Modeling Wealth Distribution
Pareto Distribution, Lorenz Curve, Gini Coefficient, and CRRA Utility
1 Introduction
The Pareto distribution has long been used to model wealth or income distributions, starting from the seminal work of Vilfredo Pareto in the late 19th century (Pareto 1964). One of the most intuitive ways to visualize the inequality within such a distribution is through the Lorenz curve, introduced by Lorenz in 1905 (Lorenz 1905). The Lorenz curve describes the cumulative proportion of total wealth (or resources) held by the bottom fraction of the population. Closely related to the Lorenz curve is the Gini coefficient, which provides a scalar measure of inequality (Gini 1912).
This document first derives the Lorenz curve for a Pareto distribution, then demonstrates how to estimate the Pareto index (shape parameter) from Lorenz curve observations. We additionally relate the Pareto distribution to Constant Relative Risk Aversion (CRRA) utility function, highlighting the same shared structure in a differential equation that governs both.
2 Deriving the Lorenz Curve of a Pareto Distribution
2.1 Definition of the Lorenz Curve
For a continuous distribution of wealth (X), the Lorenz curve \(L(p)\) is defined as the fraction of total wealth owned by the bottom \(p\) fraction of the population:
\[ L(p) \;=\; \frac{\int_{x_m}^{x(p)} x\,f(x)\,dx}{\int_{x_m}^{\infty} x\,f(x)\,dx}, \]
where
- \(p = F\bigl(x(p)\bigr)\) is the cumulative proportion of individuals with wealth below \(x(p)\),
- The numerator represents the cumulative wealth of the bottom \(p\) fraction,
- The denominator represents the total wealth in the system, given by the expected value of \(X\) over its support.
2.2 Pareto Distribution
A Pareto distribution with shape parameter \(\alpha>0\) and scale parameter \(x_m>0\) is defined by the PDF
\[ f(x) \;=\; \alpha\,\frac{x_m^\alpha}{x^{\alpha+1}}, \quad x \ge x_m, \]
and the corresponding CDF
\[ F(x) \;=\; 1 \;-\;\Bigl(\tfrac{x_m}{x}\Bigr)^\alpha, \quad x \ge x_m. \]
Equivalently, for \(0 < p < 1\), the quantile \(x(p)\) satisfying \(F\bigl(x(p)\bigr)=p\) is
\[ p \;=\; 1 \;-\;\Bigl(\tfrac{x_m}{x(p)}\Bigr)^\alpha \;\;\Longleftrightarrow\;\; x(p) \;=\; \frac{x_m}{\bigl(1 - p\bigr)^{1/\alpha}}. \]
2.3 Total Wealth
Let the total wealth be \(W_{\text{total}} = E[X]\), the expected value of \(X\). Substituting the PDF of the Pareto distribution into the definition of expectation,
\[ E[X] \;=\; \int_{x_m}^{\infty} x\,f(x)\,dx \;=\; \int_{x_m}^{\infty} x \,\alpha\,\frac{x_m^\alpha}{x^{\alpha+1}}\,dx \;=\; \alpha\,x_m^\alpha \int_{x_m}^{\infty} x^{-\alpha}\,dx. \]
For \(\alpha>1\), the improper integral converges and we obtain
\[ W_{\text{total}} \;=\; E[X] \;=\; \frac{\alpha\,x_m}{\alpha - 1}. \]
2.4 Cumulative Wealth for the Bottom \(p\) Fraction
The cumulative wealth held by the bottom \(p\) fraction is
\[ W(p) \;=\;\int_{x_m}^{x(p)} x\,f(x)\,dx \;=\;\alpha\,x_m^\alpha \int_{x_m}^{x(p)} x^{-\alpha}\,dx \;=\;\alpha\,x_m^\alpha \Bigl[\frac{x^{-\alpha+1}}{-\alpha+1}\Bigr]_{x_m}^{x(p)}. \]
Simplifying,
\[ W(p) \;=\; \frac{\alpha\,x_m}{\alpha - 1} \;\Bigl(1 - \bigl(\tfrac{x_m}{x(p)}\bigr)^{\alpha - 1}\Bigr). \]
2.5 Lorenz Curve for the Pareto Distribution
By definition,
\[ L(p) \;=\; \frac{W(p)}{W_{\text{total}}} \;=\; \frac{\frac{\alpha\,x_m}{\alpha - 1}\,\Bigl(1 - \bigl(\tfrac{x_m}{x(p)}\bigr)^{\alpha - 1}\Bigr)} {\frac{\alpha\,x_m}{\alpha - 1}} \;=\; 1 - \bigl(1 - p\bigr)^{\frac{\alpha - 1}{\alpha}}. \]
Hence, for a Pareto distribution, the Lorenz curve is
\[ L(p) \;=\; 1 \;-\; \bigl(1 - p\bigr)^{\tfrac{\alpha - 1}{\alpha}}. \]
- If \(\alpha \gg 1\), the distribution is more equal, and \(L(p)\) is closer to the 45-degree line of perfect equality.
- If \(\alpha\) is only slightly larger than 1, the distribution is more unequal, with significant concentration of wealth in the upper tail.
3 Estimating the Pareto Index from the Lorenz Curve
Suppose empirical data or external studies indicate specific points \((p, L(p))\) on the Lorenz curve. We can use
\[ L(p) \;=\; 1 - (1 - p)^{\frac{\alpha - 1}{\alpha}} \]
to solve numerically for \(\alpha\). Commonly cited examples:
Several Empirical Illustrations
- Pareto 80:20 Rule: \(p=0.80\), \(L(p)=0.20\)
\(\Rightarrow\) \(\alpha \approx \frac{\ln(4)}{\ln(5)} \approx 1.16\). - Pareto 90:10 Rule: \(p=0.90\), \(L(p)=0.10\)
\(\Rightarrow\) \(\alpha \approx 1.05\). - U.S. Stock Market: According to a report by Axios (2024), the top 10% own about 93% of total equity wealth, implying \(p=0.90\) and \(L(p)=0.07\). Solving yields \(\alpha \approx 1.03\).
