Math Problem Statement
Solution
The problem you're facing involves finding the gross output required to satisfy certain demands in a simple economy consisting of three industries: agriculture, manufacturing, and services. The problem provides a consumption matrix and demands for each sector.
The consumption matrix is given as:
0.1 & 0.2 & 0.2 \\ 0.1 & 0.2 & 0.1 \\ 0.1 & 0.1 & 0.2 \end{pmatrix}$$ The demand vector $$D$$ for the three industries (agriculture, manufacturing, and services) is: $$D = \begin{pmatrix} 190 \\ 120 \\ 170 \end{pmatrix}$$ ### Steps to solve the problem: 1. **Form the identity matrix** $$I$$, which is: $$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ 2. **Construct the matrix $$A$$** where $$A = I - C$$. This matrix accounts for how much each sector needs to produce on its own, minus internal consumption. 3. **Find the inverse** of the matrix $$A$$, i.e., $$A^{-1}$$. 4. **Multiply** the inverse of $$A$$ by the demand vector $$D$$ to find the total output $$X$$, using the equation: $$X = A^{-1} \cdot D$$ Where $$X$$ will give you the gross output required for each sector. ### Let's compute this step by step. I'll compute this for you using Python to find the inverse of the matrix and multiply it by the demand vector.The gross output required to satisfy the consumer demand is approximately: - **323.73 million dollars** worth of agricultural products, - **225.61 million dollars** worth of manufactured products, and - **281.17 million dollars** worth of services. Would you like a detailed explanation of any step? Here are five related questions to deepen your understanding: 1. How is the consumption matrix used to model interdependencies between industries? 2. Why do we subtract the consumption matrix from the identity matrix in this method? 3. What role does the inverse matrix play in calculating the required outputs? 4. How would changing one sector's demand affect the overall output calculation? 5. Can this method be extended to more than three industries? **Tip:** In linear economic models, the inverse matrix helps account for the chain reaction of production needs due to inter-industry dependencies.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Inverse Matrix
Economics Input-Output Model
Formulas
X = A^-1 * D
A = I - C
Theorems
Inverse of a Matrix Theorem
Input-Output Economic Model
Suitable Grade Level
College Level (Linear Algebra or Economics)
Related Recommendation
Gross Output Calculation Using Input-Output Matrix for Agriculture and Industry
Solving Economic Production Level Using Inverse Matrix Method
Calculating Sector Outputs in a Three-Sector Economy Using Matrix Inversion
Solving Input-Output Matrix Equations with Final Demand
Solve Input-Output Matrix Problem Using Leontief Inverse