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Lorenz curve

The Lorenz curve is a graphical representation of the cumulative distribution function of income or wealth across a population, plotting the proportion of total income received by the bottom p percent of individuals against p.^[1] It was developed by American economist Max O. Lorenz in his 1905 paper "Methods of Measuring the Concentration of Wealth," published in the Publications of the American Statistical Association.^[2] In a scenario of perfect equality, the curve coincides with the 45-degree line of equality, known as the line of perfect equality; deviations below this line quantify the degree of inequality, with greater curvature indicating higher disparity.^[3] The curve is constructed by ranking individuals or households from lowest to highest income, cumulatively summing their shares, and connecting the points to form a convex function bounded by the axes and the line of equality.^[1] This visualization facilitates empirical assessment of distributional outcomes, enabling comparisons across populations, time periods, or policy interventions through the area between the curve and the line of equality.^[4] The Gini coefficient, a scalar measure of inequality derived from the Lorenz curve, equals twice the value of this area, providing a normalized index ranging from 0 (perfect equality) to 1 (perfect inequality).^[3] While primarily applied to income and wealth, the Lorenz curve extends to other positive quantities like land ownership or market shares, underscoring its utility in first-principles analysis of concentration irrespective of normative judgments on equality.^[5] Its adoption in economics reflects a focus on observable disparities rather than prescriptive ideals, though interpretations must account for data quality and definitional choices in income measurement.^[2]

History

Origins and Initial Development

The Lorenz curve originated with American economist Max Otto Lorenz (1876–1959), who proposed it in 1905 as a graphical tool for assessing wealth concentration. In his article "Methods of Measuring the Concentration of Wealth," published in the Publications of the American Statistical Association (volume 9, issue 70, pages 209–219), Lorenz argued that traditional metrics such as averages, ratios of richest to poorest, or simple percentages obscured the underlying distribution's structure, failing to reveal how inequality varied across population segments.^[6]^[7] He introduced the curve to provide a visual summary, plotting the cumulative proportion of total wealth (y-axis) against the cumulative proportion of the population ranked from poorest to richest (x-axis), resulting in a convex function bowed below the diagonal line representing perfect equality.^[6]^[8] Lorenz derived the curve from empirical wealth data, emphasizing its utility in detecting changes in distributional shape over time or across regions. Applying it to Prussian tax records from 1848 to 1891, he observed a progressive equalizing trend in the lower half of the wealth distribution, with the curve shifting closer to the equality line for the bottom 50% of holders, while upper segments showed relative stability or slight concentration increases.^[7] Similar analysis of U.S. estate tax data circa 1890–1900 highlighted comparable patterns, underscoring the method's sensitivity to subgroup dynamics that aggregate statistics might miss. Lorenz also explored parabolic curve-fitting to approximate the graphical form for quantitative comparison, calculating coefficients to quantify deviations from equality, though he prioritized the visual diagnostic over a single index.^[6] This initial formulation focused exclusively on wealth rather than income, reflecting Lorenz's context in early 20th-century statistical economics amid Progressive Era concerns over industrial fortunes and inheritance.^[9] The approach built on prior Pareto analyses of upper-tail distributions but extended downward to the full spectrum, enabling holistic inequality appraisal without assuming specific functional forms like lognormality. Despite its innovation, the curve's adoption remained niche in Lorenz's era, partly due to computational demands for hand-plotting from tabulated data and limited data availability beyond tax or probate records.^[7]

Adoption and Theoretical Integration

Following its proposal by Max O. Lorenz in 1905 as a method to depict the concentration of wealth, the curve initially received modest attention in statistical literature but achieved greater prominence through Corrado Gini's adaptation in 1912. In his paper "Variabilità e mutabilità," Gini utilized the Lorenz curve to derive a scalar measure of inequality, defining the Gini coefficient as twice the area between the curve and the diagonal line of perfect equality, thereby transforming the graphical tool into a foundational component of quantitative inequality assessment.^[10]^[11] This linkage enabled systematic comparisons of distributions, with Gini's index—ranging from 0 for perfect equality to 1 for complete inequality—gaining traction in early empirical studies of income shares, such as those examining European tax data from the 1910s onward.^[12] Post-1912, adoption accelerated amid rising interest in empirical economics during the interwar period and after World War II, particularly in development economics where the curve visualized disparities in national income data compiled by organizations like the United Nations. By the 1950s, economists such as Simon Kuznets employed Gini-derived metrics from Lorenz curves to analyze long-term trends, hypothesizing an inverted-U pattern linking economic growth to inequality levels across U.S. and global datasets from 1913 to 1950.^[13] This practical utility extended to policy evaluations, with the curve's convex shape—reflecting the principle that poorer quintiles receive proportionally less income—informing debates on progressive taxation and redistribution in welfare states.^[6] Theoretically, the Lorenz curve integrated into welfare economics via dominance criteria, enabling unambiguous inequality rankings without parametric assumptions. A distribution exhibits Lorenz dominance over another if its curve lies weakly above the latter everywhere (with strict inequality somewhere), implying higher social welfare under any strictly increasing, concave transformation—a result formalized by Anthony B. Atkinson in 1970, linking the ordering to aversion to inequality in utilitarian frameworks.^[14] This partial ordering, robust to mean-preserving spreads, underpins comparisons in stochastic dominance theory and majorization, as distributions majorized by another possess subordinate Lorenz curves, facilitating causal inferences about inequality drivers like market structures or policy interventions. Extensions, such as the generalized Lorenz curve proposed by Anthony F. Shorrocks in 1983, incorporate mean income to rank welfare across varying average levels, broadening applicability to growth-inequality trade-offs.^[15]^[16]

Mathematical Definition

Formal Specification

The Lorenz curve for a non-negative random variable X with finite positive mean \mu = \mathbb{E}[X], cumulative distribution function F, and density f (where defined) is specified as the function L: [0,1] \to [0,1] given by

L(p) = \frac{1}{\mu} \int_{0}^{p} F^{-1}(u) \, du, \quad p \in [0,1],

where F^{-1}(u) = \inf \{ x \geq 0 : F(x) \geq u \} is the quantile function of X.^[17] Equivalently,

