# Playing with toy models

Back in 2002, Bennett McCallum did a really **nice survey piece** on contemporary monetary economics. The best parts are his insights into some of the controversial issues, but I’d like to focus on something else (in the equations I changed the style a bit—I can’t do subscripts and deltas). Here’s McCallum, with adjustments:

A striking feature of the typical models in the NBER and Riksbank conferences is that they include no money demand equations or sectors. That none is necessary can be understood by reference to the following simple three-equation system.

yt = α0 + α1Et(yt+1) + α2(Rt − Et(dpt+1)) + α3(gt − Et(gt+1)) + vt (1)

dpt = Et(dpt+1) + α4(yt − ynt) + ut (2)

Rt = µ0 + µ1(dpt − dp∗) + µ2(yt − ynt) + et (3)Here equations (1)–(3) represent an expectational IS equation, a price adjustment relationship, and a Taylor-style monetary policy rule, respectively. The basic variables are yt = log of output, pt = log of price level, and Rt = nominal one-period interest rate, so dpt represents inflation, Rt − Et(dpt+1) is the real interest rate, and yt − ynt ≡ ˜yt is the fractional output gap (output relative to its capacity or natural rate value, whose log is ynt). Also, gt represents the log of government purchases, which for present purposes we take to be exogenous. In this system, Et denotes the expectations operator conditional on information available at time t, so Et(pt+1) is the rational expectation formed at t of pt+1, the inflation rate one period in the future.

(In the original dpt was “delta” pt. I also corrected a typo in equation 2.)

Now let’s do something similar in the MM model. In equation 3 we will replace R in the previous model with NGDP futures prices (NGDPF), which is the instrument of monetary policy. (It’s not really the instrument, the base is. But then the fed funds rate is also not really the instrument, the base is. Both NGDPF and R are financial market variables that are observable and controllable in real time.) The NGDP futures price equals the target value, plus a systematic error (SE). The systematic error is the predictable part of the central bank’s policy failure.

In equation 2, actual NGDP reflects both the predicted value (previous NGDPF), and an unforecastable error term (et.) The employment gap in equation 1, more specifically the gap between actual hours worked and the natural rate of hours worked, is alpha times the NGDP gap. Alpha is probably roughly one. The hours worked gap is thus roughly equal to the difference between actual and target NGDP growth. Between mid-2008 and mid-2009, NGDP fell about 8% below trend, and hours worked also fell about 8% below trend

(Ht – Hnt) = α(NGDPt – NGDPTt) (1)

NGDPt = NGDPFt-1 + et (2)

NGDPFt-1 = NGDPTt + SEt-1 (3)

And all this boils down to:

(Ht – Hnt) = α(SEt-1 + et) (1)

Where the monetary policymaker determines SEt-1.

If they do NGDP futures targeting, then SE = 0. Let’s use an inflation targeting analogy. The ECB is targeting inflation at 1.9%, and last time I checked the 5-year inflation forecast in the German TIPS market was about -0.1%. So in the eurozone SEt-1 is roughly negative 2%. If the ECB pegged CPI futures prices at 1.9% inflation, then the SE would rise from negative 2% to zero. Actual eurozone inflation would be 1.9% plus et. Under NGDP futures targeting, SE is equal to zero and the hours worked gap is a random walk.

Of course this oversimplifies everything (but then so does the 3 equation model described by McCallum.) Hours worked would actually depend on Wage/NGDP, or even better Wage/(NGDP/person). Further refinement would include shocks to labor’s share of national income. Nominal wages depend on expected future NGDP, but are also very sticky, adjusting slowly when pushed away from the desired Wage/NGDP ratio. That would all have to be modeled.

The NGDPF market could be modeled as follows. Define the ratio of next period’s NGDP and the current monetary base as “quasi-velocity” (QV.):

Mt*QVt = NGDPt+1

Then create a futures market in QV, and tell traders that the base will be set at such a level that the base times equilibrium QV (in the futures market) is equal to target NGDP (NGDPT.) That replaces the Taylor rule. And by using a velocity futures market, you avoid the circularity problem discussed by Bernanke and Woodford (1997). QV is obviously a function of the nominal interest rate. (This is based on a 2006 *Economic Inquiry* paper I did with Aaron Jackson.)

There is nothing at all like the IS relationship, as equation 2 is simply an application of the EMH (plus the assumption that the NGDP futures price is an unbiased forecast of future NGDP.) The hours worked gap is the closest thing to a Phillips Curve. If you want output gaps, you can derive them from the hours gap equation using a variant of Okun’s Law. Once you have real output, you can also derive the price level, as NGDP is already determined. But why would you want those things? The hours gap equation measures the business cycle, and NGDP is superior to the price level as a proxy for the welfare costs of inflation. And if it’s long run economic growth you are interested in, then why mess around with monetary models?

I see several differences between the standard approach and my toy model:

1. I use NGDP futures prices, which is not subject to the ambiguity associated with nominal interest rates. NeoFisherites will not misinterpret my policy equation. And it’s more efficient, as it cuts out the middleman and uses open market operations to directly target NGDP futures, which is what you care about.

2. My “Phillips Curve” uses NGDP and not inflation (the switch from unemployment to hours is not so important.) Inflation is problematic, because it might reflect either demand or supply shocks. So the standard model needs to account for supply shocks. NGDP is better, as it only reflects demand shocks, which are what drive any Phillips Curve relationship. It simplifies things.

This is just a toy model; perhaps someone else can create a real model along market monetarist lines. As a blogger this is the approach I like best. As director of the Mercatus Monetary Policy Program, I want the model that the rest of the profession finds most convincing. I imagine that would be something more along the lines of a **Nick Rowe model**.