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

Learn more on how do we answer questions.

**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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