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2017 | OriginalPaper | Chapter

6. Marx-Sraffa Equilibria as Eigensystems

Author : Bangxi Li

Published in: Linear Theory of Fixed Capital and China’s Economy

Publisher: Springer Singapore

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Abstract

This chapter develops formal discussion of equilibrium of the Marx-Sraffa model from the angle of the eigenvalue and eigenvectors.

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previous chapter Economic Durability and Hardening Effects of Fixed Capital

next chapter Fixed Capital and China’s Economy 1995–2000

As for details of mathematical description, refer to, e.g. Gantmacher (1959), Ben-Israel et al. (2013) and Strang (1976).

Consider an $m\times n$ matrix $G=( G_1, G_2)$ with $\mathop {\mathrm {rank}}\nolimits (G)=m$, where $G_1$ and $G_2$ are of $m\times m$ and of $m\times (n-m)$ respectively. Then, G can be decomposed as $G=X(\varGamma , O)Y$, where $\varGamma $ is an $m\times m$ nonsingular matrix, and X and Y are m- and n-dimensional nonsingular matrices respectively. It is evident that triplets of $X, \, \varGamma $ and Y are chosen from infinite possible combinations. It should be noted that the choice of them is under some restrictions. Let $Y=\begin{pmatrix} Y_{11} &{} Y_{12} \\ Y_{21} &{} Y_{22} \end{pmatrix}$, and one has $Y_{11}=(X\varGamma )^{-1} G_1$ and $Y_{12}=(X\varGamma )^{-1} G_2$. This means that the upper partitions of Y are restricted, whilst the lower partitions, in particular $Y_{21}$, are arbitrary, insofar as Y is nonsingular.

Remark that $\beta $s are reciprocals of eigenvalues of $B^+M$. In arguments of economics, unbounded growth factors are usually disregarded.

$\frac{p_{2}^{2}}{p_{2}^{0}}=(1-\varphi (r, 2))(1-\varphi (r, 3)) = 0.528 \times 0.704=0.372$. For details see Kurz and Salvadori (1995).

In setting up the initial value, the price ratio portion of the fixed capital is calculated by the straight-line depreciation method, and the price ratios of brand new fixed capital of types 1 and 2 are taken as $\frac{{p^{1}}_{3}}{{p^{1}}_{1}}$, with an equilibrium ratio of $p^{1}$. On the other hand, the price ratios of current goods and consumption goods 1 and 2 are taken as $\frac{{p^{1}}_{6}}{{p^{1}}_{1}}$, $\frac{{p^{1}}_{7}}{{p^{1}}_{1}}$, and $\frac{{p^{1}}_{8}}{{p^{1}}_{1}}$, respectively, the equilibrium ratio being $p^{1}$. That is, the initial value is given as follows:

$$\begin{aligned} p(0)= \begin{pmatrix} 1&\frac{1}{2}&1 \cdot \frac{{p^{1}}_{3}}{{p^{1}}_{1}}&\frac{2}{3} \cdot \frac{{p^{1}}_{3}}{{p^{1}}_{1}}&\frac{1}{3} \cdot \frac{{p^{1}}_{3}}{{p^{1}}_{1}}&\frac{{p^{1}}_{6}}{{p^{1}}_{1}}&\frac{{p^{1}}_{7}}{{p^{1}}_{1}}&\frac{{p^{1}}_{8}}{{p^{1}}_{1}} \end{pmatrix}. \end{aligned}$$

Title: Marx-Sraffa Equilibria as Eigensystems
Author: Bangxi Li
Publisher: Springer Singapore
Book: Linear Theory of Fixed Capital and China’s Economy
Print ISBN: 978-981-10-4064-1

Electronic ISBN: 978-981-10-4065-8

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-981-10-4065-8_6