L(F(x)) = \frac{1}{\mu} \int_{-\infty}^{x} t \, f(t) \, dt

for continuity points x of F. This parametrizes the curve by the cumulative population proportion p = F(x), with L(p) denoting the corresponding cumulative income proportion. In the discrete case, consider a finite population of n units ordered by non-decreasing values $0 \leq y_1 \leq y_2 \leq \dots \leq y_n, with total sum S_n = \sum_{j=1}^n y_j > 0. The Lorenz ordinates are then

L_i = \frac{1}{S_n} \sum_{j=1}^i y_j, \quad i = 1, \dots, n,

plotted against F_i = i/n. The curve connects the points (F_i, L_i) linearly, with extensions to (0,0) and (1,1). By construction, L(0) = 0, L(1) = 1, L is non-decreasing, and L(p) \leq p for all p \in [0,1], with equality if and only if X is constant almost surely.^[17] The derivative L'(p) = F^{-1}(p)/\mu (where differentiable) is non-decreasing, implying convexity of L.

Construction and Computation

The Lorenz curve is constructed from a dataset of income or wealth holdings by first ranking the population units in ascending order of their holdings. The horizontal axis represents the cumulative proportion of the population, starting from the poorest, while the vertical axis shows the corresponding cumulative proportion of total income or wealth. In practice, empirical data such as tax records or survey responses are aggregated into percentiles or quantiles, with points plotted and typically connected by straight lines to form a piecewise linear curve.^[1] For discrete data with n equally weighted units and ordered holdings y_1 \leq y_2 \leq \dots \leq y_n, the curve passes through the points \left( \frac{i}{n}, L_i \right), where L_i = \frac{\sum_{j=1}^i y_j}{\sum_{j=1}^n y_j} for i = 1, 2, \dots, n. This formulation normalizes the cumulative sums by the total, ensuring the curve starts at the origin and ends at (1,1). Computation involves sorting the data, calculating running sums, and dividing by the grand total, often implemented in statistical software for large datasets.^[18]^[19] In the continuous case, for a non-negative random variable X with cumulative distribution function F, density f, mean \mu, and quantile function Q(p) = \inf \{ x : F(x) \geq p \}, the Lorenz curve is defined as L(p) = \frac{1}{\mu} \int_0^p Q(u) \, du for p \in [0,1], or equivalently L(F(x)) = \frac{1}{\mu} \int_{-\infty}^x t f(t) \, dt. This integral represents the expected cumulative share up to quantile p, scaled by the mean. Numerical approximation for continuous distributions may involve discretizing the density or using parametric fits to the data.^[18]^[20] For grouped or frequency-weighted data, construction adjusts by using cumulative frequencies as proportions and weighted cumulative sums of midpoints or values times frequencies, expressed as percentages of totals before plotting. This method accommodates binned survey data, ensuring the curve reflects the empirical distribution accurately.^[21]

Properties

Geometric and Functional Properties

The Lorenz curve L(p) is defined for p \in [0,1], where it satisfies L(0) = 0, L(1) = 1, and L(p) \leq p for all p \in [0,1], with equality holding throughout if and only if the underlying distribution is perfectly equal (i.e., all units receive identical shares).^[22] The curve lies entirely below or on the diagonal line of equality y = p, reflecting the cumulative nature of shares ordered from lowest to highest.^[23] Geometrically, the Lorenz curve is convex, meaning it bends upward (second derivative non-negative), which follows from the non-decreasing ordering of population and income shares.^[4] This convexity ensures that the area between the curve and the line of equality is non-negative and quantifies deviation from uniformity.^[24] Functionally, for a non-negative random variable with cumulative distribution function F, density f, quantile function x(F), and mean \mu > 0, the Lorenz curve admits the integral representation L(F) = \frac{1}{\mu} \int_{-\infty}^{x} t f(t) \, dt, where x = F^{-1}(F).^[20] The first derivative is L'(F) = \frac{x(F)}{\mu}, which is non-decreasing (as x(F) increases with F) and equals 1 on average over [0,1].^[20] The second derivative L''(F) = \frac{1}{\mu f(x(F))} is positive wherever the density f > 0, rigorously establishing convexity and smoothness under standard regularity conditions on f.^[25] These derivatives link the curve directly to the underlying distribution: the slope at any point reflects the income level at that quantile relative to the mean, while curvature inversely relates to local density, with steeper bows indicating sparser high-income tails.^[26]

Interpretive and Comparative Properties

The Lorenz curve depicts inequality by graphing the cumulative share of total income or wealth L(p) against the cumulative population proportion p, ranked from lowest to highest recipient. The point L(p) quantifies the fraction of aggregate resources controlled by the poorest p segment of the population, revealing how concentrated resources are beyond uniform distribution.^[20]^[27]^[28] The curve's convexity and position below the 45-degree line of perfect equality—where L(p) = p—illustrate deviations from equity; proximity to this diagonal signifies lower inequality, as the bottom p captures a larger share closer to p. The derivative \frac{dL(p)}{dp} = \frac{q(p)}{\mu}, with q(p) as the p-th quantile and \mu the mean, interprets the relative income at percentile p to the overall average; slopes below 1 denote sub-mean earnings, aiding assessment of distributional tiers.^[20] For comparisons, a Lorenz curve L_1(p) dominating L_2(p) if L_1(p) \geq L_2(p) for all p (strictly greater at some) indicates distribution 1 exhibits less inequality via first-order Lorenz dominance, as poorer fractions hold proportionally more.^[20]^[27] Intersecting curves yield ambiguous rankings, necessitating metrics like the Gini coefficient, derived from the area A between the curve and equality line as G = \frac{A}{A + B} where B = 1 - A, to resolve ties.^[27]^[28] Larger A universally signals heightened inequality across overlaid curves.^[28]

