Math Problem Statement

An economy is based on three sectors: agriculture, manufacturing, and energy. The production of a dollar's worth of each sector requires inputs from the other two. The goal is to find the output for each sector that satisfies a final demand of $67 billion for agriculture, $26 billion for manufacturing, and $38 billion for energy.

Solution

To solve this problem, we need to set up a system of linear equations based on the input-output coefficients provided. The economy is divided into three sectors: agriculture (A), manufacturing (M), and energy (E). The goal is to find the total output of each sector that satisfies a given final demand.

Given:

  • The final demands are dA=67d_A = 67 billion dollars, dM=26d_M = 26 billion dollars, and dE=38d_E = 38 billion dollars.
  • The production coefficients are:
    • Agriculture: 0.300.30 from agriculture, 0.300.30 from manufacturing, 0.300.30 from energy.
    • Manufacturing: 0.300.30 from agriculture, 0.300.30 from manufacturing, 0.300.30 from energy.
    • Energy: 0.300.30 from agriculture, 0.400.40 from manufacturing, 0.300.30 from energy.

Let's denote:

  • xAx_A: Total output of agriculture.
  • xMx_M: Total output of manufacturing.
  • xEx_E: Total output of energy.

The system of linear equations representing the total production required for each sector to meet the final demand is:

x_A = 0.30x_A + 0.30x_M + 0.30x_E + d_A \\ x_M = 0.30x_A + 0.30x_M + 0.30x_E + d_M \\ x_E = 0.30x_A + 0.40x_M + 0.30x_E + d_E \end{cases}$$ Now, let's express this system in a more convenient matrix form $$(I - A)x = d$$, where $$A$$ is the matrix of coefficients and $$d$$ is the vector of demands. We can solve this using matrix operations to find the missing value for the energy sector's output. Let's calculate this next.The outputs for each sector are as follows: - The output of the agricultural sector is approximately **639.571** billion dollars. - The output of the manufacturing sector is approximately **598.571** billion dollars. - The output of the energy sector is approximately **670.429** billion dollars. Would you like more details on the solution process, or do you have any questions? Here are some related questions for further exploration: 1. How do changes in the input coefficients affect the total outputs? 2. What happens to the total output if the final demand increases by 10% in each sector? 3. How would introducing a fourth sector, like technology, impact the current system? 4. Can we determine if this economic system is sustainable in the long run? 5. How can we interpret the impact of each sector's interdependence on the overall economy? **Tip:** Understanding input-output models can help analyze economic dependencies and the impact of changes in one sector on others.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Input-Output Models
Systems of Linear Equations

Formulas

x_A = 0.30x_A + 0.30x_M + 0.30x_E + d_A
x_M = 0.30x_A + 0.30x_M + 0.30x_E + d_M
x_E = 0.30x_A + 0.40x_M + 0.30x_E + d_E

Theorems

Matrix Equation Form I-Ax=d

Suitable Grade Level

Grades 10-12

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