Let's go step by step through the given problem: **Given:** - Aggregate demand: \( Q = 20,000 - 200P \) (where \( Q \) = total annual hours, \( P \) = price per hour) - 100 identical clients - Total cost: \( TC = 80Q \) (so cost per hour is $80) - Two-part tariff: annual retainer (entry fee) + hourly usage charge --- ## (a) What is the firm's MC? What is the demand curve for a representative client? ### **Marginal Cost (MC):** Given \( TC = 80Q \), so \[ MC = \frac{dTC}{dQ} = 80 \] So, **MC = $80 per hour**. --- ### **Demand Curve for a Representative Client:** Aggregate demand: \( Q = 20,000 - 200P \) There are 100 clients, so the demand per client is: \[ q = \frac{Q}{100} = \frac{20,000 - 200P}{100} = 200 - 2P \] So, for a representative client: \[ q = 200 - 2P \implies P = 100 - 0.5q \] --- ## (b) What are the profit maximizing retainer and hourly rate under a two-part tariff? What is the aggregate annual profit? **Two-part tariff:** The firm sets: - usage price per hour (\( p \)) - entry fee (\( T \)), annual retainer ### **Step 1: Set usage rate at MC** To maximize profit in a two-part tariff with identical customers, set usage price at MC: \[ p = MC = \$80 \] ### **Step 2: Find each client's yearly usage at \( p = 80 \):** From the individual demand: \[ q = 200 - 2p \\ q = 200 - 2(80) = 200 - 160 = 40 \] So, **each client will purchase 40 hours per year**. ### **Step 3: Find consumer surplus per client at \( p = 80 \):** Consumer surplus (CS) is the area under the demand curve above price, up to quantity purchased. - Choke price (where \( q = 0 \)): \( p = 100 \) - At \( p = 80 \), \( q = 40 \) So, CS per client: \[ CS = \frac{1}{2} \times (100 - 80) \times 40 = \frac{1}{2} \times 20 \times 40 = 400 \] ### **Step 4: Set entry fee (retainer) equal to CS** To extract all surplus, set entry fee \( T = \$400 \) per client. --- ### **Firm's profit:** - Each client pays: \( 40 \times 80 = \$3,200 \) in usage fees + \$400 retainer - Cost per client: \( 40 \times 80 = \$3,200 \) - Profit per client: retainer (\$400) - Number of clients: 100 \[ \text{Total profit} = 100 \times 400 = \$40,000 \] --- ## (c) Single-price monopoly: What is the profit-maximizing price, quantity, and profit? Compare to two-part tariff. ### **Step 1: Find MR and set MR = MC** Aggregate demand: \( Q = 20,000 - 200P \implies P = 100 - 0.005Q \) Total revenue: \( TR = P \cdot Q = (100 - 0.005Q)Q = 100Q - 0.005Q^2 \) Marginal revenue: \[ MR = \frac{dTR}{dQ} = 100 - 0.01Q \] Set \( MR = MC \): \[ 100 - 0.01Q = 80 \implies 0.01Q = 20 \implies Q = 2,000 \] ### **Step 2: Find monopoly price** Plug \( Q = 2,000 \) into demand: \[ P = 100 - 0.005 \times 2,000 = 100 - 10 = 90 \] ### **Step 3: Monopoly profit** - Total revenue: \( TR = P \times Q = 90 \times 2,000 = \$180,000 \) - Total cost: \( TC = 80 \times 2,000 = \$160,000 \) - Profit: \( 180,000 - 160,000 = \$20,000 \) ### **Step 4: Comparison** - **Two-part tariff profit:** \$40,000 - **Single-price monopoly profit:** \$20,000 The two-part tariff yields **twice the profit** of a single-price monopoly, because it extracts all consumer surplus as profit. --- ## **Summary Table** | Scheme | Usage Fee | Entry Fee | Q per year | Total Profit | |----------------------|-----------|-----------|------------|--------------| | Two-part tariff | \$80 | \$400 | 4,000 | \$40,000 | | Single-price monopoly| \$90 | \$0 | 2,000 | \$20,000 | - In two-part tariff, all 100 clients participate, each uses 40 hours (\( Q = 4,000 \)). - In monopoly, all 100 clients **still participate** (since demand is aggregate), but total usage is 2,000 hours (\( 20 \) hours/client on average). --- ## **Final Answers** ### (a) - The firm's **marginal cost (MC)** is \$80 per hour. - The demand curve for a representative client is \( q = 200 - 2p \) or \( p = 100 - 0.5q \). ### (b) - **Profit-maximizing hourly rate (usage fee)**: \$80 per hour. - **Profit-maximizing retainer (entry fee)**: \$400 per client per year. - **Aggregate annual profit**: \$40,000. ### (c) - **Monopoly price**: \$90 per hour - **Monopoly quantity**: 2,000 hours/year (or 20 hours/client) - **Monopoly profit**: \$20,000 - **Comparison**: The two-part tariff yields more profit (\$40,000) than the single-price monopoly (\$20,000).