Derivation of the Gini Coefficient

The Gini coefficient quantifies income inequality by measuring the deviation of the Lorenz curve from the line of perfect equality, which is the 45-degree line connecting (0,0) to (1,1) in the unit square. Geometrically, the coefficient equals twice the area A between this line and the Lorenz curve L(p), where p ranges from 0 to 1 representing the cumulative population proportion. Since the total area under the equality line is $1/2, the normalization G = 2A ensures G ranges from 0 (perfect equality, L(p) = p) to 1 (perfect inequality).^[29]^[30] Equivalently, let B denote the area under the Lorenz curve, so A + B = 1/2 and G = 1 - 2B = 1 - 2 \int_0^1 L(p) \, dp. This integral form directly derives from the geometric definition, as the area B captures the cumulative proportion of income held by the bottom p fraction of the population, integrated over all p. For a non-negative random variable Y with mean \mu > 0 and cumulative distribution function F, the Lorenz curve is L(F(x)) = \frac{1}{\mu} \int_{-\infty}^x t \, dF(t), enabling computation of G via the integral.^[29] In the discrete case, with ordered data points (X_i, Y_i) for i = 0 to k where X_0 = 0, Y_0 = 0, X_k = 1, Y_k = 1, and X_i the cumulative population shares, the area B approximates via trapezoidal integration: B \approx \sum_{i=0}^{k-1} \frac{Y_i + Y_{i+1}}{2} (X_{i+1} - X_i). Thus, G \approx 1 - \sum_{i=0}^{k-1} (Y_i + Y_{i+1}) (X_{i+1} - X_i), assuming uniform spacing or direct summation over intervals. This formula converges to the continuous case as the number of points increases.^[30] For continuous distributions, substituting the quantile function x(p) = \inf \{ y : F(y) \geq p \} yields L(p) = \frac{1}{\mu} \int_0^p x(u) \, du, so \int_0^1 L(p) \, dp = \frac{1}{\mu} \int_0^1 x(u) (1 - u) \, du. Therefore, G = 1 - \frac{2}{\mu} \int_0^1 x(u) (1 - u) \, du, linking the Gini to the expected value weighted by rank. This derivation underscores the coefficient's foundation in the Lorenz curve's cumulative structure.^[29]

Connections to Other Distribution Metrics

The Pietra index (also known as the Robin Hood or Hoover index) is directly defined using the Lorenz curve as the maximum vertical distance between the curve and the 45-degree line of equality: P = \max_{0 \leq p \leq 1} [p - L(p)]. This metric quantifies the minimal share of total resources that must be redistributed from the upper to the lower portion of the distribution to achieve perfect equality, emphasizing the point of greatest deviation in cumulative shares.^[31]^[20] Generalized entropy indices, including the Theil measures (GE(0) as mean log deviation and GE(1) as Theil's entropy index), connect to the Lorenz curve through derivations that integrate logarithmic transformations of the curve's properties or its derivatives. For instance, the Theil index can be expressed via the Lorenz function, linking the graphical representation of proportional shares to information-theoretic divergence from uniformity, with decomposability properties facilitating subgroup analysis aligned with Lorenz partial orderings.^[26]^[32] The Atkinson index relates to the Lorenz curve via consistency with Lorenz dominance: a distribution with a Lorenz curve lying entirely above another's exhibits lower Atkinson inequality for any inequality aversion parameter \varepsilon > 0, as the index is monotonically decreasing in the equally distributed equivalent income, which corresponds to the Lorenz curve's position relative to equality. This holds because the Atkinson measure respects the Lorenz order, resolving ambiguities from curve crossings in a manner sensitive to the specified aversion level.^[33]^[34]

Applications

Economic and Statistical Uses

In economics, the Lorenz curve graphically represents the distribution of income or wealth by plotting the cumulative proportion of total income received by the bottom p percent of the population against p. A curve aligning with the 45-degree line of equality signifies uniform distribution, whereas a bowed curve indicates inequality, with greater deviation reflecting higher concentration among the wealthy.^[1]^[3] This tool facilitates visual assessment of disparities across populations or over time, such as comparing national income shares where the bottom 20% might hold under 5% of total income in unequal societies.^[35] The curve underpins inequality metrics like the Gini coefficient, computed as the ratio of the area between the Lorenz curve and the equality line to the area under the equality line, providing a normalized summary of distributional skewness used in cross-country comparisons and trend analysis.^[36]^[3] In policy evaluation, it quantifies redistributive effects by contrasting pre-fiscal and post-fiscal curves, revealing how progressive taxation or transfers shift income shares toward equality; for example, such analyses demonstrate that transfers can reduce the Gini by 20-30% in welfare states.^[37] Statistically, the Lorenz curve generalizes to any non-negative random variable, characterizing concentration in probability distributions beyond income, such as firm sizes or resource allocations.^[38] In fields like epidemiology, it measures risk predictiveness in survival models, illustrating how risk scores concentrate disease incidence; for instance, advanced breast cancer prediction models using the curve show 35% of cases in the top 20% risk group, aiding targeted screening efficiency.^[5] Applications extend to scientific domains for analyzing heterogeneity in datasets, including ecological abundances or technological outputs, where it highlights deviations from uniformity.^[39]

Policy and Empirical Contexts

Lorenz curves serve as a tool for evaluating the distributional effects of public policies, particularly fiscal interventions like progressive taxation and transfer programs, by comparing pre- and post-policy income distributions. If a policy shifts the Lorenz curve upward toward the line of equality, it indicates a reduction in inequality, providing a visual and quantitative basis for assessing redistributive efficacy. For instance, policy analysts examine how changes in taxes and government expenditures alter the curve, with upward shifts signaling improved equity without requiring mean income adjustments.^[40] ^[41] In broader policy analysis, Lorenz curve dominance criteria help rank interventions; a generalized Lorenz curve that lies entirely above another's implies unambiguous welfare superiority, accounting for both inequality and average outcomes. This approach has been applied in distributional cost-effectiveness analysis for health and environmental policies, where dominance determines if targeted interventions outperform uniform ones in equity terms. Similarly, in emissions policy scenarios, Lorenz dominance ranks distributions of environmental outcomes across socioeconomic groups to inform equitable regulation design.^[42] ^[43] Empirically, Lorenz curves facilitate cross-country comparisons of income inequality, revealing stark disparities; for example, Denmark's curve lies significantly closer to equality than Namibia's, reflecting stronger redistributive mechanisms in social democratic systems versus resource-dependent economies. Global studies using parametric approximations show that while long-term trends exhibit strict Lorenz dominance indicating declining inequality, short-term (decade-scale) changes rarely do, complicating policy attributions amid data limitations. These analyses underscore causal links between institutional factors like market structures and observed curves, prioritizing empirical distributions over aggregated indices like the Gini coefficient for nuanced policy insights.^[44] ^[45]^[46]

