[The problem consists in maximizing a concave functional--given as an integral--on a class of functions. A method based on approximating optimal solutions by step functions is suggested; approximations are found by solving concave programming problems of a special type. Some convergence theorems are proved and an estimate of the error in terms of values of the objective functional is given.]
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
The exact sampling distributions of estimators of structural parameters of econometric models are unknown except for a few simple cases. In this situation two alternative approaches towards evaluating finite sample properties of various estimators have been adopted in the literature: (i) Monte Carlo experiments, and (ii) the approach pioneered by Nagar and his students in which the sampling error of an estimator is expressed as the sum of an infinite series of random variables, successive terms of which are of decreasing order of sample size in probability. It is claimed that the small sample properties of the estimator under consideration can be approximated by those of the first few terms of such an infinite series. This paper shows through examples that the Nagar approach can be misleading in the sense that it can yield an estimate for finite sample bias that differs from the true finite sample bias to the same order of sample size. And it can yield estimates of bias which are finite (infinite) while the true bias is infinite (finite). The paper also draws attention to some of the pitfalls to be avoided in studying the properties of an infinite sequence of random variables.
This paper first takes a rather pessimistic look at what has been accomplished in recent years in understanding the price mechanism. It then takes up two points in some detail. First, it is shown that stationary expectations do not ensure the convergence of all equilibrium paths on to a steady state in a neoclassical model with heterogeneous capital goods (an appendix works an example). Secondly, a tatonnement process is outlined and discussed for an economy with constant returns to scale. I AM CONCERNED on this occasion with the performance of the invisible in a number of abstract economies which have been discussed in recent years. My findings are rather pessimistic in the sense that I see no support for the view that any of the traditional methods of response of various agents to changes in their economic environment makes the hand perform as it is often taken to perform. One cannot exclude the possibility that the world behaves a good deal better than the models-but it is the models that lead people to view the economic system as they do. It certainly is hard to find a justification for the great preoccupation of both research and teaching with equilibrium economics unless one is also prepared to believe in, at least, a Marshallian tendency to equilibrium. Of course one of the reasons why so much of our effort is devoted to the study of equilibria is that they are singularly well suited to study. We all know the endless variety of adjustment models, not uncongenial to commonsense, one is capable of constrUcting. No unifying principle, such as maximization, seems available; no elegant separation theorems reduce the mass of ugly differential or difference equations to the splendid order of a chapter in Debreu. To discuss and analyze how the economy works it may be necessary to go and look. The achievements of economic theory in the last two decades are both impressive and in many ways beautiful. But it cannot be denied that there is something scandalous in the spectacle of so many people refining the analyses of economic states which they give no reason to suppose will ever, or have ever, come about. It probably is also dangerous. Equilibrium economics, because of its well known welfare economics implication, is easily convertible into an apologia for existing