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Tony Atkinson's Inequality: What Can Be Done? is a remarkable book and major contribution from the scholar who has done most to develop and drive forward the economic analysis of inequality over the last 50 years. Here we take the opportunity to delineate some of the distinctive features of this book, highlighting what may be of particular salience and interest to a sociological audience in particular but also more generally from a theoretical and policy perspective.

We apply the Atkinson (1970) inequality index to time series of asset returns to offer a novel measure of financial risk consistent with expected-utility theory. This measure is converted to a certainty-equivalent return serving as a performance measure. We extend the Atkinson index to HARA utility and derive closed-form solutions to our measures for a number of preference-return combinations. Further, we establish relationships between risk aversion and the weights assigned to the cumulants of the return distribution for our performance measure. Using data from hedge funds and asset-pricing anomalies, we find that our performance measure contains additional, economically meaningful information.

Atkinson’s Theorem (Atkinson J. Econ. Theory 2, 244–263, 1970) is a classic result in inequality measurement. It establishes Lorenz dominance as a useful criterion for comparative judgements of inequality between distributions. If distribution A Lorenz dominates distribution B, then all indices in a broad class of measures must confirm A as less unequal than B. Recent research, however, shows that standard inequality theory cannot be applied to ordinal data (Zheng Res. Econ. Inequal. 16, 177–188, 2008), such as self-reported health status or educational attainment. A new theory in development (Abul Naga and Yalcin J. Health Econ. 27(6), 1614–1625, 2008) measures disparity of ordinal data as polarization. Typically a criterion used to compare distributions is the polarization relation as proposed by Allison and Foster (J. Health Econ. 23(3), 505–524, 2004). We characterize classes of polarization measures equivalent to the AF relation analogously to Atkinson’s original approach.

Background and purpose: Utilization of health care services and food influence the health status. The food and health care expenditure ratios determine the importance level of them in household's consumption expenditures. We aimed to investigate the Iranian rural and urban food and health expenditure ratios inequality during 1998 to 2012. Materials and Methods: This is a descriptive longitudinal study, which conducted based on Iranian Statistics Centre and Central Bank annual surveys. Firstly we calculated the households food and health expenditure ratios. Then we calculated the Gini coefficients and Atkinson index by using STATA version 12. Results: The mean of rural households food and health expenditure ratios were 0.53 and 0.37 respectively. Also these were 0.53 and 0.22 for urban households. All above inequality levels are based on Gini coefficients. Conclusion: There is a high level of inequality between Iranian rural and urban income deciles for health expenditure ratio, but the food expenditure ratio inequality were less and lower.

The transformation of the energy sector in Germany is associated with a considerable increase in prices, especially for electricity, which places different burdens on households depending on income. This conflicts with the idea of a fair distribution of burdens arising from the transformation of the energy sector and might endanger the acceptance of this transformation. For methodological reasons, we do not apply the GINI index in our analysis, instead we use the Atkinson index for the first time in the energy sector to quantitatively measure the distribution of energy consumption. Furthermore, we place the epsilon parameter of this index in the context of the discussion on sustainable development and justice. The epsilon parameter gives society the opportunity to make explicit what it implicitly considers to be “just” and this concept can, if necessary, be corrected by a different choice of epsilon. If society chooses a higher epsilon value because it makes higher demands on a “just” distribution of burdens and costs then the index can identify a gap between the normative goal and societal reality. This permits the authorities to implement economic policy measures to close this gap. From our point of view, the Atkinson index is an instrument that can be used to operationalize Rawls’ theory of justice against the background of the social theory of sustainable development. For our analysis, we used the example of the burdens of transforming the energy sector in Germany. Our data basis is the German Household Expenditure Survey (EVS).

We review a variety of common measures of urban land use mix in order to understand their differences and to identify their strengths and limitations. We then apply these measures to data from a Monte Carlo simulation to ascertain statistical relationships among them, finding that they can be placed into four groups where measures within each group produce highly consistent results: Percentage and Exposure Index; all varieties of the Atkinson Index; Balance, Entropy, and Herfindahl–Hirschman indices; and the Dissimilarity and Gini indices. We find that when analyzing two dimensions of land use, generally both the Balance Index and the Dissimilarity Index should be used, and that the Dissimilarity Index should be tested at multiple scales. We provide a number of other practical recommendations about which mixed use measure to apply given the contextual information available to a researcher or analyst.