Limitations and Criticisms

Technical Shortcomings

The empirical construction of the Lorenz curve from discrete, finite population data typically involves grouping incomes and linear interpolation between points, assuming homogeneity within groups and thereby underestimating true inequality by smoothing over intra-group variations.^[47] This stepwise approximation deviates from the theoretical continuous curve, introducing bias that increases with coarser data granularity or larger group sizes.^[1] A core mathematical limitation arises when comparing distributions: if two Lorenz curves intersect, neither exhibits Lorenz dominance over the other, precluding a clear ordering of inequality without invoking additional axioms or metrics, as the curves violate complete comparability under the Lorenz criterion.^[48] This crossing phenomenon, documented in empirical studies of income distributions across countries or time periods, undermines the curve's ability to provide a total preorder on distributions solely through graphical or functional inspection.^[32] Parametric functional forms fitted to data, such as quadratic or generalized versions, often impose convexity constraints or fail to accommodate multimodal distributions, resulting in poor fits for real-world data and systematic errors in integrals derived for inequality indices like the Gini coefficient.^[49] For instance, single-parameter models yield Gini coefficients bounded below at approximately 0.418, rendering them inapplicable to low-inequality distributions observed in datasets from Nordic countries.^[50] These estimation challenges are exacerbated by incomplete or censored data, where fitting assumes unobserved portions align with observed trends, potentially overstating or understating cumulative proportions.^[1]

Conceptual and Interpretive Challenges

The Lorenz curve's reliance on the line of perfect equality as a benchmark embeds normative assumptions that equal distribution maximizes social welfare, presupposing the desirability of progressive transfers without efficiency losses, as formalized in the Pigou-Dalton principle of transfers.^[51] This framework aligns with second-order stochastic dominance criteria, which imply inequality aversion akin to concave utility functions, but such axioms introduce ethical postulates rather than purely descriptive metrics, as distributions below the diagonal are deemed inferior irrespective of incentives for productivity or innovation that may causally generate unequal outcomes.^[52] Critics, including Chateauneuf and Moyes, argue this conflates graphical deviation with intrinsic inequality measurement, questioning whether the curve captures welfare-relevant dispersion or merely relative shares detached from absolute levels or causal mechanisms.^[53] Interpretive ambiguities arise when Lorenz curves cross, precluding unambiguous dominance rankings and requiring supplementary value judgments to deem one distribution more unequal.^[54] For instance, a curve that bows more sharply at lower quantiles but converges nearer the top may reflect concentrated poverty yet limited elite capture, complicating policy inferences about redistribution's impacts without resolving whether transfers should prioritize tails or medians.^[55] This non-transitivity challenges causal realism in applications, as crossings often occur in empirical comparisons across time or populations—such as post-1980 U.S. income data where mid-distribution shifts intersect tail dynamics—demanding ad hoc criteria beyond the curve's geometry.^[56] Furthermore, the curve's scale invariance normalizes by mean income, obscuring absolute welfare changes; a more bowed curve amid rising averages can mask gains for the bottom quintile, as seen in China's 1990–2010 growth where Lorenz deviation increased while poorest deciles' real incomes rose over 500%.^[37] This relative focus invites misinterpretation in dynamic contexts, prioritizing shares over levels and ignoring first-principles drivers like market liberalization that elevate floors despite widening bows, thus complicating attributions of inequality to policy versus structural factors.^[57] Empirical estimates also face source credibility issues, with academic datasets often underreporting top incomes due to survey non-response biases favoring lower brackets, inflating apparent equality in Lorenz plots.^[58]

Recent Developments

Parametric and Extended Models

Parametric representations of the Lorenz curve provide flexible functional forms for estimating inequality from grouped or incomplete data, enabling smoother approximations and statistical inference while satisfying properties like L(0)=0, L(1)=1, and convexity.^[47] One early and widely used model is the Kakwani-Podder form, L(p) = \alpha p + (1 - \alpha) p^\gamma, where p is the population share, 0 < \alpha < 1, and \gamma > 1, which allows adjustment for varying curvature in the lower tail.^[59] Another common single-parameter variant, proposed by Rasche et al. (1980), takes L(p) = 1 - (1 - p)^c with c > 1, emphasizing behavior near perfect equality.^[47] These models have been evaluated for fit across diverse distributions, with extensions like Gupta's (1984) form incorporating additional flexibility for empirical income shares.^[60] More recent parametric advances address limitations in handling zeros, asymmetry, or extreme inequality. A 2020 single-parameter model by Paul and Shankar, L(p) = p / (1 + (1 - p) \kappa), with \kappa > 0, offers parsimonious estimation tied to the Gini coefficient and has been refined in 2025 analyses as convex combinations of benchmark curves for improved inequality ranking.^[61] ^[50] In 2023, a universal form was introduced to accommodate datasets with zero incomes and high concentration, deriving from mixtures of Pareto distributions and outperforming prior models in simulations across global inequality metrics.^[39] Such developments enhance robustness for policy simulations, though selection requires validation against non-parametric estimates to avoid bias from misspecification.^[62] Extended models generalize the Lorenz framework beyond univariate income to incorporate means, dependence, or dynamics. The generalized Lorenz curve (GLC), defined by Shorrocks (1983) as GL(p) = \mu L(p) where \mu is the mean income, scales the ordinate to enable welfare dominance comparisons that integrate average levels with distribution shape, resolving ambiguities in standard Lorenz crossings.^[63] This extension underpins theorems for ranking distributions when means differ, as verified in empirical tests for income growth scenarios.^[64] Multivariate extensions, emerging post-2020, adapt the curve using copulas or optimal transport rearrangements to capture joint inequality across attributes like income and wealth, quantifying dependence-induced disparities.^[65] Recent applications include asymmetric variants like the Lorenz-extended Rohde function (2024), which models non-convex deviations for skewed real-world data, and functional time series forecasts of evolving curves for inequality prediction up to 2030.^[66] ^[67] These build causal links to underlying densities, such as linking second derivatives of L(p) to hazard rates, aiding interpretation in high-dimensional settings.^[68]

