Questions: An industrial system involves manufacturing, transportation, and agriculture. The interdependence of the three industries is given by the input-output matrix shown on the right. At what levels must the industries produce to satisfy a demand for manufactured goods, transportation, and agricultural products, respectively, given in the demand matrix to the right? D=[ 200 140 140 ] Production levels of units of manufacturing, units of transportation, and units of agriculture are needed. (Round to the nearest whole number as needed.)

An industrial system involves manufacturing, transportation, and agriculture. The interdependence of the three industries is given by the input-output matrix shown on the right.

At what levels must the industries produce to satisfy a demand for manufactured goods, transportation, and agricultural products, respectively, given in the demand matrix to the right?

D=[
200
140
140
]

Production levels of units of manufacturing, units of transportation, and units of agriculture are needed.
(Round to the nearest whole number as needed.)

Transcript text: An industrial system involves manufacturing, transportation, and agriculture. The interdependence of the three industries is given by the input-output matrix shown on the right. At what levels must the industries produce to satisfy a demand for manufactured goods, transportation, and agricultural products, respectively, given in the demand matrix to the right? \[ D=\left[\begin{array}{l} 200 \\ 140 \\ 140 \end{array}\right] \] Production levels of $\square$ units of manufacturing, $\square$ units of transportation, and $\square$ units of agriculture are needed. (Round to the nearest whole number as needed.)

Solution

Solution Steps

To solve this problem, we need to determine the production levels of each industry to meet the given demand. This can be done using the Leontief input-output model. The model is represented by the equation $ (I - A)X = D $, where $ I $ is the identity matrix, $ A $ is the input-output matrix, $ X $ is the production vector, and $ D $ is the demand vector. We need to solve for $ X $.

Solution Approach

Define the input-output matrix $ A $.
Define the demand vector $ D $.
Compute the identity matrix $ I $.
Calculate $ (I - A) $.
Solve the equation $ (I - A)X = D $ for $ X $.

Step 1: Define the Input-Output Matrix and Demand Vector

The input-output matrix $ A $ and the demand vector $ D $ are defined as follows: \[ A = \begin{bmatrix} 0.5 & 0.2 & 0.1 \\ 0.1 & 0.4 & 0.2 \\ 0.2 & 0.1 & 0.3 \end{bmatrix}, \quad D = \begin{bmatrix} 200 \\ 140 \\ 140 \end{bmatrix} \]

Step 2: Compute the Identity Matrix

The identity matrix $ I $ of size 3 is given by: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

Step 3: Calculate $ I - A $

We compute $ I - A $: \[ I - A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} - \begin{bmatrix} 0.5 & 0.2 & 0.1 \\ 0.1 & 0.4 & 0.2 \\ 0.2 & 0.1 & 0.3 \end{bmatrix} = \begin{bmatrix} 0.5 & -0.2 & -0.1 \\ -0.1 & 0.6 & -0.2 \\ -0.2 & -0.1 & 0.7 \end{bmatrix} \]

Step 4: Solve for Production Levels $ X $

We solve the equation $ (I - A)X = D $ for $ X $: \[ X = \begin{bmatrix} 696.9697 \\ 506.6667 \\ 471.5152 \end{bmatrix} \]

Step 5: Round the Production Levels

Rounding the values to the nearest whole number, we get: \[ X_{\text{rounded}} = \begin{bmatrix} 697 \\ 507 \\ 472 \end{bmatrix} \]