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I am looking for a mathematical model which can accurately forecast what percentile ($1-p$) of a population a member with wealth $w$ would be in, based on Gini coefficient ($G$). I am planning to apply it mainly only to the top end of wealth.

I have tried to use the Pareto distribution so far. What I am unsure of there, however, is what to do with the parameter $x_{min}$ as described in Wikipedia's article (https://en.wikipedia.org/wiki/Pareto_distribution):

p = $(x_{min}/w)^P$

where $P$, the Pareto parameter, can be expressed in terms of $G$ by a known equation.

Wikipedia describes $x_{min}$ as the (necessarily positive) minimum possible value of $x$. How does one apply this to a real-world population? I can't imagine setting the value to $1 or less would allow for accurate predictions of top-end wealth.

The other problem is, this $x_{min}$ seems to me to behave like an average rather than a minimum value. For example, in a maximally unequal society, $P=1$ so $p = x_{min}/w$. This predicts that 10% of the population have wealth ten times $x_{min}$, 1% have wealth a hundred times $x_{min}$, and so forth. But while this is a pretty unequal society by real-world standards, it is definitely not the most unequal possible - imagine a system where 10% had a wealth a hundred times the "minimum" (or, say, the average of the remaining 90%), 1% had a wealth $100^2$ times the minimum, and so forth. The most unequal system would really be everyone having "minimum" wealth except one individual. But the inequality here seems to be capped much before that.

Can anyone help with my model?

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Some thoughts:

  1. accurate prediction seems ambitious without empirical data;
  2. many people have negative or zero net wealth;
  3. a continuous distribution could lead to oddities;
  4. the world may have a finite total wealth and so a finite average wealth thus excluding some distribution shapes such as Pareto distribution with $\alpha \le 1$

But you could certainly develop a model, even if it would be wrong. For example, let's try:

  • A Pareto distribution with shape parameter $\alpha$ and scaling parameter $x_{\min}$
  • The application of the Pareto 80-20 principle, so $\alpha=\log_4(5) \approx 1.160964$
  • A total global wealth of perhaps of the order of $\$ 250$ trillion and a global population of about $7.4$ billion, giving an average wealth $\mu$ of about $\$33784$ and $x_{\min}=\mu\frac{\alpha-1}{\alpha}$ would be about $\$4684$ (looks rather high to me, but $x_{\min}$ is really a scaling factor and should not be taken too seriously)
  • A Gini coefficient of $\frac{1}{2\alpha-1}$ would be about $0.76$
  • The top $1\%$ (so $74$ million people) would have over $x_{\min} / 0.01^{1/\alpha}$ each, so here about $\$250,000$
  • a person with wealth $w$ would be at the $\left(\frac{x_{\min}}{w}\right)^{\alpha}$ quantile measured from the top

Since this probably does not fit reality, you can then start adjusting it if you want: for example you can get closer to a Gini coefficient of $1$ if you reduce $\alpha$ towards $1$ from above, and this would also lower $x_{\min}$. If you use a Gini coefficient of $G$, that would make $\alpha=\frac{G+1}{2G}$ and $x_{\min}=\mu\frac{1-G}{1+G}$ so the person with wealth $w$ could be at the $\left(\frac{\mu(1-G)}{w(1+G)}\right)^{(G+1)/(2G)}$ quantile, measured from the top.

Or you could decide to give a certain share of the population a wealth of zero. If the rest keep the same $\alpha$ on a (now zero-inflated) Pareto distribution this will increase $x_{\min}$ for the rest while also increasing the Gini coefficient overall.

Or you could look for empirical information on the actual wealth distribution and then try to fit a curve to that. Or perhaps look at global income distribution instead, which has not had a particularly simple function in the past.

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Pareto distribution has an extremely simple CDF. But I don't know any example where it arises naturally (unlike abundant examples of normal, log-normal, Poisson, and some examples of stable laws). Perhaps it doesn't and indeed the parameter $x_0$ and its definition near 0 is quite arbitrary. But it suits modeling power-law tails, where one can often ignore the part for small values.

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