This problem deals with the Leontief input-output model, a type of matrix-based economic model. The system has two sectors: agriculture and oil. Each sector requires inputs from itself and the other to produce outputs. We can solve this using matrix algebra.

The data provided in the question is as follows:

• Agriculture requires: $0.40 from agriculture, $0.35 from oil.
• Oil requires: $0.20 from agriculture, $0.05 from oil.
• Final demand is $250 million for oil and $40 million for agriculture.

We can define the input-output system as: $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0.40 & 0.35 \\ 0.20 & 0.05 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} 40 \\ 250 \end{pmatrix}$ Where:

• $x_1$ = total output of agriculture,
• $x_2$ = total output of oil.

Now, rearranging the equation to solve for $x_1$ and $x_2$: $\left( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} 0.40 & 0.35 \\ 0.20 & 0.05 \end{pmatrix} \right) \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 40 \\ 250 \end{pmatrix}$

The matrix equation becomes: $\begin{pmatrix} 1 - 0.40 & -0.35 \\ -0.20 & 1 - 0.05 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 40 \\ 250 \end{pmatrix}$ $\begin{pmatrix} 0.60 & -0.35 \\ -0.20 & 0.95 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 40 \\ 250 \end{pmatrix}$

To solve this, we can invert the matrix and multiply it by the final demand vector. I'll proceed with these steps to calculate the outputs $x_1$ and $x_2$:

Step 1: Find the inverse of the matrix

$A = \begin{pmatrix} 0.60 & -0.35 \\ -0.20 & 0.95 \end{pmatrix}$

The determinant of this matrix $\text{det}(A)$ is: $\text{det}(A) = (0.60)(0.95) - (-0.35)(-0.20) = 0.57 - 0.07 = 0.50$

The inverse of matrix $A$ is: $A^{-1} = \frac{1}{0.50} \begin{pmatrix} 0.95 & 0.35 \\ 0.20 & 0.60 \end{pmatrix} = \begin{pmatrix} 1.90 & 0.70 \\ 0.40 & 1.20 \end{pmatrix}$

Step 2: Multiply the inverse matrix by the final demand vector

Now, multiply the inverse matrix by the final demand vector $\begin{pmatrix} 40 \\ 250 \end{pmatrix}$:

$\begin{pmatrix} 1.90 & 0.70 \\ 0.40 & 1.20 \end{pmatrix} \begin{pmatrix} 40 \\ 250 \end{pmatrix}$

Calculating the multiplication: $x_1 = (1.90)(40) + (0.70)(250) = 76 + 175 = 251$ $x_2 = (0.40)(40) + (1.20)(250) = 16 + 300 = 316$

Conclusion:

• The output of agriculture ($x_1$) is 251 million dollars.
• The output of oil ($x_2$) is 316 million dollars.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would the result change if the input from agriculture to oil production was higher?
2. Can this method be extended to an economy with more than two sectors?
3. How is the inverse matrix calculation important in economic models like this one?
4. What happens if the demand for oil changes to $300 million?
5. How can this model be applied in real-world economic forecasting?

Tip: When dealing with matrix inversions, always ensure the determinant is non-zero. If the determinant is zero, the matrix is singular and cannot be inverted.