Question 9.20: Given the input/output table for the three-sector economy:...

Step 1: Use the underlying assumption total input=total output to complete the input/output table

Step 2: Calculate the matrix of technical coefficients, A, by dividing each column of inputs by total input.

Step 3: Get the inverse of the matrix (I − A), since this inverse matrix is required in the equation X = (I – A)^{–1} d:

(I − A) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}  – \left\lgroup\begin{matrix} \frac{150}{600} & \frac{225}{900} & \frac{125}{300} \\ \frac{210}{600} & \frac{250}{900} & \frac{140}{300} \\ \frac{170}{600} & \frac{0}{900} & \frac{30}{300} \end{matrix} \right\rgroup  = \begin{pmatrix} 0.75 & −0.25 & −0.42 \\ −0.35 & 0.72 & −0.47 \\ −0.28 & 0.00 & 0.90 \end{pmatrix}

To calculate the inverse of (I –A), (i) use the elimination method, (ii) use the cofactor method. Set out the table of cofactors:

The inverse of (I – A) = C^T/|I – A|

= \frac{1}{0.289 678} \left(\begin{matrix} 0.648 & 0.4466 & 0.2016 \\ 0.225 & 0.5574 & 0.070 \\ 0.4199 & 0.4995 & 0.4525 \end{matrix} \right)^T

= \frac{1}{0.289 678} \left(\begin{matrix} 0.648 & 0.255 & 0.4199 \\ 0.4466 & 0.5574 & 0.4995 \\ 0.2016 & 0.070 & 0.4525 \end{matrix} \right)

Step 4: Finally, state the column of new external demands, d, and solve for X, by equation (9.34):

X = (I − A)^{−1} d