To solve this problem, let’s go through each part step by step.

Problem Setup

We are given:

1. Matrix $A$, representing the interdependence of three industries in an economy. Each entry $a_{ij}$ denotes the output from industry $j$ required by industry $i$.
2. Final demand vector $\mathbf{d}$, which consists of $d_1$, $d_2$, and $d_3$, representing the demand for each industry’s output in billions of dollars.

The Leontief model for an economy is given by: $(I - A) \mathbf{x} = \mathbf{d}$ where:

• $I$ is the identity matrix.
• $\mathbf{x}$ is the vector of total output levels for each industry.
• $\mathbf{d}$ is the final demand vector.

Solution Steps

(a) Write the Leontief matrix and the input-output matrix equation.

1. Leontief Matrix: The Leontief matrix is $(I - A)$.

   • Let $I$ be the identity matrix: $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
   • The matrix $(I - A)$ is then: $I - A = \begin{bmatrix} 1 - a_{11} & -a_{12} & -a_{13} \\ -a_{21} & 1 - a_{22} & -a_{23} \\ -a_{31} & -a_{32} & 1 - a_{33} \end{bmatrix}$

2. Matrix Equation: Using the Leontief model, we write the input-output matrix equation for this economy: $(I - A) \mathbf{x} = \mathbf{d}$ This is the equation that links the total output $\mathbf{x}$ with the demand $\mathbf{d}$.

(b) Find the solution output levels by the matrix inverse method.

1. Solve for $\mathbf{x}$: To find $\mathbf{x}$, we rearrange the equation as follows: $\mathbf{x} = (I - A)^{-1} \mathbf{d}$
2. Inverse Calculation: Compute $(I - A)^{-1}$ and then multiply it by $\mathbf{d}$ to obtain the output levels $\mathbf{x}$.

To proceed with a numerical solution, values for $A$ and $\mathbf{d}$ would be necessary. Without specific values, the solution can only be expressed symbolically.

Would you like further details on the matrix inversion process, or do you have any questions?

Related Questions to Explore:

1. How does the Leontief model apply to real-world economies with more than three industries?
2. What happens if the matrix $(I - A)$ is singular, meaning it doesn’t have an inverse?
3. How can the Leontief model be extended to include external factors such as imports and exports?
4. In what scenarios could the Leontief model fail to provide accurate predictions of economic output?
5. How can sensitivity analysis be applied to study changes in the demand vector $\mathbf{d}$?

Tip:

Remember, for matrix inversion, the matrix $(I - A)$ must be non-singular (invertible). This typically means the diagonal dominance in $A$ should not be too high.