I was interested to read Meyer and Kim [5], where I learned a little about sugar futures, but regret to say that I found the attempted Box-Jenkins analysis singularly lacking in expertise. It is my intention, here, to discuss a few of its most obvious shortcomings. My list will not be exhaustive, but will include just five points.

This paper presents results from an unusual microdata set assembled by the authors and researchers at the Social Security Administration. The data set pools information from three sources: death certificates for residents of Washington, D.C. dying in 1967; Washington, D.C. estate tax returns; and Social Security earnings records. Under an arrangement worked out by Smith with the city of Washington and the National Center for Health Statistics, all (about 2,500) estate tax returns for 1967 decedents were matched with their death certificates. The match provided information on age, sex, race, place of birth, marital status, cause of death and assets and liabilities. Washington, D.C. has its own estate tax, which unlike the federal estate tax, starts at a very low ($1,000) filing level. A full description of this part of the data base and an estate multiplier estimate of the distribution of wealth in Washington, D.C. has been published elsewhere (Smith). This year, thanks to our colleagues, Frederick Scheuren and Wendy Alvey of the Social Security Administration, a procedure was worked out which permitted us to turn over to them our files and to obtain from them analytical results from matched records from our files and their records of covered earnings under the Social Security Act. The intended use of this data base is to estimate a lifetime savings model with earnings as a key determinant. We still may be able to do so, but the prospects look rather grim. In the spirit that science is advanced by knowing what doesn't work as well as wuiat does, we present below a few initial findings which show some promise and of a lot of statistical husbandry which bore little fruit. We shall proceed by first looking at differences in the levels of covered income reported by black and white workers, then at the wealth levels of blacks and whites, and finally at an attempt to predict the wealth of black and white workers using demographic variables and earnings records.

I. Monopoly versus competition, under perfect information, 420.—II. The competitive production of public goods, 425.—III. Afterword, 428.—Appendix: The publicness of the characteristics of products, 428.

This study introduces ordinally additive, ordinally linear, and ordinally Cobb-Douglas utility functions for the analysis of risky decisions when the uncertainty affects several attributes. Practical algorithms for the determination of utility functions with these forms are provided. Further, the study offers several risk invariance axioms on choice behavior under multidimensional risk. These axioms, for the first time, extend to the multidimensional context the heuristic correspondence between risk aversion and subjective wealth, heretofore familiar in only one dimension. In addition, the consequences of these new risk invariance axioms for utility functional forms in the multi-dimensional context are investigated. The result is a sequence of theorems which show that ordinally linear, ordinally Cobb-Douglas, and ordinally additive von Neumann-Morgenstern utility functions are characterized by the risk invariance axioms. IN THE STUDY OF DECISION MAKING under uncertainty, it is well known that plausible sets of axioms imply that the decision maker acts as if he maximizes his expected von Neumann-Morgenstern utility. (See [1], for example.) If the uncertain outcomes are multidimensional, then the appropriate utility concept is a function of many variables. This is the case, for example, for a firm choosing marketing policies which will affect sales and profits, for an individual faced with investment choices which will affect consumption during several years, and for a government deciding among projects which differ in their costs, outputs, and environmental impacts. In grappling with such problems, decision analysts have found it impossibly difficult to proceed with utility measured by a general function of the outcome variables. Instead, they have used multi-attribute utility functions with special forms, and found that their conclusions are sensitive to the particular form utilized. (See [13], for example.) Thus, it falls to theorists to develop testable hypotheses about risky choice which are equivalent to special (and, hopefully, convenient) functional forms for multiattribute utility. Fishburn [2, 3, and 4], Keeney [5 and 6], and Pollak [9, 10, and 11] have made contributions in this vein. This study introduces ordinally additive von Neumann-Morgenstern utility functions (i.e., those which are a monotonic transformation of a sum of functions, each of one variable) to the literature. I show that they should be well suited to practical decision analysis by presenting algorithms for their use. I propose several risk invariance axioms which plausibly extend to the multidimensional context the intuitive link between risk aversion and wealth in one dimension. These axioms are shown to characterize (in the presence of some other assumptions) ordinally additive, ordinally linear, and ordinally Cobb-Douglas von NeumannMorgenstern utility functions.

[A kernel of a set of alternative actions over which there is a partial order is defined in terms of optimality properties. It is shown to be the same as the generalized efficient set. A variety of theorems such as uniqueness, existence, and composition in terms of other sets are established. Related sets, such as quasi kernels and weak kernels, are also considered.]

This paper explores a new approach to the Nash bargaining problem in which the axiom of symmetry is dropped and it is assumed that the final allocation depends on both the status quo and the threat point. The resulting final allocation, unlike that formalized by Nash, cannot be represented by a simple analytic expression; rather, it leads to a whole class of solutions. Properties of the final allocation are analyzed. It is shown that for every initial allocation there exists a Nash fiber, corresponding to the Nash allocation, that it is possible to determine the sign of the derivatives of the final allocation with respect to changes in the threat point, and that a Slutsky-like equation relates these derivatives to the derivatives with respect to the initial allocation. It is also shown that, under certaiia conditions, as play is repeated the final allocation asymptotically converges to the Nash allocation.