Consider the production model ${\bf x} = {\bf C x} + {\bf d}$ for an economy with two sectors, where $$ {\bf C} = \begin{bmatrix} 0.0 & 0.5 \\ 0.6 & 0.2 \end{bmatrix}, \qquad {\bf d} = \begin{bmatrix} 50 \\ 30 \end{bmatrix} $$ Use an inverse matrix to determine the production level necessary to satisfy the final demand.
Answer:
The necessary production level is $$ {\bf x} = \begin{bmatrix} 110 \\ 120 \end{bmatrix} $$
Currently, I am studying for a test and this question and answer were both in the book. However, I can't seem to figure out how they got the answer. I saw an example of a similar problem in Alistair Savage's solutions to Leontief input-output model exercises. For some reason, that method wasn't bringing me to the correct answer. If you could please provide step-by-step instruction on how to do this, it'd be a huge help.
Equation: ${\bf x} = ({\bf I} - {\bf A})^{-1} {\bf d}$
[1 0] [0 .5] [1 -.5] [1.6 1] minus = multiply by ^-1 (inverse) = [0 1] [.6 .2] [-.6 .8] [1.2 2][1.6 1] [50] [110] x(by D) = [1.2 2] [30] [120]Answer: [110, 120]
Thorough Explanation Of Similar Problem Here!