Sunday, October 5, 2014

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


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


  1. Hello,

    I looked this over fairly quickly. The problem I see is that the "velocity of transactions" is somewhat limited, as people are paid on a periodic basis. This limits how fast money can circulate in the economy, and would impact your series expansion.

    The models in Godley & Lavoie - Monetary Economics - start off at a similar place. You get a multiplier from fiscal policy that is based on tax rates. It's been awhile since I read their book, so I am not sure how it relates to your model.

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  3. Brian,

    Thanks for taking a look!

    The equation GDP = MS/TR is a limit never reached even at at infinite number of transactions. So yes, by itself it is useless to predict how fast money can circulate in the economy.

    It seems to me that there is value in knowing that a limit exists. The limit indicates that once money is introduced into the economy by government, taxes take the money away quickly at first, then slower as less money remains.

    You probably noticed that the exact amount of money remaining can be found for each stimulus event by using the full version of the equation (equation 5 in the post). Knowledge of this process, and the limit, offers a common reference system. I hope to expand on this possibility in a future post.

    I have a copy of Monetary Economics in my library. I found, on page 72, text that indicates the authors were aware that a limit existed for the expansion. They seem to have concentrated on the GDP that would be reached if there were no taxes but there may be savings by consumers. This is similar to the use of the Keynesian Multiplier so far as I can see. There may be more in their book along the nature of a limit but I have not yet found it.

    This idea of a limit relating GDP, money supply and taxation seems so basic (to me) that others must have already expanded upon it. I would welcome comments that provided such references.

    One graph that I find interesting is for the term K in the equation

    GDP = new money supply/TR + K * (remaining new MS +old money supply)/TR.

    which follows directly from equation 5. It is a puzzle to decide exactly which numbers from the Federal Reserve Economic Data Series to use for old money supply and new money supply. This is a velocity equation but modified to recognize that new money is almost certainly spent to increase GDP but old and remaining new money supply is significantly delayed by savers.

    Thanks again for looking at this post.

    (c) Roger Sparks 2014