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. <xref ref-type="bibr" onclick="findReferencesById(event)" class="bli2011" rid="bli2011">Li and Fujimori (2011)</xref> and <xref ref-type="bibr" onclick="findReferencesById(event)" class="bli2017" rid="bli2017">Li (2017)</xref> 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 <i>pB</i>=<i>αpM</i> and activity equilibrium <i>Bx</i>=<i>βMx</i>. They also proved that the solution of the equilibrium price system is equivalent to the characteristic system of the matrix <i>MB</i>⁺. 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.