Math Problem Statement

An industrial system has two industries with the following input requirements. (a) To produce $1.00 worth of output, Industry A requires $0.10 of its own product and $0.50 of Industry B's product. (b) To produce $1.00 worth of output, Industry B requires $0.40 of its own product and $0.30 of Industry A's product. Find D, the input-output matrix for this system. A B D = Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. A B

Solve for the output matrix X in the equation X = DX + E, where E is the external demand matrix E = 10,000 20,000 . (Round to the nearest whole number.)

Solution

Let's break down the problem step-by-step.

Step 1: Formulate the Input-Output Matrix DD

Given the input requirements:

  • For Industry A:
    • Requires $0.10 of its own product
    • Requires $0.50 of Industry B's product
  • For Industry B:
    • Requires $0.30 of Industry A's product
    • Requires $0.40 of its own product

The input-output matrix DD can be structured as follows:

0.10 & 0.50 \\ 0.30 & 0.40 \end{bmatrix}$$ Where: - $$D_{11} = 0.10$$ (A requires 10% of its own output) - $$D_{12} = 0.50$$ (A requires 50% of B's output) - $$D_{21} = 0.30$$ (B requires 30% of A's output) - $$D_{22} = 0.40$$ (B requires 40% of its own output) ### Step 2: Set up the Equation The equation to find the output matrix $$X$$ is: $$X = DX + E$$ Where: - $$X$$ is the output vector - $$E$$ is the external demand matrix, given as $$E = \begin{bmatrix} 10000 \\ 20000 \end{bmatrix}$$ ### Step 3: Solve for $$X$$ Rewriting the equation, we get: $$X - DX = E$$ Factor out $$X$$: $$(I - D)X = E$$ Where $$I$$ is the identity matrix: $$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ So, $$I - D$$ is: $$I - D = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 0.10 & 0.50 \\ 0.30 & 0.40 \end{bmatrix} = \begin{bmatrix} 0.90 & -0.50 \\ -0.30 & 0.60 \end{bmatrix}$$ ### Step 4: Calculate $$X$$ To find $$X$$, we need to multiply the inverse of $$I - D$$ by $$E$$: $$X = (I - D)^{-1} E$$ Let's compute this using Python to get the precise numerical values.The output matrix $$X$$ for this system, rounded to the nearest whole number, is: $$X = \begin{bmatrix} 41026 \\ 53846 \end{bmatrix}$$ This means that: - Industry A needs to produce approximately 41,026 units. - Industry B needs to produce approximately 53,846 units. Would you like a more detailed explanation of any part of this process or have any questions? Here are 5 related questions you might find interesting: 1. How would the output change if the external demand $$E$$ were different? 2. What happens if one of the industries becomes more self-sufficient (e.g., Industry A requiring more of its own product)? 3. How does the stability of this economic system depend on the values in the input-output matrix $$D$$? 4. Can this method be extended to more than two industries? 5. How does this model relate to real-world economic systems? **Tip:** In linear algebra problems involving economic models, understanding the structure of matrices and their inverses is crucial for interpreting solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Economic Models

Formulas

X = DX + E
(I - D)X = E
X = (I - D)^{-1} E

Theorems

Inverse Matrix Theorem
Input-Output Economic Model

Suitable Grade Level

Undergraduate Level (Economics, Linear Algebra)

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