Finding the Exponent in the Fiat Decay Model

In macroeconomics, the term “velocity of money” is often used. The concept is that money is reused as it passes from person to person so it must pass with some “velocity” (measured in exchanges per period).

A similar (but not identical) relationship is found in the model described in my post “Mapping Stimulus to GDP” . [In that post, the model had no name. In this post, the model will be label the Fiat Decay Model (FD Model).]  In the FD Model, the number of transactions is critical to establishing the limit of GDP expansion possible with any new fiat money supply. After each transaction, part of the new fiat issue is returned to government by taxation which leaves a diminished amount for further GDP expansion. Each transaction is treated as an increment of an exponent.

It is easy to find the velocity term in the common usage. Use the formula

Velocity = GDP/ Money Supply

where GDP is Gross Domestic Product.

It is not so easy to find the exponent from the formula

(1)    - TR*GDP = - MS + MS*(1-TR)^(n+1)

(which is the way I left the equation in “Mapping Stimulus to GDP”.  MS represents Money Supply, TR is Tax Rate).

Frankly, I did not realize how useful the entire equation could be. I simply looked at it as being an intermediate step to finding the limit of possible GDP growth. It was somewhat later that I realized that the second term incrementally reduced to zero, one exponential step at a time, which made the second term VERY useful for finding the amount of potential money supply used during any period.

If using the FD Model is harder than simply finding velocity, why would we want to expend the extra effort? First the results are NOT the same. The FD Model considers POTENTIAL GDP from an existing amount of money supply and and can calculate the number of transactions that actually occurred to account for a data driven GDP number. In addition, the FD Model includes taxation. As a result of these two important differences, the FD Model allows additional insight into macroeconomic events.

To solve the equation for the exponent, we will first rewrite equation (1) to read (Note 1.)

.

Then divide both sides of the equation by MS to get

.

Now, use the logarithmic form of the equation to get

.

Finally, rearrange to write

(2)      

The reader is invited to compare curves generated by Equation (2) and velocity using data from the American economy. The term TR will be found using the Federal Reserve series FGRECPT divided by GDP. The money supply used will be the series FDHBPIN added to series FDHBFRBN. The term  TR*GDP/MS will simplify(?) to FGRECPT/(FDHBPIN+FDHBFRBN). After inserting all the data series, we can write

    

The term n+1 is plotted in the graph below.

The trace of Velocity and the Exponent from the Fiat Decay Model

This model is very much a work in progress. I am unaware if others have already written using a similar framework, but if they have, I would very much like to have references to any prior similar work.

The Fiat Decay Model seems to be a new tool for macro-economic study. I hope to have future posts that will further explore macroeconomics from a Fiat Decay Model perspective.

Note 1. We will use Google Docs and the Add-on formula editor to improve the formula presentation.