Tuesday, April 5, 2016

A Master Equation for SIM Models Using the GDP Object DRAFT


There seems to be a path from the barter economy to the money economy.  The SIM spreadsheet model is a simple emulation of a stable money beginning but we need a better path from start to monetary stability, Lets's see if we can find that path and equations for building a SIM style of model.

The Barter Economy

First, we will build an economy measured in transactions and described by GDP (Gross Domestic Product). This is a barter economy without money. We label GDP as GDPt (using the sub t as a identifier). There is a private sector (PSt) and a government sector (Gt). Government is allowed to levy tax to enable spending. The tax is applied at a rate of TRt on a period basis.

We can write three equations that will describe this eonomy:

(1)     Gt + PSt = GDPt
(2)     GDPt*TRt = Gt
(3)     GDPt*(1-TRt) = PSt

With these three equations and an assigned (or discovered) value of TRt, we can describe this economy in a spreadsheet environment. Years can be represented by columns marching across the page. These three equations are a simple measurement with no memory between periods (There is no need for memory because transactions are events.)

Introducing Money

Modern economies are money based, not transaction based. How can we introduce money into this spreadsheet model?

Figure 1. A barter economy can be converted into a money economy. The GDP curves are considered to be square or rectangular hyperbolas, with the property xy =1.                        
From Figure 1, we can see that GDP can be represented by transactions or money. How would we make and display the transition from transactions to money?

Unlike transactions, money has a characteristic of duration. Money is an object with a lifespan. The three transaction equations will need to be supplemented with additional terms and equations that record the money carried between periods.

We will define money that is held from one measurement period into a second period as wealth (H). Wealth from a previous period will be labeled H-1.

Wealth held in more than one period must be dynamic. There must be a method of creation, holding, and destruction. Money can be created by borrowing [Exactly how is an ongoing debate among economist and philosophers.] In this simple model we will assume that government can borrow from itself, spend this borrowed money into existence, and finally destroy money with taxation. While the money is in existence, the private sector will be allowed to store money as wealth.

There is no way to know when the private sector will spend it's stored wealth. There is no way to know how much money the private sector will save from each period. We do know that the private sector cannot save more than that part of GDP left after government extracts it's share. The private sector gets a remainder. This sequential nature of events will be important later when we design equations.

We are ready to write equations that describe the entry of money into a transaction economy. We will use three assumptions to simplify the equation construction:
  1.  Government will have no savings.
  2.  Government will spend new money into existence unless it has tax money to spend. This is similar to government beginning a new program and using money it borrows from itself to fund the program.
  3.  Equations 1, 2 and 3 are still valid but they are missing wealth terms. We  add the necessary wealth terms and drop the sub t label-modifier.
We will take an empirical approach. Because we cannot hope to predict the actions of people, we will build in adjustable parameters that can be calibrated to reasonable values discovered by measurements of existing actual economies. What we need is a mathematical system that repeats at every replication with some memory of the past. We are ready to begin.

Building the Dynamic Money Equation

If government spends (G) and collects taxes (T) at rate (TR), we can write

(4)     GDP*TR = G = T

For the private sector share of GDP,  write

(5)     GDP*(1-Tr) = PS

Here is a critical point. These equations describe a stable economy.  If we stopped this analysis using only equations 4 and 5, we would be assuming that government recovered the entire initial amount spent. [In equation 4, we wrote that the tax collected equals the amount initially spent. This describes a stable economy, not a dynamic economy with wealth remaining at the end of the period.] The GDP for any period can only be as big as allowed by the remaining wealth circulating completely expanded in the economy.

The act of saving money from the present period will be treated as second tax.   This savings rate tax (α1) will be applied to the private sector share of annual GDP (AGDP) to fund the annual amount saved H. We can write

(6) AGDP*(1-TR)*α1 = H

We assume that government will tax the annual GDP so we write

(7) AGDP*TR = AT

where AT is the annual tax collected by government.

We assume that the amount saved added to the amount collected in taxes equals the amount spent by government added to the amount spent from savings during the period. We can write

(8) H + AT = G + H-1.

We can combine equations 6, 7, and 8 to write

(9) AGDP*(1-TR)*α1 + AGDP*TR = G + H-1.

Rearrange equation 9 to write the master equation for period GDP

(10) AGDP = (G + H-1) / ((1-TR)*α1 + TR).

Now we can examine equation 10 to see that if we assign values to terms G, TR, and α1, we have defined AGDP. The values for these three parameters will be discovered by empirical methods.

Tie to the SIM Model

Wynne Godley and Marc Lavoie in their book "Monetary Economics (2007) , chapter 3, describe an economic model SIM. Several versions of these models are available on the Internet. The parameters used in this post can be directly converted to the SIM parameters.

