Math Problem Statement

Calculate the total output for the Industry and Services sectors based on the Input-Output transaction matrix, given final demands of 270 for Industry and 240 for Services.

Solution

The question provides an Input-Output (IO) transaction table for Central Java Province over five years and asks for the total output for each sector, given specific demand values. The sectors in question are "Industri" (Industry) and "Jasa" (Services), with the target final demands being 270 for Industry and 240 for Services.

We can model this using an Input-Output matrix, where the total output for each sector (XIX_I and XJX_J) is given by the following linear system of equations based on the coefficients from the matrix:

Let XIX_I be the total output for the Industry sector and XJX_J be the total output for the Services sector. The matrix is structured as follows:

0.290909 & 0.26087 \\ 0.2 & 0.23913 \end{pmatrix} \begin{pmatrix} X_I \\ X_J \end{pmatrix} + \begin{pmatrix} Y_I \\ Y_J \end{pmatrix} = \begin{pmatrix} X_I \\ X_J \end{pmatrix}$$ Where $$Y_I = 270$$ and $$Y_J = 240$$ are the final demands for Industry and Services respectively. The equation becomes: $$\begin{aligned} X_I &= 0.290909 X_I + 0.26087 X_J + 270 \\ X_J &= 0.2 X_I + 0.23913 X_J + 240 \end{aligned}$$ To solve this, we can simplify it into a system of linear equations and find $$X_I$$ and $$X_J$$. Let me calculate that for you.The total output for each sector is approximately: - For the Industry sector ($$X_I$$): **550.00** - For the Services sector ($$X_J$$): **460.00** These are the calculated outputs given the demand values of 270 for Industry and 240 for Services. Would you like further details or explanations on how this was computed? Here are 5 follow-up questions that may help expand your understanding: 1. How is the Input-Output matrix used in economic modeling? 2. Why do we consider final demand in calculating total output? 3. What assumptions underlie the use of a static Input-Output model? 4. How could changes in the coefficients affect the total output? 5. Can this model account for external economic shocks or policy changes? **Tip:** In economic models like this, feedback effects between sectors can significantly influence total output, which is why these systems of equations are essential in planning.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Input-Output Models
Economic Modeling

Formulas

X_I = 0.290909 X_I + 0.26087 X_J + 270
X_J = 0.2 X_I + 0.23913 X_J + 240

Theorems

System of Linear Equations

Suitable Grade Level

College level (Economics or Mathematical Modeling)

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