A Complement to Pearce on Complements

Abstract

There are almost as many measures of substitutability/complementarity as there are economists working in demand analysis — indeed, perhaps more since several eminent economists (Hicks, Samuelson, Pearce) have more than a single measure. There are cardinal measures based on the direct utility function (Auspitz and Leiben, Edgeworth-Pareto); on the indirect utility function (McKenzie) and ordinal measures based on behaviour of the direct marginal rate of substitution (Hicks-Allen) and of the marginal rate of substitution of the expenditure function (Morishima). There are measures based on properties of the compensated demand functions (Hicks-Allen, Pearce, Samuelson); on properties of the inverse compensated demand functions (Hicks) and on the ordinary demand functions. Typically these measures all involve some common elements: they are defined locally in terms of partial derivatives of the relevant functions; they involve only pairs of goods but beyond those factors they have little in common. One of the reasons for such a plethora of measures appears to be that economists have not really decided what it is that they want to measure. At least five strands of argument can be found: