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It is possible to employ either income or expenditure as the base for personal taxation. A considerable literature has developed that investigates the relative efficiency of these bases. The answer is usually in favor of the expenditure tax since it does not distort the choice between consumption and saving. In contrast, the literature is almost silent on the relative equity of the two bases. We investigate the redistributive consequences of the choice in models with two sources of heterogeneity: skill in employment and lump-sum endowment. The Gini coefficient is used to measure the degree of equity achieved by the tax bases in static and dynamic settings. Income taxes and expenditure taxes that generate equal welfare or equal revenue are compared. In the static economy the income tax leads to lower inequality except when skill and endowment are negatively correlated. Inequality is always lower with the income tax in the dynamic economy. These results support the choice of income as the base for personal taxation if reduction in inequality is a priority of policy.

The optimal location of plants by a global firm is analyzed for the first time using measures of distance along the spherical surface of Planet Earth. With a uniform distribution of customers an optimal location strategy will normally seek a space-filling configuration of identical areas that are as near circular as possible. The hexagonal space-filling solution for location on an infinite plane cannot be generalized to the surface of a sphere. Different spatial patterns are required for different numbers of plants; these may be based on triangles, squares, or pentagons. The chapter reviews the current state of knowledge on the topic, drawing on theories of spherical geometry and regular convex polyhedra, and on applications in physics, chemistry, and medicine. Overall, there appears to be no general solution to the problem; only a set of quite different solutions for various special cases. The lack of any general solution to this central problem in international business illustrates the “impossibility” referred to in the title of this chapter.

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.