To find the output for each sector $X$ in this economy, we can use the formula:

$X = (I - M)^{-1} \cdot D$

where:

• $(I - M)^{-1}$ is the inverse matrix provided,
• $D$ is the demand matrix.

Given Data:

1. The inverse matrix $(I - M)^{-1}$ is: 4.4 & 2.4 & 0.8 \\ 1.6 & 3.6 & 1.2 \\ 1 & 1 & 2 \\ \end{bmatrix}$$
2. The demand matrix $D$ is: 50 \\ 38 \\ 32 \\ \end{bmatrix}$$

Solution:

To find $X$, multiply $(I - M)^{-1}$ by $D$: $X = \begin{bmatrix} 4.4 & 2.4 & 0.8 \\ 1.6 & 3.6 & 1.2 \\ 1 & 1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 50 \\ 38 \\ 32 \end{bmatrix}$

Carrying out this matrix multiplication:

Calculations

1. For $X_A$: $X_A = (4.4 \times 50) + (2.4 \times 38) + (0.8 \times 32)$

2. For $X_B$: $X_B = (1.6 \times 50) + (3.6 \times 38) + (1.2 \times 32)$

3. For $X_E$: $X_E = (1 \times 50) + (1 \times 38) + (2 \times 32)$

Let's compute these values:

1. $X_A = 220 + 91.2 + 25.6 = 336.8$
2. $X_B = 80 + 136.8 + 38.4 = 255.2$
3. $X_E = 50 + 38 + 64 = 152$

Final Answer:

$X = \begin{bmatrix} 336.8 \\ 255.2 \\ 152 \end{bmatrix}$

Thus, the output for each sector is approximately:

• $X_A \approx 336.8$ billion dollars
• $X_B \approx 255.2$ billion dollars
• $X_E \approx 152$ billion dollars

Would you like additional clarification on any step?

Related Questions

1. What would the output be if the demand increased by 10% in each sector?
2. How would the outputs change if the technology matrix $M$ were slightly altered?
3. What is the significance of each element in the inverse matrix $(I - M)^{-1}$?
4. How would the results differ if there was an additional sector introduced?
5. What does each sector's output indicate about its dependency on demand?

Tip:

In economic models, the inverse matrix $(I - M)^{-1}$ can reveal how changes in demand impact total output in each sector.