sumner says that the central bank sets expectations for NGDP with monetary policy. Instead of a single NGDP futures market target (his preferred policy), the Fed sets this with a combination of a de facto inflation target, IOR and its internal forecasts of NGDP and inflation.

Additionally, Sumner says his model is consistent with rational expectations, therefore the expected value of NGDP at time period t+1 is the actual value of NGDP at time period t+1 plus an (unbiased) error. He also adds in the possibility of a systematic error term stemming from the difference between a targeting an ideal liquid NGDP futures market price/growth rate and whatever the central bank does in practice.

However, we can frame this in terms of a Taylor rule. We start with:

i = π + r* + a (π - π*) + a (y - y*)

Sometimes the parameter a is taken to be different for the two terms, but let's keep them the same. Now let's re-arrange:

i - π - r* = a (π + y - π* - y*)

Using n ≡ π + y and n* ≡ π* + y*, we have

i - π - r* = a (n - n*)

The LHS is the difference between the short term nominal interest rate target and the equilibrium nominal interest rate i* = π + r*. We can call r* the equilibrium Wicksellian rate if we'd like. Let's define this deviation to be Sumner's systematic error (SE) plus random unbiased error σ:

i - π - r* ≡ SE + σ

We don't really know what SE is (except in the case of a central bank targeting an NGDP futures market), so this is completely valid. We have:

SE + σ = a (n - n*)

If a = 1, this is Sumner's equation (2) and (3) shown at this link. That means monetary policy in Sumner's model can be represented as a Taylor rule that is always off (in the long run) by some amount SE unless the central bank targets an NGDP futures market.

This of course should make a lot of sense from Sumner's view. If a central bank perfectly targeted an NGDP futures market, then whatever the nominal interest rates was, it would be the equilibrium nominal interest rate. If there was some other less efficient monetary policy target, then that policy could be expressed as a deviation of the observed nominal interest rate from the equilibrium nominal interest rate (whatever it was).

The systematic error can be expressed as a function of the interest rate SE = SE(i). Interest rates being "high" are a sign of loose monetary policy, rates being "low" are a sign of tight monetary policy (per Sumner's invocation of Milton Friedman). Therefore SE should be positive when monetary policy is "loose", negative when it is "tight" and zero when it is 'right on' as per an NGDP futures market.

Putting this all together:

1. Okun's law in the short run
2. A Taylor rule: i = π + r* + (n - n*) = π + r* + SE(i)
3. Okun's law in the short run
4. Wages are a price. There is an equilibrium in the labor market.

Overall, 1, 3 and 4 are basically true of any economic model (unless it is inconsistent with the data). Therefore you could probably represent Sumner's model as a New Keynesian model with a particular Taylor rule containing a systematic error term.

Note: n - n* is never zero unless you target an NGDP futures market, and on average n - n* = SE(i). This doesn't really pin the model down completely since SE(i) is determined by some vague judgment calls about whether interest rates are "high" or "low" relative to where they "should be". However if someone (John Handley?) wanted to, someone could generically take SE to be a negative value approximately equal to the current (nominal) output gap.

Update 11/3/2015

I wanted to make clearer that the point of this post was that Sumner's musical chairs model as described is so generic that it is likely any number of DSGE models (New Keynesian or otherwise) could fit the 4 listed characteristics while having wildly different policy implications.

Analogous to the need for at least three points to define a plane, we need more than these 4 characteristics to define an specific economic model.


  1. Hi Jason,

    I'm glad you liked my take on Sumner's post.

    In a lot of ways, Market Monetarist 'models' are pretty conventional, except for the gratuitous usage of NGDP and the focus on nominal wage rigidity. Market Monetarists need nominal wage rigidity to explain why the divine coincidence isn't always true and that an NGDPLT is better than an inflation target or a price level target. The gratuitous usage of NGDP is just annoying and unnecessary. Sumner could have given his entire model without referencing nominal gross domestic product until he suggested the optimal policy in the model. I suppose I should read Sumner as "nominal wage rigidity exists, therefore divine coincidence fails, therefore we need to target NGDP." I just wish he wouldn't present his models as being so unconventional just so he can use the word "NGDP" ten times more than necessary.

    1. I think failure of the divine coincidence is a good way to characterize it. And yes it is fairly conventional. However, my point (and I will add a bit at the end to emphasize it) is that it's not just conventional but so incredibly vague that the 4 "characteristics" of the model can be satisfied by just about any framework that can have totally different additional consequences but still conform to the 4 characteristics.

      For a physics analogy (sorry), it's like saying:

      1. It's a gauge theory
      2. It conserves energy
      3. It has matter

      The theories that conform to those can be as different as the strong force, the weak force and electromagnetism.

      I'm sure there are several NK theories that have Sumner's 4 characteristics that have completely different policy implications.

      Actually, the IT model satisfies those basic characteristics! And it reduces to the IS-LM model in its low inflation limit.

    2. I agree that Sumner's model is incredibly generic. However, due to the nature of DSGE, I think it would be hard to come up with a DSGE model with the same frictions as Sumner's 'model' that give with different policy implications than he suggests.

      Part of the problem I see is that a lot of Sumner's model relies on a very high degree of assumption. He assumes that monetary policy works at the zero lower bound. That assumption needs to at least have some economic logic behind it - something that I don't believe Sumner has provided (although maybe I can coax him into it by continuing to comment on his post).

  2. Jason, this is the first time I enter your blog, congrats on the rigorous analysis you try to do with your models. I have been following Sumner's blog for a few months now, and I find most of what he says very reasonable. For various reasons, I find his focus on NGDP his most interesting (and useful) proposition. But, in your analysis above, I was wondering, how do you interpret the SE you include in the model ?

    1. Welcome Jose.

      I learned much of what I know of monetary economics from Sumner, and he is fairly reasonable when discussing the mechanisms of monetary economics. However, he tends to see data (or evidence in general) through the lens of confirmation bias and doesn't seem to understand other models (see e.g. here) -- a problematic combination as it makes you unable to see how the same evidence might be consistent with another model.

      As far as the SE term -- that was entirely Sumner's doing. It is a Systematic Error that was his way of representing the difference between a central bank that targets a NGDP futures market (SE = 0) and one that targets inflation, interest rates or unemployment (in which case SE ≠ 0).

      Here is the reference:

      And here was my take on that model:


Comments are welcome. Please see the Moderation and comment policy.

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.

Note: Only a member of this blog may post a comment.