Math Problem Statement

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 $17 billion for agriculture, $62 billion for manufacturing, and $64 billion for energy.

Solution

The problem here involves solving for the economic output in a Leontief input-output model with three sectors: agriculture, manufacturing, and energy. Each sector’s output depends on inputs from the others as described.

Step-by-Step Solution

Let:

  • xax_a be the output for agriculture,
  • xmx_m be the output for manufacturing, and
  • xex_e be the output for energy.

The system of equations can be set up using the information about each sector's required inputs and the final demand:

  1. Agriculture sector requires: xa=0.3xa+0.3xm+0.3xe+17x_a = 0.3x_a + 0.3x_m + 0.3x_e + 17
  2. Manufacturing sector requires: xm=0.3xa+0.3xm+0.3xe+62x_m = 0.3x_a + 0.3x_m + 0.3x_e + 62
  3. Energy sector requires: xe=0.3xa+0.4xm+0.3xe+64x_e = 0.3x_a + 0.4x_m + 0.3x_e + 64

Converting to Standard Form

Rewrite these equations to isolate terms:

  1. xa−0.3xa−0.3xm−0.3xe=17x_a - 0.3x_a - 0.3x_m - 0.3x_e = 17 0.7xa−0.3xm−0.3xe=170.7x_a - 0.3x_m - 0.3x_e = 17

  2. −0.3xa+0.7xm−0.3xe=62-0.3x_a + 0.7x_m - 0.3x_e = 62

  3. −0.3xa−0.4xm+0.7xe=64-0.3x_a - 0.4x_m + 0.7x_e = 64

Now, we have a system of linear equations:

0.7x_a - 0.3x_m - 0.3x_e = 17 \\ -0.3x_a + 0.7x_m - 0.3x_e = 62 \\ -0.3x_a - 0.4x_m + 0.7x_e = 64 \\ \end{cases}$$ ### Solving the System I will solve these equations to find the output for each sector.The solution for the output required for each sector is as follows: - The output of the agricultural sector ($$x_a$$) is approximately **656.429 billion dollars**. - The output of the manufacturing sector ($$x_m$$) is approximately **701.429 billion dollars**. - The output of the energy sector ($$x_e$$) is approximately **773.571 billion dollars**. Would you like a breakdown of each step in more detail, or do you have any other questions? Here are some related questions for further exploration: 1. How would the output change if the final demand for one sector increases? 2. What effect would altering the input coefficients have on the economy? 3. Can this model apply to an economy with more than three sectors? 4. What assumptions are made in the Leontief input-output model? 5. How does interdependence among sectors influence overall economic stability? **Tip:** When working with economic models, pay close attention to rounding conventions, as they can significantly impact large-scale results.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Input-Output Model
Systems of Linear Equations

Formulas

x_a = 0.3x_a + 0.3x_m + 0.3x_e + 17
x_m = 0.3x_a + 0.3x_m + 0.3x_e + 62
x_e = 0.3x_a + 0.4x_m + 0.3x_e + 64

Theorems

Leontief Input-Output Model

Suitable Grade Level

Grades 11-12

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