Modern Empirical Applications

Lorenz curves have been employed in empirical analyses of income inequality dynamics during the COVID-19 pandemic, enabling visualization of distributional shifts through cumulative shares. A 2024 study on China and Italy constructed Lorenz curves from household survey data to quantify pandemic-induced changes, finding a Gini coefficient increase from 0.38 to 0.42 in Italy by 2021, attributed to disproportionate job losses among low-income groups, while China's rose modestly from 0.46 to 0.48 due to targeted fiscal transfers.^[69] Similarly, U.S. analyses from 2020-2022 used Lorenz curves derived from Current Population Survey data to show temporary inequality reductions via stimulus payments, with the bottom quintile's income share rising 1.5 percentage points before reverting post-2021.^[70] In global and regional contexts, recent applications forecast inequality trajectories via time-series Lorenz curves. A 2025 functional time series model applied to European regional data predicted convex deviations from equality lines, estimating a 5-10% Gini rise in high-disparity areas like Southern Italy by 2030 under baseline growth scenarios, validated against Eurostat distributions from 2015-2022.^[71] Global updates to Lorenz curves, incorporating 2020s household surveys and national accounts, reveal between-country inequality declines since 2000 but within-country rises, with the top 1%'s share exceeding 20% in the U.S. by 2022 per updated plots.^[72] Advanced parametric fits address empirical challenges like zero incomes in developing datasets. A 2023 universal Lorenz model, tested on World Bank microdata from 50+ countries (2010-2020), accurately captured extreme tail behaviors in distributions with 10-20% zero earners, yielding Gini estimates within 2% error margins compared to nonparametric benchmarks, particularly for African nations like Namibia where traditional curves understate concentration.^[39] In U.S. regional policy evaluations, 2022 New England Lorenz curves from IRS tax data highlighted Massachusetts' bottom 20% share at 3.1% versus the line of equality, informing progressive tax reforms amid post-pandemic recovery.^[73] Beyond income, Lorenz curves constrain policy models for sustainable consumption. A 2024 ecological economics framework integrated decent living standards as Lorenz bounds on global caloric and energy distributions (2015-2020 data), showing excessive top-decile shares (over 30% of energy) necessitate redistribution for carbon neutrality, with empirical fits from FAO and IEA datasets.^[74] These applications underscore the curve's utility in causal policy simulations, though sensitivity to data granularity persists across studies.^[50]

Illustrative Examples

Theoretical Constructions

The Lorenz curve, introduced by economist Max O. Lorenz in his 1905 paper on measuring wealth concentration, is constructed theoretically by ordering economic units (such as individuals or households) from lowest to highest income or wealth levels and plotting the cumulative proportion of total income against the cumulative proportion of the population. In the discrete case applicable to finite datasets, assume a sample of n non-negative observations y_1 \leq y_2 \leq \dots \leq y_n with total sum S_n = \sum_{j=1}^n y_j > 0. The points are (F_i, L_i) for i = 0, 1, \dots, n, where F_i = i/n represents the population share and L_i = (\sum_{j=1}^i y_j)/S_n the corresponding income share, with (0,0) and (1,1) as endpoints; the curve connects these linearly, yielding a convex, non-decreasing function bounded below the line of equality L(F) = F.^[18]^[75] This discrete formulation aligns with Lorenz's original method for grouped statistical data, where intervals of population shares map to aggregated income shares, ensuring the curve starts at the origin and ends at unity while reflecting the sorted distribution's partial sums. For theoretical illustration, under perfect equality (y_j constant for all j), the points lie on L(F) = F; under perfect inequality (all income held by one unit), L(F) = 0 for F < 1 and jumps to 1 at F=1.^[10] In the continuous case, extending the construction to probability distributions, let X \geq 0 be a random variable with cumulative distribution function F, probability density f, and finite mean \mu = \mathbb{E}[X] = \int_0^\infty x f(x) \, dx > 0. The Lorenz curve is L(p) = \frac{1}{\mu} \int_0^p Q(u) \, du for p \in [0,1], where Q(u) = F^{-1}(u) is the quantile function, or equivalently L(F(x)) = \frac{1}{\mu} \int_{-\infty}^x t f(t) \, dt; the curve remains convex, lies below the diagonal, and satisfies L(0)=0, L(1)=1, L'(p) = Q(p)/\mu \geq 0 with increasing derivative.^[18]^[75] Theoretical constructions for parametric families yield explicit forms: for an exponential distribution with rate \lambda > 0 (\mu = 1/\lambda), L(p) = 1 - (1-p)^2; for a Pareto distribution with shape \alpha > 1 and minimum x_m > 0 (\mu = \alpha x_m /(\alpha-1)), L(p) = 1 - \left(1 - p^{1/\alpha}\right)^\alpha. These derive from inverting the cdf and integrating the quantile function, demonstrating how heavier tails (lower \alpha) bow the curve further from equality.^[75]^[76]

