Knowledge that Transforms

Duality in nonlinear programming is investigated via the usual Lagrangian function in the absence of assumptions concerning convexity or differentiability of the underlying functions. Equivalent forms of the primal and dual problems are discussed along with relations between the respective optimal values. A theorem is presented which gives a weak sufficient condition for equality of primal and dual optimal values. Geometric and economic implications of these results are explored.

The central idea of the business cycle is of a pervasive cyclical movement of economic indicators. This paper shows that the concept of such simultaneous movements can be given a precise meaning by performing a principal component analysis of spectral density matrices or, with a different shade of meaning, coherence matrices. It suggests also a new method of computing spectral approximations for models that are nonlinear in their variables. The methods are applied to the Klein-Goldberger model for the United States.

[A utility tree is specified yielding a complete set of demand equations that subsumes the linear expenditure system as a special case. In contrast to the Stone-Geary system, it is shown that our S-branch model allows for Hicks-Allen complements, and it does not restrict the own-price elasticity. Moreover, it is not costly in terms of the additional parameters required. Maximum-likelihood estimates of the S-branch system are presented; these are derived by means of the Bard version of the Gauss-Newton algorithm. In this application to food expenditure data in the United States the use of the S-branch system avoids a potential misspecification which would have resulted from the application of the linear expenditure system.]

[The choice of development period and consequent introduction time for a single innovation by an expected profit maximizing firm operating under conditions of rivalrous competition is studied. Factors taken into account by the firm are the increasing cost with compression of the development period, the reduction of profit opportunities with prolongation of the development period, and the probability of rival innovation and imitation which affect the potential rewards available to the firm. Comparisons is made with the timing that would be selected in the absence of rivalry. The effects of intense rivalry are also examined.]

In this note we shall show that we can further restrict the coalitions that are allowed to form and still have the above identity. Let the commodity space be of finite dimension 1, and let a > 0. The result is that an allocation is a Walras allocation if and only if it cannot be blocked by a coalition which is the union of at most 1 + 1 coalitions, each of which has measure and diameter less than a. That a coalition has measure and diameter less than a intuitively means that the coalition consists of relatively few agents, and that the agents in the coalition resemble one another in chosen characteristics, e.g., initial allocation, production possibilities,

[A social decision rule is one that produces a social decision for each configuration of individuals' decisions. Such a rule is representative if it produces a social decision that is the result of repeatedly applying the rule of simple majority decision to decisions obtained by that rule. We give necessary and sufficient conditions for a social decision rule for two alternatives to be representative.]

[It is shown that every voting procedure determines a partition of the set of all voters into separate committees, the latter forming a hierarchical structure.]