# Step-by-Step Solution Let's break down the problem step by step: --- ## **1. Model Setup and Given Data** - **Aggregate Capital (K):** 900 - **Aggregate Labor (L):** 400 - **Production Function:** \( Y = K^{0.5} L^{0.5} \) - **Consumption Function:** \( C = 70 + 0.7Y_d \) - **Investment Function:** \( I = 84 - 100r \) - **Government Spending (G):** 100 - **Taxes (T):** 100 - **Marginal Products:** - \( MPL = 0.5 K^{0.5} L^{-0.5} \) - \( MPK = 0.5 K^{-0.5} L^{0.5} \) --- ## **2. Find Output (Y)** Plug in \( K = 900 \), \( L = 400 \): \[ Y = 900^{0.5} \times 400^{0.5} = 30 \times 20 = 600 \] --- ## **3. Find Factor Prices (Wage and Rental Rate)** **Wage (\( w \)) = \( MPL \):** \[ MPL = 0.5 \times 900^{0.5} \times 400^{-0.5} = 0.5 \times 30 \times \frac{1}{20} = 0.5 \times 1.5 = 0.75 \] **Rental Rate (\( r \)) = \( MPK \):** \[ MPK = 0.5 \times 900^{-0.5} \times 400^{0.5} = 0.5 \times \frac{1}{30} \times 20 = 0.5 \times 0.6667 = 0.3333 \] --- ## **4. Income and Disposable Income** **Total Output (Income):** \( Y = 600 \) **Taxes (T):** 100 **Disposable Income (\( Y_d \)):** \[ Y_d = Y - T = 600 - 100 = 500 \] --- ## **5. Consumption (C):** \[ C = 70 + 0.7 \times 500 = 70 + 350 = 420 \] --- ## **6. Investment (I):** \[ I = 84 - 100r \] We don't know \( r \) yet; let's find equilibrium. --- ## **7. Goods Market Equilibrium** \[ Y = C + I + G \] \[ 600 = 420 + (84 - 100r) + 100 \] \[ 600 = 420 + 100 + 84 - 100r \] \[ 600 = 604 - 100r \] \[ 100r = 604 - 600 = 4 \] \[ r = 0.04 \] --- **Now plug \( r \) back into the Investment function:** \[ I = 84 - 100 \times 0.04 = 84 - 4 = 80 \] --- ## **8. Verify Walras' Law (Goods Market Clears)** \[ Y = C + I + G \] \[ 600 = 420 + 80 + 100 = 600 \] **Goods market clears.** --- ## **9. Competitive Equilibrium Summary** | Variable | Value | |----------|-------| | Output (\( Y \)) | 600 | | Consumption (\( C \)) | 420 | | Investment (\( I \)) | 80 | | Government Spending (\( G \)) | 100 | | Taxes (\( T \)) | 100 | | Wage (\( w \)) | 0.75 | | Rental Rate (\( r \)) | 0.04 | --- ## **10. Market Diagrams (Sketch Description)** **Labor Market:** - **Supply:** Vertical at \( L = 400 \) - **Demand:** Downward sloping from \( MPL \) as function of \( L \) - **Equilibrium Wage:** \( w = 0.75 \) **Capital Market:** - **Supply:** Vertical at \( K = 900 \) - **Demand:** Downward sloping from \( MPK \) as function of \( K \) - **Equilibrium Rental Rate:** \( r = 0.04 \) **Goods Market:** - **Supply:** Horizontal at \( Y = 600 \) - **Demand:** \( C + I + G = 600 \) - **Equilibrium at \( Y = 600 \)** --- ## **11. Summary Table** | Market | Supply | Demand | Equilibrium Price | Equilibrium Quantity | |-------------|---------------|------------------|-------------------|---------------------| | Labor | 400 | From \( MPL \) | \( w = 0.75 \) | 400 | | Capital | 900 | From \( MPK \) | \( r = 0.04 \) | 900 | | Goods | 600 | 600 | — | 600 | --- ### **Conclusion** - **Equilibrium allocations:** \( Y = 600 \), \( L = 400 \), \( K = 900 \) - **Equilibrium prices:** \( w = 0.75 \), \( r = 0.04 \) - **Goods market clears, verifying Walras' Law.** --- If you need hand-drawn diagrams, let me know and I can describe how to sketch them!

## Hand-Drawn Diagrams Description ### **1. Labor Market Diagram** - **Axes:** - X-axis: Quantity of Labor (L) - Y-axis: Wage Rate (w) - **Supply Curve:** - Vertical line at \( L = 400 \) (perfectly inelastic supply). - **Demand Curve:** - Downward sloping from \( MPL \) values. Start high on the Y-axis and slope downwards toward the right. - **Equilibrium Point:** - Mark the intersection of the demand curve and the vertical supply line. - Label equilibrium wage \( w = 0.75 \). ### **2. Capital Market Diagram** - **Axes:** - X-axis: Quantity of Capital (K) - Y-axis: Rental Rate (r) - **Supply Curve:** - Vertical line at \( K = 900 \). - **Demand Curve:** - Downward sloping from \( MPK \) values. Start high on the Y-axis and slope downwards toward the right. - **Equilibrium Point:** - Mark the intersection of the demand curve and the vertical supply line. - Label equilibrium rental rate \( r = 0.04 \). ### **3. Goods Market Diagram** - **Axes:** - X-axis: Total Output (Y) - Y-axis: Aggregate Demand - **Supply Curve:** - Horizontal line at \( Y = 600 \) (perfectly elastic supply). - **Demand Curve:** - Downward sloping curve based on \( C + I + G \) values, intersecting the supply line at \( Y = 600 \). - **Equilibrium Point:** - Mark the intersection of the demand curve and the horizontal supply line. - Label total output \( Y = 600 \). ### **Visual Elements to Include:** - Use different colors for supply and demand curves. - Clearly label axes, equilibrium points, and values. - Add arrows to indicate the direction of shifts if needed. --- This description provides a guide to creating hand-drawn diagrams for each market. If you need further details or modifications, please let me know!