Finding a Production Vector From a Consumption Matrix and Demand Vector
\[C\]
and a demand vector \[\mathbf{D}\]
we can construct a production vector \[P\]
using the relationship \[(I-C)\mathbf{P}=\mathbf{D} \rightarrow \mathbf{P}=(I-C)^{-1} \mathbf{D}\]
.Example:
\[C=\left( \begin{array}{cc} -1 & 3 \\ 1 & -1 \end{array} \right), \mathbf{D}=\begin{pmatrix}1\\3\end{pmatrix}\]
.\[\begin{equation} \begin{aligned} \mathbf{P}&=(I-C)^{-1} \mathbf{D} \\ &= (\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)-\left( \begin{array}{cc} -1 & 3 \\ 1 & -1 \end{array} \right))^{-1}\begin{pmatrix}1\\3\end{pmatrix} \\ &= \left( \begin{array}{cc} 2 & -3 \\ -1 & 2 \end{array} \right)^{-1} \begin{pmatrix}1\\3\end{pmatrix} \\ &= \left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \begin{pmatrix}1\\3\end{pmatrix} \\ &= \begin{pmatrix}11\\7\end{pmatrix} \end{aligned} \end{equation}\]
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