Certainly! Here is a step-by-step rewrite of your answer, with explanations and all final answers filled in. --- ## **Step 1: Restating the Problem** - **Total Greaze supply:** 400 million gallons (to be allocated over 4 years) - **Annual demand curve:** $$ Q_t = 225 - P_t $$ - **Marginal cost (MC):** 0 (production is costless) - **Interest rate:** 0% (no discounting over time) - **Implication:** By Hotelling’s rule, the scarcity rent is constant, so the optimal allocation is smooth (equal each year) unless something disrupts the market. - **Twist:** In Year 3, a perfect substitute ("Poodz") enters, available in **unlimited supply at \$75** per gallon. --- ## **Step 2: Perfect Competition, No Poodz** **Goal:** Allocate the 400 million gallons for maximum efficiency under perfect competition, before Poodz arrives. - **With no interest:** Spread supply evenly over 4 years. $$ Q_t = \frac{400}{4} = 100 \text{ million gallons per year} $$ - **Find price using the demand curve:** $$ Q = 225 - P \implies P = 225 - Q $$ $$ P = 225 - 100 = 125 $$ - **Conclusion:** ✅ **Year 1 sales = 100 million gallons at \$125 per gallon** --- ## **Step 3: Monopoly, No Poodz** **Goal:** Find monopoly allocation over 4 years (before Poodz). - **With 0% interest:** The monopolist wants to smooth out extraction (no incentive to delay). - **So:** $$ Q_t = 100 \text{ million gallons per year}, \quad P = 125 $$ - **Conclusion:** ✅ **Year 1 sales = 100 million gallons at \$125 per gallon** --- ## **Step 4: Perfect Competition, Poodz Arrives in Year 3** **Scenario:** Poodz arrives in Year 3, offering unlimited substitute at \$75. - **From Year 3 onward:** Greaze can only sell at \$75 or less. - **At \$75, demand =** $$ Q = 225 - 75 = 150 \text{ million gallons} $$ - **But:** With unlimited Poodz, Greaze earns no scarcity rent in Years 3–4 (price capped at \$75). - **Result:** Competitive Greaze sellers prefer to sell **all 400 million gallons before Year 3** (no incentive to wait). - **Evenly split over Years 1 and 2:** $$ Q_1 = Q_2 = \frac{400}{2} = 200 \text{ million gallons per year} $$ - **Price in each of Years 1 and 2:** $$ P = 225 - 200 = 25 $$ - **Conclusion:** ✅ **Year 1 sales = 200 million gallons at \$25 per gallon** --- ## **Step 5: Monopoly, Poodz Arrives in Year 3** **Goal:** Monopolist maximizes profit, knowing price will be capped at \$75 in Years 3–4. - **In Years 3–4:** Can't charge more than \$75 (Poodz sets price). - **So, in Years 1–2:** Monopolist will sell at a quantity where marginal revenue (MR) = \$75. - **Demand curve:** \( Q = 225 - P \) - **Total Revenue (TR):** \( TR = PQ = (225 - Q)Q = 225Q - Q^2 \) - **Marginal Revenue (MR):** $$ MR = \frac{dTR}{dQ} = 225 - 2Q $$ - **Set MR = 75:** $$ 225 - 2Q = 75 \implies 2Q = 150 \implies Q = 75 $$ - **Corresponding price:** $$ P = 225 - 75 = 150 $$ - **So, in Years 1 and 2:** Sell 75 million gallons/year at \$150. - **Total sold in Years 1–2:** \( 75 \times 2 = 150 \) million gallons. - **Leftover for Years 3–4:** \( 400 - 150 = 250 \) million gallons, but can only sell at \$75/gal. - **But at \$75, total demand is 150/year. With 250 left, all can be sold (125/year each in Years 3–4).** - **Conclusion:** ✅ **Year 1 sales = 75 million gallons at \$150 per gallon** --- ## **Step 6: Summarizing Final Answers** | Scenario | Year 1 Sales (million gallons) | | ---------------------------------------- | ------------------------------ | | Perfect Competition (no Poodz) | **100** | | Monopoly (no Poodz) | **100** | | Perfect Competition (with Poodz looming) | **200** | | Monopoly (with Poodz looming) | **75** | --- Would you like to see a **4-year allocation table** for each scenario, showing the quantity and price for every year?