Real-World Empirical Instances

Empirical Lorenz curves are derived from national household survey data, plotting cumulative income shares against cumulative population shares ordered by income. In the United States, U.S. Census Bureau data for 2022 reveal that the lowest income quintile accounted for 2.8% of total household income, the second quintile 6.7%, the middle quintile 14.6%, the fourth 23.0%, and the highest 52.9%.^[77] These shares translate to Lorenz curve coordinates where the bottom 20% of households hold 2.8% of income, the bottom 40% hold 9.5%, the bottom 60% hold 24.1%, the bottom 80% hold 47.1%, and the full 100% hold 100%.^[77] In South Africa, marked by high inequality, World Bank estimates from 2014 household surveys show the lowest quintile receiving 2.4% of total income, the second 4.8%, with the top quintile capturing over 65%.^[78]^[79] This results in a Lorenz curve bowed far from the equality line, with the bottom 40% holding under 7.2% of income, reflecting structural factors including historical apartheid legacies and labor market disparities.^[80] Recent Statistics South Africa data from 2023 confirms persistent skew, with average household income at R204,359 but highly concentrated, as the top 20% dominates aggregate earnings.^[81] Comparatively, in Norway, income distribution data yield a Lorenz curve closer to equality, with a Gini coefficient of 27.7 in 2019 derived from surveys showing the bottom 20% receiving around 9-10% of income, supported by progressive taxation and social transfers.^[82] Empirical constructions from historical Norwegian tax records (1875-2017) demonstrate long-term shifts, with inequality peaking pre-WWII before declining due to welfare policies.^[83] Globally, World Bank analyses of household surveys across 1988-2008 plot Lorenz curves indicating moderate between-country convergence but persistent within-country inequality, where the bottom 50% worldwide held about 8.5% of income in recent estimates.^[84] These instances underscore the curve's utility in visualizing deviations from equality, informed by primary data sources like censuses and surveys.^[85]

