The wealth threshold

The explanation for growing wealth inequality proposed by Thomas Piketty in his iconic book Capital in the Twenty-First Century, is that the rate of growth from capital exceeds that of the entire economy in general. Thus, the wealth of owners of capital (i.e. investors) will increase faster than everyone else. However, even if the rate of growth were equal, any difference in initial conditions or savings rate, would also amplify exponentially. This can be seen in this simple model. Suppose w is the total amount of money you have, I is your annual income, E is your annual expense rate, and r is the annual rate of growth of investments or interest rate. The rate of change in your wealth is given by the simple formula

\frac{dw}{dt} = I(t) - E(t)+ r w,

where we have assumed that the interest rate is constant but it can be easily modified to be time dependent. This is a first order linear differential equation, which  can be solved to yield

w = w_0 e^{r t} + \int_{0}^t (I-E) e^{r(t-s)} ds,

where w_0 is your initial wealth at time 0. If we further assume that income and expenses are constant then we have w = w_0 e^{r t} +  (I-E)( e^{rt} -1)/r. Over time, any difference in initial wealth will diverge exponentially and there is a sharp threshold for wealth accumulation. Thus the difference between building versus not building wealth could amount to a few hundred dollars in positive cash flow per month. This threshold is a nonlinear effect that shows how small changes in income or expenses that would be unnoticeable to a wealthy person could make an immense difference for someone near the bottom. Just saving a thousand dollars per year, less than a hundred per month, would give one almost a hundred and fifty thousand dollars after forty years.


One thought on “The wealth threshold

  1. One could add a few more nonlinearities.

    For a concrete (or applied) example (i tend to think in terms of pictures or real world examples) one could define ‘wealth’ as your weight or size of family–single, or 1 kid, or 10 kids…

    so this is a dynamic not static issue.

    so your E ( expenses) will be E (w). (if you want to keep your weight up or increasing you may incur more expenses–of course with a large family maybe then you can get your kid to work for more income and no more expenses–they pay for their own needs and pay rent.).

    But if you have more dependents then your E goes up too.

    (I live in an apt –rent controlled but goes up every year by like 10%– and if you miss a rent payment they charge extra so that might be whatever you have saved unless you want to be evicted).

    If w (wealth) stands for weight, then maybe you eventually get obesity so then you have more health expenses (E) or need a tummy tuck.

    The model above also assumes it appears a constant money value. If the value of the money changes then you might be rich overnite (eg bitcoins) or poor overnite (have alot of wealth in useless money (zimbabwe)).
    Maybe if you move to a different country with a good exchange rate then a US dollar would be worth for example 1 1/3rd canadian dollar or you can go to a place like india ( i been there and discovered that certain things sold in DC sold for 40$ were 2$ there).

    I have low ‘impulse conrol’ in my environment. I could and did live on much less –when iwas out in the mountain s in west va. I could save too. There was nothing to buy and nowhere to go except where i could walk. I was happier and healthier too. For food i could get what i needed for 1-3 $/day (lentils and veggies and tea) and find the rest. For excercuize i could walk outside and if i wanted to walk hundreds of miles on trails (i usually just did a daily little walk of 5 miles—climb almot 1600 feet up a mountain, watch the sunset (i also kept a few cheap beers up there–just needed one) and walk or run back down.

    Around here if i want a beer its well over 8$ including tip if i go to a nice place like busboys and poets. If i want to go on a nice hike up in mtns its minimum 26$. I could just walk around here but i like variety.
    Also some people save forever, live frugally (don buy vegetables–too expensive–just eat canned soup)
    and end up wealthy and sick or dead.

    Piketty’s (r.g) reduces to ‘what the employers get, versus what their employees get’. if it was slavery it’d be (100%, , 0%) not exactly rocket science, and a relative of mine is graduating from MIT this year and has a job at FB doing data analytics for FB in UK).

    my view is you might need something more like a lotka volterra equation (eg Goodwin model) or a langevin equation.

    w(dot) = E(w, t,r, I)I(w,t,E, r) + w(E,I,r,t)r(t, politics)


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