This paper discusses an extended version of the Hawkins and Simon condition which constitutes a synthetic formulation of the mathematical properties that viable economies must satisfy in single production models. The new version is implicit in the economic interpretations offered by them of the Hawkins and Simon condition, once a correction is introduced in one of those interpretations. Moreover, the paper details the meaning of the extended version following the interpretation of the original version proposed by Dorfman, Samuelson, and Solow. It also introduces a characteristic property of indecomposable matrices that has not previously been published.
When studying the economies capable of reproducing themselves, it is useful to distinguish between first- and second-class economies. The former ones do not produce a surplus while the latter ones do produce a surplus. Furthermore, both classes may be viable or not, depending on whether they comply or not with certain requirements which, in turn, are not the same for different authors (e.g., Benítez Sánchez and Benítez Sánchez [
Including this introduction, the paper is divided into seven sections. Section
Benítez Sánchez and Benítez Sánchez [
Regarding the economic interpretation of HS, it is important to distinguish the two approaches introduced, respectively, by Dorfman et al. [
The first part of this section presents the model of a single production economy and the second one presents the definition of viable techniques.
The reference economy is integrated by
A square matrix
In first-class economies,
Let
In accordance with this, Definition
This section presents two properties of indecomposable matrices and uses one of them to expose EHS.
Let HS: EHS:
Therefore, EHS states that all the principal minors of [
Let In every
The equivalences (i)
The following theorem relates viable economies and EHS.
Let EHS.
(i)
(ii)
It follows from this theorem that a square matrix
This section exposes the economic interpretation of HS proposed by Dorfman et al. [
According to Dorfman et al. ([
An economy is self-sustaining if, for each
To visualize the relation between HS and this definition, it is useful to consider the next equation system that, for any given D-set, can be formulated for each
System (
An economy is self-sustaining if, given any D-set, the quantity of any good
Therefore, an economy is self-sustaining if any set of industries producing a unit of a good consumes in this process, directly and indirectly through the goods produced by the set, less than one unit of the same good. Moreover, it is possible to observe that Definitions
The preceding analyses help us to visualize the following interpretation of EHS.
A technology is viable if any set of industries producing a unit of a good consumes in this process, directly and indirectly through the goods produced by the set, less than one unit of the same good. The only possible exception is the set of all goods, in which case that quantity is equal to one unit for every good.
In a viable economy of the second class, the corresponding coefficient matrix satisfies HS. As this situation has already been treated in the preceding subsection, I will discuss here only the case of a viable economy of the first class. On the one hand, it is important to observe that the analysis developed in the previous subsection is valid for any D-set such that
The first part of this section discusses two economic interpretations of HS offered by Hawkins and Simon ([
The following economic interpretation refers to second-class economies.
The condition that all principal minors must be positive means, in economic terms, that the group of industries corresponding to each minor must be capable of supplying more than its own needs for the group of products produced by this group of industries.
The next economic interpretation refers to first-class economies. We have made some minor changes, adapting the original text to the notation followed here.
Let
Regarding this proposition, it is necessary to consider first the situation when matrix
However, apparently Hawkins and Simon ([
If
Matrix (
The techniques satisfying the conditions indicated in Propositions
This fact has not been previously pointed out, as far as I know. It follows from Proposition
In this regard, it is important to mention that Georgescu-Roegen ([
Given a technique
A technique
Convertibility, thus defined, is reflexive when, for a technique
Let
(i) ⇒ (ii). If
(ii)
Some closely related results may be found in Fisher [
Proposition
A technique is viable if and only if every group of industries is capable of supplying at least its own needs for the group of products produced by this group of industries and, with the possible exception of the group of all the industries, it can supply more of at least one of these goods.
As indicated in Section
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the anonymous referee and the editor for the very useful comments and suggestions.