# Refining a NGDP target for a policy rate rule

This post will lay out a way to incorporate a NGDP target into a policy rate rule, so that the rule steers inflation back to a NGDP target, not to an inflation target.

First, the equation for a NGDP target.

NGDP target, Nt = real rate of output growth, Lr + desired long-run average core inflation target, Ct

Nt = Lr + Ct

Ct = Nt – Lr

For example, if Lr = 1.8% and a Ct of 2.5% is desired, then Nt will be established at 4.3%. Nt is meant to be constant over time. If actual NGDP diverges from Nt, then inflation will need to over-compensate to bring NGDP back to its target over time. When actual core inflation falls below Ct, then, NGDP will fall below target. Then eventually core inflation will have to rise above Ct in order for the NGDP target to return back to its target. In this way, the NGDP target is maintained. For example, if NGDP drops below 4.3%, then NGDP will have to go above 4.3% in the future to compensate the drop. David Beckworth has estimated that maybe 5% would be a good NGDP target, but he recognizes that it may be above or below 5%.

The Federal Reserve is actually able to set the long-run average core inflation target, Ct. This then will determine Nt.

I use core inflation for the long-run inflation target instead of the headline inflation, even though nominal GDP is based on headline inflation. Monetary policy should not be based on headline inflation.

Now, I bring in a policy rate rule.

Policy rule, PR = short-term real rate, Sr + desired long-run average core inflation target, Ct + (1+a)*(current core inflation, C – Ct)

PR = Sr + Ct + (1+a)*(C – Ct)

The problem with this equation is that Ct is a constant. The equation makes inflation return to Ct, instead of leading it beyond Ct when necessary to compensate for previous divergences from Ct. This equation does not work for NGDP targeting.

(a is a weighting coefficient for how strongly the Fed rate should move inflation back to target (Ct).)

So, an equation for adjusting the inflation target, Ct, is needed. Replace Ct with an adjustable core inflation target, Ca.

Policy rate rule with adjustable inflation target, PRa.

PRa = Sr + Ca + (1+a)*(C – Ca)

Now, I need to define Ca for how to adjust the core inflation target (Ct) for NGDP targeting.

Ca = Ct + Ct – average of past core inflation(Cv)

Ca = Ct + Ct – Cv

Ct = Ca – Ct + Cv

Cv depends on how far back core inflation is averaged. Cv is a “NGDP targeting” adjustment of Ct for when average core inflation (Cv) has been different from Ct. If Cv = Ct, then Ca = Ct. When Cv has been below Ct, then Ca will rise to a higher inflation target, which allows NGDP to rise above Nt so that NGDP returns to its Nt average target over time.

The idea of Ca is this… If Cv has been averaging 2.0% (below the desired Ct of 2.5% for NGDP targeting), then Ca will rise to 3.0% until you are able to bring NGDP back to a long-term average of 4.3%.

Now, substitute in for Ca in the PRa equation. Ca = Ct + Ct – Cv

PRa = Sr + Ca + (1+a)*(C – Ca)

PRa = Sr +ย Ct + Ct – Cv + (1+a)*(C – (Ct + Ct – Cv))

Simplify equation…

PRa = Sr – 2aCt + aCv + (1+a)C

This is the final equation that I use as a policy rate rule to move NGDP toward its long-term NGDP targeting goal. (a = weighting coefficient)

The equation embodies the equation for the NGDP target (Nt). Substitute in for Ct = Nt – Lr.

PRa = Sr – 2a(Nt – Lr) + aCv + (1+a)C

PRa = Sr + 2a(Lr – Nt) + aCv + (1+a)C

So now to determine the short-term real rate (Sr), I use my effective demand monetary rule. You could use another method.

Effective Demand Monetary rule for short-term real rate, Sr = z(T^{2} + L^{2}) – ( 1 – z)*(T + L)

z = (2*L + long-term natural real rate, Lr)/(2*(L^{2} + L))

T = capacity utilization * (1 – unemployment rate)

L= effective demand limit function on the utilization of labor and capital. (see Synopsis of Effective demand.)

OK… everything is in place to test the equation. Now I compare the PRa equation to the actual Fed rate assuming a core inflation average (Cv) based on the previous 2.5 years. Core inflation target (Ct) of 2.5% is kept constant through time series for simplicity. Natural real rate (Lr) has changed over time from 3% in the 1970’s to 1.8% currently. And finally a =0.5.

The lines do not match perfectly for many reasons, but that is not the present goal. The goal is to start forming a policy rate rule where the Fed rate steers inflation to a nominal GDP target, not to an inflation target.

The PRa equation for NGDP targeting is saying that the Fed rate should be around 2.5% currently based on the parameters given above. Those parameters can be changed.

Update: Another graph with different parameters to show that the lines could match better. (a = 0.3, Cv based on previous 3 years, Ct = 3% and the Lr same as in first graph.

What is Ct?

And what does this calculation have to do with NGDP targeting (which requires neither an inflation measure nor indeed a real GDP measure) and doesn’t deal with rates of change in the short term either.

Look the logic of NGDP targeting is simple. If capacity utilitisation is low then growth in NGDP will tend to be real with lower inflation and if capacity utilization is high then growth in NGDP will tend to be inflation with lower real growth. So maintaining steady NGDP will tend to supply balanced growth. (Look at another way, if long term real interest rates are low that implies that low real growth is expected so a flat inflation target will be too low). Remember also that debt is repaid from nominal income, so a NGDP target will fairly reliably combat debt deflation, if there are unexpected negative real growth surprises, which an inflation target would not do.

