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.