Knowledge that Transforms

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.

The distributions of the LIML and TSLS estimates of the coefficient of an endogenous variable in a single equation can be approximated by asymptotic expansions. This paper relates the expansions in terms of the noncentrality parameter and the sample size going to infinity, the noncentrality parameter going to infinity with the sample size held fixed, and the standard deviation of the disturbance going to zero (small-o). 1. INTRODUCriON RECENTLY, ASYMPTOTIC EXPANSIONS of the distributions of estimates of coefficients of a single equation in a system of simultaneous equations have been made by Anderson [1], Anderson and Sawa [2], Mariano [6 and 7], and Sargan and Mikhail [11]. The expansions have usually been carried out on the basis that the sample size increases and that the effect of the exogenous variables (the noncentrality parameter) increases along with the sample size. In this paper we consider the case of the covariance matrix of the disturbances known and alternatively the case of the sample size fixed. We relate these three cases to the approach of letting the disturbance decrease (the small-o- approach). The estimates treated are two-stage least squares (TSLS) and limited information maximum likelihood (LIML).

[A bargaining situation is described by a set of alternative which are feasible to n individuals when they do cooperate, and an alternative which comes about when they do not cooperate. The paper addresses the question of which cooperative outcome will be chosen. A Nash-type approach is used to prove that, under plausible axioms describing the underlying bargaining process, the individuals must be doing interpersonal comparison of utility. The model and the solution overcome some difficulties recently described by Nydegger and Owen.]

[This paper investigates an operational inconsistency between the constrained priority rationing scheme introduced by Manove in 1973 and the criterion used to determine an "optimal" priority matrix. A truncation of the optimality criterion is suggested in order to resolve this inconsistency and also reduce a potential source of bias against final demand.]

[In recent years, stability analysis has been extended in two directions, which are useful for economic applications. The first direction concerns differential equations with discontinuous right-hand sides. The second direction concerns difference equations with multivalued right-hand sides. The present paper reviews some of these contributions (in Section 5, 6, and the Appendix), brings out their similarities, and illustrates their applications to economic problems. The illustration concerns an economy with both private and public goods (Section 2). It is shown (Section 3) how an efficient allocation for that economy can be reached through a globally stable process, combining a price-guided market allocation of private goods and a quantitative planning procedure for public goods. A discrete version of the planning procedure, using an internally defined variable speed of adjustment, is also shown (Section 4) to be quasi-stable.]

This paper deals with the theoretical development of some aspects of the trend removal problem. The objective is to show the difference between the two most popular trend removal methods: first differences and linear least squares regression. On the one hand, we show that if first differences are used to eliminate a linear trend, the series of residuals would be stationary but would not be white noises as they contain a first lag autocorrelation of -0.50. Furthermore, the spectral density function (SDF) of these residuals relative to that of a white noise series would be exaggerated at the high frequency portion and attenuated at the low frequency portion. On the other hand, we show that the regression residuals from the linear detrending of a random walk series would contain large positive autocorrelations in the first few lags. Relative to that of white noises, the SDF of the regression residuals would be exaggerated at the low frequency portion and attenuated at the high frequency portion.

[A recent paper by D. J. Roberts and A. Postlewaite [1] shows a possible justification for the assumption that participants in a market behave as price-takers when the number of participants becomes large. The present note compares this justification with other approaches to the same question and argues that the various approaches are complementary rather than alternative since, in order to demonstrate the viability of a system with price-taking behavior, all types of feasible deviations from such behavior must be explored. The note introduces a taxonomy for such deviations which may be useful in comparing various approaches and contributions found in the literature.]

[This paper describes a market in which firms vary their quantities of production according to a new adjustment process. Each firm bases its new production entirely upon a knowledge of its own previous productions and profits. It has no knowledge of the payoff functions of the market. Numerical analysis of the process indicates an approach to equilibrium for all initial states. The set of allowed limit points is rigorously characterized, and determined explicitly in the case of two firms. Some exact solutions are found. The process can be regarded as a way of playing a continuous game with a minimum of information.]

This paper examines the durability of capital goods produced under different market structures when tax considerations are included. Since investment tax credit and depreciation allowances are realized by the owner of the durable good, the durability of products produced by an industry which sells its output differs from that of an industry which rents. For each of these two commercial forms we consider both monopolistic and competitive market structure. Potential gains from different forms of regulation are discussed.