P.S. The only issue I have with NGDP is that it might have problems with once off price shocks (like an oil shock). So it may respond to sudden inflation because of say a terms of trade change by slamming on the breaks when it shouldn’t. But note that it responds to ngDp (i.e. only domestic production a large increase in pure input prices won’t effect it much). But a resource produced mostly domestically but traded internationally is another issue and here it will respond to the detriment of the majority of the population.

P.P.S. But of course in the above case a price targeting rule is even worse. The problem is that inflation is not the same as price change (because of sudden readjustments) and telling the two apart with a simple rule is impossible.

Hello Reason,

Reply to your first comment… Ct is the inflation needed above real growth of output (Lr) to get your NGDP target (Nt). If real growth of output changes and you want to maintain the same Nt, then you adjust Ct. It is simple and the equation works perfectly.

And the equation does look at rates of change of short term real rates through the Sr variable.

Reason,

Reply to second comment… the PRa equation above does exactly what you are defining as a NGDP target.

Reply to third comment… price shocks are dealt with by the equation through the Sr variable which encompasses the affect of headline inflation on demand and the affect of core inflation on monetary policy. Your concern here is resolved by the intrrnal mechanics of the equation.

Reply to fourth comment… Again the internal mechanics of the equation address the differences between headline inflation and core inflation. A rise in headline inflation lowers effective demand, while a rise in core inflation raises effective demand by making monetary policy more accommodative.

Also the inflation target in the equation Ct is adjusted by the Cv variable. So in effect the equation does not have a constant inflation target which is the main problem with the Taylor rule. This equation is better than the Taylor rule.

The coefficient a determines the intensity of the Fed rate to drive inflation.

If a=0, then the Fed rate has no response to core inflation. As a rises, the response.gets stronger.

No I am not talking about the difference between core inflation and headline inflation. That assumes there are volatile prices. I’m talking about a once off change in the terms of trade that make one sector richer at the expense of the others.

With Ct the post just loses me – what are you trying to achieve here? With NGDP targeting you ignore inflation (in fact you only know real growth with a delay, NGDP is available much faster). You consider only NGDP versus NGDP target. If you are below target and not growing fast enough to catch up you reduce you set your policy instruments to expand, or if you are ahead of target, or growing so fast that you soon will be you adjust them to slow down. The actual rate required depends on other things (such as productivity growth and population growth and foreign developments).

Help me to understand you better… are you wanting a polucy rule that distributes money to combat inequalities in the economy? Or are you wanting a policy rule that does not over react to supply and price shocks? Or are you wanting a policy rule that incorporates exhange ratrs?

I need to see clearly what you are seeing.

Look I think you are fundamentally misunderstanding NGDP targeting. You have to make up not just for the lower than desired output, but also the lower than desired output. And you have ignored that. We still have to close the output gap.

Lambert,

what I was getting at is that a shock that massively changed internal distribution (i.e. an oil price shock) may result in a NGDP target that is too low for most of the population. So I just think there may be some circumstances when a NGDP target needs to be interpreted flexibly.

Reason,

I am careful to put NGDP in terms of core inflation, not headline inflation. If you go to FRED graphs, GDP is based on headline inflation. The volatility of headline inflation is conducive to good monetary policy.

Ct is your best estimate of the core inflation needed for your NGDP target based on your best estimate of real growth. These estimates change as you say with delays, but there are methods to forecast them too.

The equation above does exactly what needs to be done to adjust an inflation target toward a price level target.

The problem I think that David Beckworth has is that he does not have a way to calculate Sr. He tweeted the other asking if anyone knew of any central banks that publish these short term real rate estimates. I have a way to do it, but he is looking for a way to do it apparently.

Edward

You don’t need inflation for NGDP targeting!!!!! Why do you keep insisting that you do.

And you haven’t understood my point about changes in terms of trade. Consider what happens if there is a massive oil price increase. Then Texas and the Dakotas are cheering and everybody else in trouble. But what does our rule say we do? It may well see a quite good growth in NGDP, but most of the country is depressed. Do you get it now? It has nothing to do with price volatility but with a structural change in income distribution.

If you like think about this wrt what happened in the EUR area.

Reason,

you say… ” We still have to close the output gap.” … my equation takes that into consideration by using the limit function within the Sr variable. The output gap is the difference between effective demand and the composite utilization of labor and capital.

you say… “So I just think there may be some circumstances when a NGDP target needs to be interpreted flexibly.” … David Beckworth would disagree with you. NGDP price level targeting needs to stay firm.

you say… “It may well see a quite good growth in NGDP, but most of the country is depressed.” … Monetary policy has to respond to the aggregate data. There are other policies that will deal with the imbalances between different sectors of the economy.

and to add… NGDP price level targeting is a good idea. You will see convergence on this idea as time goes by. The model needs to be worked out. I am offering a model to the cause.

David Beckworth may disagree with me, but I disagree with him. Yes, one possibility is to tax the texans and spread a dividend to the rest of the country. But I wonder if that was not done (because the political system is disfunctional, or in the case of Europe there is no global fiscal authority) whether the central bank could then maintain its independence.

But seriously NGDP price level targeting IS A CONTRADICTION IN TERMS. NGDP targeting IGNORES the price level.

Besides which, trying to fit what the Fed has done under price level targeting backwards to a NGDP regime (which has completely different dynamics – think of the the lagged responses of prices to growth) is not a valid excercise.

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Reason,

You are correct that putting a NGDP regime on the past Fed rate is like mixing apples and oranges. But you get a baseline of comparison. Once you align the NGDP regime with past Fed data, you see the parameters that would have identified a NGDP regime. This is a way to confirm the shape and scope of the model.

The key now is to recognize the parameters and determine how those parameters should be adjusted now to make NGDP targeting work.