The first part of this article integrates the concept of (relative) risk aversion with respect to income (r) with the static analysis of demand for many commodities. Alternative representations of preferences and demand functions, using duality, give rise to many alternative representations and interpretations of r, and to theorems regarding attitudes towards risk in bundles of quantities and in prices. In the second part, a previous analysis by Deschamps is corrected and completed by specifying the general form of preferences and demands such that r is a function of the utility level only, independent of relative prices. Finally, preferences and demand functions associated with constant r (previously analyzed by Stiglitz and Deschamps) are specified more explicitly and completely. A general conclusion emerging is that demand behavior under certainty can hardly throw any light on the nature of attitudes towards risk. THE CONCEPT OF THE relative risk aversion function as a unit-free measure of individual aversion to income risk under expected utility maximization, was defined by Arrow [1] and Pratt [15], and has proved useful in various applications. Stiglitz [21] studied relations between an individual's aversion to income risk, and his indirect utility and demand functions for many commodities, obtained under certainty in competitive markets. In particular, he analyzed the cases of risk indifference and constant relative risk aversion (r). Deschamps [4] extended this analysis to study implications of alternative assumptions: (i) That absolute risk aversion R = rly is independent of prices, nominal or relative, for given income. This is equivalent, however, to constant r (which is the case analyzed by Stiglitz) for both cases. (ii) That R or r are constant on each indifference surface. This is shown to be impossible for R, but meaningful and interesting for r = r(u). Unfortunately, however, Deschamps could not show the utility and demand functions for this case, and conducted an indirect analysis based on the second-order differential equation implied, failing to note that any such r(u) is compatible with homothetic preferences. In addition, his analysis contains errors (e.g., the case r = 1 constant) and may be subject to misleading interpretations. The purpose of this article is twofold: (i) To complete the analysis of risk aversion with many commodities, by using various alternative formulations of the relative risk aversion function to study general relations between income risk aversion and attitudes towards risk with respect to quantities (e.g., when both relative prices and income are subject to risk), or with respect to prices. (ii) To complete the analysis of Deschamps by showing the general forms of utility and demand functions when r = r(u), and to correct some errors of analysis and interpretation.

Direct or indirect additivity of production or utility functions implies dependence of substitution effects on income effects. This dependence is eliminated by implicit additivity, or strong separability along isoquants or indifference surfaces. The present study proposes and analyzes two models, with direct and indirect implicit additivity, respectively, which are generally non-homothetic, non-CES, and include less than 3n parameters for n goods. They give rise to log-linear systems of estimable demand relations. Many other models, such as Cobb-Douglas, CES, Direct and Indirect Addilog, CRESH, CDE, and Non-homothetic CES, are simple, testable special cases of either or both of these models.

[The paper defines and analyzes a functional form for a one-output, many-factórs production function, which is homothetic (or homogeneous), and exhibits CRES; that is, its ES (Allen-Uzawa elasticities of substitution) vary along isoquants and differ as between pairs of factors, but the ES stand in fixed ratios everywhere. Given data on factor prices, quantities, and output, and assuming competitive costminimization, the parameters of CRESH are estimable from a system of log-linear equations, each containing at most three independent variables. The CES function, as well as its limiting forms (the Cobb-Douglas (σ = 1), Leontief (σ = 0), and linear (σ = ∞) functions) are special cases of CRESH. The Mukerji CRES function has an identical unit-isoquant surface, but it is not homothetic. Appendix A analyzes the Mukerji function. Appendix B derives the (implicit) CRESH cost function.]