Marx-Sraffa均衡と固有値問題 : Moore-Penrose擬似逆行列の応用

抄録

In this paper, we inherited Hua LooKeng's problem consciousness, and we investigate the stability of price equilibria of Marx-Sraffa joint-production system, where equilibrium is defined in terms of equalities with rectangular matrices, from the angle of eigensystems. We transform the problem of searching equilibiria of Marx-Sraffa joint-production systems into eigenvalue problems by means of Moore-Penrose inverses. First, we give an overview of Sraffa-Okishio-Nakatani (SON) reduction from the angle of nonsingular transformation. SON dealt with a limitted joint production system in which only aged fixed capital is produced jointly. In SON, elimination of terms related to aged fixed capital leads to the determination of profit rates and prices of brandnew commodities. It is shown that this elimination in SON gives a decomposition of coefficient matrix into three parts; the first one concerns only brandnew commodities, the second aged fixed capital to brandnew fixed capital, and the third part zeros. Second, it is shown that the second part of the above decomposition is exactly the companion matrices in renewal dynamics of fixed capital discussed by Yamada-Yamada. Thirdly, by assuming the rank condition on input and output matrices, we demonstrate that price equilibria of general joint production system of the Marx-Sraffa style can be transformed into an equivalent eigenvalue problem by means of MPinverses. We give a numerical example after the theory. Since the resulting square matrices of the eigensystem are no longer necessarily nonnegative, the dominant eigenvalue is not the one corresponding to equilibria. In this sense, one cannot assert that price equilibrium is stable. In SON, price equilibrium is dynamically stable, because the relative prices of aged fixed capital are determined separated from the determination of profit rates. The rate of depreciation which depends on profit rates is applied therein. In this sense, the accounting system of amortisation is an stabilizing effect in the economy.