Benefit-cost analyses of health regulations traditionally evaluate their economic efficiency—ignoring equity. To help address the importance of equity, we develop a survey to elicit respondents’ preferences towards equality in health risks stemming from environmental causes. Survey responses are used to parameterize an Atkinson index over environmental health risks. We compare these results to similar questions in the income context and find that respondents are significantly more averse to inequality in health risks than in income. The mean respondent is willing to accept a 22% increase in average health risk if risks are equally distributed in the population, but willing to accept a decrease of only 5% in average income if incomes are equally distributed in the population. We find that 30% of respondents answer health risk questions lexicographically—always preferring an equal distribution of risks to an unequal distribution, even if the latter makes everyone better off.Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

In a population consisting of heterogeneous types, whose income factors are indicated by nonnegative vectors, policies aggregating different factors can be represented by coalitions in a cooperative game, whose characteristic function is a multi-factor inequality index. When it is not possible to form all coalitions, the feasible ones can be indicated by a graph. We redefine Shapley and Banzhaf values on graph games to deduce some properties involving the degrees of the graph vertices and marginal contributions to overall inequality. An example is finally provided based on a modified multi-factor Atkinson index.

When a sensitive quantitative variable is under study and the Randomized Response Theory is adopted, a great deal of literature has been devoted to the estimation of the population mean (or total) or – at most – simple functions of population totals. However, in many real surveys the main interest might rely on the estimation of a complex parameter, usually a nonlinear combination of population totals. Hence, in order to face with this problem, we suppose to collect data by means of the well-known unrelated question method proposed by Greenberg et al. (1971) , and under the design-based framework, we propose to handle such a complex parameter as a population functional by suitably extending the linearization approach proposed by Deville (1999) . The considered strategy permits to obtain parameter estimation by means of the substitution method based on the empirical functional, and to achieve the corresponding variance estimator. Some selected illustrative examples are provided mostly concerning the estimation of two inequality indices, namely the Gini concentration index and the Atkinson index, widely discussed in the social and economic literature.

The Atkinson index is an inequality measure based on a social welfare function. We will use the Atkinson index as an analytical tool to examine the interactions between economic activity and energy services. The Atkinson index has a specific feature for the calculation of distribution. The index uses the epsilon parameter to explicitly reveal the inequality aversion of society. Epsilon defines how sensitively the Atkinson index should react to income inequalities. The results of applying the Atkinson index could make a significant contribution to science and policy debates on income and energy equity in the context of sustainable development.

This paper investigates the performance of accessibility-based equity measurements in transportation and proposes a multiobjective optimization model to simulate the trade-offs between equity maximization and cost minimization of network construction. The equity is defined as the spatial distribution of accessibilities across zone areas. Six representative indicators were formulated, including GINI coefficient, Theil index, mean log deviation, relative mean deviation, coefficient of variation, and Atkinson index, and incorporated into an equity maximization model to evaluate the performance sensitivity. A bilevel multiobjective optimization model was proposed to obtain the Pareto-optimal solutions for link capacity enhancement in a stochastic road network design problem. A numerical analysis using the Sioux Falls data was implemented. RESULTS verified that the equity indicators are quite sensitive to the pattern of network scenarios in the sense that the level of equity varies according to the amount of overall capacity enhancement as well as the assignment of improved link segments. The suggested multiobjective model that enables representing the Pareto-optimal solutions can provide multiple options in the decision making of road network design. Copyright © 2012 John Wiley & Sons, Ltd. Language: en

The main purpose of this study is to examine the relationship between income inequality and property crime rate, including auto theft, burglary and larceny, in Iran from 1984 to 2009. Income inequality measures include Gini coefficient and Atkinson index, with 1 and 2 degrees of inequality aversion (e=1, 2). Unrestricted error correction model (UECM)is employed to do bounds test and to examine the presence of cointegration. The results indicate that there is not any long-run relationship between income inequality and property crime rate.

The Atkinson index of income inequality is based on a comparison of the average income with the equivalent income, where the equivalent income is defined as the level of income that, if given to everyone, would generate the same social welfare as the existing distribution of income. This paper explores the possibility of extending this approach to the measurement of socioeconomic inequality of health. It assumes a social evaluation function that depends upon two variables: socioeconomic status as well as health status. With a general form of this function, an Atkinson measure is derived, which gives exactly the same result when applied to the socioeconomic variable and when applied to the health variable. The paper examines the properties of the index and suggests various extensions.

