The Distance Function in Consumer Behaviour with Applications to Index Numbers and Optimal Taxation

The use of duality concepts has now become widespread in production and consumption theory. In both of these areas, one of the central and most useful concepts is the cost or expenditure function which represents tastes or technology through the minimum cost to consumer or producer of reaching a particular utility or output level at given prices. Under utility maximizing or cost minimizing assumptions, the value of this function is actual outlay, so that the cost function defines the relation between expenditure, prices and utility or production as the case may be. For many problems, prices and outlay are the natural variables with which to work, and it is this that makes the cost function such a convenient representation of preferences. Even so, the mathematical properties of the cost function, particularly its homogeneity and concavity, give it decisive advantages over either direct or indirect utility functions even in situations where quantities are the more natural variables. For this reason, it is useful to consider the dual of the cost function itself, retaining its mathematical properties, but defined on primal, rather than dual variables. This dual is the distance function, sometimes also referred to as the transformation function, the gauge function, or the direct cost function. This function has made a number of distinguished but infrequent appearances in the literature. Wold (1943) uses it to relate quantity bundles to a given reference vector and thus to define a utility function. Debreu (1951) defines a " coefficient of resource utilization" through the distance function while Malmquist (1953) develops a systematic theory of quantity indices based upon it. In production theory, the distance function is discussed by Shephard (1953) and more recently is systematically and extensively used in the forthcoming monograph by Fuss and McFadden (1978), especially in the contributions by McFadden and by Hanoch. In the demand context, the function is briefly discussed by Diewert (1974), but the main contributions are in an unpublished paper by Gorman (1970) and, more briefly, in Gorman (1976). The function is used in a number of recent publications, notably by Blackorby and Russell (1975), Hanoch (1975), Blackorby and Donaldson (1976), Blackorby, Lovell and Thursby (1976) and Diewert (1976a) and (1976b). The aim of the present paper is to present a reasonably systematic, if informal presentation of the distance function in the context of consumer behaviour. Much of what follows is derived from one or more of the contributions listed above. However, none of these provides anything like a complete treatment, and in view of the wide range of potential applications of the analysis, in particular to demand studies, to rationing theory and to welfare economics generally, a synthesis of such useful material is overdue. Section 1 of the paper defines the distance function and discusses its properties. Particular attention is focused on the duality between the distance and cost functions and