Let's solve the input-output problem **step by step**. --- ## **Step 1: Define Variables** Let: - \( x_1 \): production level of **equipment** (in million dollars) - \( x_2 \): production level of **components** (in million dollars) --- ## **Step 2: Construct the System of Equations** Given external demands: - Equipment: \( d_1 = 80,000 \) million - Components: \( d_2 = 91,000 \) million From the table: - Equipment used by Equipment: 6,200 - Components used by Equipment: 600 - Equipment used by Components: 23,000 - Components used by Components: 30,000 Total outputs: - Equipment: \( x_1 = 90,000 \) - Components: \( x_2 = 140,000 \) But for the required year, **we need to find new \( x_1 \) and \( x_2 \) for the given external demand.** --- ### **General Input-Output Equations** \[ \begin{align*} x_1 &= \text{(equipment used by equipment)} + \text{(components used by equipment)} + \text{(external demand for equipment)} \\ x_2 &= \text{(equipment used by components)} + \text{(components used by components)} + \text{(external demand for components)} \\ \end{align*} \] But, the "used by" values are **fractions** of the total output. So, let's compute the input coefficients: #### **Input Coefficients Matrix** \[ \begin{align*} \text{Equipment to Equipment:} &\quad \frac{6,200}{90,000} = .0689 \\ \text{Components to Equipment:} &\quad \frac{600}{140,000} = .00429 \\ \text{Equipment to Components:} &\quad \frac{23,000}{90,000} = .2556 \\ \text{Components to Components:} &\quad \frac{30,000}{140,000} = .2143 \\ \end{align*} \] But, each column represents an output, so: - **For equipment (\(x_1\)) output:** - Equipment required: 6,200 (by equipment) from total 90,000 output = \( \frac{6,200}{90,000} = .0689 \) - Components required: 23,000 (by equipment) from total 90,000 output = \( \frac{23,000}{90,000} = .2556 \) - **For components (\(x_2\)) output:** - Equipment required: 600 (by components) from total 140,000 output = \( \frac{600}{140,000} = .00429 \) - Components required: 30,000 (by components) from total 140,000 output = \( \frac{30,000}{140,000} = .2143 \) So, the **input coefficient matrix \(A\)** is: \[ A = \begin{bmatrix} .0689 & .00429 \\ .2556 & .2143 \end{bmatrix} \] But we need the coefficients such that: - For \(x_1\): what fraction comes from **equipment and components**? - For \(x_2\): what fraction comes from **equipment and components**? So, correct coefficients (from "From" to "To"): \[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} \frac{6,200}{90,000} & \frac{600}{140,000} \\ \frac{23,000}{90,000} & \frac{30,000}{140,000} \end{bmatrix} = \begin{bmatrix} .0689 & .00429 \\ .2556 & .2143 \end{bmatrix} \] --- ## **Step 3: Set Up the Input-Output Equations** The **Leontief input-output model**: \[ x = Ax + d \] \[ (I - A)x = d \] Where: - \( x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \) - \( d = \begin{bmatrix} 80,000 \\ 91,000 \end{bmatrix} \) --- ## **Step 4: Write the Equations** \[ (I - A)x = d \] \[ \begin{bmatrix} 1 - .0689 & -.00429 \\ -.2556 & 1 - .2143 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 80,000 \\ 91,000 \end{bmatrix} \] \[ \begin{bmatrix} .9311 & -.00429 \\ -.2556 & .7857 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 80,000 \\ 91,000 \end{bmatrix} \] --- ## **Step 5: Solve the System** Let’s solve using Cramer’s rule or matrix inversion. ### **Write as equations:** 1) \( .9311 x_1 - .00429 x_2 = 80,000 \) 2) \( -.2556 x_1 + .7857 x_2 = 91,000 \) --- ### **Step 5.1: Solve for \(x_1\) and \(x_2\)** #### **Express \(x_1\) from equation 1:** \[ .9311 x_1 - .00429 x_2 = 80,000 \\ .9311 x_1 = 80,000 + .00429 x_2 \\ x_1 = \frac{80,000 + .00429 x_2}{.9311} \] #### **Substitute in equation 2:** \[ -.2556 x_1 + .7857 x_2 = 91,000 \\ -.2556 \left( \frac{80,000 + .00429 x_2}{.9311} \right) + .7857 x_2 = 91,000 \] Calculate \( -.2556 / .9311 \): \[ -.2556 / .9311 \approx -.2746 \] So: \[ -.2746 (80,000 + .00429 x_2) + .7857 x_2 = 91,000 \\ -21,968 - .001178 x_2 + .7857 x_2 = 91,000 \\ .7857 x_2 - .001178 x_2 = 91,000 + 21,968 \\ .7845 x_2 = 112,968 \\ x_2 = \frac{112,968}{.7845} \approx 144,048 \] --- #### **Now find \(x_1\):** \[ x_1 = \frac{80,000 + .00429 \times 144,048}{.9311} \] \[ .00429 \times 144,048 \approx 617. \] \[ x_1 = \frac{80,000 + 617.}{.9311} = \frac{80,617}{.9311} \approx 86,599 \] --- ## **Step 6: Final Answers (Rounded to Two Significant Digits)** \[ \boxed{ \begin{align*} \text{equipment:} & \quad \$87,000 \text{ million} \\ \text{components:} & \quad \$140,000 \text{ million} \end{align*} } \] (rounded: \(86,599 \to 87,000\), \(144,048 \to 140,000\)) --- ### **Summary Table** | Sector | Production Required (\$ millions) | |--------------|:---------------------------------:| | Equipment | **87,000** | | Components | **140,000** | --- **Let me know if you need further explanation for any step!**