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

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

If the final demands from each sector are changed to 500 from agriculture, 550 from industry, 300 from financial services, calculate the total output from each sector.

 Input to Agric. Industry Services Other demands Total output Output from Agric. → 150 225 125 100 600 Industry → 210 250 140 300 900 Services → 170 0 30 100 300
Step-by-Step
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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 = $(I − A)^{−1} d$

$\left(\begin{matrix} T_{agri.} \\ T_{ind.}\\ T_{serv.} \end{matrix} \right) = \frac{1}{0.289 678} \left(\begin{matrix} 0.648 & 0.255 & 0.4199 \\ 0.4466 & 0.5574 & 0.4995 \\ 0.2016 & 0.070 & 0.4525 \end{matrix} \right) \left(\begin{matrix} 500 \\ 550\\ 300 \end{matrix} \right) = \left(\begin{matrix} 1980.5 \\ 2346.5\\ 949.5 \end{matrix} \right)$
 Input to Agric. Industry Services Other demands Total output Output from Agric.→ 150 225 125 100 600 Industry→ 210 250 140 300 900 Services→ 170 0 30 100 300 Other inputs 70 425 5 Total input 600 900 300

 Input to Agric. Industry Services Other demands Total output Output from Agric.→ 150/600 225/900 125/300 100 600 Industry→ 210/600 250/900 140/300 300 900 Services→ 170/600 0/900 30/300 100 300 Total input 600/600 900/900 300/300

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