Exact Measures of Welfare and the Cost of Living

Abstract

1. To evaluate the effects of any economic policy upon a single individual, it is necessary to be able to write some well-defined cardinal index of that individual's ordinal utility solely in terms of observable information: prices pi, quantities consumed Xi, money expenditure Y and the parameters of his demand function. Consumer's surplus measures and index numbers all represent attempts at achieving this aim, but all suffer from wellknown weaknesses which render their general application rather suspect.' Many economists have argued that since no other techniques are available, such measures, principally consumer's surplus, must nevertheless be resorted to if any practical policy recommendations are to be made at all. However, an alternative does exist. In this paper, we show how an exact measure may be derived, free of all undesirable restrictions, capable of defining, for the individual consumer, a precise ordering of social states, however close in the ordering any alternative states may be. A numerical example illustrates the superiority of the new approach over the more traditional welfare indicators. 2. In essence the problem is as follows. Suppose that the individual consumer possesses a preference function U(X1, ..., XJ), that we do not know its functional form, but that we do know the commodity demand functions associated with it. Utilizing the first-order maximizing conditions, we can derive expressions for local changes in consumer satisfaction, dU = A)pidXi, or equivalently dU = A(dY-ZXidpi) where A is the marginal utility of money, a non-observable. Since A is always positive, for small changes we can simply use as a welfare index, YpidXi or dY-ZXidpi. Either version can be interpreted as measuring an infinitesimal change in consumer's surplus. Alternatively, the former expression may be called a quantity whereas the latter, assuming that expenditure is unchanged, may be labelled a price variation . For discrete changes in the variables, however, such measures are no longer applicable. Quantity or price variations written in Laspeyres and Paasche form are subject to regions of ignorance . The addition of the well-known consumer's surplus triangle in an attempt to construct an approximate discrete welfare indicator suffers from the same drawback.2 3. We could, however, do the following. Substitute into the above differential equations, the observed inverse and ordinary demand functions for pi and Xi respectively. Then the problem becomes one of finding some integrating factor A, equal to the marginal utility of money, such that the differential equations are integrable.3 In other words, if we can find such a A, we can retrieve the underlying preference function from our knowledge of the consumer's demand functions. Since we have assumed U to exist in the first place, we know that A will also exist and that it will be a function of the variables and parameters describing the individual's demand functions. Although i itself is not directly observable, it is, in principle, derivable from information that is known. Further, an infinity of i's could be derived, each one generating a preference function that is a monotonic transformation of any other in the family. The drawback to this procedure is that there exist no general rules for deriving even one of the many integrating factors. As a result, a long and tedious search procedure may have to be endured.