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Let's walk through the problem step-by-step.

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### (a) **Leontief Matrix and Input-Output Equation**

The **Leontief model** describes the relationships between industries in an economy. The core idea is that each industry requires inputs from other industries, and the system is represented using **matrix algebra**.

The general **Leontief input-output equation** is:

\[
X = AX + D
\]

Where:
- \(X\) = Output vector (the levels of output we need to find)
- \(A\) = Input-output matrix (with elements \(a_{ij}\) representing the input from industry \(i\) to industry \(j\))
- \(D\) = Final demand vector \(\begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}\)

To simplify the equation, we rearrange it to:

\[
(I - A)X = D
\]

Where:
- \(I\) = Identity matrix (3x3)
- \(I - A\) = Leontief matrix

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### (b) **Solution Output Levels Using Matrix Inversion Method**

To solve for the output levels \(X\), we need to isolate \(X\). The equation becomes:

\[
X = (I - A)^{-1} D
\]

#### Steps:
1. **Construct the Leontief matrix**: \(I - A\)
   \[
   I - A = \begin{bmatrix} 
   1 - a_{11} & -a_{12} & -a_{13} \\ 
   -a_{21} & 1 - a_{22} & -a_{23} \\ 
   -a_{31} & -a_{32} & 1 - a_{33} 
   \end{bmatrix}
   \]

2. **Invert the matrix**: Find \((I - A)^{-1}\).

3. **Multiply** the inverse matrix with the demand vector \(D\) to get the solution for output levels \(X\):

   \[
   X = (I - A)^{-1} \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}
   \]

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### **Summary of Required Outputs**

- **Leontief matrix**: \(I - A\).
- **Equation for output levels**: \(X = (I - A)^{-1} D\).
- Once \(A\) and \(D\) are provided with numerical values, you can calculate the output vector \(X\) using the inverse of \(I - A\).

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Would you like me to demonstrate a numerical example for the matrix inversion? Or do you have further questions about these steps?

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### **5 Follow-up Questions for Deeper Understanding:**

1. What happens if the matrix \((I - A)\) is not invertible?
2. How do the elements \(a_{ij}\) of the input-output matrix reflect interdependencies among industries?
3. Can the Leontief model handle more than 3 industries, and what changes in the matrix equations?
4. What are the economic implications if one of the industries has a very high demand value?
5. How sensitive is the output vector \(X\) to changes in the final demand \(D\)?

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**Tip:** When working with real-world data, ensure the matrix \(I - A\) is invertible by checking if its determinant is non-zero.

If the final demand of three industries is d_1, d_2, and d_3 (in billion dollars), 
(a) Write the Leontief matrix, and the specific input-output matrix equation for this economy.
(b) Find the solution output levels by matrix inverse method.

Learn how to set up and solve an input-output model for an economy with three industries using the Leontief matrix. This problem guides you through forming the matrix equation and solving for output levels using matrix inversion. Suitable for advanced high school or undergraduate students.

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