Journal Article The Impact of Technical Change on the Holotheticity of Production Functions Get access Ryuzo Sato Ryuzo Sato Brown University Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 47, Issue 4, July 1980, Pages 767–776, https://doi.org/10.2307/2296942 Published: 01 July 1980 Article history Received: 01 October 1975 Accepted: 01 January 1980 Published: 01 July 1980

Journal Article Fiscal Policy in a Neo-Classical Growth Model: An Analysis of Time Required for Equilibrating Adjustment Get access Ryuzo Sato Ryuzo Sato Honolulu, Hawaii Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 30, Issue 1, February 1963, Pages 16–23, https://doi.org/10.2307/2296026 Published: 01 February 1963

Recent papers by Houthakker [3 and 4], Samuelson [8 and 9], Sir John Hicks [2], and others deal with the question of the existence of a nontrivial preference ordering which exhibits the mathematical properties in terms of both the direct and indirect utility functions. It is shown that homogeneity and separability are compatible with both the direct and indirect utility functions, but that direct and indirect additivity is consistent with only limited classes of utility functions. Samuelson has raised the question of whether there exists a nontrivial self-dual preference ordering which requires more stringent conditions than homogeneity and separability. By a self-dual preference is meant a preference ordering such that the direct utility function is identical with the corresponding indirect utility function. The purpose of this paper is to present a complete solution to the problem of self-duality. First, we elaborate on the'necessary and sufficient conditions for an exactly or strongly self-dual utility function (in Samuelson's sense). Then, using some well-known concepts of the continuous group theory of transformations, we study the case of weakly self-dual preference orderings and give a precise formulation of the concept of the same mathematical form. Some special classes of self-dual preferences are subjected to detailed analysis. RECENT PAPERS BY Houthakker [3 and 4], Samuelson [8 and 9], Sir John Hicks [2], Pollak [6], and Lau [5] deal with the question of the existence of a nontrivial preference ordering which exhibits the mathematical properties in terms of both the direct and indirect utility functions. Some of the mathematical properties investigated in these works are homogeneity, separability, and additivity. It is shown that homogeneity and separability are compatible with both the direct and indirect utility functions, but that direct and indirect additivity is consistent with only limited classes of utility functions ([2, 8, and 9]). Samuelson has raised the question of whether there exists a nontrivial self-dual preference ordering which may require more stringent conditions than homogeneity and separability [8]. By a self-dual preference is meant a preference ordering such that the direct utility function is identical with the corresponding indirect utility function [8].2 Although partial answers given by Houthakker [4], Pollak [6], and Russell [7] are very illuminating, the solution is far from complete. The purpose of this paper is to present a more complete solution to the problem of self-duality. First, we elaborate on the necessary and sufficient conditions for an exactly self-dual utility function (in Samuelson's sense). Then, using some well-known concepts in the theory of the continuous family of transformations, we study the case of weakly self-dual preference orderings and give a precise formulation of the concept of the same mathematical form. Some special classes of self-dual preference orderings which are convenient for empirical estimation are subjected to detailed analysis.

We are gratified that mention of our paper on Relative Shares has been made in the preceding note by Paul Samuelson. Consistent with his past contributions, Samuelson has provided a simplified treatment of the various concepts of elasticity of substitution. We would like to take this occasion to make some further observations. With regard to Samuelson's statement about the usefulness of the elasticity concepts, first, it is hard to imagine that Samuelson would go so far as to belittle the theoretical contributions of all the studies done in such areas as growth, production functions, and technical progress which have greatly benefited by the use of the elasticity of substitution concept. In these studies the elasticity of substitution concept was not used in a trivial manner. Secondly, and this is more relevant to the type of problems we address, there is the matter of substitutes and complements which is an integral aspect of the multifactor analysis. In problems dealing with multifactor production, dual partial elasticity concepts developed by Allen, Hicks, Sato-Koizumi, etc.1 serve several useful functions. Besides providing a taxonomy of effects that need to be measured, these various concepts provide us with alternative frameworks of conceiving of substantive problems. Although it can be shown that the partial concepts are related to some of the other elasticitv concepts and hence need not be glorified bv fancy names, there do exist particular problems where it is crucial to understand how the total effect is divided into its component effects. For example, in the case where we are dealing with derived demand with some factors (subsitutes or complements) limited in supply, it is important to know exactly how the existence of substitutability or coomplementarity among factors works for or against a particular factor share.

[This paper is an attempt to study optimal savings policy in a world where the growth rate of labor responds to economic factors. This modification makes the form of society's social welfare function important--its elasticity affects "real" economic variables. In addition, a rule for direct population control is investigated.]