Part 1 of 3
A simplified economy involves just three commodity categories - agriculture, manufacturing, and transportation, all in appropriate units. Production of 1 unit of agriculture requires $1 / 2$ unit of manufacturing and $1 / 3$ unit of transportation; production of 1 unit of manufacturing requires $1 / 3$ unit of agriculture and $1 / 3$ unit of transportation; and production of 1 unit of transportation requires $1 / 5$ unit of agriculture and $1 / 3$ unit of manufacturing. If the demand is 607 units of each commodity, how many units of each commodity should be produced?
Find the input-output matrix, $A$, and the demand matrix, $D$, for this economy.
The input-output matrix is
(Type an integer or fraction for each matrix element.)
Answer
\(\boxed{\text{Final Answer: The input-output matrix, } A, \text{ is } \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{5} & \frac{1}{3} & 0 \end{bmatrix} \text{ and the demand matrix, } D, \text{ is } \begin{bmatrix} 607 \\ 607 \\ 607 \end{bmatrix}}\)
Step 1 ：Define the problem in terms of an input-output matrix, $A$, and a demand matrix, $D$. The input-output matrix represents how different sectors within an economy interact with each other. Each row of the matrix represents the inputs required from each sector to produce one unit of output in the sector corresponding to that row. The demand matrix represents the demand for each sector's output.
Step 2 ：Given that production of 1 unit of agriculture requires $1 / 2$ unit of manufacturing and $1 / 3$ unit of transportation; production of 1 unit of manufacturing requires $1 / 3$ unit of agriculture and $1 / 3$ unit of transportation; and production of 1 unit of transportation requires $1 / 5$ unit of agriculture and $1 / 3$ unit of manufacturing, we can construct the input-output matrix, $A$, as follows: \[ A = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{5} & \frac{1}{3} & 0 \end{bmatrix} \]
Step 3 ：Given that the demand is 607 units of each commodity, we can construct the demand matrix, $D$, as follows: \[ D = \begin{bmatrix} 607 \\ 607 \\ 607 \end{bmatrix} \]
Step 4 ：\(\boxed{\text{Final Answer: The input-output matrix, } A, \text{ is } \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{5} & \frac{1}{3} & 0 \end{bmatrix} \text{ and the demand matrix, } D, \text{ is } \begin{bmatrix} 607 \\ 607 \\ 607 \end{bmatrix}}\)
