Hoover index
The Hoover index, also known as the Robin Hood index, is a measure of inequality that represents the minimum proportion of total aggregate income or wealth that must be redistributed from individuals above the mean to those below it to achieve perfect equality across a population.[1][2][3] Ranging from 0, indicating complete equality, to 1 (or 100%), signifying maximum inequality where one entity holds all resources, the index is valued for its simplicity and intuitive interpretation as the share of resources requiring transfer for equalization.[1][2] Proposed by economist Edgar M. Hoover in his work on economic localization and distribution, the index is computed as half the relative mean absolute difference in incomes, or equivalently, half the L1 distance between the empirical distribution function and the uniform distribution in the Lorenz curve representation.[4][1] It exhibits properties of anonymity, population independence, and scale invariance, making it suitable for comparing inequality across diverse datasets, though it lacks decomposability unlike more complex measures such as the Theil index.[1][5] Primarily applied in economics to assess income disparities within populations or regions, the Hoover index provides a straightforward metric for policy analysis, highlighting the extent of redistribution needed without delving into normative judgments on inequality itself.[2][3]Definition and Interpretation
Core Concept
The Hoover index measures the extent of inequality in the distribution of income, wealth, or other resources across a population by calculating the proportion of total resources that would require redistribution from those with above-average holdings to those below average to attain perfect equality.[3] This metric, ranging from 0 for complete equality to a maximum of 0.5 for extreme inequality where one entity holds all resources, provides a direct economic interpretation as the minimal transfer needed for equalization.[1] Known alternatively as the Robin Hood index due to its connotation of "robbing the rich to give to the poor," it emphasizes the practical scale of disparity in absolute terms rather than relative sensitivities.[2] Mathematically, for a set of values x_i representing individual or group holdings, the index H is given by H = \frac{1}{2} \frac{\sum_i |x_i - \bar{x}|}{\sum_i x_i}, where \bar{x} is the mean value; the factor of \frac{1}{2} accounts for the fact that transfers occur only from surplus to deficit without double-counting.[1] This formulation derives from the L1 norm of deviations from the mean, normalized by total sum, yielding a scale-invariant measure applicable to various distributional analyses.[6] In population terms, it equates to half the sum of absolute differences between actual and equal shares in proportions: H = \frac{1}{2} \sum_{i=1}^N \left| \frac{E_i}{E_{\text{total}}} - \frac{A_i}{A_{\text{total}}} \right|, where E_i and A_i denote equal and actual amounts for unit i, and totals sum to unity in shares.[5] Its simplicity facilitates computation from grouped data, such as census income brackets, without requiring full microdata, though it is sensitive to the granularity of partitioning, potentially understating inequality in coarse aggregates.[7] Empirical applications, including regional population concentration or sectoral economic effects, underscore its versatility beyond income, provided the underlying distribution permits a meaningful equal-share benchmark.[8]Intuitive Meaning and Redistribution Interpretation
The Hoover index, also referred to as the Robin Hood index, intuitively quantifies income inequality as the proportion of aggregate income that lies above the line of perfect equality in a Lorenz curve diagram. This value represents the minimal share of total income required to be transferred from higher-income individuals to lower-income ones to attain complete equality across the population.[3][9] The index thus provides a straightforward, actionable insight into the degree of deviation from uniformity, where a value of 0 indicates perfect equality and higher values reflect greater disparity, bounded practically by 0.5 for distributions with at least two agents.[1] In its redistribution interpretation, the Hoover index directly translates inequality into the scale of fiscal or policy interventions needed for equalization. It measures the exact fraction of total resources—such as income or wealth—that must be reallocated from the richer subset of the population to the poorer subset, akin to the archetypal transfer from affluent to destitute.[10][11] For example, an index value of 0.1 signifies that redistributing 10% of total income would suffice to eliminate all inequality, offering a concrete benchmark for assessing the feasibility of egalitarian policies.[12] This interpretation distinguishes it from abstract metrics, emphasizing causal implications for resource transfers without assuming normative judgments on such actions.[2]Historical Background
Origins in Inequality Measurement
