# 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.