How Certain People Get Rich, mathematically speaking
1. Assume that people accumulate wealth linearly at the start of their careers
Income = k * tRead this as: if you do work steadily, you will get rewarded
2. Assume that at some point, the wealthy start accumulating assets at an exponential rate, mainly via investments.
Income = b * exp(a*t)In other words, once you reach a threshold, you will get exponentially rewarded with little extra work.
3. Total income combines the two.
Income = k*t + b*exp(a*t)You would expect that linear income starts out more strongly than that of the compounded growth exponential. Unfortunately, this turns into a transcendental function to solve for t. This will get partitioned instead of solved in a later step.
4. Assume an exponential distribution of income dispersion at any point in time. We assume a wide dispersion such that the standard deviation equates to the mean. The accumulated income at a point disperses in rate R as:
N = N0 exp (-C/R)This basically follows the same distribution as runners finishing a marathon race as I posted before (or searching for oil discoveries). The fastest runners occur rarely, and so do the fastest income earners.
5. We need to change this from a rate distribution to a time distribution to compare it to readily available data. First we take the derivative in (4)
dN/dR = C*N0 exp (-C/R)/R^26. Next convert this to a time dependence, R ~ 1/T
dN/dt = dN/dR * dR/dT ~ exp (-C*T)7. The integral of this equation becomes the histogram of the income distribution, which has some dependence on the effort T put into your work
N(T) ~ exp (-C*T)We have not set the term T yet, but in this case it relationally equates time to Income (see update at the end of the post).
8. Plotting this for linear effort T, where income directly relates to effort expended, you get the red curve for the cumulative histogram below. Note that 100% of earners make at least $0 and then it decreases rapidly for higher incomes.
9. But from assumption (3) we know that the scale of T needs to "accelerate" to meet the needs of the exponential growth term. So if we weakly include that term near the intersection
T ~ ln(k*Income/b)If we plot this term in step 7, replacing T, it shows up as the green curve above.
I could iterate and solve the transcendental equation in (3) numerically but for now, it shows the relationship quite clearly.
The idea that this derivation conveys predicts that the dispersion of efforts from many income-earners (via hard-work or luck) will result in a certain segment of the population having enough assets to trigger the compound growth dependence which supplements and finally outpaces their regular income.
This paper analyzes the USA's census income data in the sense of picking out some heuristics from physics and trying to fit the curves.
It tries to cast an understanding in terms of hand-wavy statistical mechanics, but I think I have provided a better and intuitive derivation. The paper also notes that the green curve likely has to do with the stock market -- a real compound growth situation if profits get reinvested. Realize that the bottom 97 to 98% of the population never get to invoke the strong compound growth that the elite 2% to 3% do. And the dispersion continues in this term leading to the Pareto law long tails.
I have a feeling that this break in the curve represents the separation between the "haves" and "have-nots" that exist in the stock market or in investing in general. In the current market, mutual funds no longer work to create the strong compound growth for anybody but the most committed and financially backed investors. I have a feeling that the market simply "pumps" the buy/sell cycle to attract investors. I recall discussing this stuff back here.
Wall Street learned well that too many individual investors during the stock run of the 90's used ordinary mutual funds and ended up reducing the "entry barrier" and thus removing the distinction between the "haves" and "have-nots". And it clearly could not have sustained itself as you could see how much capital it would introduce into the market. They have thus tipped the scales back to a flat ratchety growth curve designed to ensnare ordinary investors (not to mention 401Ks and Social Security). If you blink like many people did the past few weeks, you miss it. The last 10 years basically came out flat. I didn't lose anything on my own investments (as I tend to stay away from risk) but I managed a fund for a relative and screwed up royally, getting out only on the downslide and losing 16%. Even the fixed income mutual funds got hammered (don't do growth bonds). I feel bad for everyone involved and it will take me a while to get over this, and hope the world economy rights itself. I'd like to say I could learn from this and put it to good use next time, but that won't happen for me. These people think the money in your wallet belongs to them, and they will do anything to get at it. And the cretins who defaulted on the bond funds probably realize the retirees that depend on this don't go around wielding baseball bats and shotguns.
What a mess!
Update: The set of two equations is actually very easy to solve. Simply parametrically plotting the two equations (3) and (7) using time (t) as the independent parameter gives this result:
This formulation does wonders for understanding what causes what. I find it interesting that I can actually extract the compound growth rate of Wall Street investments from the curve. The long tail above comes about from a 30% yearly growth rate, matching some of the high end returns in the 90's.
I will likely update this post at a later time since it combines the two growth rates, constant and exponential, that we use in the Dispersive Discovery models. The big distinction between the that model and the income model, is that the latter uses both types of growth in the same model! This suggests that it likely happens in oil discovery, of which the empirical absence has puzzled me for awhile. I assume that if we look hard enough we may see it for oil too.
Update 2: After studying the dynamics a bit more, I have decided that the compound growth part does not play as big a role as I first thought. The idea for using this adjunct form of compensation first surfaced in the referenced paper where the authors saw effects somewhat correlated to the stock market. I believe that the high income part of the curve maps to the Dispersive Discovery model with a stochastic "well of knowledge" or "depth of wisdom". We don't actually reach a specific point at which we determine our salary, but instead grow that salary over a range of years that we would willingly try to advance. For a dispersion with a damped exponential well, this leads to a straight line on the log-log plot, which matches that of the high income part of the curve. The low income part of the curve has a more rapidly declining profile, indicative of a income earner that trains for a fixed period of time and becomes satisfied with the salary at that point. This period may match the length of time for a high school education or a trade school job. On the other hand, for the higher-income earner, the training may become a life-long process, so that the goal-posts continuously expand until they reach retirement age. So I don't believe that the change in slope has as much to do with stock market fortunes as it has to do with effort expended. To first order, this effort relates to education time, continuing education, and overtime worked, leading to increased wages with experience.
The decline rate under the conditions of a variable academic stint plus a job maturation period should lead to a slope of ~ 1/t2, close to that observed (usually above 1/t1.5).
Dispersion would indicate that only a few billionaire salaries would exist, and you can see that in the extrapolated curve. Other than that, the difficulty remains in absolutely aligning and correlating time effort with an average level of salary. Empirically it looks like a power-law somewhere around 1.