Let's break down and solve each part step by step: --- ## Given Data - **Marginal Revenue (MR):** \( MR = 700 - 4Q \) - **Production Function:** \( Q = L^{.5} K^{.25} \) - **Fixed Costs:** $10 - **Price of Labor (\(w\)):** $8 - **Price of Capital (\(r\)):** $12 --- ## a) **Isoquant and Isocost for Producing 5 Units** ### **Isoquant for \( Q = 5 \):** Given: \( Q = L^{.5} K^{.25} \) Set \( Q = 5 \): \[ 5 = L^{.5} K^{.25} \] Solve for \( K \): \[ L^{.5} K^{.25} = 5 \implies K^{.25} = \frac{5}{L^{.5}} \implies K = \left(\frac{5}{L^{.5}}\right)^4 = \frac{625}{L^2} \] **Isoquant Equation:** \[ K = \frac{625}{L^2} \] --- ### **Isocost Line:** Total cost for labor and capital: \[ C = wL + rK \] Let total variable cost for producing 5 units be \(C\): \[ C = 8L + 12K \] --- ### **Sketching:** - **Isoquant**: Plot \( K = \frac{625}{L^2} \) (downward sloping, convex to origin). - **Isocost**: For a given cost \(C\), \( K = \frac{C}{12} - \frac{8}{12}L \) (straight line with slope \(-\frac{8}{12}\)). **[You would draw these two curves on the provided grid, with the isoquant as a convex curve and the isocost as a straight line. The tangency point gives the optimal combination for 5 units.]** --- ## b) **Price Charged and Quantity Produced by the Monopolist** ### **Step 1: Find Cost Function** From production function \( Q = L^{.5} K^{.25} \): Let’s find the cost-minimizing input mix for any Q. Write in terms of K: \[ Q = L^{.5} K^{.25} \implies K = \left(\frac{Q}{L^{.5}}\right)^{4} \implies K = Q^4 L^{-2} \] **Cost minimization:** Set up Lagrangian: \[ \mathcal{L} = 8L + 12K + \lambda(Q - L^{.5} K^{.25}) \] **First Order Conditions:** \[ \frac{\partial \mathcal{L}}{\partial L} = 8 - .5\lambda L^{-.5}K^{.25} = \] \[ \frac{\partial \mathcal{L}}{\partial K} = 12 - .25\lambda L^{.5}K^{-.75} = \] Divide the two FOCs: \[ \frac{8}{12} = \frac{.5\lambda L^{-.5}K^{.25}}{.25\lambda L^{.5}K^{-.75}} \] \[ \frac{2}{3} = \frac{.5}{.25} \cdot \frac{L^{-.5}}{L^{.5}} \cdot \frac{K^{.25}}{K^{-.75}} \] \[ \frac{2}{3} = 2 \cdot L^{-1} \cdot K^{1} \] \[ \frac{2}{3} = 2 \cdot \frac{K}{L} \implies \frac{2}{3} = 2 \cdot \frac{K}{L} \implies \frac{1}{3} = \frac{K}{L} \implies K = \frac{L}{3} \] **Plug into production function:** \[ Q = L^{.5} (L/3)^{.25} = L^{.5} L^{.25} 3^{-.25} = L^{.75} 3^{-.25} \] \[ Q = 3^{-.25} L^{.75} \implies L = (Q 3^{.25})^{1/.75} \] \[ L = Q^{4/3} \cdot 3^{1/3} \] \[ K = \frac{L}{3} = Q^{4/3} \cdot 3^{1/3} \cdot \frac{1}{3} = Q^{4/3} \cdot 3^{-2/3} \] **Total Variable Cost:** \[ TVC = 8L + 12K = 8(Q^{4/3} \cdot 3^{1/3}) + 12(Q^{4/3} \cdot 3^{-2/3}) \] \[ = Q^{4/3} \left( 8 \cdot 3^{1/3} + 12 \cdot 3^{-2/3} \right) \] **Total Cost:** \[ TC = TVC + FC = Q^{4/3} \left( 8 \cdot 3^{1/3} + 12 \cdot 3^{-2/3} \right) + 10 \] --- ### **Step 2: Find Marginal Cost (MC)** Let’s call \( a = 8 \cdot 3^{1/3} + 12 \cdot 3^{-2/3} \): \[ MC = \frac{dTC}{dQ} = \frac{4}{3} a Q^{1/3} \] --- ### **Step 3: Set \( MR = MC \) to find \( Q^* \)** \[ 700 - 4Q = \frac{4}{3} a Q^{1/3} \] To solve, we need to compute \( a \): \[ 3^{1/3} \approx 1.4422 \] \[ 3^{-2/3} = (3^{2/3})^{-1} = (2.0801)^{-1} \approx .4807 \] \[ a = 8 \times 1.4422 + 12 \times .4807 = 11.5376 + 5.7684 = 17.306 \] So, \[ MC = \frac{4}{3} \times 17.306 \times Q^{1/3} \approx 23.0747 Q^{1/3} \] Set equal to MR: \[ 700 - 4Q = 23.0747 Q^{1/3} \] This is a cubic equation, solve numerically. Let \( x = Q^{1/3} \implies Q = x^3 \): \[ 700 - 4x^3 = 23.0747x \implies 4x^3 + 23.0747x - 700 = \] Use trial and error for \( x \): Try \( x = 5 \): \[ 4(125) + 23.0747(5) - 700 = 500 + 115.3735 - 700 = 615.3735 - 700 = -84.6265 \] Try \( x = 6 \): \[ 4(216) + 23.0747(6) - 700 = 864 + 138.4482 - 700 = 1002.4482 - 700 = 302.4482 \] So between \( x = 5 \) and \( x = 6 \) the sign changes. Try \( x = 5.4 \): \[ x^3 = 157.464 4(157.464) + 23.0747(5.4) - 700 = 629.856 + 124.6034 - 700 = 754.4594 - 700 = 54.4594 \] Try \( x = 5.2 \): \[ x^3 = 140.608 4(140.608) + 23.0747(5.2) - 700 = 562.432 + 119.9884 - 700 = 682.4204 - 700 = -17.5796 \] Interpolate: From \( x = 5.2 \) (-17.58) to \( x = 5.4 \) (54.46): \[ x^* = 5.2 + \frac{17.58}{54.46+17.58} \times (5.4-5.2) = 5.2 + \frac{17.58}{72.04} \times .2 = 5.2 + .0488 = 5.25 \text{ (approx.)} \] So, \[ Q^* = (5.25)^3 \approx 144.7 \] --- ### **Step 4: Find Price** Demand curve (since MR = 700 - 4Q, linear demand): \[ MR = P + Q \frac{dP}{dQ} \implies P = MR + Q \frac{dP}{dQ} \] But more directly, if MR is \( 700 - 4Q \), then demand is \( P = 700 - 2Q \). So, \[ P^* = 700 - 2Q^* \] \[ P^* = 700 - 2 \times 144.7 = 700 - 289.4 = 410.6 \] --- ### **Final Answers for (b):** - **Quantity produced:** \( Q^* \approx 145 \) - **Price charged:** \( P^* \approx \$411 \) --- ## c) **Graph and Calculate CS, PS, DWL, Profit** ### **Step 1: Find Competitive Output \( Q_C \)** Set \( P = MC \): \[ 700 - 2Q_C = MC \] \[ 700 - 2Q_C = 23.0747 Q_C^{1/3} \] Let \( x = Q_C^{1/3} \implies Q_C = x^3 \): \[ 700 - 2x^3 = 23.0747x \] \[ 2x^3 + 23.0747x - 700 = \] Try \( x = 5.5 \): \[ x^3 = 166.375 2(166.375) + 23.0747(5.5) - 700 = 332.75 + 126.911 - 700 = 459.661 - 700 = -240.339 \] Try higher \( x \): Try \( x = 7 \): \[ x^3 = 343 2(343) + 23.0747(7) - 700 = 686 + 161.523 - 700 = 847.523 - 700 = 147.523 \] Try \( x = 6.25 \): \[ x^3 = 244.14 2(244.14) + 23.0747(6.25) - 700 = 488.28 + 144.217 - 700 = 632.497 - 700 = -67.503 \] Try \( x = 6.6 \): \[ x^3 = 287.50 2(287.50) + 23.0747(6.6) - 700 = 575.00 + 152.293 - 700 = 727.293 - 700 = 27.293 \] So between \( x = 6.25 \) and \( x = 6.6 \): \[ x^* = 6.25 + \frac{67.5}{67.5+27.3} \times (6.6-6.25) = 6.25 + \frac{67.5}{94.8} \times .35 = 6.25 + .249 = 6.50 \] So, \[ Q_C = (6.5)^3 = 274.6 \] \[ P_C = 700 - 2Q_C = 700 - 2 \times 274.6 = 700 - 549.2 = 150.8 \] --- ### **Step 2: Calculate Areas** #### **Consumer Surplus (CS):** Area between demand and price: \[ CS = \frac{1}{2} (P_{max} - P^*) Q^* \] Where \( P_{max} = 700 \) (when \( Q = \)), \( P^* \approx 411 \), \( Q^* \approx 145 \): \[ CS_M = \frac{1}{2} (700 - 411) \times 145 = \frac{1}{2} \times 289 \times 145 = 20,952.5 \] #### **Producer Surplus (PS):** Area between price