- Credit Suisse Global Wealth Report: In 2013, it was reported that the top 1% control about 50% of global wealth, which implies \(p=0.99\) and \(L(p)=0.50\). Solving gives \(\alpha \approx 1.18\). Additionally, the top 10% were said to own about 85% of global wealth (\(p=0.90\), \(L(p)=0.15\)), giving \(\alpha \approx 1.08\). Comparing such estimates across years (e.g., 2013 vs. 2020) can reveal the time dynamics of the global wealth distribution (Credit Suisse 2013, 2020).
4 The Gini Coefficient
The Gini coefficient is a measure of wealth or income inequality that is closely related to the Lorenz curve. The Gini coefficient is defined as the ratio of the area between the Lorenz curve and the 45-degree equality line to the total area under the 45-degree line. Mathematically, the Gini coefficient ( G ) is given by:
\[ G = 1 - 2 \int_0^1 L(p) \, dp. \]
Substituting the Lorenz curve for a Pareto distribution:
\[ L(p) = 1 - (1 - p)^{\frac{\alpha - 1}{\alpha}}, \]
we obtain:
\[ G = 1 - 2 \int_0^1 \big[1 - (1 - p)^{\frac{\alpha - 1}{\alpha}}\big] \, dp = \frac{1}{2\alpha - 1}. \]
Thus, for a Pareto distribution, the Gini coefficient is:
\[ G = \frac{1}{2\alpha - 1}, \quad \text{for } \alpha > \frac{1}{2}. \]
Some features of the Gini coefficient:
- As \(\alpha \to 1^+\), the Gini coefficient approaches 1, indicating extreme inequality (a few individuals hold nearly all the wealth).
- As \(\alpha \to \infty\), the Gini coefficient approaches 0, indicating perfect equality.
- For typical empirical values of \(\alpha\) in wealth distributions (e.g., 1.1 to 1.8), the Gini coefficient ranges from 0.83 to 0.38, reflecting significant inequality
- Relative measure: \(G\) compares the distribution to perfect equality, but does not capture absolute differences.
- Non‐additivity: One cannot simply average the Gini coefficients of subpopulations to obtain an overall Gini coefficient.
- Sensitivity: The Gini coefficient is sensitive to changes in the middle of the distribution, but less so at the tails.
5 Pareto Distribution and CRRA Utility
In microeconomic theory, a widely used utility specification is the Constant Relative Risk Aversion (CRRA) form (Arrow et al. 1974; Pratt 1978). The CRRA utility function \(u(x)\) satisfies
\[ -\frac{x\,u''(x)}{u'(x)} \;=\; \gamma, \]
where \(\gamma>0\) is the coefficient of relative risk aversion. Solving for \(u(x)\) under boundary conditions such as \(u(1)=0\) yields:
- If \(\gamma=1\), \(u(x)=\ln x\). (using the L’hopital’s Rule)
- If \(\gamma\neq 1\), \(u(x)=\frac{x^{\,1-\gamma}-1}{\,1-\gamma\,}\).
Larger \(\gamma\) indicates higher risk aversion, while \(\gamma=0\) corresponds to risk neutrality (\(u(x)=x\)).
A Pareto PDF can also be derived from a differential equation with similar form. If \(f(x)\) is the PDF of a Pareto random variable on \(x\ge x_m>0\), one can write:
\[ -\frac{x\,f'(x)}{\,f(x)\!}\;=\;(1+\alpha)\,x_m^{\alpha}, \]
which likewise has a “power‐law” solution structure. Thus, Pareto distributions and CRRA utilities each emerge from a linear differential equation of analogous form, underscoring a conceptual parallel in how “power‐type” functional solutions can appear in both economic choice models (through marginal utility) and in heavy‐tailed probability distributions.
Furthermore, in mainstream economic theory, marginal utility \(u'(x)\) is assumed to be strictly positive, and \(u''(x)\) typically negative (diminishing marginal utility). In probability theory, any valid PDF \(f(x)\) must be positive, and for heavy‐tailed distributions like Pareto, \(f(x)\) decreases for large \(x\). These parallels lead to a one‐to‐one analogy between certain types of declining utilities and distributions whose density functions also decline in \(x\).
Remark: There is a well‐known relationship via logarithmic transforms: if \(X\) is Pareto(\(x_m,\alpha\)), then \(Y=\ln(X/x_m)\) is exponentially distributed with rate \(\alpha\). This exponential distribution also arises from a first‐order linear differential equation, reinforcing these structural similarities.
6 Conclusion
Because the Pareto distribution has only two parameters (\(x_m\) and \(\alpha\)), even minimal distributional data—such as “the bottom 80% own 20% of total wealth”—enables one to solve directly for the Pareto index \(\alpha\). This simplicity makes the Pareto distribution a convenient model or approximation for global wealth distribution, although in practice the estimated \(\alpha\) can vary greatly depending on the dataset, sampling, and specific segment of the population observed.
In wealth and income distribution analysis, pairing empirical Lorenz curves with Pareto modeling remains a powerful—if simplified—approach to gauging inequality. For both theoretical and practical reasons, it continues to be integral in economic research, policy discussions, and broader studies of social welfare. Meanwhile, connections to CRRA utility function illustrate that core economic principles and certain types of heavy‐tailed probabilistic behavior can share similar mathematical underpinnings.