An economy is based on three​ sectors, agriculture,​ manufacturing, and energy. Production of a​ dollar's worth of agriculture requires inputs of ​$0.30 from​ agriculture, $ 0.30 from​ manufacturing, and ​$0.30 from energy. Production of a​ dollar's worth of manufacturing requires inputs of ​$0.30 from​ agriculture, ​$0.30 from​ manufacturing, and ​$0.30 from energy. Production of a​ dollar's worth of energy requires inputs of ​$0.30 from​ agriculture, ​$0.40 from​ manufacturing, and ​$0.30 from energy. Find the output for each sector that is needed to satisfy a final demand of ​$70 billion for​ agriculture, ​$11 billion for​ manufacturing, and ​$28 billion for energy.
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Part 1
The output of the agricultural sector is

enter your response here billion dollars.

To solve this problem, we can set it up as a system of linear equations using input-output analysis. Let's denote the output levels required for each sector by $ x_a $ (agriculture), $ x_m $ (manufacturing), and $ x_e $ (energy). We have the following information:

1. **Agricultural Sector Requirement:** Each dollar of agriculture requires:
   - $0.30 from agriculture
   - $0.30 from manufacturing
   - $0.30 from energy

2. **Manufacturing Sector Requirement:** Each dollar of manufacturing requires:
   - $0.30 from agriculture
   - $0.30 from manufacturing
   - $0.30 from energy

3. **Energy Sector Requirement:** Each dollar of energy requires:
   - $0.30 from agriculture
   - $0.40 from manufacturing
   - $0.30 from energy

4. **Final Demand:** The economy needs:
   - $70 billion of agriculture
   - $11 billion of manufacturing
   - $28 billion of energy

### Step 1: Set Up the System of Equations

The total output needed for each sector must satisfy both the inter-sectoral dependencies and the final demand. The equations are set up as follows:

\[
x_a = 0.3x_a + 0.3x_m + 0.3x_e + 70
\]
\[
x_m = 0.3x_a + 0.3x_m + 0.3x_e + 11
\]
\[
x_e = 0.3x_a + 0.4x_m + 0.3x_e + 28
\]

### Step 2: Rearrange Each Equation

Isolate $ x_a $, $ x_m $, and $ x_e $ on the left side:

1. For agriculture:
   \[
   x_a - 0.3x_a - 0.3x_m - 0.3x_e = 70
   \]
   Simplifying, we get:
   \[
   0.7x_a - 0.3x_m - 0.3x_e = 70
   \]

2. For manufacturing:
   \[
   x_m - 0.3x_a - 0.3x_m - 0.3x_e = 11
   \]
   Simplifying, we get:
   \[
   -0.3x_a + 0.7x_m - 0.3x_e = 11
   \]

3. For energy:
   \[
   x_e - 0.3x_a - 0.4x_m - 0.3x_e = 28
   \]
   Simplifying, we get:
   \[
   -0.3x_a - 0.4x_m + 0.7x_e = 28
   \]

### Step 3: Matrix Representation

We can represent this system of equations as a matrix equation $ AX = B $:

\[
\begin{bmatrix}
0.7 & -0.3 & -0.3 \\
-0.3 & 0.7 & -0.3 \\
-0.3 & -0.4 & 0.7 \\
\end{bmatrix}
\begin{bmatrix}
x_a \\
x_m \\
x_e \\
\end{bmatrix}
=
\begin{bmatrix}
70 \\
11 \\
28 \\
\end{bmatrix}
\]

### Step 4: Solve for $ X $

Solving this system will yield values for $ x_a $, $ x_m $, and $ x_e $. I'll calculate these values for you.

The output needed for each sector to satisfy the given final demand is as follows:

- **Agriculture sector**: approximately **$541.86 billion**
- **Manufacturing sector**: approximately **$482.86 billion**
- **Energy sector**: approximately **$548.14 billion**

Would you like more details on how the calculations were done, or do you have any questions?

Here are some related questions to explore further:

1. How would an increase in demand for one sector affect the output requirements for others?
2. What happens if the input-output coefficients change for a sector?
3. How does this model account for indirect effects between sectors?
4. Can this method be used to analyze other types of economies?
5. How sensitive are the sector outputs to changes in final demand?

**Tip:** Input-output analysis is useful in economic planning as it helps determine the interdependencies among sectors and the impact of final demand on total output.

An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a dollar's worth of agriculture requires inputs of $0.30 from agriculture, $0.30 from manufacturing, and $0.30 from energy. Production of a dollar's worth of manufacturing requires inputs of $0.30 from agriculture, $0.30 from manufacturing, and $0.30 from energy. Production of a dollar's worth of energy requires inputs of $0.30 from agriculture, $0.40 from manufacturing, and $0.30 from energy. Find the output for each sector that is needed to satisfy a final demand of $70 billion for agriculture, $11 billion for manufacturing, and $28 billion for energy.

Explore how to calculate the output required for agriculture, manufacturing, and energy sectors to meet final demands using input-output analysis. This problem involves setting up a system of linear equations and solving with matrix methods, covering key linear algebra concepts. Ideal for college-level economics and mathematics students.

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