TR -> θ || G -> G || α1->α1 || α2->α2 || H->H || AGDP->Y

We have not yet used the term α2. We used the term H-1 (wealth) to provide the memory between periods. Empirically, we find that there is no need for all of the wealth saved in one period to be used in a second period. We use the factor α2 ( Propensity to consume wealth) to modify the amount of wealth used in a later period. Therefore, we can write (for example)

H-0 = α2*H-1
$H_0 = \alpha_2 H_{-1}$

and use H-0 as the starting wealth in the next replication. Term α2 is also empirically determined.

Using the Equation in a Spreadsheet

We can use equation 10 to build a spreadsheet model of an economy without using spreadsheet iteration. This makes the model much easier to understand. Each of the three empirical parameters can be adjusted to create a unique model. The wealth carried between periods can be adjusted by changing term α2. 

Spreadsheet columns can represent time periods. Each period can be adjusted to introduce "jump" changes in later periods. [Later adjustments require a second empirical entry table and decision logic in the spreadsheet column structure. This is easy to do but tedious in construction]


We have found one possible path from the barter economy to a simulated monetary economy. This is a very simple model but flexible to allow inclusion of additional parameters. The ability to construct a mathematically satisfying simple model encourages further use and development of this mechanical method and theory.


Friday, April 1, 2016

For Tom: The GDP Object


[4/3/2016 1:30 update
Tom comments that the drawing shows a "square hyperbola", also known as a "rectangular hyperbola". This is a hyperbola with the relation xy = 1. This is useful in creating a model economy. We can assign one axis to represent money, the second axis to represent transactions. Assuming every transaction can be represented by money, the sum of all transactions multiplied by the sum of all money values will form a square hyperbola if every-possible-combination-that-forms-the-same-constant-value is plotted. This gives us the "GDP Object". 

The GDP Object will be useful in writing a SIM style of spreadsheet model that avoids iteration but yields similar results. This results in easier to understand equations. (I hope!)
(This concept and the enabling equations have not yet passed peer review.)]]

[4/2/2016 10:00 AM update    
"The GDP Object" may not be the best way to characterize the nature of GDP.

GDP (Gross Domestic Product) can be related to at least three different concepts:

1)   A measure of economic activity. It can be considered as the sum of all transactions with a price value. Here, GDP is a defined measurement. If government expenditures are also known, an average tax rate can be calculated.

2)   A theoretical limit. Money supplied by government can be taxed every time it is received. If only one issue is made, money disappears from the private sector and returns to government. Eventually, the entire issue is recovered. Limit GDP is the maximum possible GDP if the tax rate is known.

3)  A  PERIOD theoretical limit. The theoretical limit can be divided into time periods. Each time period will have a different GDP limit based on a period common tax rate and rate for parallel money collectors, and a period-unique beginning money supply.

A characterization of GDP as an "object" nears becoming misdirection. Perhaps we should characterize GDP as a "limiture" (where we combine the words "limit" and "measure"), giving GDP an unique characterization.]

This is for Tom. It is quick and dirty. I am bogged down with detail in another attempt to present the same material.

Figure 1. The GDP Object is the value on the GDP curve at any point in money-transaction space. 

GDP can be considered a limit defined by G and T as in


where T is a dimensionless pure number. G is money and GDP is money. GDP is constant when G and T are defined. Now GDP is an object.

Find GDP for a period

We can use T with a time period dimension to find the GDP expansion for that period. We can count on T being less than one because it takes an infinite time period with infinite transactions to find the GDP limit by series expansion.

If we assume that we have TWO taxing authorities, one authority can be government using the assigned tax rate FOR A PERIOD. The second authority will be assigned a tax rate that captures the remainder of the potential GDP FOR A PERIOD. The remaining GDP potentially available for capture is GDP*(1-T) .

This gives us two equations that capture the entire GDP expansion to limit.

Convert into a series of annual events 

We can convert GDP to annual events (AGDP) by considering every step is a division between two taxing authorities. The primary authority will receive the Annual Tax (AT) share and the second authority (savers) will receive the remainder (AR). Write this in two equations.

(1) AGDP*Tr = AT


(2) AGDP*(1-Tr)*Rr = AR

where Rr is the Remainder "tax" rate.

Notice that the sequence of events is important here. Tax is removed from GDP before a remainder can be calculated.
We know that the sum of the two tax divisions is the original injection by government (G):

(3) AT + AR = G

Substitute  1 and 2 into 3 to get

(4) AGDP*Tr + AGDP*(1-Tr)*Rr = G

Simplify 5 to get

(5) AGDP*( Tr + (1-Tr)*Rr) = G

Now we can define parameter Rr just as we defined the government tax rate. AGDP is constant for the period just as GDP is the constant GDP Object. Once we know AGDP, we can find wealth and every other term as you did using Linear System Analysis.

We can next add wealth to the next period by assuming that wealth is also all spent to create a new GDP Object.  It now becomes repetition to complete the table for as many years as desired.

At this point, I think we may be in correlation with Linear System Analysis

Does this make sense now?