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.]

different concepts of returns to scale. The first, which is most widely used for defining returns to scale,' is the relative increase in output as all input quantities are increased proportionally along a ray through the origin. The second, which is the more relevant concept for micro-economic analysis, is the increase in output relative to costs for variations along the expansion path where input prices are constant and costs are minimized at every output.2 As it turns out, these two concepts yield equal measures for the Elasticity of Scale3 e at every minimum-cost point in the input space. However, the change in E with output is generally different for the two concepts,4 since expansion paths do not coincide with rays, unless the production function is homothetic.5 But it is the change of E along the expansion path-and not along the raywhich determines the shape of the average cost curve (AC), and which is therefore relevant for the analysis of firms and industry behavior under competition.6 These relations are stated and analyzed below. It is shown that assumptions made in the literature7 which are related to the behavior of E along rays, or to the slope of the technically optimal surface (where returns to scale are locally constant), are irrelevantbeing neither necessary nor sufficient for implying classical U-shaped average cost curves. The following definitions and results are used here:

A model of labor force participation, market wages, and labor supply of annual hours and weeks of work is estimated for married males and unmarried females in a sample from the retirement history survey, 1969-75, matched with social security earnings records. Emphasis is placed on the effects of variables associated with retirement and old age and on the estimation of net age profiles purged of biases associated with specific years and cohorts and with the other included variables. The effects of most of the usual variables appear to be consistent with the theory and with other empirical findings for both sexes. In addition, age itself has a strong net autonomous effect on each of the dependent variables due to biological, societal, or institutional factors.

Giora Hanoch, Marjorie Honig, "True" Age Profiles of Earnings: Adjusting for Censoring and for Period and Cohort Effects, The Review of Economics and Statistics, Vol. 67, No. 3 (Aug., 1985), pp. 383-394

Individual decisions about investment may be regarded as choices among alternative probability distributions of net returns. It is assumed that these distributions are completely known and independent of initial wealth positions, and that individuals determine the preferred portfolio of investment in accordance with a given, consistent set of preferences.