Where Wa is average wealth, N is the total number of households in the population, and n are integers starting at one and counting away from the top 20%. Households below the bottom 17.6375% have zero wealth.
So using the First General Wealth Equation, one can predict the wealth distribution knowing only average wealth and population size. So in the hypothetical economy of one million households, the evaluation begins above and below the 200,000th richest household in the ranking. Average wealth happens at the 166328th richest household, and zero wealth begins at the 176375th poorest household, making those with average wealth and above at 16.3% approximately equal to those with zero wealth at 17.6% (1.3% difference) considering the entire range of values.
Examining the First General Wealth Equation term,
one can see that as rank decreases (R starts at.398942 and always goes lower for each iteration until the end of the tail is reached), the term increases in value. Depending on the sign of the multiplier M, small linear changes in rank cause large differences in wealth either increasingly or decreasingly since e is being raised to a power which includes rank.
It is worth noting that the multipliers were evaluated using one million members in the economy. If evaluating with a higher number of members, slight adjustment of the multipliers will be necessary.
Almost by accident I noticed that if one takes the log of both the wealth and population axes, and then reverses the axes, a curve much resembling a first order band pass filter (used in electronics)can be seen over the top 80% of the distribution. Here is the Second General Wealth Equation:
The symbol Pw represents the number of households owning wealth and Pi is an individual household starting with the wealthiest at 1 and going to 80% of the population. The bottom 20% when using this equation has no wealth. Taking the log of a predicted Wealth distribution using this equation gives a very similar S shape curve and very similar percentages as compared to the curve of the First General Wealth Equation. The Second General Wealth Equation is also scalable for average wealth and population:
Comparison of First and Second General Wealth Equations. Similar curves produce similar results from independently derived equations.
The Second General Wealth Equation places those with average wealth and above at 18.2% and those with no wealth at 20.0% (1.8% difference) again confirming the approximate equality of the two. Here an exponentiation term with a ratio including rank causes large differences in wealth, although e is not being raised to a power as in the First General Wealth Equation.
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