Monday, May 2, 2016

An Alternative SIM Equation

Wynne Godley and Marc Lavoie's book "Monetary Economics" provides an excellent introduction to SFC analysis. Chapter 3 introduces model SIM (for simplest) and introduces the reader to a system of equations suitable for entry into a spreadsheet. Unfortunately, the system introduced is interlaced with the premise that people have a tendency to spend part of their income [1] . While undoubtedly true, this premise leads the reader down a difficult path. This path leads to the arithmetic of diminishing series which requires enabling the iteration capabilities of the spreadsheet and complicated, interactive calculations.

The possibility that an alternate path may exist is inferred from the knowledge that a diminishing series (such as we have here) will converge to a limit. The fact that a limit does exist should give us confidence that any path found can be coequal with series-based-pathways .

A Monetary Relationship

A relationship between taxes, value exchanged and money is easily found. Ask yourself "If I have money, say $ \\{$100} $, how much can I and all the people I trade with buy before the government takes the $ \\{$100} $ away with taxes?"

We can answer that question easily. Assume a tax rate of 20% (like the income tax) that is extracted at each exchange. We set up an equation in the form of unknown amount times tax rate equals $ \\{$100} $ . We can write that in a conventional format by letting X = the unknown amount and θ = the tax rate . We would write

(1)     $ \theta X = 100 $

The reader can see that if θ is 20%, X will be $ \\{$500} $ .

We can relate this to the general economy. Government is the source of much of the money that powers our economy. When government pays wages, the wages count toward the Gross National Product (GDP). Government typical collects taxes on those wages in the form of an income tax collected before the worker actually holds those wages in his hands. (The tax collected does not have to be an "income tax" but the concept is easier to accept when we use familiar terms and conditions.) If we let T represent any tax repetitively collected at each transaction, we can write

(2)    $ GDP = \frac{T}{θ} $

Equation 2 gives us a simple, intuitive way to relate GDP, taxes and a tax rate. It also provides a relationship to the money supply.

The Money Supply

We began this analysis by assuming we had $ \\{$100} $. This is a supply of money. We continued with another supply of money, $ \\{$500} $ which the government used to pay wages and collect taxes. Now we notice something peculiar. Government needed $ \\{$500} $ to pay the wages and taxes, but we only needed $ \\{$100} $ to generate $ \\{$500} $ in GDP. This is a multiplier effect found when translating between wealth and GDP. This effect is a result of the durable nature of money which can be reused until removed from circulation by taxation.

We have assumed that government used it's money supply to pay wages. What happened to the money once it left government hands? It became wealth held by the private sector [2]. The money supply, at least what is left after taxation, is then all held by the private sector as wealth. We will assign this wealth the general term H (following G&L who considered this to be High-powered-money), a term we will frequently use for the remainder of this analysis. We assign the term G to mean government expenses (such as paying wages) and we can write

(3)    $ G = T + H $

We previously said that government expenses are counted as GDP. This will be important later as we develop the analysis. For now, we will modify Equation 3 to add GDP and write

(4)    $ \\{part GDP} = G = T + H $

In Equation 4, we can foresee a logical extension to use GDP as a substitute for the T and H terms. This extension opens the door to building a simple equation that can be used to build a simple spreadsheet model.

[In Equation 4, we have both parts of the flow-of-funds method of measuring economies. The G represents the expense or spending method while the T and H represent the income method. The two methods should give the identical result but practical data collection considerations will typically result in small differences in reported numbers.]

We will write the government  and private GDP contribution equations as

(5)    $  \theta GDP = T $

(6)    $ GDP (1 - \theta) = H $

where GDP is recognized as just being part of the entire GDP.

We can combine Equations 4,5 and 6 to write

(7)    $ G =  \theta GDP + GDP (1 - \theta)  $

Further Spending Within the Period

Equation 7 describes a single transaction and the wealth distribution following the transaction. There is no reason to think that the economy stops here. Instead, the economy can be expected to move ahead and people will spend their newly earned money. Re-spending will expose the income to additional taxes and can be expected to increase the reported GDP as well as the taxes collected by government. We have a situation where we need to develop an equation using a parameter controlling the expected reuse of money by the private sector during the time period under consideration. G&L use the concept of "propensity to spend" [1] but here we use the concept of "propensity to save". We will label propensity to save as $ \alpha_4 $ .

 [Both "propensity to spend" and "propensity to save" are terms related to the expected behavior of people as they act in the market economy. As such, they can be expected to change from period to period. Later in this analysis, we will examine the USA flow-of-funds data and attempt to relate this term to actual data.]

We will assume that the propensity to save acts the same as a government tax. Part of each transaction is set aside by the private sector for the duration of the period under consideration. (Equation 4 will now be considered to describe an earlier time in the economy) Using the term $ \alpha_4 $, we write

(8)    $ G =  \theta GDP  + GDP (1 - \theta) \alpha_4  $

Equation 8 only includes government spending as the initial economic driver. We need to add provision for past wealth, held over from previous periods, which will also be an economic driver. We will assign wealth from a previous period the label $ H_{-1} $ and use the term $ \alpha_2 $ to modify H to conform with various descriptions of money supply. We will leave the term $ \alpha_4 $ unchanged but recognize that it may vary as previously described. Increasing the money supply is expected to increase GDP.

With the addition of $ \alpha_2 H_{-1} $, we write

(9)    $ G + \alpha_2 H_{-1} =  \theta GDP  + GDP (1 - \theta) \alpha_4  $

Equation 9 can be rearranged to write

(10)     $ GDP = \frac{G + \alpha_2 H_{-1}}{ \theta + (1 - \theta) \alpha_4} $

which is the result we are seeking.

Find   $ \alpha_4 $ from flow-of-funds Data

If we allow $ \alpha_2 = 1 - \alpha_4 $ , Equation 10 can be rearranged to write

(11)    $ \alpha_4 = \frac{G + H_{-1} - T}{GDP - T + H_{-1}} $

Using Equation 11, we build Figure 2 which is the chart of $ \alpha_4 $ since 1947.
Figure 2. Chart of term $ \alpha_4 $ since 1947.  $ \alpha_2 = 1 - \alpha_4 $

Equation 10 represents a considerable divergence from the SIM_0 model. The focus is upon wealth, not consumption.

In limited testing, using equivalent factors, both the SIM_0 model and SIM_MECH_0 give similar results. This is a requirement if the initial premise (two pathways to the same result) is to be fulfilled.

The existence of two SFC methods, both giving substantially identical results allows crosschecking of methodology, there-by bolstering the validity of the model.

I would like to acknowledge the contributions of Tom Brown to this post. He asked a number of questions, made many suggestions for improved appearance and raised the standard for consistent mathematics. His electrical circuit  representation of the SIM model gives us confidence that a wealth memory effect is important.

[1] In their book "Monetary Economics", page 66, G*L make a subtle choice in definition of terms. They choose to consider household consumption rather than household wealth. This leads them to write $ C_d = \alpha_1 YD + \alpha_2 H_{-1} $ where $ C_d $ is household consumption. In this analysis, making the same G&L choice, we would write $ C_d =(1-  \alpha_4 ) YD + \alpha_2 H_{-1} $ Our emphasis is on wealth, not consumption.

[2] The private sector is the counter-party to most government exchange examples. Government to government exchanges become important when both governments have the ability to create money. The ability to create money is powerful economic tool usually carefully controlled by government.


  1. Hi Roger, very good. I will certainly have a few questions for you at some point on this.

    1. One thing that strikes me is that your $\alpha_4$ seems more well behaved on your chart than a previous similar chart showed.