Price, Activity Equilibrium, and Stability of Marx-Sraffa Joint Production: A Further Proof Based on the Moore-Penrose Pseudo Inverse

In 1984, Hua Luogeng proposed an analytical framework for the global optimal solution under large-scale economic planning. Li and Fujimori (2011) and Li (2017) extended it to the case of joint production containing fixed capital, and used the properties of Moore-Penrose (MP) pseudo inverse to calculate the optimal price equilibrium pB=αpM and activity equilibrium Bx=βMx. They also proved that the solution of the equilibrium price system is equivalent to the characteristic system of the matrix MB⁺. On the basis of the above literature, this thesis proposes a more effective way to calculate equilibrium solutions and extends the classical Perron-Frobenius (PF) theorem under the Marx-Sraffa joint production model. The research results indicate that in the case of joint production, both the price and activity system cannot achieve steady-state equilibrium.