The Hoover index emerged from early 20th-century statistical methods aimed at quantifying disparities in economic distributions, particularly income inequality. Italian statistician Gaetano Pietra first proposed the measure in 1915 as a simple indicator of income unevenness, defining it as the minimal proportion of total income that would need to be redistributed from higher to lower earners to achieve perfect equality; this formulation captured the L1 distance between actual and equal distributions, halved and normalized.[13] Pietra's index, later termed the Pietra ratio, provided an intuitive bound on inequality, representing the maximum cumulative deviation in Lorenz curve terms and serving as a precursor to more complex metrics like the Gini coefficient.[13] American economist Edgar Malone Hoover independently formulated an equivalent index in 1936, initially applying it to measure industrial localization—the concentration of manufacturing activity across regions—rather than personal income. In his analysis, Hoover computed the index as one-half the relative mean absolute deviation from uniformity, using it to evaluate how much of total industrial output deviated from an even spatial distribution, with values ranging from 0 (perfect evenness) to approaching 1 (extreme concentration). This adaptation extended the measure's utility beyond income to any partitioned aggregate, such as sectoral or geographic allocations, while retaining its core interpretation as the fraction requiring reallocation for equality; Hoover's work in regional economics thus bridged inequality assessment with spatial economics.[14] Subsequent economists, including William R. Schutz in 1951, reinforced its application to income inequality by deriving it from absolute deviations in empirical distributions, emphasizing its computational simplicity and direct policy relevance—namely, the "Robin Hood" transfer needed for equalization. Schutz's contributions, building on Hoover's framework, integrated the index into broader inequality analysis, distinguishing it from relative measures by its absolute scale insensitivity and focus on total redistribution volume.[15] These origins underscore the index's evolution from a tool for descriptive statistics in uneven distributions to a standardized metric for empirical inequality studies, predating more parametric indices like those of Theil or Atkinson.[7]Development and Alternative Names
The Hoover index was devised in 1936 by Edgar Malone Hoover Jr., an American economist specializing in regional economics and location theory.[16] Hoover initially developed the metric to quantify concentration in spatial distributions, such as population or industrial activity across regions, where it measures the minimum proportion of total units that must be relocated to achieve uniformity.[17] This formulation proved adaptable to income inequality analysis, capturing the share of aggregate income requiring transfer from above-average to below-average recipients for perfect equality, thereby providing an intuitive proxy for redistribution needs without assuming specific welfare functions.[3] The index gained broader application in inequality studies during the mid-20th century. William R. Schutz extended its use to personal income distributions in empirical work published in 1955, emphasizing its computational simplicity and interpretability compared to more complex measures like the Gini coefficient.[18] Schutz's contributions, including validations against U.S. census data, helped establish the index's role in econometric assessments of disparity, though it remained less prevalent than parametric alternatives due to its sensitivity to extreme values. Commonly referred to as the Robin Hood index for evoking the legendary redistribution from rich to poor, the metric underscores its practical interpretation as the "robbery" fraction needed for equity.[1] It is also known as the Schutz index in recognition of Schutz's influential applications.[2] Less frequently, it overlaps with the Pietra index (proposed in 1915), which shares a similar absolute deviation basis but differs in normalization.[7] These aliases reflect its evolution from a tool in economic geography to a staple in inequality measurement, though primary sources consistently credit Hoover's 1936 formulation as foundational.[3]Mathematical Formulation and Computation
Formula and Derivation
The Hoover index H for a discrete distribution of incomes \{x_i\}_{i=1}^N is given by where \bar{x} = \frac{1}{N} \sum_i x_i is the mean income.[1][2] This expression derives from the minimum volume of transfers needed to achieve income equality. The term \sum_i |x_i - \bar{x}| quantifies the aggregate absolute deviation from the mean across all units; positive deviations (excesses above the mean) exactly balance negative deviations (deficits below the mean) in total magnitude. Each unit transferred from an excess to a deficit reduces the overall mismatch by two units (one from excess, one from deficit), so the total transfer volume required equals half the sum of absolute deviations. Dividing by total income \sum_i x_i normalizes this to the proportionate share of aggregate resources that must be reallocated, ranging from 0 (perfect equality) to approaching 0.5 (maximum inequality in the limit).[1][2][19] Equivalently, grouping observations into k categories with population shares p_j = \frac{n_j}{N} (where n_j is group size) and corresponding income shares q_j = \frac{\sum_{i \in j} x_i}{\sum_i x_i}, the index simplifies to This form represents half the L1 (Manhattan) distance between the population-share vector \{p_j\} and income-share vector \{q_j\}, reflecting the symmetric transfer interpretation across groups.[7][20]Step-by-Step Calculation with Example
The Hoover index H for a distribution of non-negative values \{x_i\}_{i=1}^n (such as incomes) is computed as H = \frac{1}{2} \frac{\sum_i |x_i - \bar{x}|}{\sum_i x_i}, where \bar{x} is the mean value \bar{x} = \frac{\sum_i x_i}{n}.[1][2] To calculate it step by step:- Sum the values to obtain the total S = \sum_{i=1}^n x_i.
- Compute the mean \bar{x} = S / n.
- For each x_i, determine the absolute deviation |x_i - \bar{x}|, then sum these to get D = \sum_{i=1}^n |x_i - \bar{x}|.
- Divide D by S to find the relative mean absolute deviation, and multiply by $1/2 to yield H = \frac{1}{2} (D / S), which ranges from 0 (perfect equality) to a maximum approaching 1 (one recipient holds all value).[1]