# How many dollars does it take to be rich?

This is probably a dumb economics question since I don't know anything about that subject beyond a few buzzwords (but I do know a little math). I'm trying to figure out how many dollars it takes to be "rich" by a notion of "rich" that I think is more fundamental than "top 1%" or "can afford a yacht", etc. I expected it to be a simple matter of writing some equations and solving them, but I'm having trouble formalizing it properly, so I'll describe it informally then ask for help.

The basic notion is to assume as an axiom that wealth follows a Pareto distribution (a standard observation in economics). This is the distribution $\rho(x)=cx^\alpha$ where the multiplicative factor $c$ is a scale parameter and the exponent $\alpha$ is a shape parameter. The total amount of money in the economy is "of course" just the area under the curve (part of the problem I'm having is that the integral diverges unless somehow truncated, which I'll get back to).

Suppose there is some change in national economic policy (perhaps a revenue-neutral adjustment to the tax code) that doesn't create or destroy wealth, but has the effect of favoring the rich, i.e., it pushes money from the lower end of the graph to the upper end. By our axiom, the policy change preserves the invariant that after the dust settles, there is still a Pareto distribution. We'd say the change results in the old distribution's shape parameter $\alpha$ increasing slightly, while the scale parameter $c$ changes correspondingly to keep the integral the same. Conversely, a policy change that favors the poor decreases $\alpha$ and makes the inverse change to $c$. In the former case, someone at the upper end of the income distribution finds themselves with more money, while someone at the lower end finds themselves with less; and in the latter case, it's the other way around. Either way, for given values $(c,\alpha)$, there is a wealth level $w$ such that someone at that exact level ends up with exactly what they had before.

So my idea is to define "rich" as any wealth level above that equilibrium amount $w$, so people there benefit from one kind of change, and "non-rich" is any wealth level below $w$, whose holders benefit from the opposite direction of change. That means that the "rich" boundary depends very much on the $(c,\alpha)$ parameters, and I'd like to know that boundary value in the current and historical US economies. Can it be that for sufficient $\alpha$, $w$ is high enough that further increases in it benefit almost nobody (the cdf at $w$ is close to 1), and people somehow sense this enough that it starts to affect politics (Occupy Wall St etc.)? That would correspond to Pareto's original observation that $\alpha$ tends to be in a certain range, like 2 to 3.5 or something like that.

Anyway I'll skip the boring algebra I did, that didn't work. What I'm wondering is 1) what to do about the improper integral and 2) whether this overall approach is standard or interesting.

Thanks.

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What are $\rho$ and $x$ in your equation $\rho(x)=cx^\alpha$? According to my brief skimming of the Wikipedia article, the probability density of having $x$ dollars is $f(x) = \alpha x_{\mathrm m}^\alpha x^{-(\alpha+1)}$ for $x \ge x_{\mathrm m}$. So the total amount of money in the economy, $\int_{x_{\mathrm m}}^\infty x f(x)\,\mathrm dx$, shouldn't diverge as long as $\alpha > 1$. –  Rahul Nov 20 '12 at 4:16
It would seem that the Pareto distribution (type I, but the same holds for its generalisations) is defined by a (complementary) cumulative distribution function, not by a probability density function (the latter is derived from the former by differentiating). As such the integral of the probability density function is convergent by construction: integrating just gives you back the cumulative distribution you started with. Moreover, it is the complementary cumulative distribution that is given by a power law. Your question therefore seems based on mistaken starting assumptions. –  Marc van Leeuwen Aug 2 '13 at 6:30

The Pareto principle for wealth is that the proportion of the population having wealth $\geq x$ is

$$y=\left(\frac{x_m}{x}\right)^\alpha$$

for constants $x_m > 0$ and $\alpha > 1$, and restricting the domain to $x > x_m$. Inverting this function to find the wealth of the top $y$'th quantile ($100y$'th percentile), we get

$$x=x_my^{-\frac{1}{\alpha}}$$.

The mean wealth is

$$\frac{\alpha x_m}{\alpha -1 }$$.

So, we can phrase your question as follows: given old parameters $(\alpha, x_m)$ and new parameters $(\alpha', x_m')$, find the break-even point $x_0$ such that

$$\left(\frac{x_m}{x_0}\right)^\alpha = \left(\frac{x_m'}{x_0}\right)^{\alpha'}$$.

We can solve this for $x_0$ and get

$$x_0 = \exp \left[\frac{\alpha\log x_m - \alpha'\log x'_m}{\alpha-\alpha'}\right]$$, requiring $x_0 > x_m$ and $x_0 > x'_m$.

Generally, if $\alpha' > \alpha$ then the change will benefit those above this break-even point, and if $\alpha' < \alpha$ then the change will benefit those below it.

If we're instead trying to find the break even point for relative wealth, we can solve

$$x_my_0^{-\frac{1}{\alpha}} = x_m'y_0^{-\frac{1}{\alpha'}}$$

for $y_0$ and get

$$y_0 = \exp\left[{\frac{\alpha\alpha'(\log x_m' - \log x_m)}{\alpha - \alpha'}}\right]$$.

We're not quite finished answering your question, though, because we need to address the constraint that the total amount of wealth in the economy remains constant. Assuming that the population holds constant (no Stalinist purges allowed!), this is the same as requiring that the mean hold constant. So,

$$\frac{\alpha x_m}{\alpha -1 } = \frac{\alpha' x_m'}{\alpha' - 1}$$.

Let's take $x_m'$ to be our dependent variable and solve for it:

$$x_m' = x_m \cdot \frac{\alpha(\alpha'-1)}{\alpha'(\alpha-1)}$$ .

Then you can perform that substitution into the equations for $x_0$ and $y_0$.

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