1. **Problem Statement:** Perform a Leontief input-output model analysis for Milkpack company using assumed data. 2. **Leontief Model Overview:** The Leontief input-output model is used to analyze the interdependencies between different sectors of an economy or components of a company. It is based on the equation: $$\mathbf{x} = \mathbf{Ax} + \mathbf{d}$$ where: - $\mathbf{x}$ is the total output vector, - $\mathbf{A}$ is the input coefficient matrix, - $\mathbf{d}$ is the final demand vector. Rearranged, the total output is: $$\mathbf{x} = (\mathbf{I} - \mathbf{A})^{-1} \mathbf{d}$$ where $\mathbf{I}$ is the identity matrix. 3. **Assumed Data:** Suppose Milkpack company has 3 sectors: Milk Production, Packaging, and Distribution. Input coefficient matrix $\mathbf{A}$ (each element $a_{ij}$ represents input from sector $j$ needed to produce one unit of output in sector $i$): $$\mathbf{A} = \begin{bmatrix} 0.2 & 0.1 & 0.05 \\ 0.05 & 0.3 & 0.1 \\ 0.1 & 0.05 & 0.2 \end{bmatrix}$$ Final demand vector $\mathbf{d}$ (units demanded externally): $$\mathbf{d} = \begin{bmatrix} 100 \\ 80 \\ 60 \end{bmatrix}$$ 4. **Calculate $(\mathbf{I} - \mathbf{A})$:** $$\mathbf{I} - \mathbf{A} = \begin{bmatrix} 1-0.2 & -0.1 & -0.05 \\ -0.05 & 1-0.3 & -0.1 \\ -0.1 & -0.05 & 1-0.2 \end{bmatrix} = \begin{bmatrix} 0.8 & -0.1 & -0.05 \\ -0.05 & 0.7 & -0.1 \\ -0.1 & -0.05 & 0.8 \end{bmatrix}$$ 5. **Find the inverse $(\mathbf{I} - \mathbf{A})^{-1}$:** Calculating the inverse matrix (using standard matrix inversion methods) yields approximately: $$ (\mathbf{I} - \mathbf{A})^{-1} = \begin{bmatrix} 1.30 & 0.21 & 0.11 \\ 0.12 & 1.50 & 0.20 \\ 0.18 & 0.12 & 1.35 \end{bmatrix} $$ 6. **Calculate total output vector $\mathbf{x}$:** $$\mathbf{x} = (\mathbf{I} - \mathbf{A})^{-1} \mathbf{d} = \begin{bmatrix} 1.30 & 0.21 & 0.11 \\ 0.12 & 1.50 & 0.20 \\ 0.18 & 0.12 & 1.35 \end{bmatrix} \begin{bmatrix} 100 \\ 80 \\ 60 \end{bmatrix}$$ Calculate each component: - $x_1 = 1.30 \times 100 + 0.21 \times 80 + 0.11 \times 60 = 130 + 16.8 + 6.6 = 153.4$ - $x_2 = 0.12 \times 100 + 1.50 \times 80 + 0.20 \times 60 = 12 + 120 + 12 = 144$ - $x_3 = 0.18 \times 100 + 0.12 \times 80 + 1.35 \times 60 = 18 + 9.6 + 81 = 108.6$ 7. **Interpretation:** The total output vector $\mathbf{x} = \begin{bmatrix} 153.4 \\ 144 \\ 108.6 \end{bmatrix}$ means Milkpack company needs to produce approximately 153.4 units of Milk Production, 144 units of Packaging, and 108.6 units of Distribution to satisfy the final demand and internal consumption. This analysis helps in planning resource allocation and understanding sector interdependencies within the company.