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For over 60 years, Lerner's (1944) probabilistic approach to the welfare evaluation of income distributions has aroused controversy. Lerner's famous theorem is that, under ignorance regarding who has which utility function, the optimal distribution of income is completely equal. However, Lerner's probabilistic approach can only be applied to compare distributions with equal means when the number of possible utility functions equals the number of individuals in the population. Lerner's most controversial assumption that each assignment of utility functions to individuals is equally likely. This paper generalizes Lerner's probabilistic approach to the welfare analysis of income distributions by weakening the restrictions of utilitarian welfare, equal means, equal numbers, and equal probabilities and a homogeneous population. We show there is a tradeoff between invariance (measurability and comparability) and the information about the assignment of utility functions to individuals required to evaluate expected social welfare.

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.

Given the rise of individuals who have complex communication needs (CCN), it is important to identify appropriate assistive technology systems that can support the individual's communication needs. Augmentative and alternative communication (AAC) systems can serve as a means to assist individuals to communicate independently. The goal of AAC is to enhance or replace the individual's current and limited verbal or written communication skills. This chapter focuses on feature matching, aided communication and the selection process for aided communication systems including low to high technology systems. It also emphasises other key considerations pertaining to person-centred planning such as conducting preference assessments and trial periods to minimise system abandonment.

The position of these Irish agitators is illogical and untenable; the remedy they propose is no remedy at all – nevertheless they are talking about the tenure of land and the right to land; and thus a question of worldwide importance is coming to the front.³