An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a dollar’s worth of
agriculture requires inputs of Taka 0.20 from agriculture, Taka 0.20 from manufacturing, and Taka 0.20 from
energy. Production of a dollar’s worth of manufacturing requires inputs of Taka 0.40 from agriculture, Taka 0.10
from manufacturing, and Taka 0.10 from energy. Production of a dollar’s worth of energy requires inputs of
Taka 0.30 from agriculture, Taka 0.10 from manufacturing, and Taka 0.10 from energy. Find the output for each
sector that is needed to satisfy a final demand of Taka 10 billion for agriculture, Taka 15 billion for
manufacturing, and Taka 20 billion for energy.

This problem describes an input-output model for an economy, which can be solved using matrix algebra. Here's how the solution can be approached:

### Step-by-Step Solution:

1. **Define the Input-Output System:**
   The problem gives us the input-output coefficients. Let \( x_1 \), \( x_2 \), and \( x_3 \) represent the total outputs of agriculture, manufacturing, and energy, respectively. The inputs for each sector can be described using a matrix.

   The input-output matrix \( A \) is:
   \[
   A = \begin{bmatrix}
   0.20 & 0.20 & 0.20 \\
   0.40 & 0.10 & 0.10 \\
   0.30 & 0.10 & 0.10 \\
   \end{bmatrix}
   \]
   This matrix indicates how much input from each sector is required to produce a unit output of each sector.

2. **Final Demand:**
   The final demand is given as:
   \[
   D = \begin{bmatrix}
   10 \\
   15 \\
   20 \\
   \end{bmatrix}
   \text{ (in billions of Taka)}
   \]

3. **Equilibrium Condition:**
   To satisfy the final demand, the total output \( X \) must satisfy the equation:
   \[
   X = AX + D
   \]
   This can be rearranged to:
   \[
   (I - A)X = D
   \]
   where \( I \) is the identity matrix, and \( (I - A) \) is the Leontief matrix.

4. **Set Up the Leontief Matrix:**
   The identity matrix \( I \) is:
   \[
   I = \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1 \\
   \end{bmatrix}
   \]
   Now, calculate \( I - A \):
   \[
   I - A = \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1 \\
   \end{bmatrix}
   -
   \begin{bmatrix}
   0.20 & 0.20 & 0.20 \\
   0.40 & 0.10 & 0.10 \\
   0.30 & 0.10 & 0.10 \\
   \end{bmatrix}
   = \begin{bmatrix}
   0.80 & -0.20 & -0.20 \\
   -0.40 & 0.90 & -0.10 \\
   -0.30 & -0.10 & 0.90 \\
   \end{bmatrix}
   \]

5. **Solve the System:**
   We now solve the system of equations:
   \[
   (I - A)X = D
   \]
   That is:
   \[
   \begin{bmatrix}
   0.80 & -0.20 & -0.20 \\
   -0.40 & 0.90 & -0.10 \\
   -0.30 & -0.10 & 0.90 \\
   \end{bmatrix}
   \begin{bmatrix}
   x_1 \\
   x_2 \\
   x_3 \\
   \end{bmatrix}
   =
   \begin{bmatrix}
   10 \\
   15 \\
   20 \\
   \end{bmatrix}
   \]
   Solving this system will give us the outputs \( x_1 \), \( x_2 \), and \( x_3 \) for each sector.

Let me solve this system.The outputs for each sector required to meet the final demand are:

- \( x_1 = 30 \) billion Taka for agriculture,
- \( x_2 = 34 \) billion Taka for manufacturing,
- \( x_3 = 36 \) billion Taka for energy.

These values represent the total output needed from each sector to satisfy the final demands of Taka 10 billion for agriculture, Taka 15 billion for manufacturing, and Taka 20 billion for energy.

Would you like more details or have any questions?

### Related Questions:
1. How would changes in the demand for one sector affect the overall outputs?
2. What is the economic interpretation of the Leontief inverse matrix in this model?
3. How would the system change if one sector became more self-reliant (e.g., requiring fewer external inputs)?
4. Can this model be extended to include more sectors or different types of industries?
5. How sensitive is the output to changes in the input coefficients?

### Tip:
In input-output models, small changes in demand or input coefficients can significantly affect the total outputs, especially in highly interconnected economies.

An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a dollar’s worth of agriculture requires inputs of Taka 0.20 from agriculture, Taka 0.20 from manufacturing, and Taka 0.20 from energy. Production of a dollar’s worth of manufacturing requires inputs of Taka 0.40 from agriculture, Taka 0.10 from manufacturing, and Taka 0.10 from energy. Production of a dollar’s worth of energy requires inputs of Taka 0.30 from agriculture, Taka 0.10 from manufacturing, and Taka 0.10 from energy. Find the output for each sector that is needed to satisfy a final demand of Taka 10 billion for agriculture, Taka 15 billion for manufacturing, and Taka 20 billion for energy.

This detailed example explains how to solve a Leontief input-output model for an economy comprising agriculture, manufacturing, and energy sectors. Using matrix algebra, we calculate the required outputs for each sector to meet a final demand of Taka 10 billion for agriculture, Taka 15 billion for manufacturing, and Taka 20 billion for energy. Suitable for college-level students.

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