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The chapter investigates inequality reducing taxation for various inequality views. Using the general definition of an inequality concept (Ebert, 2004), corresponding definitions of Lorenz dominance, inequality reduction, and measures of tax progression are provided. The framework allows us to simplify and clarify the different approaches found in the literature, to extend this analysis, and to present brief and transparent proofs.

The paper examines social welfare of and inequality within a finite homogeneous population consisting of identical individuals. These individuals form groups. All members of a group possess the same income; groups may differ in size and per-capita income. The framework is a generalization of the usual one to a situation in which the weight assigned to an income level can be arbitrary (but is, of course, fixed). It is in particular relevant for the measurement of welfare and inequality when households having different needs or size are investigated. Several classes of orderings which depend on the ranking of incomes are completely characterized.

Our purpose is to examine the “envy” within the context of income inequality measurement.

We use a simple axiomatic structure that takes into account “envy” in the income distribution. The concept of envy incorporated here concerns the distance of each person's income from his or her immediately richer neighbour.

We derive two classes of inequality indices – absolute and relative. The envy concept is shown to be similar to justice concepts based on income relativities.

This is the first time a complete characterisation has been provided for envy-related inequality.

There is a consensus in the general public that income taxes should be everywhere progressive. Starting from the basic properties normally required, we examine the possibilities of designing everywhere progressive income tax schedules. An axiomatic analysis investigates the (in)consistency of these requirements with further restrictions on the degree of progression. It turns out that everywhere progressive tax schedules have to be maximally progressive or almost proportional in some income range.

Volume 18 of Research on Economic Inequality contains selected papers from the Society for the Study of Economic Inequality's Third Annual Meeting (July, 2009) in Buenos Aires, Argentina. The volume collects eleven papers, five of which focus on inequality and poverty in Latin America. The Latin American papers address basic needs and poverty, multidimensional poverty, educational mobility, poverty dynamics, and the role of cash transfer programs in addressing inequality. The second half of Volume 18 collects other papers by ECINEQ members. The topics covered include taxation and inequality, evaluating poverty orderings, testing for mobility dominance, measuring relative deprivation, estimation of the costs of maintaining a child, and evaluating nutritional inequality.

For equity, societies may wish to eliminate certain forms or manifestations of inequality. Horizontal equity and vertical equity in the income tax are topics which have interested me for some years. Although any shortfall from each of these objectives can be measured in terms of unwanted inequalities, equity per se is a different concept from equality. Equity relates to fairness, justice and other societal norms which give expression to the best aspirations of our collective social conscience. For example, equal access to health care for those in equal need is an accepted norm for horizontal equity in the health field. Vertical equity in this context means treating appropriately differently those who have different needs. When offered the opportunity to be Guest Editor of this volume of Research on Economic Inequality, I decided to define the focus simply as “equity”, without placing any further restriction on topics. The papers which were ultimately included in this volume are the ones, from among those offered, which survived a rigorous refereeing process. Each has its own “take” on the concept of equity, and its link with equality. I hope that you, the reader, will gain from reading all of these contributions and pondering their significance.