PurposeMeasures of inequality determine the effectiveness of social and economic policies aimed at reducing inequality and to design effective intervention policies. The purpose of this paper is to focus on poverty reduction and welfare improving impacts of reducing income inequality in the case of Fiji. Using Fiji's Household Income and Expenditure Survey 2002‐2003, a comprehensive analysis is used to measure the level of inequality by household income, quintile income distribution, decomposition of inequality by ethnicity and regional groups, and the household income inequality by source of income.Design/methodology/approachSeveral statistical techniques have been applied to investigate the degree of inequality in the household income. These include the Gini coefficient, the Nelson ratio, the concentration index and the Atkinson index. An evaluation by ethnicity, regions and household income sources reflects the level of inequality, and concerns for policies and governance.FindingsThe results show that urban households, in particular, experience greater inequalities, in both positive and normative terms. The Indo‐Fijian households experience greater income inequalities than the Fijian households. Decomposition results for the separate factor income components also indicate major sources of inequality. These findings clearly establish that Fiji still has a long way to go in reducing the income gaps between the rich and the poor in both rural and urban households.Originality/valueThe paper is a first study that estimates various measures of inequality in the case of Fiji. The implication of the empirical findings suggests that Fiji is unlikely to achieve its Millennium Development Goal of halving poverty rate by 2015 due to the large income differentials by ethnicity and in the urban‐rural areas.

This paper studies the economic inequality as a phenomenon resulting from two variables: The variable of responsibility which depends on the agents’ performance in the economy, and the variable of opportunity which is independent of the individuals’ performance in the economy. Identifying the contribution of the inequality of opportunity in the total inequality as the problem to be measured, we propose a decomposition of the Atkinson inequality index, by using one of the central solution concepts of cooperative game theory, called the Shapley value.

Several recent papers have sought to apply inequality measures from economics, such as the Atkinson Index (AI) for inequality of income distributions, to compare the risk inequality of different mortality risk distributions in an effort to help promote efficiency and environmental justice in pollution-reducing interventions. Closer analysis suggests that such applications are neither logically coherent nor necessarily ethically desirable. Risk inequality comparisons should be based on axioms that apply to probabilistic risks, and should consider the multidimensional and time-varying nature of individual and community risks in order to increase efficiency and justice over time and generations. In light of the limitations of the AI applied to mortality risk distributions, it has not been demonstrated to have ethical or practical value in helping policymakers to identify air pollution management interventions that reduce (or minimize) risk and risk inequity.

The U.S. Environmental Protection Agency undertook a case study in the Detroit metropolitan area to test the viability of a new multipollutant risk-based (MP/RB) approach to air quality management, informed by spatially resolved air quality, population, and baseline health data. The case study demonstrated that the MP/RB approach approximately doubled the human health benefits achieved by the traditional approach while increasing cost less than 20%--moving closer to the objective of Executive Order 12866 to maximize net benefits. Less well understood is how the distribution of health benefits from the MP/RB and traditional strategies affect the existing inequalities in air-pollution-related risks in Detroit. In this article, we identify Detroit populations that may be both most susceptible to air pollution health impacts (based on local-scale baseline health data) and most vulnerable to air pollution (based on fine-scale PM(2.5) air quality modeling and socioeconomic characteristics). Using these susceptible/vulnerable subpopulation profiles, we assess the relative impacts of each control strategy on risk inequality, applying the Atkinson Index (AI) to quantify health risk inequality at baseline and with either risk management approach. We find that the MP/RB approach delivers greater air quality improvements among these subpopulations while also generating substantial benefits among lower-risk populations. Applying the AI, we confirm that the MP/RB strategy yields less PM(2.5) mortality and asthma hospitalization risk inequality than the traditional approach. We demonstrate the value of this approach to policymakers as they develop cost-effective air quality management plans that maximize risk reduction while minimizing health inequality.

Equality of opportunity is a political ideal which requires that ex-ante inequalities, and only those inequalities, should be eliminated. Justice requires leveling the playing field by rendering everyone's opportunities equal in an appropriate sense, and then letting individual choices and their effects dictate further outcomes. In this paper we propose a methodology to decompose the path-independent Atkinson index of equality through Shapley value seeking for a measure of overall inequality produced by the marginal contributions of the opportunity and the responsibility component.

The effect of income inequality on health has been a contested topic among social scientists. Most previous research is based on cross-sectional comparisons rather than temporal comparisons. Using data from the General Social Survey and the U.S. Census Bureau, this study examines how rising income inequality affects individual self-rated health in the U.S. from 1972 to 2004. Data are analyzed using hierarchical generalized linear models. The findings suggest a significant association between income inequality and individual self-rated health. The dramatic increase in income inequality from 1972 to 2004 increases the odds of worse self-rated health by 9.4 percent. These findings hold for three measures of income inequality: the Gini coefficient, the Atkinson Index, and the Theil entropy index. Results also suggest that overall income inequality and gender-specific income inequality harm men's, but not women's, self-rated health. These findings also hold for the three measures of income inequality. These findings suggest that inattention to gender composition may explain apparent discrepancies across previous studies.

