Welfare comparisons with bounded equivalence scales

The paper considers the problem of comparing income distributions for heterogeneous populations. The first contribution of this paper is a precise dominance criterion combined with a simple algorithm for implementing the criterion. This criterion is shown to be equivalent to unanimity among utilitarian social planners whose objectives are compatible with given intervals of equivalence scales. The second contribution of the paper is to show that this criterion is equivalent to dominance for two different families of social welfare functions, one inspired by Atkinson and Bourguignon (in: G.R. Feiwel (Ed.), Arrow and Foundation of the Theory of Economic Policy, Macmillan, London, 1987), in which household utility is a general function of income and needs, and a second family inspired by Ebert (Soc. Choice Welfare 16 (1999) 233), in which household utility is a function of equivalent incomes. Finally, we extend our results to the case where the distributions of needs differ between the two populations being compared.     

Multidimensional inequality comparisons: A compensation perspective

We provide a unified treatment of the two approaches pioneered by Atkinson and Bourguignon (1982, 1987) [3], [4] by resorting to compensation principles in the bivariate case. We treat the attributes of individual utility asymmetrically by assuming that one attribute can be used to compensate another. Our main result consists of two sufficient second-order stochastic dominance conditions. In the case where the compensated variable has a discrete distribution, the distribution of the compensating variable must satisfy a condition which degenerates to the Sequential Generalized Lorenz test for identical marginal distributions of the compensated variable. Furthermore, the distributions of the compensated variable must satisfy the Generalized Lorenz test.     

Comparisons of heterogeneous distributions and dominance criteria

We are interested in the comparisons of standard-of-living across societies when observations of both income and household structure are available. We generalise the approach of A.B. Atkinson and F. Bourguignon (1987) [3] to the case where the marginal distributions of needs can vary across the household populations under comparison. We assume that a sympathetic observer uses a utilitarian social welfare function in order to rank heterogeneous income distributions. Insofar as any individual can play the role of the observer, we take the unanimity point of view according to which the planner?s judgements have to comply with a certain number of basic normative principles. We impose increasingly restrictive conditions on the household?s utility function and we investigate their effects on the resulting rankings of the distributions. This leads us to propose four dominance criteria that can be used for providing an unambiguous ranking of income distributions for heterogeneous populations.     

Direct Versus Indirect Taxation: The Design of the Tax Structure Revisited

This article studies the optimal direct/indirect tax mix problem when individuals differ in several unobservable characteristics (productivity and endowments). It presents general expressions for the optimal commodity tax rates and proves that contrary to Atkinson and Stiglitz's (1976) result, differential commodity taxation remains a useful instrument of tax policy even if preferences are separable between labor and produced goods. When cross substitution effects are zero, the expressions resemble traditional many households Ramsey rules. In a Cobb–Douglas illustration, where endowments differ only in good 1 (interpreted as “wealth”), the tax on good 2 provides an indirect way to tax the unobservable wealth.     

Education versus cash redistribution: The lifetime context

Education looks more powerful relative to cash redistribution when viewed in the context of lifetime rather than an annual income. The reason is that transitory income and family size affect the pattern of annual cash redistribution and both of them vary less across lifetimes than in individual years. Education may look even more powerful if the lifetime context is taken to include the period of childhood. For it is easier to target education to people whose childhood incomes are low than towards people whose adult incomes will be low. To analyse these issues one needs to compare the equalising power of a policy across income distributions that differ in their initial level of equality. We develop a method of doing this, using the Atkinson equality measure.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

A Bayesian Measure of Poverty in the Developing World

We propose a new methodology to revise the international poverty line (IPL) after Ravallion et al . (2009) using the same database, but augmented with new variables to take into account social inclusion in the definition of poverty along the lines of Atkinson and Bourguignon (2001). We provide an estimation of the world income distribution and of the corresponding number of poor people in the developing world. Our revised IPL is based on an augmented two‐regime model estimated using a Bayesian approach, which allows us to take into account uncertainty when defining the reference group of countries where the IPL applies. The influence of weighting by population is discussed, as well as the IPL revision proposed in Deaton (2010). We also discuss the impact of using the new 2011 PPP and the recent IPL revision made by the World Bank.     

