[A general definition of majority decision in terms of a hierarchy of voting councils has been given by Murakami [3, 4]. The present article establishes a set of necessary and sufficient conditions for Murakami's majority decision or representative system in terms of properties of a group decision function for two alternatives. One corollary of the general theorem is Murakami's conjecture, which says that if a group decision function is dual, strongly monotonic, and nondictatorial, then it is a representative system.]

A typical proof of the existence of a perfectly competitive market equilibrium employs an appropriately continuous price-to-price mapping that depends on excess demands. If trade tax-subsidy distortions are introduced into the model, the excess demand mappings may have disconnected image sets and destroy the continuity of the price-to-price mapping. This difficulty is overcome by developing a technique which explicitly takes account of the dependence of demand on both income and prices and simultaneously solves for equilibrium prices and income levels for all agents. This technique is then applied to establish two existence theorems for models of international trade with trade tax-subsidy distortions.

[The results of A. K. Sen and P. K. Pattanaik on sufficient conditions for the transitivity of simple majorities are extended to the case where it is not assumed that the individual's indifference relations are transitive.]

This paper surveys the results of mostly recent research on optimal aggregate economic growth models, and comments on the difficulties encountered and on desirable directions of further research.

or vice versa will inevitably yield incomplete if not erroneous results. SUMMARY THE PRESENT investigation is a part of a study of the stock market which is motivated by three basic considerations: (1) Existing demand theory is inadequate for analyzing the problem of speculative prices and thus incapable of providing a valid prediction theory for this type of price mechanism. (2) The volume of sales in a commodity market has an economic significance in its own right and thus deserves much more attention by economists than it has received so far. (3) Prices and volumes of sales in the stock market arejoint products of a single market mechanism, and any model that attempts to isolate prices from volumes or vice versa will inevitably yield incomplete if not erroneous results. While the theoretical model and its implications will be presented elsewhere, the empirical results reported here serve as a basis for the proposed theory. Among the significant results so far found are: (1) A small volume is usually accompanied by a fall in price. (2) A large volume is usually accompanied by a rise in price. (3) A large increase in volume is usually accompanied by either a large rise in price or a large fall in price. (4) A large volume is usually followed by a rise in price. (5) If the volume has been decreasing consecutively for a period of five trading days, then there will be a tendency for the price to fall over the next four trading days.