Objectives: To examine whether relaxation of control of entry regulations for community pharmacy contracts in England, introduced in 2005, affected the distribution of community pharmacies relative to population need indicators. Methods: Community pharmacy locations and population need indicators were used to calculate three summary measures of distributional equity across Primary Care Trust (PCT) areas (n = 152): the Gini coefficient, Atkinson Index and community pharmacies per PCT population. The indicators were adjusted for need using data from NHS GP contract Quality and Outcomes Framework disease registers, deprivation, all-cause mortality and elderly population rates. Results: Numbers of community pharmacies increased by 397 (4%) between 2005 and 2007 with three supermarket chains accounting for 152 (38%) of new pharmacies. Over one-quarter of PCTs experienced increases of 5% or more in community pharmacies per capita between 2005 and 2007. Gini and Atkinson indicators showed small increases in distributional equity across all population needs indicators. Conclusion: Deregulation was associated with more community pharmacies per capita and a small increase in geographic equity of community pharmacy distribution at PCT level. Future research should continue to monitor how pharmacy distribution changes over time and assess the extent to which the new regulatory framework has allowed clustering of pharmacies which could result in increased inequity below PCT level.

Background: Private psychiatric services are produced and consumed on a fee-for-service (FFS) basis in Australia. The Commonwealth Government subsidises these (and all) medical services via Medicare, a universal, comprehensive, tax-financed medical and hospital financing mechanism. A key purpose of Medicare is to improve equality of access to medical services.Aims: To measure the distribution of “access”, as measured by utilization, to private FFS psychiatric services at a regional level; and to determine the temporal trend in equality in regional access to these services during the Medicare period.Method: Conventional measures of statistical dispersion and economic inequality (the coefficient of variation, Gini coefficient and the Atkinson measure) are applied to quarterly time-series data on quantities of private psychiatric services for Australia's regions since 1984. Equations are modelled statistically on the distributional data generated by applying these measures. Lorenz curves are also constructed...

An opportunity egalitarian view focus not only on equality of outcomes but essentially on the means or opportunities to realize such outcomes. In this paper we propose a methodology to decompose an opportunity egalitarian Atkinson index by income sources seeking for a measure of overall equality into an opportunity and a responsibility component.

This article analyzes the mathematical connections between two kinds of inequality: inequality between persons (e.g., income inequality) and inequality between subgroups (e.g., racial inequality). The authors define a general inequality parameter in two-parameter continuous distributions. This parameter governs all measures of personal inequality (e.g., the Gini coefficient) and governs as well the gap (difference or ratio) between the means of subdistributions. It is thus established that in the distributions analyzed here, as personal inequality increases, so does inequality between subgroups. This general inequality parameter also governs Lorenz dominance and all quantities in the decomposition of Theil's mean logarithmic deviation into between-subgroup and within-subgroup components in the Pareto case. Thus, the general inequality parameter captures the ``deep structure'' of inequality. Finally, a whole-distribution graphical tool is introduced for assessing personal and subgroup inequality. Substantively, this work suggests that in societies characterized by special income distributions, whenever inequality disrupts social cohesion, it attacks on two fronts, via subgroup inequality as well as personal inequality.

Constructing tests for exponentiality has been an active and fruitful research area, with numerous applications in engineering, biology and other sciences concerned with life-time data. In the present paper, we construct and investigate powerful tests for exponentiality based on two well known quantities: the Atkinson index and the Moran statistic. We provide an extensive study of the performance of the tests and compare them with those already available in the literature.

This paper examines how the human demand on bioproductive lands, as measured by the Ecological Footprint, is distributed across the globe using methods commonly used to measure the distribution of income. Measuring the distribution of natural resource use will be necessary to achieve an economy that is sustainable, just and efficient. Currently, different nations place different demands on the environment, i.e. the Ecological Footprint is not evenly distributed across the globe. Calculation of the Gini coefficient for total Ecological Footprint and its components explains the sources of inequality in overall resource use. Calculation of the Atkinson index shows how inequality in the Ecological Footprint is related to the inequality of income, and environmental intensity.

