The production processes for three goods, C1, C2, C3 are interlinked. To produce one dollar's worth
of C1 requires the input of $0.2 worth of C1, $0.4 of C2 and $0.1 of C3. To produce one dollar's
worth of C2 requires $0.3 worth of C1, $0.1 worth of C2 and $0.3 worth of C3, and to produce one
dollar's worth of C3 requires $0.2 worth of each of Cl, C2 and C3. Suppose that, in a given time
period, there is an external demand for d1 dollars' worth of CI, d2 of C2 and d3 of C3. What are the
production levels Xl, X2, X3 of C1, C2, C3 required to satisfy all demands in the given period?

To solve this problem, we use **Leontief Input-Output Analysis**. The equations are based on the production system represented by a matrix where the coefficients indicate the dependencies between goods.

### System Representation

The problem can be represented as:
\[
X = (I - A)^{-1}D
\]
where:

- $ X $ is the production vector $[X_1, X_2, X_3]$ (output levels of $C_1, C_2, C_3$).
- $ I $ is the identity matrix.
- $ A $ is the input-output matrix.
- $ D $ is the demand vector $[d_1, d_2, d_3]$.

The input-output matrix $A$ based on the problem statement is:
\[
A = \begin{bmatrix}
0.2 & 0.4 & 0.1 \\
0.3 & 0.1 & 0.3 \\
0.2 & 0.2 & 0.2
\end{bmatrix}
\]

The demand vector $D$ is:
\[
D = \begin{bmatrix}
d_1 \\
d_2 \\
d_3
\end{bmatrix}
\]

### Steps to Solve:

1. Compute the matrix $I - A$, where $I$ is the $3 \times 3$ identity matrix:
\[
I = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

2. Find $(I - A)^{-1}$ (the inverse of $I - A$).

3. Multiply $(I - A)^{-1}$ by $D$ to find $X = (I - A)^{-1}D$.

Let’s perform these calculations. numpy as np

# Define the input-output matrix A
A = np.array([
    [0.2, 0.4, 0.1],
    [0.3, 0.1, 0.3],
    [0.2, 0.2, 0.2]
])

# Define the identity matrix I
I = np.eye(3)

# Calculate (I - A)
I_minus_A = I - A

# Invert (I - A)
I_minus_A_inv = np.linalg.inv(I_minus_A)

# Display the result for further calculations
I_minus_A_invThe inverse of $ I - A $ is:
\[
(I - A)^{-1} = \begin{bmatrix}
1.71875 & 0.88542 & 0.54688 \\
0.78125 & 1.61458 & 0.70313 \\
0.625 & 0.625 & 1.5625
\end{bmatrix}
\]

Now, we can find the production levels $ X = (I - A)^{-1} D $. Please provide the external demand values ($ d_1, d_2, d_3 $) to proceed with the calculation.

The production processes for three goods, C1, C2, C3 are interlinked. To produce one dollar's worth of C1 requires the input of $0.2 worth of C1, $0.4 of C2 and $0.1 of C3. To produce one dollar's worth of C2 requires $0.3 worth of C1, $0.1 worth of C2 and $0.3 worth of C3, and to produce one dollar's worth of C3 requires $0.2 worth of each of C1, C2 and C3. Suppose that, in a given time period, there is an external demand for d1 dollars' worth of C1, d2 of C2 and d3 of C3. What are the production levels X1, X2, X3 of C1, C2, C3 required to satisfy all demands in the given period?

Explore how to determine production levels (X1, X2, X3) of goods C1, C2, C3 using the Leontief Input-Output model. This involves solving matrix equations with given input-output coefficients and external demand. Suitable for students studying economics or applied mathematics.

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