This paper presents the necessary and sufficient conditions for determining the signs of the solution variables of a system of linear equations based only upon a knowledge of the signs of the coefficient matrix and the signs of the right hand side variables. This problem was initially formulated in economics due to the idea that the signs of an equation's derivatives might have a stronger empirical basis than that of a particular functional form. A new interest in qualitative problems has arisen in connection with the need to develop analytic measures in order to better manage the understanding and use of large, computer-based mathematical systems. The conditions for the qualitative determinancy of nonhomogeneous systems are developed in terms of a small number of necessary conditions which are jointly sufficient. Algorithmic approaches are given for testing a given system for qualitative determinancy. For nonhomogeneous systems algorithms are given for constructing all possible qualitatively determinate systems of a given size. For the homogeneous case conditions are also given for the qualitative invertibility of the (irreducible) coefficient matrix. These conditions are then related to the problem of partially qualitatively determinate systems and the signs in the qualitative inverse of a matrix.

Gorman [2] has derived necessary and sufficient conditions for the existence of category expenditure functions which yield the optimal allocation of a consumer unit's income to each of a number of groups of commodities as functions of total income and group price indices. These conditions take the form of certain restrictions on the structure of the utility function. Gorman, however, did not address the problem of how these functions are derived. In this paper, we construct an algorithm (a budgeting procedure) for deriving the category expenditure functions and show that the necessary and sufficient condition for this procedure to be consistent is that the utility function be separable into homothetic parts. CASUAL OBSERVATION REVEALS that many consumers budget; that is, they first allocate their total expenditure among broad commodity categories and then decide upon the precise allocation of category expenditure to each of the commodities within the group. This type of consumer behavior is especially interesting if it is possible to carry out the broad category allocation with reference only to price indices for each of the budgeting categories, and then decide upon the intracategory allocation with reference only to commodity prices within that group. R. H. Strotz [4, 5] and W. M. Gorman [2] have examined the relationship between this type of consumer behavior and the form of the consumer unit's utility function. More precisely, they have shown that the necessary and sufficient conditions for the existence of group price indices (which depend only upon commodity prices within the group), such that category expenditures are functions only of these price indices and total expenditure, are that the utility function be (a) homothetically separable2 or (b) strongly (additively) separable (with a certain restriction on the polar form of the utility function).3 Strotz and Gorman do not address the issue of how these functions could be derived. This paper discusses a method of deriving the category expenditure functions-a budgeting procedure-which requires stronger constraints on the utility function than does the mere existence of the functions. Price indices are first derived for each budgeting category. Then the