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The Lorenz curve as an archetype: A historico-epistemological study
Aug 9, 2025 · ... Methods of measuring the concentration of wealth. According to Derobert and Thieriot (2003) , Lorenz was opposed to the idea of representing ...
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Methods of Measuring the Concentration of Wealth - ADS
Methods of Measuring the Concentration of Wealth. Lorenz, M. O.. Abstract. Publication: Publications of the American Statistical Association.Missing: Max | Show results with:Max
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Max O. Lorenz - The History of Economic Thought Website
Max Otto Lorenz was the son of German immigrants in Iowa. He studied at the University of Iowa, completing his BA in 1899.Missing: origins | Show results with:origins
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[PDF] Measuring Income Inequality: The Lorenz Curve and Gini Coefficient
The Gini Coefficient was developed by the Italian statistician and sociologist Corrado Gini. (and published in his 1912 paper Variability and Mutability)—it ...
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The Discovery of the Gini Coefficient | History of Political Economy
Feb 1, 2021 · It suggests that Gini's efforts to perfect the Lorenz curve may well have facilitated his discovery of what came to be known as the Gini ...
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https://www.tandfonline.com/doi/full/10.1080/03610926.2025.2488895
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The Measurement and Trend of Inequality: A Basic Revision - jstor
Lorenz curve and the related Gini concen- tration ratio, both of which ... small and could be overlooked, but now they exceed the cash allotments and ...<|separator|>
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Welfare Functions: Unanimity and Dominance - Oxford Academic
Dec 13, 1973 · Atkinson's result shows that for all welfare functions satisfying these properties, if the Lorenz curve ... Welfare Economics · Books.
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CONCENTRATION INDICES AND CONCENTRATION CURVES
eralized Lorenz Curve (Shorrocks, 1983b)). For a general discussion of the ... Gini's mean difference rediscovered. Biometrika 55,. 573-575. DAVID ...
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[PDF] Working Paper Series Multidimensional Lorenz dominance - ECINEQ
This paper seeks to develop an economic approach to the problem. Towards that end it formulates a definition of a multidimensional Lorenz dominance relation ( ...
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A General Definition of the Lorenz Curve - IDEAS/RePEc
Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.<|control11|><|separator|>
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Lorenz Curve -- from Wolfram MathWorld
The Lorenz curve is used in economics and ecology to describe inequality in wealth or size. The Lorenz curve is a function of the cumulative proportion of ...
[19]
[PDF] The Discrete Lorenz Curve - ITAM
L(t) = 1. µ. ∫ t. 0. Y (s)ds = ∫ t. 0. Y (s)ds. ∫ 1. 0. Y (s)ds . Additionally, we can (continously) extend it to the extremes of the interval by setting L(0) = ...Missing: _0^ du
[20]
[PDF] Lorenz curves, the Gini coefficient and parametric distributions
The mean income in the population is found at that percentile at which the slope of L(p) equals 1, that is, where q(p) = µ and thus at percentile F(µ) (as shown ...Missing: ∫ _0^
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Lorenz Curveb : Meaning, Construction, and Application
Apr 15, 2025 · Lorenz Curve is the measure of the deviation of the actual distribution of a statistical series from the line of equal distribution.
[22]
[PDF] LORENZ CURVES - myweb - Texas Tech University
Lorenz curves measure distribution of income in a group, society or country. A Lorenz curve L(x) bears the following properties: • The domain of L is [0,1].
[23]
Lorenz Curves, Size Classification, and Dimensions of Bubble Size ...
If there is any inequality in size, then the Lorenz curve is convex and lies below the line of equality, see Figure 1. As a measure of inequality (and for a ...Missing: convexity | Show results with:convexity<|separator|>
[24]
Quantile versions of the Lorenz curve - Project Euclid
For each F ∈ F0 the Lorenz curve of F is defined by L0(F; p) ≡ pμp/μ, for 0 ≤ p ≤ 1. The lowest proportion of incomes p have proportion L0(p) of the total ...
[25]
[PDF] Modelling Lorenz curve
Nov 30, 2012 · d L p. k e dp e. = >. - . The second derivative is positive and hence the Lorenz curve is convex. Consequently, the first derivative is ...
[26]
[PDF] Derivation of Theil's Inequality Measure from Lorenz Curves
If expressed as a continuous function, the Lorenz curve is typically given as. (5). )(π η f. = where π is the cumulative population share and. 1. 0. ≤. ≤ π η is ...
[27]
Measurements of Inequality — Data 88E: Economic Models Textbook
The Lorenz Curve visually presents income inequality by plotting household income percentile on the x -axis, and the cumulative income share that the bottom x ...
[28]
Measuring inequality: what is the Gini coefficient? - Our World in Data
Jun 30, 2023 · The Lorenz curve is the “line of equality” where incomes are shared perfectly equally. Area A is 0, and hence so is the Gini coefficient. Where ...
[29]
[PDF] Gini coefficient Calculation
It was developed by the Italian statistician Corrado Gini and published in ... income in the Lorenz curve and therefore not taken into account in the Gini.
[30]
[PDF] Statistical Consulting Report Derivation of the Gini Index
The Gini index is the area between the equality and Lorenz curves, divided by the area under the equality curve, or G = 1 − (Yi+1 + Yi)(Xi+1 − Xi).
[31]
Income Inequality Measures - Scientific Research Publishing
The Pietra index P is defined as the maximum Dmax=F(μX)−LX(F(μX)) , presented above. It can be graphically represented as the longest vertical distance between ...<|control11|><|separator|>
[32]
[PDF] Not all inequality measures were created equal
These indicators can be linked to the Lorenz curve, which we will use to understand and compare distributions.Missing: metrics | Show results with:metrics
[33]
[PDF] Choosing an inequality index
For a measure to be. Lorenz consistent the following two conditions must hold: First, inequality rises (declines) when the Lorenz curve lies everywhere below. ( ...
[34]
A simple method for measuring inequality - Nature
Jun 4, 2020 · The closer the index is to one (where the area A is large), the more unequal the distribution of income. Fig. 1: Lorenz curve. figure 1. The ...
[35]
2.2 Measuring the economy: Inequality – Macroeconomics
Using the Lorenz curve to measure the Gini coefficient ... The Lorenz curves in Figure 2.3 show that more unequal distributions have a greater area between the ...Missing: integration | Show results with:integration
[36]
Lorenz Curve - Economics Help
Lorenz in 1905 for representing wealth distribution. The Lorenz curve shows the cumulative share of income from different sections of the population. If there ...Missing: origins | Show results with:origins
[37]
[PDF] Measuring the Redistributive Impact of Taxes and Transfers
This paper provides a theoretical foundation for analyzing the redistributive effect of taxes and transfers for the case in which the ranking of individuals ...
[38]
An ordered family of Lorenz curves - ScienceDirect.com
The Lorenz curve is one of the most popular and intuitive methods. For any given data batch, the Lorenz curve can be derived in several ways. One method is to ...Missing: history invention
[39]
A universal model for the Lorenz curve with novel applications for ...
Mar 23, 2023 · This study shows that, the proposed model fits not only the data whose actual Lorenz plots have a typical convex segment but also the data whose actual Lorenz ...Missing: convexity | Show results with:convexity
[40]
[PDF] for Lorenz Curves - Holy Cross logo
The functional form of Lorenz curves is also relevant to the measurement of tax progressivity. Several global measures of progressivity use the Lorenz-Gini ...
[41]
[PDF] assessing the distributional impact of public policy
This principle offers a unifying framework for analyzing the socioeconomic impact of public policy by using a wide variety of evaluation functions, inequality ...
[42]
Distributional Cost-Effectiveness Analysis: A Tutorial - PMC - NIH
A distribution is dominated if the generalized Lorenz curve lies wholly below that of an alternative intervention. Under this criterion, both the targeted and ...
[43]
Ranking Emissions Distributions Generated by Different Policy ...
Generalized Lorenz (GL) dominance has the advantage of imposing minimal preference structure but has the disadvantage of being a partial ordering; it cannot ...
[44]
Applications of Lorenz curves in economic analysis (English)
In this paper the Lorenz curve technique is used as a tool to introduce distributional considerations in economic analysis. The concept has been...
[45]
[PDF] WIDER Working Paper 2021/61-Trends in global inequality using a ...
Mar 31, 2021 · Strict Lorenz dominance occurs only in the long term, pointing to an overall decline but is rare when comparing ten-year periods, implying a ...