In many settings there are tensions between efficiency and equity in deciding on optimal pollution control strategies. Within the context of benefit-cost analysis, efficiency may be related to implementing the least-cost control strategy to achieve a given health benefit, or alternatively, to maximizing net benefits. Similarly, equity can involve procedural fairness (i.e., equal involvement in public proceedings) or equity in the distribution of outcomes (Jacobson et al. 2005). Inequity consists of those inequalities that may be considered unjust or unfair (Macinko and Starfield 2002). Although there are multiple interpretations of these terms, we focus here on efficiency as maximizing the public health benefits of a control measure, and on equality in the distribution of those benefits across at-risk individuals as the dimension of equity that can be included in quantitative analysis. Given these definitions, although efficiency is incorporated into any health benefits analysis, equity and related distributional issues are often omitted (Yitzhaki 2003). Most regulatory impact analyses have focused exclusively on aggregate benefits [U.S. Environmental Protection Agency (EPA) 1999a, 1999b] without formally considering the geographic or demographic distributions of these benefits. In parallel, many studies of equity or environmental justice did not quantify health risks, instead focusing on proximity to sources (Burke 1993; Pollack and Vittas 1995; Sheppard et al. 1999), emissions (Millimet and Slottje 2002a, 2002b; Perlin et al. 1995), or concentrations (Lopez 2002). Studies that quantified risk inequality (Apelberg et al. 2005; Morello-Frosch and Jesdale 2006) or proposed a framework to do so (Finkel 1990, 1997) focused on characterizing baseline distributions of risk rather than the benefits of control strategies, and the appropriate methodology may differ in this context. The lack of a systematic framework to simultaneously consider efficiency and equity in a decision context may imply that decisions are based largely on maximization of societal benefits without formal consideration of equity implications. To address these limitations, we developed a framework by which risk inequality could be formally quantified within health benefits analysis (Levy et al. 2006). Briefly, we proposed that quantitative indicators of inequality, similar to those used to measure income inequality, could allow decision makers to construct an optimal efficiency–equality frontier and avoid policies that are dominated across both dimensions. Based on an axiomatic approach, we selected the Atkinson index (Atkinson 1970) as the most appropriate indicator for health benefits analysis, focusing on the change in this indicator under different control scenarios. Other indicators were considered useful for sensitivity analyses (the Gini coefficient, mean log deviation, and the Theil entropy index). Quantitative measures of risk-based efficiency and equality may be useful in many contexts, including the evaluation of national-level policies to control emissions from power plants in the United States. In theory these policies could involve site-specific control requirements or cap-and-trade programs. Cap-and-trade programs are designed primarily for economic efficiency but operate under the presumption that health benefits would be similar regardless of the distribution of emissions (Farrell and Lave 2004). However, given differences in atmospheric conditions and population patterns, how emission controls are distributed geographically could influence the magnitude and distribution of benefits. Sulfur dioxide (SO2) emission trading related to the Title IV Acid Rain Program (U.S. EPA 2007) resulted in greater health benefits than a hypothetical program without trading, based on the geographic distribution of controls (Burtraw and Mansur 1999). Regardless of efficiency claims, environmental justice advocates and communities housing power plants have expressed concern that unrestricted emission trading does not decrease and may exacerbate environmental inequities (Solomon and Lee 2000). Previous analyses (Corburn 2001; Swift 2001) focused on the possibility of emissions hot spots associated with Title IV and whether low-income or minority populations tended to have lesser emission reductions in proximate facilities. While these studies concluded that there were no hot spots, they used a procedural rather than an outcome-based concept of equity and therefore did not address the question of changing patterns of health risks. The benefits analysis of Title IV (Burtraw and Mansur 1999) indicated that certain geographic areas received health benefits while others had health disbenefits. However, without a more formal analysis, it is difficult to determine whether health inequality increased, decreased, or stayed the same, or to ascertain the potential impacts of future policies. Given the framing of the debate about national power plant controls, an outcome-based focus implies that an evaluation of how various distributions of emission controls correspond to changes in health benefits and in the spatial inequality of health risk would be informative for the design of future emission control programs. In this analysis, we focus on the various ways by which emissions reductions for power plants in the United States could be distributed to meet hypothetical national emissions caps for SO2, nitrogen oxides (NOx), and primary fine particulate matter (particulate matter with a diameter < 2.5 μm; PM2.5). For each control scenario, we estimate both the public health benefits and the change in the spatial inequality of health risk. We consider the sensitivity of our conclusions to the pollutants evaluated, the inequality indicators selected, and other factors.

Inequality within the household is neglected in wealth inequality re-search due to paucity of data and established theoretical frames for accounting for intrahousehold distribution. We develop a framework for welfare theoretic interpretation of intrahousehold wealth inequality. Illustrative data from India shows that 32% of total wealth inequality is attributable to intrahousehold inequality that results in a median welfare loss of as much as 80% for plausible values of inequality aversion.

