[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
GDP = G/T
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?