[46]
Global inequality: How large is the effect of top incomes?
In this study, we propose a fully-parametric approach to approximate the Lorenz curve of the entire income distribution for each country and year. ... The BE ...2. Measuring Global Income... · 6. Results · Appendix A
[47]
A simple method for estimating the Lorenz curve - Nature
Nov 16, 2021 · The Lorenz curve was devised by an American economist named Max O. ... Lorenz MO (1905) Methods of measuring the concentration of wealth.
[48]
Problems With Measuring Inequality - Economics from the Top Down
Jun 26, 2019 · Here's the problem with the Gini index. It tells us about the size of the area between the given Lorenz curve and the line of perfect equality.Missing: shortcomings | Show results with:shortcomings
[49]
About some difficulties with the functional forms of Lorenz curves
Aug 22, 2022 · We study to what extent some functional form assumption on the Lorenz curve are amenable to calculating headcount poverty, or poverty threshold.
[50]
Estimating Income Inequality Using Single-Parameter Lorenz Curves
May 22, 2025 · This paper explores new properties of their model, offering a refined representation in terms of convex linear combinations of Lorenz curves.<|control11|><|separator|>
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Normative Economics and Economic Justice
May 28, 2004 · This approach, in particular, justifies the widespread use of Lorenz curves in the empirical studies of inequality. The Lorenz curve depicts the ...
[52]
[PDF] Theoretical Foundations of Lorenz Curve Orderings - EconStor
Abstract. This paper is concerned with distributions of income and the ordering of related. Lorenz curves. By introducing appropriate preference relations ...
[53]
Does the Lorenz curve really measure inequality? - EconPapers
By Patrick Moyes and Alain Chateauneuf; Does the Lorenz curve really measure inequality?
[54]
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1977483
[55]
Limitations of using the Lorenz curve framework to... : AIDS - LWW
Mar 13, 2017 · First, in some circumstances, Lorenz curves can cross, yielding identical or similar Gini coefficient values. These scenarios would complicate ...Missing: crossings interpretation challenges
[56]
[PDF] Why Measure Inequality - Harvard DASH
shortcomings, because of cultural inertia and the ability to visualize its interpretation from the Lorenz curve. 18Many of the problems mentioned here that ...
[57]
[PDF] Why Measure Inequality? - Harvard Law School
The literature on equivalences between properties of the Lorenz curve, social welfare assessments, and certain other criteria draws on the close analytical.
[58]
[PDF] The Balance of Inequality, the Gini Index ... - LIS Working Paper Series
Mar 10, 2022 · Applying the usual definition of the Lorenz curve in terms of the quantile function, i.e., as the inverse of the cumulative distribution ...
[59]
Notes on a new functional form for the Lorenz curve - ScienceDirect
A new functional form for the Lorenz curve was proposed by Kakwani and Podder (1976). n = aπα (√2−π)β. This letter applies this form of the Lorenz curve ...
[60]
An alternative functional form for estimating the Lorenz curve
The most popular single parameter Lorenz curves are the forms proposed by Kakwani and Podder (1973), Gupta (1984) Chotikapanich (1993) and a form implied by ...
[61]
An investigation of the performance of parametric functional forms ...
Jun 23, 2023 · This study investigates the performance of a single-parameter functional form proposed by Paul and Shankar (2020) who use income data of Australia.
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https://www.roiw.org/2014/n4/14.pdf
[63]
Ranking Distributions with Generalized Lorenz Curves - jstor
' Shorrocks and Kakwani independently develop the "generalized" Lorenz curve as a device for simultaneously taking account of differences in mean incomes and ...
[64]
[PDF] Ranking Income Distributions with Generalised Lorenz Curves
Steps 3 and 4 require that standard Lorenz curves be built and checked as to whether they cross or whether the dominating distribution has a lower mean income. ...Missing: interpretation challenges<|separator|>
[65]
Lorenz map, inequality ordering and curves based on ... - arXiv
Apr 11, 2024 · We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory.
[66]
The lorenz-extended rohde function for asymmetric lorenz curve ...
This study aimed to develop the Lorenz-Extended Rohde function that could effectively accommodate the asymmetry of the Lorenz curve.
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[PDF] Forecasting a time series of Lorenz curves - arXiv
Apr 6, 2025 · The Lorenz curve is a fundamental tool for analysing income and wealth distribution and inequality at national and regional levels.
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[PDF] Parametric Lorenz Curves and the Modality of the Income Density ...
By Definition (1), the first and second derivatives of an LC are positive. If the third derivative is positive as well, the density is zeromodal, but if its ...<|separator|>
[69]
Impact of Covid-19 on Income Inequality; a Comparative Analysis for ...
Apr 2, 2024 · PDF | This study examines the impact of covid 19 on income inequality and comparison between China and Italy. Trends in Gini coefficient ...<|separator|>
[70]
Trends in Income and Well-Being Inequality During the COVID-19 ...
First, we examine the trends in income inequality before and during the COVID-19 pandemic using the Gini coefficient. We calculate the Gini coefficients ...
[71]
Forecasting a time series of Lorenz curves: one-way functional ...
The Lorenz curve is an example of a constrained functional time series, which has received increasing attention in the functional data analysis literature.
[72]
Revisiting the trends in global inequality - ScienceDirect.com
The Lorenz curves show that the trend of inequality within countries is clear and does not depend on a specific sensitivity to different parts of the ...
[73]
[PDF] An Empirical Analysis of the Status of Income Inequality New ...
May 31, 2024 · The Lorenz Curve was developed by economist Max Lorenz with the intention to describe inequality in respect to wealth distribution. Another ...
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Decent living standards, prosperity, and excessive consumption in ...
Decent living standards are introduced as constraints to the Lorenz curve. Consumption is split into basic needs satisfaction, prosperous, and excessive.
[75]
[PDF] Measurement of Inequality - University of Minnesota
Figure 7: Construction of the Lorenz curve. 9. Page 10. The four-quadrant diagram in Figure 7 shows, for the case of the Pareto distri- bution, how the Lorenz ...
[76]
[PDF] Estimating Pareto's constant and Gini coefficient of the Pareto ...
The Lorenz curve is a tool used to evaluate the share of income of sub-populations or to measure the inequality of individual income distributions.
[77]
[PDF] Income in the United States: 2022 - U.S. Census Bureau
Measures of income dispersion. Mean household income of quintiles. Shares of household income quintiles. Summary measures. Gini index of income inequality. Mean.
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Income share held by lowest 20% - South Africa - IndexMundi
Dec 28, 2019 · The value for Income share held by lowest 20% in South Africa was 2.40 as of 2014. As the graph below shows, over the past 21 years this indicator reached a ...
[79]
South Africa - Income distribution - IndexMundi
Dec 28, 2019 · The value for Income share held by second 20% in South Africa was 4.80 as of 2014. ... Percentage shares by quintile may not sum to 100 because of ...
[80]
Six Charts Explain South Africa's Inequality
Jan 28, 2020 · The top 20 percent of the population holds over 68 percent of income (compared to a median of 47 percent for similar emerging markets). The ...
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[PDF] Income and Expenditure Survey of Households 2022/2023, P0100
The statistical release present results of the Income and Expenditure Survey (IES) 2022/2023 conducted by. Statistics South Africa (Stats SA) from 7 ...
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Norway Gini inequality index - data, chart | TheGlobalEconomy.com
Norway: Gini income inequality index: The latest value from 2019 is 27.7 index points, an increase from 27.6 index points in 2018. In comparison, the world ...
[83]
[PDF] Lorenz curves for Norway 1875-2017
In each figure the horizontal axis shows the proportion of the population and the vertical axis shows the proportion of total gross income.
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[PDF] Global income inequality by the numbers: In history and now
reflected in the data. The Lorenz curves, which plot the percentage of cumulative income (running from 1 to 100) on the vertical axis against the percentage ...
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Measuring income inequality - IZA World of Labor
The Lorenz curve is a commonly used metric that allows for the quick and visual comparison of inequality across countries. The Gini coefficient uses information ...