# Political Policy and the Wealth Gap

Consider the US income distribution in 2005.

Image by Catherine Mullbrandon (1)

It demonstrates that a clear majority of the population earns an income at the relatively low end of possible incomes in our economy.  This sort of curve isn’t too surprising since those who acquire relatively more resources initially can in turn acquire additional resources easily since they can afford costly investments with large payoff.

This presents a problem of endgame capitalism.  The idea is that if a system has a finite amount of resources and a finite number of people attempting to acquire as many resources as they can such that none have 0 resources, then the instability of socialism (where everyone has the same amount of resources) will force some to have more, and in turn make it easier for them to get more by contracting others (i.e. forming a corporation).  Slowly those with less lose more and more over time until several “corporations” remain.  Then the process repeats, and eventually we are left with a single owner, or at least one person getting arbitrarily close to possessing all of the resources.  To combat this endgame, we are left with political policy.

The highest tax bracket in the US is 35% and applies to households making over \$373,000 [2].  That is less than 2% of the population.  There will come a point in our future where we will have to decide between “fairness” and efficiency.  It would be far more efficient for very low incomes to have 0% (or at least very low) tax rates and for the upper echelons pay bigger percentages.  They could still rank far above the rest after taxes.  During previous time periods taxes were raised to combat the costs of the federal government (in particular WWII in 1945).

Image by Greg Hollingsworth (3)

Also of interest (as Greg points out) is the correlation between the decrease in the top tax rate and increase in the national debt.

# A New Tax Policy

Consider a simple tax policy where you plug your income into a function, and it spits your tax rate out.  The idea is to do away with the discrete tax bracket system and instead have a continuous system.  Consider the generic function

$t_0(x)=1-e^{-\alpha x^2}.$

We have $t(0)=0$ and $t(\infty)=1$.  This satisfies the boundary conditions we want on a continuous progressive tax policy where $t(x)$ represents the tax rate ($1$ being $100$%) if your income is $x$.  To gain some control over this generic function, we will wish to fix a point on it (i.e. assign a value for $\alpha$).  Suppose the average income is $\sigma$, and that we want $t(\sigma)=.25$.  Then we have that $\alpha=\sigma^{-2}\log\frac{4}{3}$.  Substituting this gives

$\displaystyle t(x)=1-e^{\log(\frac{3}{4})\sigma^{-2}x^2}=1-\left(\frac{3}{4}\right)^{\sigma^{-2}x^2}.$

Differentiating once shows the slope starts and “ends” at $0$.  Differentiating again shows that the maximal rate of tax rate increase is at $x=\sigma/\sqrt{\log(\frac{16}{9})}$ (that is, tax rates increase the fastest around this income).

We may now attempt to construct another function $\mu$ where $\mu(x)$ is the number of people in a system who have income $x$.  We can generalize to define $\mu(A)$ for some $A\subseteq [0,\infty)$, where it will denote the number of people whose income falls in $A$$\mu([0,\infty))=P$ where $P$ is a constant called the population.  We could attempt to start with a generic function

$\mu_0(x)=P_0e^{-\alpha x^2}$

where $P_0$ is called the unemployment number ($\mu_0(0)=P_0$).  In particular we could attempt to determine $\alpha$ by requiring

$\displaystyle\sigma=\frac{1}{P}\int_0^\infty\mu_0(x)\,dx.$

But evaluating the integral poses a problem; and hence so does expressing $\alpha$ in terms of the mean (or even median).  Thus it is difficult to start with the mean or median.  Let $m$ denote the mode income and $P_m$ be the number of people with income $m$.  Consider the function

$\displaystyle\mu(x)=P_m\left(\frac{P_m}{P_0}\right)^{-m^{-2}(x-m)^2}.$

Here we have $\mu(m)=P_m$, $\mu(0)=P_0$, and $\lim_{x\to\infty}\mu(x)=0$ since $P_m\geq P_0.$  With tax and distribution functions $t$ and $\mu$ we can define the revenue function as the Riemann-Stieltjes integral

$\displaystyle R(t,\mu)=\int_0^\infty xt(x)\,d\mu(x).$

Similarly we have the gross product

$\displaystyle\Pi=\int_0^\infty\mu(x)\,dx=P\sigma.$