ATKINSON AND BOURGUIGNON'S DOMINANCE CRITERIA: EXTENDED AND APPLIED TO THE MEASUREMENT OF POVERTY IN FRANCE

We demonstrate that the criteria put forward by Atkinson and Bourguignon (1987) can be satisfactorily applied to the analysis of poverty even when the demographic composition of society and the definition of poverty change over time. We show that they provide both sufficient and necessary conditions for any robust conclusions concerning the changes in welfare and poverty. We apply these criteria to a set of data tracking annual income distribution movements in France from 1977 to 1994. The Atkinson and Bourguignon criteria posit that recent changes in poverty are a function of macroeconomic fluctuations in activity.     

On the optimal income taxation: Some further numerical results

Using numerical calculations we show in the optimal income tax model that a realistic value for the elasticity of substitution between consumption and leisure (?=0.5) leads to conclusions different from some of those drawn by Mirrlees (1971) and Atkinson (1973). The marginal tax rates are not so low and the shape of the tax schedule is for a great majority of the population substantially nonlinear. A new feature in our calculations is that both results come out without going to maximin, as done by Atkinson (1973). We study in special cases the locality of the zero limit of the marginal tax rate at the upper end of the ability distribution and we achieve a very definite conclusion: this result is really very local. We also present an interesting approximation result concerning the relationship between the level of the government revenue and the marginal tax rate for individuals with high ability. An important methodological remark analogous to Sen's critique on the ethical measures of inequality is made on the interpretation of optimal income tax calculations.     

INEQUALITY MEASUREMENT USING ‘NORM INCOMES’: WERE GARVY AND PAGLIN ONTO SOMETHING AFTER ALL?

The "norm income" approach to inequality measurement is based on a comparison of the observed income distribution with a reference distribution consistent with the socially desired minimum degree of inequality (and not the equal shares distribution). Garvy and Paglin suggested such an approach, and we show that their methods, suitably modified, are closely related to the multivariate methods recently proposed by Atkinson and Bourguignon. The advantage and disadvantages of a norm income approach are analyzed in detail.     

Consumption externalities: a representative consumer model when agents are heterogeneous

We examine a growth model with consumption externalities where agents differ in their initial capital endowment and their reference group. We show under which conditions the aggregate equilibrium with heterogeneous agents replicates that obtained with a representative consumer, despite the fact that different individuals have different consumption levels. Next we consider the implications of the presence of consumption externalities for the long-run distributions of income and wealth. We find that, in a growing economy, “keeping up with the Joneses” results in less inequality than would prevail in an economy with no consumption externalities.     

Sequential Generalized Lorenz Dominance and Transfer Principles*

In an important contribution in 1987, Atkinson and Bourguignon proposed dominance criteria which allow us to compare income distributions for heterogeneous populations. They are defined by the unanimous preference of the dominating distribution by a set of welfare functions which satisfy some conditions on the derivatives of utility‐of‐income functions. This paper contributes to an understanding of some of the orderings considered. In particular it provides a characterization of the sequential generalized Lorenz ordering by transfer principles. The axiomatization illuminates the criterion.     

Ecological footprint inequality across countries: The role of environment intensity,income and interaction effects

Recently, White (2007) analysed the international inequalities in ecological footprints per capita (EF hereafter) based on a two-factor decomposition of an index from the Atkinson family (Atkinson, 1970). Specifically, this paper evaluated the separate role of environment intensity (EF/GDP) and average income as explanatory factors for these global inequalities. However, in addition to other comments on their appeal, this decomposition suffers from the serious limitation of the omission of the role exerted by probable factorial correlation (York et al., 2005). This paper proposes, by way of an alternative, a decomposition of a conceptually similar index like Theil's (1967) which, in effect, permits clear decomposition in terms of the role of both factors plus an inter-factor correlation, in line with Duro and Padilla (2006). This decomposition might, in turn, be extended to group inequality components (Shorrocks, 1980), an analysis that cannot be conducted in the case of the Atkinson indices. The proposed methodology is implemented empirically with the aim of analysing the international inequalities in EF per capita for the 1980–2007 period and, amongst other results, we find that, indeed, the interactive component explains, to a significant extent, the apparent pattern of stability observed in overall international inequalities.     

Portfolio selection in a lognormal securities market

Conclusion This paper, therefore, once again establishes the usefulness of the two parameter distributions in the analysis of portfolio selection and taxation. Assuming a lognormal securities market and a chance-constrained portfolio choice model, we derive the well known results of portfolio separation and the effects of taxation which were earlier obtained under more restrictive mean-variance assumptions.This paper, an earlier version of which was presented at the 8th International Symposium on Mathematical Programming, Stanford University, August 27–31, 1973, is based on a chapter of the author's Ph. D. thesis (Ahsan, 1974). The author is grateful to S. Ahmad, A. B. Atkinson, C. J. Bliss, D. W. Butterfield, A. L. Robb, T. Russell and W. M. Scarth for helpful comments.     

