Robert Willig, Session Topic: Recent Theoretical Developments in Financial Theory: Discussion, The Journal of Finance, Vol. 33, No. 3, Papers and Proceedings of the Thirty-Sixth Annual Meeting American Finance Association, New York City December 28-30, 1977 (Jun., 1978), pp. 792-794

This study introduces ordinally additive, ordinally linear, and ordinally Cobb-Douglas utility functions for the analysis of risky decisions when the uncertainty affects several attributes. Practical algorithms for the determination of utility functions with these forms are provided. Further, the study offers several risk invariance axioms on choice behavior under multidimensional risk. These axioms, for the first time, extend to the multidimensional context the heuristic correspondence between risk aversion and subjective wealth, heretofore familiar in only one dimension. In addition, the consequences of these new risk invariance axioms for utility functional forms in the multi-dimensional context are investigated. The result is a sequence of theorems which show that ordinally linear, ordinally Cobb-Douglas, and ordinally additive von Neumann-Morgenstern utility functions are characterized by the risk invariance axioms. IN THE STUDY OF DECISION MAKING under uncertainty, it is well known that plausible sets of axioms imply that the decision maker acts as if he maximizes his expected von Neumann-Morgenstern utility. (See [1], for example.) If the uncertain outcomes are multidimensional, then the appropriate utility concept is a function of many variables. This is the case, for example, for a firm choosing marketing policies which will affect sales and profits, for an individual faced with investment choices which will affect consumption during several years, and for a government deciding among projects which differ in their costs, outputs, and environmental impacts. In grappling with such problems, decision analysts have found it impossibly difficult to proceed with utility measured by a general function of the outcome variables. Instead, they have used multi-attribute utility functions with special forms, and found that their conclusions are sensitive to the particular form utilized. (See [13], for example.) Thus, it falls to theorists to develop testable hypotheses about risky choice which are equivalent to special (and, hopefully, convenient) functional forms for multiattribute utility. Fishburn [2, 3, and 4], Keeney [5 and 6], and Pollak [9, 10, and 11] have made contributions in this vein. This study introduces ordinally additive von Neumann-Morgenstern utility functions (i.e., those which are a monotonic transformation of a sum of functions, each of one variable) to the literature. I show that they should be well suited to practical decision analysis by presenting algorithms for their use. I propose several risk invariance axioms which plausibly extend to the multidimensional context the intuitive link between risk aversion and wealth in one dimension. These axioms are shown to characterize (in the presence of some other assumptions) ordinally additive, ordinally linear, and ordinally Cobb-Douglas von NeumannMorgenstern utility functions.

The purpose of this paper is to settle the controversy surrounding consumer's surplus' and, by so doing, to validate its use as a tool of welfare economics. I will show that observed consumer's surplus can be rigorously utilized to estimate the unobservable compensating and equivalent variations-the correct theoretical measures of the welfare impact of changes in prices and income on an individual. I derive precise upper and lower bounds on the percentage errors of approximating the compensating and equivalent variations with consumer's surplus. These bounds can be explicitly calculated from observable demand data, and it is clear that in most applications the error of approximation will be very small. In fact, the error will often be overshadowed by the errors involved in estimating the demand curve. The results in no way depend upon arguments about the constancy of the marginal utility of income. Consequently, this paper supplies specific empirical criteria which can replace the apologetic caveats frequently employed by those who presently apply consumer's surplus. Moreover, the results imply that consumer's surplus is usually a very good approximation to the appropriate welfare measures. To preview, below I establish the validity of these rules of thumb: For a

A recent line of research has exposed some technological determinants of the structure of industries that produce more than one good. The analyses of both multiproduct perfect competition and natural monopoly require a generalized notion of average cost and, in addition, several newly identified technological characteristics pertinent only to joint production. This paper provides an overview of these new results, and suggests a unifying framework in which the theory can be further developed.

I began my article Consumer's Surplus Without Apology (henceforth, CSWA) with the words The purpose of this paper is to settle the controversy surrounding consumer's surplus... (p. 589). That is my purpose here, as well. However, George McKenzie's published comments have taught me that if articles and careful analyses settle controversies, they only do so very slowly. McKenzie makes sweeping and attacking statements about CSWA but fails to substantiate them. He scatters birdshot criticisms at CSWA that are based on misreadings of rather clear material. He offers an example in which he miscalculates multiproduct consumer's surplus. Finally, he contends that his own (with Ivor F. Pearce) approach to welfare analysis is preferable to the consumer's surplus approach. In this reply, to keep the record straight, I show in Section I that each of McKenzie's strongly worded attacks is unsubstantiated and invalid, and that each of his more technical sounding criticisms rests only on misreadings of CSWA. More interestingly, in Section II, I summarize some of the theory of multiproduct consumer's surplus that is needed to understand the calculatiop error in and proper interpretation of the example that McKenzie proffers. In Section III, I argue that the approach to welfare analysis advocated by McKenzie and Pearce is far less useful than the consumer's surplus methodology.