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.

Estimating GDP Expansion from Additional New Money

In a well written post, Brian Romanchuk provides the example of an author who had the good fortune of earning an extra $10,000 after taxes in year 2014. He then saves that money, never uses it, and leaves it to heirs.  The heirs 100 years hence (in 2114) spend the money.

I would like to know the amount of GDP expansion that can be expected in year 2114. 

In comments on the article, Brian correctly suggests that the amount of GDP expansion will depend upon the model used. I would like to use the model developed in my previous post "Mapping Stimulus to GDP".

This could be called the Fiat Decay Model (FD Model). It is a universal model for any fiat currency issued by a government with taxing authority.

This model recognizes that a one time injection of money into an economy can be entirely recovered by government through taxation. It makes the assumption that the money re-captured will not be re-spent but instead will disappear. This would be consistent with a Modern Monetary Theory (MMT) position.

We will ignore interest and the effects of inflation.

From "Mapping Stimulus to GDP", the maximum amount of GDP possible is given by the formula

          (1)  GDP = MS/TR

where GDP is Gross Domestic Product, MS is Money Supply, and TR is the Tax Rate.

If we assume that the tax rate is 20% and the amount of new money in 2114 is $10,000, the maximum amount of GDP would be

Maximum GDP = 10,000/0.20 = $50,000.

(Notice that if the tax rate were zero, the amount of possible GDP expansion would be infinite.)

(Notice also that to attain the full $50,000 GDP expansion, an infinite number of transactions would need to occur. The $50,000 is a limit, not an exactly attainable amount despite the fact that we can come as close to the exact limit as we desire.)

To predict the amount of GDP we might actually expect to see in 2114 when the $10,000 is spent, we need to use the full equation used originally to find equation (1). The full equation is

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

where n is the number of taxable exchanges using the number zero as the first exchange. (When x = 0, the exponent for the 1-TR term is 1, which is correct for a single exchange.)

Before we use equation (2) to predict future GDP, we need to examine past economic performance to learn what value of n is presently achieved. A result from previous work for the American economy reveals a present value of n to be 0.4. The value of the exponent n+1 will then be 1.4.

Now we can solve equation (2) to find the expected GDP. We will use a money supply of $10,000, a tax rate of 0.20, and an exponent of 1.4.  The result is 

Expected GDP = (10000/0.2)*(1-(1-0.2)^1.4) = $13,415.60.

(In my comment to Brian's post, I incorrectly suggested that the GDP increase would be $14,000. I incorrectly (in haste) multiplied the exponent by $10,000 to arrive at the $14,000 figure.)

We would all see that the accuracy (to the penny) is purely mechanical. GDP is an estimate and nothing more. 

The Fiat Decay Model does much more than simply make possible a prediction of future GDP expansion. It formalizes a link between new money, taxes and potential GDP. At least one of these three components will exist at the foundation of every economic discussion.

Mapping Stimulus to GDP

Current economic thinking is driven by the concept that expansion of the money supply will increase economic growth (as measured by GDP). Expansion of the money supply is accomplished by expanded debt.

One ongoing question relates to the effectiveness of debt in expanding GDP. How much expansion of GDP does theory allow and what are the determining factors?

No question such as this can be answered without setting up a model listing the assumptions and relevant factors. The focus in this post will be to make real world assumptions using realistic parameters and players.

The concept of stimulus (such as QE by the U.S. Central Bank) is that something new is added to the economy that would not exist except for the efforts of the central banks. The new thing is money which is added to the economic system and which can be measured by increases in money supply and increased debt by (mostly) government.

Our model will assume that the new money is created by a central bank, used by government to create a new program, and then is received by the private sector. In this model, the private sector exchanges goods and services for government money.  All exchanges at this level count towards increased GDP.

We will also assume that government taxes each exchange of money. For simplicity, we assume that the tax is in the form of an income tax (The actual method of tax makes no difference in the final result but may make a difference in the timing of the effect on GDP). The long term effect of taxation is that government can initiate action by spending money and then recover ALL the money by future taxation.

During the time the new money remains in the economy, this additional new money can and will be circulated within the private economy. Each transfer will be assumed to be taxed as an income tax with the result that each subsequent transaction will involve less money, continuing until all the new money is returned to government.

This series of events can be modeled easily as a series of transactions. Solving the resulting equation is a little tricky but not hard once the correct sequence is applied. Our model will NOT consider the time interval, if any, between transactions.

We will use the terms GDP for Gross Domestic Product,  TR for Tax Rate, and MS for Money Supply. The money supply (MS) would be the new money (or new debt) no matter how first created.

Assuming government makes the first exchange, the first increase in GDP would be represented by

      GDP = MS.

GDP would increase with the second exchange but amount of increase will be less by the amount of tax removed. The sum of original GDP increase and second exchange increase could be represented by

      GDP = MS + MS*(1-TR)

(We could represent this second GDP calculation as GDP2 to distinguish it from other GDPs but the object is to find the maximum GDP that can be obtained from one initial injection of new money supply.)

The term MS*(1-TR) represents the remaining portion of the money supply after tax is extracted.

