= Pareto top shares and the 80/20 rule
{c}
{title2=$T(q)=q^{1-1/\alpha},\quad\alpha>1$}
For a nonnegative <random variable> with finite positive <mean>, the share of its total amount carried by the largest fraction $q$ of outcomes can be defined by dividing their contribution to the <expected value> by the whole <expected value>. For a <Pareto distribution>, let $0<q\leq1$ and choose $x_q=\beta q^{-1/\alpha}$, so that $\mathbb P(X\geq x_q)=q$. If $\alpha>1$, direct integration gives
$$
T(q)=\frac{\mathbb E[X\mathbf1_{\{X\geq x_q\}}]}{\mathbb E[X]}
=\frac{\alpha\beta^\alpha x_q^{1-\alpha}/(\alpha-1)}{\alpha\beta/(\alpha-1)}
=q^{1-1/\alpha}.
$$
This is a share of amount, distinct from the fraction of outcomes. It depends on $\alpha$, but not on the unit of measurement $\beta$.<Power laws, Pareto distributions and Zipf's law by Mark Newman>[Section III.D, pp. 10–11, Eqs. (27)–(28): upper-population share; Newman's density exponent equals alpha+1 in the present tail-exponent convention]{ref}
For the chosen $\alpha=2$, the top 20% starts at $x_{0.2}=10\sqrt5\simeq22.361$ and carries $T(0.2)=\sqrt{0.2}\simeq44.72\%$ of the amount. To make the top 20% carry exactly 80%, solve
$$
(1/5)^{1-1/\alpha}=4/5
\quad\Longrightarrow\quad
\alpha=\frac{\log5}{\log4}\simeq1.160964.
$$
The resulting <Pareto distribution> has a finite <mean> and infinite <variance>. The 80/20 rule selects one <shape parameter>; it is not a universal law, or even a property of every <Pareto distribution>. For $\alpha\leq1$, the population-share definition above is unavailable because its denominator is infinite; substituting such an $\alpha$ into the finite-mean formula is invalid.
\Image[/pareto-alpha-quantiles-and-top-share.png]
{title=Changing Pareto alpha changes quantiles, moment existence and the top 20 percent share}
The left panel compares three <quantiles> in units of the minimum. The right panel shows $T(0.2)$ only where the <mean> is finite, and marks the chosen 80/20 value. The thresholds $\alpha=1$ and $\alpha=2$ separate the <moment> regimes; they do not separate valid and invalid <probability distributions>.
A straight-looking empirical <log-log plot> does not by itself establish a <Pareto distribution> or a reliable tail exponent; estimating an exponent and assessing <goodness of fit> are separate tasks.<Power laws, Pareto distributions and Zipf's law by Mark Newman>[Section II, pp. 4–5, Eqs. (3)–(5) and discussion: survival versus density exponent, biases from fitting plotted slopes, and the lower threshold for power-law behaviour]{ref} If a <Pareto distribution> is used only above a fitted lower threshold, $T(q)$ describes that conditional tail population, not automatically the whole population. A genuine finite upper cutoff gives a <truncated distribution> with finite nonnegative <moments>; its <survival function>, <quantiles> and top shares must be recalculated. Truncation is a modelling change, distinct from merely observing a finite sample. None of the chosen numbers here estimates an actual wealth distribution.
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