I have found an equation to determine the lower limit of the Fed funds rate using Effective demand. Let's just get started...

We start with the equation for the lower boundary of the Fed funds rate as a function of the TFUR (total factor utilization rate), which is simply multiplying capital utilization rate by the employment utilization rate (capacity utilization * (1 - unemployment rate)).

Fed funds rate, F ≥ 0.61 * (TFUR) - 0.438

The Fed funds rate can simply be over the number given by the right side. Now we solve for TFUR...

TFUR ≤ 1.64 * F + 0.719

Now we substitute this equation into the effective demand equation...

Effective demand, ED = real GDP * els / TFUR

Effective demand, ED = real GDP * els / (1.64 * F + 0.719)

Now we solve for the Fed funds rate...

F ≥ 0.61 * real GDP * els/ED - 0.438

Ok... let's put this equation to the test. The numbers as of 4Q-2012 are... real GDP=$13.665 trillion, ED=$14.204 trillion, els=74.3%. We can then calculate the lower bound of the Fed funds rate.

F ≥ 0.61 * $13.665 trillion * 74.3%/$14.204 trillion - 0.438 = -.20%

The result is slightly negative, which confirms the Fed funds rate being at the zero lower bound.

Another useful way to arrange the equation is with the inflation rate on the right side. This way we could eventually graph the relationship between the Fed funds rate and inflation (all else constant). Let's put inflation in...

Fed funds rate, F ≥ 0.61*real GDP * ulc_{e}/(ED*(1+i)) - 0.438

ulc_{e} is effective unit labor costs which is calculated by taking the index of unit labor costs and multiplying by 0.78, the conversion factor for labor share of income.

i is the inflation rate.

Let's test out this equation for data from 4Q-2012...

F ≥ 0.61*13.665*75.6%/(14.204*(1.018)) - 0.438 = -0.22%

Ok, it checks out... and of course if this result is less than zero (0%), the Fed funds rate still has to be more than zero (0%). This describes a liquidity trap situation.

It seems to me that the Fed funds rate is waiting for the lower bound to turn positive before rising. My question is what will happen if the Fed funds rate goes below this lower bound. It hasn't happened since 1988. Something to watch.

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