Dayioglu M. and Baslevent C. (2006) Imputed rents and regional income inequality in Turkey: a subgroup decomposition of the Atkinson index, Regional Studies 40, 889–905. The purpose of this study is to examine the regional differences in the effect of home ownership on income distribution in urban Turkey. The earlier finding that income distribution improves with the inclusion of imputed rents from owner-occupied housing has raised the question about whether this pattern is unique to a small number of densely populated areas where squatting is more widespread or whether it is more of a widely observed phenomenon. A decomposition analysis – based on the Atkinson index – that treats Turkey's geographical regions (and, alternatively, provinces) as the subgroups that make up the whole population reveals that the main contributors to the overall improvement are the regions in which squatting is more prevalent. Dayioglu M. et Baslevent C. (2006) Les loyers imputes et l'inegalite du revenu regional en Turquie: u...

We use three conventional inequality indices—the Gini, the coefficient of variation of income, and the relative mean deviation of income—and the Atkinson index to examine the effect of income tax rates, the minimum wage, and all the major government welfare and transfer programs on the evolution of income inequality for rural and urban areas by state from 1981 to 1997. We find that these programs have qualitatively similar but quantitatively different effects on urban and rural areas. Most importantly, taxes are more effective in redistributing income in urban than in rural areas, while welfare and other government transfer programs play a larger role in rural areas.

It is often argued that the inequality in the world is increasing. We study the global income and resource inequality between 1960 and late 1990s. Income is measured in GDP/capita in terms of both current market exchange rate and purchasing power parity (PPP). Consumption of paper, energy, electricity, food, animal food, as well as emissions of carbon dioxide, is also studied. Changes in the absolute and relative gap between the top and bottom 20% consumers as well as the Atkinson measure are used as inequality indicators. Some remarks are made on how to compare income across countries and how inequality should be measured. We find that the inequality in terms of GDP/capita measured in PPP terms is rather stable in relative terms, but increasing in absolute terms. We conclude that inequality as measured by the Atkinson index is decreasing for all resources but that the gap in resource consumption in absolute terms is increasing for paper and electricity consumption.

In a recent paper in this journal, Lambert et al. [J. Public Econ. 87 (2003) 1061] sought to establish the natural rate of subjective inequality (NRSI) hypothesis. Their statistical analysis, however, may have simply identified a relationship between the Gini coefficient and the Atkinson index, rather than confirm the existence of a NRSI. In this note, an approximate relationship between the Gini coefficient and Atkinson index is identified using simulation analysis. This finding does not appear to overturn the NRSI hypothesis and indicates the need for further investigation of this insightful and potentially far-reaching hypothesis.

This paper examines several Bayesian methods of obtaining posterior probability density functions of the Atkinson inequality measure and its associated social welfare function, in the context of grouped income distribution data. The methods are compared with asymptotic standard errors. The role of the number of income classes is investigated using a simulated distribution. If only a small number of groups is available in published data, there is a clear gain from generating the posterior probability density function when using an explicit income distribution assumption. Even with a small number of groups, the Bayesian approach gives results that are close to the sample values obtained using the corresponding individual observations.

Inequality can be analyzed from various aspects. In this paper our attention is drawn to economic inequality, most frequently manifested through income and wealth. The measurement of economic inequality is a complex task. The Lorenz curve and a number of numerical indices are applied, and let us mention the following ones: the Gini coefficient, the coefficient of variation, the Theil index and the Atkinson measure. These indices do satisfy the criteria (principles) presenting, according to general consent an appropriate measure of economic inequality: anonymity (symmetry) principle, population principle, relative income principle and the Dalton principle of transfer. In recent times, the problem of inequality has been attracting a lot of attention. The explanation should be sought in the widening of income differences (within individual countries and between them) and also in new knowledge about the relationship between inequality and development. The attitude to inequality being determined mainly by the economic development level (as presented in the Kuznets hypothesis) is gradually being replaced by the attitude to inequality being the determinant of income and its growth. Contrary to previous beliefs about the stronger income inequalities being favorable to the economic growth, more recent research has pointed to the fact that a more equal distribution of income through various channels, can possibly act as an efficient stimulus of growth.

This paper analyzes the effect of regional income disparity on the allocation of investments using a dynamic multiregional Social Accounting Matrix (SAM) model. The optimal shares of total investments required to achieve the most balanced regional growth under the constraint of a 5% fixed annual economic growth rate were 37.74% for the Capital region, 18.83% for the Southeast region, 15.92% for the Central region, 13.90% for the East region and 13.61% for the West region. This policy requires an additional investment of 1.37% of the GDP, bringing about a 52.2% reduction in regional income disparity in terms of the Atkinson index. This implies that the decentralization of investment expenditures to less developed regions would lead to a substantial gain in regional income equity, on the one hand, and to the loss of cost effectiveness in investments, on the other.

