Takao Fujimoto

Two characterizations of inverse-positive matrices: the Hawkins-Simon condition and the Le Chatelier-Braun principle.

Dedicated to the late Professors David Hawkins and Hukukane Nikaido Abstract. It is shown that (a weak version of) the Hawkins-Simon condition is satisfied by any real square matrix which is inverse-positive after a suitable permutation of columns or rows. One more characterization of inverse-positive matrices is given concerning the Le Chatelier-Braun principle. The proofs are all simple and elementary.

Asymptotic Properties for Inhomogeneous Iterations of Nonlinear Operators

The theorems on weak and strong ergodicity for inhomogeneous products of nonnegative matrices are extended to inhomogeneous iterations of nonlinear positive operators on Euclidean space. In particular some concave version of the Coale–Lopez theorem is presented and applied to a density-dependent Leslie model. The results are obtained, via Hilbert’s projective pseudometric, from general theorems on inhomogeneous iterations of operators mapping a metric space into itself.

Non-linear leontief models in abstract spaces

Abstract This paper is to elaborate, using a functional analysis approach, the mathematical aspects of scale-dependent input-output models developed by Evans, Sandberg, Lahiri and Chander. All the basic results are established in abstract spaces of any dimension and in the case of existence theorems we do not require the continuity of the input function. A further generalization is made to a model with alternative processes. An application is also made to a model with indivisible commodities.

The Frobenius theorem, its Solow-Samuelson extension and the Kuhn-Tucker theorem

Abstract In economics, the Frobenius theorem and its extension are often used in the theory of linear models and the theory of balanced growth. In these contexts optimization does not appear explicitly. Our proof by the Kuhn-Tucker theorem may lead to some interesting economic reinterpretations of the Frobenius root and eigenvectors in these models. As for economic applications, see, for example, Morishima (1964); the extended bibliography of Debreu and Herstein (1953) is also useful.

Commodity Content in a General Input–Output Model

This technical paper provides a definition of ‘commodity content’ in extended von Neumann–Morishima models of production. Our definition admits as special cases the classical definition of ‘labour values’ in Leontief models, as well as Morishimas definition of ‘optimal values’ in a von Neumann model. Proper joint production, heterogeneous labour, durable consumption goods, household activities and disposal processes are all allowed for. We also propose a tentative mathematical criterion for defining bads and for distinguishing between unskilled and skilled labour.

Nonlinear generalization of the Frobenius theorem: A symmetric approach

Abstract A further generalization of the Samuelson-Solows results for the Frobenius theorem is presented. We ask the existence of a semi-positive vector x with a non-negative scalar λ for an equation F ( x ; λ )=0, where F is a vector function from R n into itself with the parameter value λ. The nature of our extension as well as economic interpretation of our assumptions is also discussed in the paper.

Strong ergodicity for strictly increasing nonlinear operators

Abstract As a further generalization of the Perron-Frobenius theorem from linear to nonlinear operators, we prove uniqueness of the solution as well as ergodicity for nonlinear operators which are strictly increasing and weakly homogeneous on a certain subset of the Euclidean space. We also discuss higher order difference equations which involve this type of operators.

A sensitivity analysis for linear systems involving M-matrices and its application to the Leontif model

A sensitivity analysis is made for solutions to linear equation systems involving M-matrices. We present a theorem which tells about relative changes of elements of the solution vector when the coefficients of a given M-matrix shift. The Metzler theorem and the Morishima theorem are generalized, and applied to the Leontief model.

A Refutation of the Commodity Exploitation Theorem

This note is to show that the generalized commodity exploitation theorem put forward by Bowles and Gintis and Roemer is nothing but an alternative form of the Hawkins-Simon condition for given technical data, and that it has nothing to do with exploitation. That is, the Hawkins-Simon condition means a mere possibility of an economic system to produce a surplus in each commodity, and as such does not guarantee the existence of positive profits. To consider exploitation or the existence of positive profits, we need to introduce prices at which unequal exchanges may be carried out. Copyright

Nonlinear Complementarity Problems in a Function Space

A class of nonlinear complementarity problems defined by compact operators is considered in the space of continuous functions on a compact Hausdorff space, and an existence theorem is established. The method of proof is based on Schauders fixed-point theorem. Related topics such as a least element and indifferent opimization problems are also discussed.