Question

Exercises 1-4 refer to an economy that is divided into three sectors - manufacturing, agriculture, and services. For each unit of output, manufacturing requires .10 unit from other companies in that sector, .30 unit from services. For each unit of output, agriculture uses .20 unit of its own output, .60 unit from manufacturing, and .10 unit from services. For each unit of output, the services sector consumes .10 unit from services, .60 unit from manufacturing, but no agricultural products.

1. Construct the consumption matrix for this economy, and determine what intermediate demands are created if agriculture plans to produce 100 units.

Short answer

The intermediate demand is \(Cx = \left[ {\begin{array}{*{20}{c}}{60}\\{20}\\{10}\end{array}} \right]\).

Step-by-step solution

Construct the consumption matrix for the economy

The important point is that each column represents the unit consumption vector for the corresponding vector. When you order sector manufacturing, agriculture, and services, the consumption matrix is

\(C = \left[ {\begin{array}{*{20}{c}}{.10}&{.60}&{.60}\\{.30}&{.20}&0\\{.30}&{.10}&{.10}\end{array}} \right]\).

Determine the intermediate demand created

The total intermediate demand from all three sectors is

\(\begin{array}{c}\left\{ {intermediate demand} \right\} = {x_1}{c_1} + {x_2}{c_2} + {x_3}{c_3}\\ = Cx.\end{array}\)

Here, C is theconsumption matrix\(\left[ {\begin{array}{*{20}{c}}{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\).

\(Cx\)represents the intermediate demands produced by the production vector x. If agriculture plans to produce 100 units, then the intermediate vector is

\(\begin{array}{c}Cx = \left[ {\begin{array}{*{20}{c}}{.10}&{.60}&{.60}\\{.30}&{.20}&0\\{.30}&{.10}&{.10}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}0\\{100}\\0\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0.1 \times 0 + 0.6 \times 100 + 0.6 \times 0}\\{0.3 \times 0 + 0.2 \times 100 + 0 \times 0}\\{0.3 \times 0 + 0.1 \times 100 + 0.1 \times 0}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0 + 60 + 0}\\{0 + 20 + 0}\\{0 + 10 + 0}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{60}\\{20}\\{10}\end{array}} \right].\end{array}\)

Thus, the intermediate demand is \(Cx = \left[ {\begin{array}{*{20}{c}}{60}\\{20}\\{10}\end{array}} \right]\).