We describe spatial patterns in environmental injustice and inequality for residential outdoor nitrogen dioxide (NO2) concentrations in the contiguous United States. Our approach employs Census demographic data and a recently published high-resolution dataset of outdoor NO2 concentrations. Nationally, population-weighted mean NO2 concentrations are 4.6 ppb (38%, p2.5 hours/week of physical activity). Inequality for NO2 concentration is greater than inequality for income (Atkinson Index: 0.11 versus 0.08). Low-income nonwhite young children and elderly people are disproportionately exposed to residential outdoor NO2. Our findings establish a national context for previous work that has documented air pollution environmental injustice and inequality within individual US metropolitan areas and regions. Results given here can aid policy-makers in identifying locations with high environmental injustice and inequality. For example, states with both high injustice and high inequality (top quintile) for outdoor residential NO2 include New York, Michigan, and Wisconsin.

This paper uses the Atkinson method, which was initially developed to measure the shape of the social welfare function (SWF) in the domain of income, to measure the shape of the SWF with respect to the distribution of health benefits. Two separate studies were conducted involving a total of 71 respondents. A comparison of the results across the two studies suggests that reference point effects play an important role in determining responses. Thus, more research is needed on the role that reference point effects ought to and do play in determining the nature and extent of the efficiency – equity trade-off before the results of studies of this kind can be interpreted as ‘equity parameters’ which may simply be ‘plugged into’ an appropriately specified SWF.

Fields (The Economic Journal, vol. 103, 1993, pp. 1228–35) provides a forceful argument in favor of a U-shaped path for inequality during a period of high-income-sector enlargement in a dual-economy model. This paper explores the assumptions necessary to derive Fields’s controversial result and demonstrates that in fact a U-shaped pattern is possible using the Atkinson index, but some bizarre assumptions about inequality aversion are required.

This paper presents a method of examining the effects of macroeconomic variables on the personal distribution of income over time. The approach involves modeling the complete distribution of income in each year using a flexible functional form from the generalized exponential family of distributions. The parameters of the distribution are specified as functions of the macroeconomic variables. It is shown how comparative static analysis, involving the modes and Atkinson inequality measures, can be performed. The method is applied to male New Zealand income distribution data for the period 1985-1994. The rate of unemployment is found to be the primary influence on the form of the distribution. Higher unemployment is found to decrease the modal income and increase the Atkinson inequality measure. {Copyright 2001 Elsevier Science Inc. All rights reserved}.

This paper explores a method of examining the effects of macroeconomic variables on the personal distribution of income over time. The approach involves modelling the complete distribution of income in each year using a conditional mixture distribution. The parameters of the distribution are specified as functions of the macroeconomic variables. The paper shows how comparative static analysis, involving the Atkinson inequality measure, can be performed. The method is applied to male New Zealand income distribution data for wage and salary earners over the period 1985 to 1994. It appears that cyclical variations in unemployment and GDP contributed substantially to the observed increase in inequality over the period 1987 to 1991 and the reduction in inequality during the years 1993 and 1994.

The purpose of this paper is to propose theoretical foundations on the impact of transfer scheme, e.g. Community Agricultural Policy, on income inequality within European Countries. First, we show that ex-post inequality (in the after-transfer distribution) may increase if either initial aggregate income or the amount of fiscal contributions are sufficiently high. Second according to welfare ordering, we characterize a multi-factor decomposition of the Atkinson index to gauge the impact of each income source on the inequality profile. Third, we introduce a methodology to construct a cooperative game played by different income factors (as net incomes and/or incoming transfers) explicitly measuring the cost of inequality across the population in terms of welfare loss. We finally rely on Banzhaf and Shapley values to determine the marginal contributions of each factor to overall inequality.