Certainly! Here's a more organized and clear presentation of the solution, structured for better readability and comprehension: --- # **Problem Overview** - **Total Greaze supply:** 400 million gallons (to be allocated over 4 years) - **Demand function:** \( Q_t = 225 - P_t \) - **Marginal cost (MC):** 0 - **Interest rate:** 0% (implying no discounting, Hotelling’s rule leads to smooth allocation unless disrupted) - **Event in Year 3:** Introduction of **Poodz**, a perfect substitute available in unlimited supply at \$75 per gallon --- # **Part 1: Scenario Without Poodz** ### **A. Perfect Competition (No Poodz)** - **Allocation:** Equal over 4 years due to zero interest rate \[ Q_t = \frac{400}{4} = 100 \text{ million gallons/year} \] - **Price:** Using demand curve \( P = 225 - Q \) \[ P = 225 - 100 = \$125 \] - **Result:** | Year | Quantity (million gallons) | Price (\$) | |--------|------------------------------|------------| | 1 | 100 | 125 | --- ### **B. Monopoly (No Poodz)** - **Since no interest and no Poodz:** Monopolist also chooses to allocate equally - **Same as perfect competition:** | Year | Quantity (million gallons) | Price (\$) | |--------|------------------------------|------------| | 1 | 100 | 125 | --- # **Part 2: Scenario With Poodz (Arriving in Year 3)** ### **A. Perfect Competition with Poodz** - **In Years 3–4:** - Price ceiling at \$75 (due to Poodz) - Demand at \$75: \[ Q = 225 - 75 = 150 \text{ million gallons} \] - **Implication:** - Since Poodz supply is unlimited at \$75, Greaze earns **no scarcity rent** in Years 3–4 - Rational sellers will **sell all 400 million gallons before Year 3** to avoid low prices - **Distribution of all 400 million gallons in Years 1 and 2:** \[ Q_1 = Q_2 = \frac{400}{2} = 200 \text{ million gallons} \] - **Prices in Years 1 and 2:** \[ P = 225 - 200 = \$25 \] | Year | Quantity (million gallons) | Price (\$) | |--------|------------------------------|------------| | 1 | 200 | 25 | | 2 | 200 | 25 | --- ### **B. Monopoly with Poodz** - **In Years 1–2:** - The monopolist maximizes profit, knowing that in Years 3–4: \[ P_{\text{max}} = \$75 \] - **In Years 1–2:** - Demand: \( Q = 225 - P \) - Revenue: \( R = P \times Q = (225 - Q)Q = 225Q - Q^2 \) - Marginal Revenue (MR): \[ MR = \frac{dR}{dQ} = 225 - 2Q \] - Set MR equal to the capped price (\$75): \[ 225 - 2Q = 75 \implies 2Q = 150 \implies Q = 75 \] - Corresponding price: \[ P = 225 - 75 = \$150 \] - **Sales in Years 1 and 2:** \[ Q_{1,2} = 75 \text{ million gallons per year} \] - **Total sold in Years 1–2:** \[ 75 \times 2 = 150 \text{ million gallons} \] - **Remaining 250 million gallons are sold in Years 3–4 at \$75**, which is feasible given demand of 150 million gallons/year. | Year | Quantity (million gallons) | Price (\$) | |--------|------------------------------|------------| | 1 | 75 | 150 | | 2 | 75 | 150 | --- # **Final Summary of Key Quantities in Year 1** | Scenario | Year 1 Sales (million gallons) | |--------------------------------------------|------------------------------| | **Perfect Competition (no Poodz)** | **100** | | **Monopoly (no Poodz)** | **100** | | **Perfect Competition (with Poodz)** | **200** | | **Monopoly (with Poodz)** | **75** | --- # **Additional Notes** - The **allocation strategies** differ significantly once Poodz arrives, especially under perfect competition where sellers prefer to sell all supply early at low prices. - Under monopoly with Poodz, the monopolist maximizes profit in Years 1–2 by setting MR = \$75, leading to lower sales in Year 1 than in the no Poodz case. --- Would you like me to prepare a **full 4-year table** with annual quantities and prices for each scenario?