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 allow NGDP to rise above Nt so that NGDP returns to its Nt target averaging 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 + Ct + Ct - Cv + (1+a)*(C - (Ct + Ct - Cv))
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(T2 + L2) - ( 1 - z)*(T + L)
z = (2*L + long-term natural real rate, Lr)/(2*(L2 + 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 make a policy rate rule where the Fed rate steers inflation to a nominal GDP target, not just an inflation target.
The PRa equation implies that the Fed rate was very loose in the 1970's. Since the 1970's, the PRa equation has trended with the Fed rate.
In order to get the lines to match up perfectly, the parameters would have to coincide with each specific data point in time. What was the inflation target? What is the best time period for averaging core inflation? What was the estimated short-term real rate in each period? What method was used to determine the Fed rate at each data point.
I will change the parameters so that the lines match better. I make a = 0.3, Cv based on previous 3 years, Ct = 3% and the same Lr.
The PRa equation for NGDP targeting is saying that the Fed rate should be around 2.5% to 3% currently based on the parameters given above. The parameters can be changed.