Application Leontief input-output analysis

In a Leontief model, $M_{ij}$ is input from sector $i$ required per unit output of sector $j$. Total output $x$ satisfies $x=Mx+d$, hence $(I-M)x=d$. Invertibility gives an algebraic solution; nonnegative productive output needs additional conditions. For nonnegative $M$ with spectral radius $\rho(M)<1$, $(I-M)^{-1}=\sum_{k\ge0}M^k$ is nonnegative.

For $M=\begin{bmatrix}0.2&0.1\\0.3&0.2\end{bmatrix}$ and $d=(10,20)^T$, $\det(I-M)=0.61$, giving $x=(1000/61,1900/61)^T\approx(16.3934,31.1475)^T$. Substitution verifies that final demand is exactly $(10,20)^T$.Copy