# Step-by-Step Solution to the Monopolist Problem ## Given Data - **Marginal Revenue (MR):** \( MR = 700 - 4Q \) - **Production Function:** \( Q = L^{0.5} K^{0.25} \) - **Fixed Costs (FC):** $10 - **Price of Labor (w):** $8 - **Price of Capital (r):** $12 --- ## 1. Isoquant and Isocost for Producing 5 Units ### Isoquant for \( Q = 5 \) Set the production function equal to 5: \[ 5 = L^{0.5} K^{0.25} \] Rearranging gives: \[ K = \frac{625}{L^2} \] ### Isocost Line The total cost function is: \[ C = wL + rK \] For a given total cost \( C \): \[ C = 8L + 12K \] ### Sketching - **Isoquant:** Plot \( K = \frac{625}{L^2} \) (downward sloping, convex to the origin). - **Isocost:** For a given cost \( C \), \( K = \frac{C}{12} - \frac{8}{12}L \) (a straight line). --- ## 2. Price Charged and Quantity Produced by the Monopolist ### Step 1: Find Cost Function **Total Variable Cost (TVC):** Assuming input combinations yield the same output, we express TVC as: \[ TVC = 8L + 12K \] Using \( K = \frac{L}{3} \): \[ TVC = 8L + 12\left(\frac{L}{3}\right) = 8L + 4L = 12L \] ### Step 2: Find Marginal Cost (MC) \[ TC = TVC + FC = 12L + 10 \] \[ MC = \frac{d(TC)}{dQ} = \frac{d(TVC)}{dQ} = 12\frac{dL}{dQ} \] ### Step 3: Set \( MR = MC \) to Find \( Q^* \) 1. **Find Marginal Cost Function:** From the earlier analysis, we know: \[ MC = \frac{dTC}{dQ} = 23.0747 Q^{1/3} \] 2. **Set MR equal to MC:** \[ 700 - 4Q = 23.0747 Q^{1/3} \] This cubic equation can be solved numerically. ### Step 4: Calculate Price Using the demand curve derived from \( MR \): \[ P = 700 - 2Q \] Substituting \( Q^* \) into the demand equation gives the price. --- ## 3. Competitive Output \( Q_C \) To find \( Q_C \): 1. **Set \( P = MC \):** \[ 700 - 2Q_C = 23.0747 Q_C^{1/3} \] This is another cubic equation to be solved numerically. 2. **Find Price at Competitive Output:** Use \( P = 700 - 2Q_C \). --- ## 4. Calculate Areas for CS, PS, and DWL ### Consumer Surplus (CS) \[ CS = \frac{1}{2} (P_{max} - P^*) Q^* \] Where \( P_{max} = 700 \) and \( P^* \) is the monopolistic price. ### Producer Surplus (PS) \[ PS = \text{(Price received - MC) * Quantity produced} \] ### Deadweight Loss (DWL) The area of lost consumer and producer surplus due to monopolistic pricing compared to competitive equilibrium. --- ## Final Answers Summary 1. **Quantity produced by the monopolist:** Approx. \( Q^* \) 2. **Price charged by the monopolist:** Approx. \( P^* \) 3. **Competitive output:** \( Q_C \) 4. **Consumer Surplus:** \( CS \approx \) calculated value 5. **Producer Surplus:** \( PS \approx \) calculated value 6. **Deadweight Loss:** \( DWL \approx \) calculated value (Note: The exact numerical values for \( Q^*, P^*, Q_C, CS, PS, \) and \( DWL \) need to be calculated based on the numerical solutions from the cubic equations derived.)