Tjalling C. Koopmans

Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms

Zusammenfassung Die von Fock im Rahmen seiner Naherungsmethode zur Behandlung des quantenmechanischen Mehrelektronenproblems aufgestellten Gleichungen werden auf etwas allgemeinerer Grundlage diskutiert. Es wird angegeben, wie man in eindeutiger Weise den einzelnen Elektronen bestimmte Wellenfunktionen und Eigenwerte zuordnen kann. Diese Eigenfunktionen genugen einer Gleichung, die in einem etwas anderen Zusammenhang von Fock abgeleitet wurde. Die Eigenwerte sind bis auf kleinen Korrektionen den Ablosungsarbeiten der einzelnen Elektronen entgegengesetzt gleich. Das erreichte Ergebnis hat nur Bedeutung in denjenigen Fallen, wo der Ansatz einer einzigen Slaterschen Determinante fur die Wellenfunktion sinnvoll ist.

Stationary Ordinal Utility and Impatience

This paper investigates Bohm-Bawerks idea of a preference for advancing the timing of future satisfactions from a somewhat different point of view. It is shown that simple postulates about the utility function of a consumption program for an infinite future logically imply impatience at least for certain broad classes of programs. The postulates assert continuity, sensitivity, stationarity of the utility function, the absence of intertemporal complementarity, and the existence of a best and a worst program. The more technical parts of the proof are set off in starred sections.

Optimum Utilization of the Transportation System

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.

Objectives, Constraints, and Outcomes in Optimal Growth Models

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.

Additively decomposed quasiconvex functions

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX1, ⋯,Xm.Assume thatf is quasiconvex and is the sum of nonconstant functionsf1, ⋯,fm defined on the respective factor sets. Then everyfi is continuous; with at most one exception every functionfi is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function.We define the convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2, or, equivalently, in terms of the separation of the graphs off1 andf2 by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.

Maximizing Stationary Utility in a Constant Technology

Abstract : This paper is concerned with a problem in the optimal control of a nonstochastic process over time. It can also be looked on as a problem in convex programming in a space of infinite sequences of real numbers. The literature on optimal economic growth contains several papers in which a utility function of the form (1) U(x1,x2,...) = Summation, t=1 to t=infinity, of alpha (superscript(t-1)) u(x sub t), Oalpha1, is maximized under given conditions of technology and population growth. Here xt is per capita consumption in period t, and u(x) is a strictly concave, increasing, single-period utility function. Alpha is called a discount factor. A generalization of (1) has been proposed under the name stationary utility, and is definable by a recursive relation (2) U(x1, x2, x3,...) = V(x1, U(x2, x3,...)). One obtains (1) by V(x, U) = u(x) + alpha U. The natural generalization of alpha in (1) to stationary utility is the function (2a) alpha(x) = (the partial derivative of V(x,U) with respect to U) subscript U = U(x,x,x,...). In this paper we study the maximization of (2) under production assumptions.

Economic Growth at a Maximal Rate

In 1936, John von Neumann published, in an Austrian mathematical periodical little known to economists, a paper (von Neumann, 1937)3 that has greatly influenced economic theory up to the present time, and of which all the ramifications have perhaps not yet become fully apparent.

Examples of Production Relations Based on Microdata

The view has been expressed by many that a meaningful capital theory can and should be developed without ever defining an aggregate capital index. A fine prototype of this approach is Malinvaud’s now classical paper of 1953. The same banner has been unfurled, though not with full identity of views, in Cambridge, England and in Cambridge, Massachusetts.

Concepts of optimality and their uses

Lecture to the memory of Alfred Nobel, December 11, 1975(This abstract was borrowed from another version of this item.)

The Transition from Exhaustible to Renewable or Inexhaustible Resources

Allow me to begin with some simple and rather obvious remarks on the nature of the transition problem from exhaustible to renewable or inexhaustible resource use. First, a shift in resource use means also a shift in technology, because in this age resources go together with technologies that process them and put them to use. Secondly, while I have used the word ‘exhaustible’, the term ‘depletion’ is a more suitable word, in that it suggests a more gradual process. The later stages of depletion will then whenever possible call forth a substitute resource that allows society to meet the same or a similar need to that met by the resource being depleted. Finally, I will follow the model of price as a regulator that will touch off the substitution, smoothly if the degree and rate of depletion are foreseen sufficiently in advance.