GDP would increase again with the third exchange and could be represented by

      GDP = MS + MS*(1-TR)  + (MS*(1-TR))*(1-TR)
               = MS + MS*(1-TR) + MS*(1-TR)^2.

GDP would increase again with the forth exchange and could be represented by

      GDP = MS + MS*(1-TR) + MS*(1-TR)^2 +MS*(1-TR)^3.

We can see a pattern developing here. Each additional increment of spending is money supply (MS) multiplied by the term (1-TR) and then multiplied by (1-TR) another time. Thus, after four exchanges, the term (1-TR) is multiplied by itself three times ( (1-TR)^3 ). Another feature of the developing pattern is that each new term is smaller.

As we make more and more transactions, GDP will keep increasing, each term getting smaller, until each new term becomes a value too small to matter to a real economy. At that point we will write

(1)         GDP = MS +MS*(1-TR)  + MS*(1-TR)^2 +
                                                  MS*(1-TR)^3 + + + MS*(1-TR)^n

where the term n represents the nth +1 transaction.

This equation is difficult to solve due to having so many terms. Fortunately, equation (1) is easily transformed into a much easier-to-use equation. We can multiply each term by (1-TR) and re-arrange to get

(2)         GDP - TR*GDP = MS*(1-Tr) + MS*(1-TR)^2 + MS*(1-TR)^3 +
                                                  MS*(1-TR)^4 + + + MS*(1-TR)^(n+1).

Now we can subtract equation (1) from equation (2) to get

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

The last term raised to the n+1 power can be made as small as we wish (nearly zero) for accuracy. The equation can then be re-written to get

(3)         GDP = MS/TR

which is an easily used equation.

Skeptics may be worried by what seems to be a cavalier dropping of the term

         MS*(1-TR)^(n+1)

on the justification that it approaches zero. Please notice that the term is negative (if included in equation 3) which gives the result that adding additional terms by raising n reduces the error.

While it is true that the term MS*(1-TR)^(n+1) may be too small to be significant, the fact that it exists at all IS significant. The existence of the term reminds us that equation (3) is not an exactly equal representation of the terms GDP, MS and TR. Instead, equation (3) is a limit that is approached with continuing successive transactions using the original money supply.

We can relate equation (3) to the Flow of Funds data series made available by the U.S. Federal Reserve. GDP and government receipts are reported. Government receipts can be considered as a reduction of money supply available to the private economy (most tax payers will agree with this). The measured tax rate (TR) would be government receipts divided by GDP.

If government can remove money from the economy, it can also put it back in. Government can spend the tax receipts. If government spends the money and the tax rate stays the same, then we can expect the GDP to again come back as

  (4)     GDP = tax receipts/Tax Rate (TR),

 the same as we derived in Equation (3).

While this comparison is very useful, it is at the same time, misleading. Equation (4) is a statistical relationship while equation (3) is a predicted limit. Yes, the two equations look identical, but, the context in which each can be used is very different. Equation (4) is a statistical relationship. Equation (3) is a prediction that becomes accurate after MANY transactions (that may take place only after many years of economic activity have passed, or may occur very quickly under hyperinflation conditions).

It sometimes helps if economic equations can be compared to puzzles from every day life. Here is a simple puzzle involving cars and the distance between cities:

A couple traveling by car between two cities decides to leisurely travel 20 percent of the remaining distance each day. The first day they travel 300 miles. What is the distance between the two cities?

The answer: set up an equation that defines the first days travel. 300 miles equals 20% of unknown distance.

        300 = 0.2 * X.

Re-arrange and solve to write

        X = 300/.2  = 1500 miles.

If distance between cities in miles (X) was GDP, the money supply would be $300 and the tax rate 20%.

Stimulus really does map to GDP!

This puzzle followed the logic of equation (4), not equation (3). Here we have relied on the subtle fact that the couple must have known the distance between the cities and therefore only traveled 300 miles the first day. Knowing this, we solved the problem as we did.

Here is a second puzzle to illustrate the logic of equation (3): On a second trip, a couple decides to begin the trip with one hard days travel, then drive less by 20% each succeeding day. On the first day they travel 500 miles. How many miles will their trip cover and how long will it take?

From equation (3), we can see that they will cover no more than 500/0.20 or 2,500 miles. From knowledge that a remainder will always exist, we can see that we have defined the puzzle in a manner that prevents the couple from ever completing the trip. Each day, they travel, but each day they only advance 1 - 0.20 of the amount traveled the previous day. There is no end to this journey!

Finally, we will go back to equation (3) and then replace the term we cavalierly (but correctly) dropped. We do this to show that we can find the GDP sum after every transaction. The entire equation (3) is

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

If you try to use equation (5), remember that the number of transactions begins with zero. Thus, the first transaction is identified by n = 0. Following the first transaction, GDP = MS.

GDP, money supply and tax rate ARE intimately connected. Unfortunately, the connection is of two varieties with the same formula. It should be no surprise that economist may disagree over the relationships of GDP, money supply and tax rates. They may each be discussing different concepts!

(c) Roger Sparks 2014