Abstract
The Hawkins-Simon condition and the Perron-Frobenius theorem constitute the essence of the theory of production in Marxian economics. Actually, the two are not found in standard textbooks of mathematics. However, in textbooks and papers of Marxian economics, they are often treated as propositions already proven. In order to understand the proof, it is necessary to refer to textbooks on economic mathematics, which are quite challenging for many students of economics. Moreover, even if one intends to follow only the properties of the Hawkins-Simon condition and the proof of the Perron-Frobenius theorem, many textbooks require thorough reading.
As a result, despite their high importance in Marxian economics, the Hawkins-Simon condition and the Perron-Frobenius theorem are difficult to learn. This paper explains and proves the two, assuming only very basic knowledge of linear algebra at the university level. In Section 1 (authored by Nagahara), mathematical discussions are developed, which constitute the main contributing part of this paper as mentioned above.
Section 2 (authored by Ehara) describes the economic significance of the Hawkins-Simon condition and the Perron-Frobenius theorem, thereby reinforcing the contributions of Section 1 from the perspective of Marxian economics. Furthermore, it provides explanations of the economic properties of the Hawkins-Simon conditions and the proofs of “Fundamental Marxian Theorem” and the Perron-Frobenius theorem in the general case of the two sector model.