One of four overarching goals of Healthy People 2020 (HP2020) is to achieve health equity, eliminate disparities, and improve the health of all groups. In health disparity indices (HDIs) such as the mean log deviation (MLD) and Theil index (TI), disparities are relative to the population average, whereas in the index of disparity (IDisp) the reference is the group with the least adverse health outcome. Although the latter may be preferable, identification of a reference group can be affected by statistical reliability. To address this issue, we propose a new HDI, the R\'{e}nyi index (RI), which is reference-invariant. When standardized, the RI extends the Atkinson index, where a disparity aversion parameter can incorporate societal values associated with health equity. In addition, both the MLD and TI are limiting cases of the RI. Also, a symmetrized R\'{e}nyi index (SRI) can be constructed, resulting in a symmetric measure in the two distributions whose relative entropy is being evaluated. We discuss alternative symmetric and reference-invariant HDIs derived from the generalized entropy (GE) class and the Bregman divergence, and argue that the SRI is more robust than its GE-based counterpart to small changes in the distribution of the adverse health outcome. We evaluate the design-based standard errors and bootstrapped sampling distributions for the SRI, and illustrate the proposed methodology using data from the National Health and Nutrition Examination Survey (NHANES) on the 2001-04 prevalence of moderate or severe periodontitis among adults aged 45-74, which track Oral Health objective OH-5 in HP2020. Such data, which use a binary individual-level outcome variable, are typical of HP2020 data., Comment: Published in at http://dx.doi.org/10.1214/12-AOAS621 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The concept of economic and environmental security is explored in a broad sense as a series of necessary entitlements to forest and other resources. Traditional measures of inequality—such as the GINI coefficient or the Atkinson index of inequality which rely on measures of the distribution of formal cash income or wealth—are regarded as flawed to the extent that they do not adequately reflect the value and distribution of access to non-marketed goods and services. An “ecologically-sensitive” Atkinson or GINI index is proposed as a more relevant measure of inequality and of economic and environmental security. To improve security, policies must attempt to improve the value of these indices. An empirical example is provided based on household survey data for 357 households in 24 villages in a rainforest zone of Cameroon. Analyses of data relating to forest use, incomes, and demographics provides insights into the distribution of ecological entitlements across various income groups and into the impacts of these entitlements on economic and environmental security (through measures related to population migration). The empirical analysis demonstrates a number of important conclusions: (a) use of ecologically adjusted indices provides a more complete picture of inequality-inequality in the forest zone is less pervasive than traditional measures would have us believe; (b) traditional forest entitlements contribute positively to reducing inequality and improving security; and (c) income support programs targeted to conventional cash crops (cocoa and coffee) increase inequality and reduce security. The analyses also demonstrate that, when other economic factors have been taken into account, population movement can be a useful indicator of environmental security. An analysis of migration, demographic and income patterns suggests that low population movement in this zone can be construed as an indicator of higher environmental security. The analysis also provides direct evidence that a decline in ecological entitlements increases the probability of migration out of an area.

Diversity index is a static measure for local members and consists of various types of objects. It is used to measure the local members of a set, and many fields can be studied to evaluate the diversity of any population. For instance, it is used in ecology to measure biodiversity in an ecosystem, in demography to measure the distribution of population of various demographic groups, in economics to measure the distribution over sectors of economic activity in a region and in information science to describe the complexity of a set of information. There are many income inequality indices such as Gini index, Theil index, Hoover index, Robin Hood index, Atkinson index, Suits index and generalised entropy index, that increase as diversity decreases, so that they can be used to measure diversity if it is understood that a smaller value represents higher diversity. The values of these indices can be viewed as representing a lack of diversity (isolation or segregation), redundancy, inequality, non-randomness or compressibility in the data.

Measuring the income inequality is a major concern of the economists. Therefore, numerous indices have been devised to show different features of the income inequality. In general, the simple random sampling procedure is commonly utilized to estimate the inequality measures, while the ranked set sampling is a more cost saving method which increases the precision and the efficiency of the inequality estimators. In this paper the advantages of the ranked set sampling when measuring the amount of the income inequality are examined. Through using Monte Carlo simulation technique, this paper proves that the ranked set sampling, increases the precision of inequality indices estimations. In the end, a real income data set is analyzed to illustrate the obtained results.

The purpose of this article is to propose a theoretical foundation on the impact of a transfer scheme on income inequality in the redistribution process among participants in a related agreement. Our example involves the study of the Common Agricultural Policy implemented by EU Countries. First, we show that ex-post inequality (after the distribution process) may increase if either initial aggregate income or the amounts of fiscal contributions are sufficiently high. Second, we characterize a multifactorial methodology according to Palestini and Pignataro (2014) to gauge the impact of redistribution and the effects of different income sources to the inequality profile. Finally, we propose an exercise where a hypothetical policy is implemented and we apply the Banzhaf and Shapley values to determine the marginal contributions of each factor to overall inequality.

A group of agents must defend their individual income from an external threat by pooling their efforts against it. The winner of this confrontation is determined by a contest success function where members’ efforts may display different degrees of complementarity. Individual effort is costly and follows a convex isoelastic function. We investigate how the success of the group in the conflict and its members’ utilities vary with the degree of within-group inequality. We show that there is a natural relationship between the group’s probability of victory and the Atkinson index of inequality. If members’ efforts are complementary or the cost function convex enough, more egalitarianism within the group increases the likelihood of victory against the external threat. The opposite holds when members’ efforts are substitutes and the cost linear enough. Finally, we obtain conditions under which richer members of the group are willing to make transfers to poorer membres in order to enhance their final payoff.