Comparing Income Distributions Using Atkinson's Measure of Inequality

This article is aimed at undergraduate and graduate economics students, and public sector economists, who are interested in inequality measurement. It examines the use of the Atkinson inequality measure to compare income distributions. A major feature of this measure is that distributional value judgements are made explicit, via the use of a particular form of social welfare function. Emphasis is given to the interpretation of changes in inequality and the role of the relative inequality aversion parameter, which reflects an important feature of those value judgements.     

Shadow directional distance functions with bads: GMM estimation of optimal directions and efficiencies

Because of its greater flexibility, the directional distance function (DDF) has been employed with increasing frequency to estimate multiple-input and multiple-output production, where inputs and outputs can be good or bad. However, typically researchers make three restrictive assumptions. First, they assume a direction of movement of firm production toward the frontier. Second, they assume that actual quantities of inputs and outputs are allocatively or price efficient. Third, they assume exogeneity of all inputs and all outputs, except for the normalized one. The first contribution of this paper is to include parameters to estimate optimal directions which correspond to the firm’s profit-maximizing (PM) position. The second contribution is to generalize the DDF to a shadow-quantity DDF. This entails adding distortion parameters to each input and output quantity of the DDF, creating shadow quantities. To estimate the shadow quantities and the structural parameters, we form the shadow DDF system, which includes the shadow DDF and all the first-order price equations from the shadow-PM problem. These include prices for bad inputs and bad outputs, where we approximate their missing prices for use in their first-order price equations. The third contribution is that we estimate the shadow DDF system using a Generalized Method of Moments approach, where all variables are potentially endogenous. This approach is simpler than the Bayesian one employed in Atkinson et al. (Estimating efficient production with bad inputs and outputs using latent prices and optimal directions. Working paper, University of Georgia, Athens, 2016), which estimated shadow prices and optimal directions. Using the same data set, both sets of results are qualitatively very similar, although they differ somewhat quantitatively.     

Second‐Order Discrimination and Generalized Lorenz Dominance

We propose a definition of second‐order discrimination that does not require the reference distribution to first‐order dominate the comparison one, and allows rankings of discrimination patterns when both the reference and the comparison distributions differ. It involves comparing the probabilities that randomly selected individuals in the reference and comparison distributions belong to subgroups having the same cumulative mean income, yields orderings of distributions equivalent to those from generalized Lorenz dominance, and allows orderings of discrimination patterns, partial or complete, across pairs of distributions. We compare discrimination against U.S. seniors (inter‐distributional inequality between seniors and non‐seniors) by ethnicity.     

On the undesirability of commodity taxation even when income taxation is not optimal

An important result due to Atkinson and Stiglitz (1976) [Atkinson, A.B., Stiglitz, J.E., 1976. The design of tax structure: Direct versus indirect taxation. Journal of Public Economics 6, 55–75.] is that differential commodity taxation is not optimal in the presence of an optimal nonlinear income tax (given weak separability of utility between labor and all consumption goods). This article demonstrates that this conclusion holds regardless of whether the income tax is optimal. In particular, given any commodity tax and income tax system, differential commodity taxation can be eliminated in a manner that results in a Pareto improvement. Also, differential commodity taxation can be proportionally reduced so as to generate a Pareto improvement. In addition, for commodity tax reforms that neither eliminate nor proportionally reduce differential taxation, a simple efficiency condition is offered for determining whether a Pareto improvement is possible.     

The Intensity and Shape of Inequality: The ABG Method of Distributional Analysis

Inequality is anisotropic: its intensity varies by income level. We here develop a new tool, the isograph, to focus on local inequality and illustrate these variations. This method yields three coefficients which summarize the shape of inequality: a main coefficient, α, which measures inequality at the median; and two correction coefficients, β and γ, which pick up any differential curvature at the top and bottom of the distribution. The analysis of a set of 232 microdata samples from 41 different countries in the LIS datacenter archive allows us to provide a systematic overview of the properties of the ABG (α β γ) coefficients, which are compared to a set of standard indices including Atkinson indices, generalized entropy, Wolfson polarization, and the GB2 distribution. This method also provides a smoothing tool that reveals the differences in the shape of distributions (the strobiloid) and how these have changed over time.