Give step-by-step solution with explanation and final answer:
Scientists in the US have developed a new fuel extracted from the combs and hairbrushes of teenage boys during COVID lockdowns. They call

Question

Give step-by-step solution with explanation and final answer: 
Scientists in the US have developed a new fuel extracted from the combs and hairbrushes of teenage boys during COVID lockdowns. They call this new fuel Greaze.

With lockdowns in the US over, scientists only have 400 million gallons of Greaze on hand but at zero marginal cost. Assume there is no hope of creating any more. Also assume that the current supply will spoil and be simply too gross to use at the end of 4 years. You estimate that the market demand for Greaze is Qt = 225 - Pt., where t = 1...4, represents the year in which the Greaze is sold and Q is in units of millions.

The interest rate (e.g. rate of return) is zero.

If the market for Greaze were perfectly competitive, what quantity of Greaze would be sold year 1? There will be _______ million gallons sold in year 1.
Instead assume you were a monopolist who controlled the entire supply of Greaze. Now how much would you sell in year 1? You would sell _______ million gallons.
Now assume that Europeans are working on a product, Poodz, derived from poodle hair. It will be a perfect substitute for Greaze.  They will be able to produce an unlimited amount of P

VIPSolutions · Accepted Answer

Let's solve each part step by step.

### **Given:**
- Total Greaze available: **400 million gallons** (fixed supply for all 4 years)
- Demand in year t: $ Q_t = 225 - P_t $ (in millions)
- Supply must be exhausted by the end of year 4 (spoilage).
- **Interest rate is 0.**
- Years: $ t = 1, 2, 3, 4 $
- **No more Greaze can be produced.**
- **Marginal cost = 0.**
- **Poodz** (perfect substitute) enters in **year 3** with a marginal cost of $75/million gallon, unlimited supply.
---

## **1. Perfect competition, no Poodz yet: Year 1**

**Perfect competition:** Price = Marginal cost = $0

**Demand in year 1:**  
$ P_1 = 0 \implies Q_1 = 225 - 0 = 225 $ (million gallons)

But total supply over 4 years is only **400 million**, so if 225 is sold in year 1, only 175 is left for years 2, 3, and 4.

But under perfect competition, with zero marginal cost and no future competition or storage value (interest rate zero), producers will sell **all** at once. Why wait? Future prices can't be higher with zero marginal cost and no storage incentive.

**Quantity sold in year 1 = 400 million gallons**

---

## **2. Monopoly, no Poodz yet: Year 1**

Monopolist chooses quantities for each year to maximize total profit over 4 years, subject to the constraint that:

$$
Q_1 + Q_2 + Q_3 + Q_4 = 400
$$
and in each year:
$$
P_t = 225 - Q_t
$$

Since interest rate is zero, the monopolist is indifferent to which year the Greaze is sold (as consumers' willingness to pay does not change across years). The optimal policy is to **spread out sales to maximize total revenue**, i.e., set $ Q_1 = Q_2 = Q_3 = Q_4 $.

Let $ Q_t = q $ for all $ t $. Then $ 4q = 400 \implies q = 100 $.

For each year,

$$
P_t = 225 - Q_t = 225 - 100 = 125
$$

**So, the monopolist will sell 100 million gallons in year 1.**

---

## **3. Poodz enters at $75 in year 3 (perfect competition), what changes?**

Now, after year 2, price will not be able to exceed $75 due to competition from Poodz (perfect substitute).

### **A. Perfect competition (Greaze), Poodz coming in year 3**

**Perfect competition:** Sell at marginal cost until supply is exhausted.

- In years 3 and 4, the market price cannot exceed $75 due to Poodz.
- In years 1 and 2, market price is $0 (if sold at all), but if too much is held for years 3 and 4, the price is capped at $75.

**So what happens?**

In years 3 and 4, **nobody would pay more than $75** for Greaze, since Poodz is a perfect substitute.

Demand at $75:
$$
Q = 225 - 75 = 150 \text{ million gallons per year}
$$

So, in years 3 and 4, a **maximum of 150 million gallons per year** can be sold, for a total of 300 million gallons.

Any **remaining Greaze** must be sold in years 1 and 2. Since the price is $0 (marginal cost), all of it is sold immediately under perfect competition (since sellers can't get more by waiting).

But, to maximize revenue, sellers might want to "save" Greaze for years 3 and 4 to get the $75 price. However, in perfect competition, sellers can't coordinate and will sell immediately at $0 (since keeping it for later doesn't yield a higher price due to zero interest rate and lack of coordination).

But since only **300 million gallons** can be sold in years 3 and 4 at a price above $0, the **first 100 million gallons** must be sold in years 1 and 2 at $0.

So, **in year 1, there is nothing preventing all 100 million from being sold at $0** (could be split between years 1 and 2, but with zero interest rate, either is fine). Usually, if no storage cost or interest, it could be split evenly.

**Final answer:**  
**There will now be 50 million gallons sold in year 1** (assuming split evenly between years 1 and 2), but if all is sold in year 1, it would be 100 million gallons.  
However, conventionally, with no reason to prefer one over the other, it is split evenly:

$$
\text{Year 1: } \frac{400 - 150 - 150}{2} = 50 \text{ million gallons}
$$

---

### **B. Monopolist, Poodz coming in year 3**

Monopolist wants to maximize profit, knowing that in years 3 and 4, the most they can charge is $75 per million gallons, for up to 150 million gallons per year.

So, in years 3 and 4:
- Sell 150 million gallons at $75 each year = 300 million gallons total, revenue = $75 × 150 × 2 = $22,500 million.

The remaining **100 million gallons** must be sold in years 1 and 2, where demand is $ Q = 225 - P $.

To maximize profit, the monopolist sets quantity for years 1 and 2 to maximize total revenue, knowing that total for both years is 100 million gallons. With zero interest, this is split evenly:

Let $ q $ = amount sold in year 1 (and also in year 2, as optimal split).

So,
$$
q + q = 100 \implies q = 50
$$

Price in each year:
$$
P = 225 - 50 = 175
$$

Revenue in each year = $ 50 \times 175 = 8,750 $

Total revenue for years 1 and 2: $ 8,750 \times 2 = 17,500 $

Total revenue for years 3 and 4: as above, $ 22,500 $

**So, in year 1, the monopolist sells 50 million gallons.**

---

## **Summary Table**

| Scenario

VIPSolutions · Answer

Let's analyze the problem carefully, step by step, to verify the correctness of your reasoning and the final answers.

---

## **Part 1: Perfect Competition, No Poodz, Year 1**

- **Given:**
  - Total supply: 400 million gallons (fixed)
  - Demand: $Q_t = 225 - P_t$
  - Zero marginal cost
  - No discounting (interest rate = 0)
  - No future competition

- **Implication:**
  - In perfect competition with zero marginal cost, the supply will be sold at the lowest possible price, which is zero.
  - Since total supply is 400 million gallons available over 4 years, and there's no reason to delay sales (no interest), the entire amount will be sold immediately or evenly over the years.

- **Question:**
  - *What quantity is sold in Year 1?*

- **Solution:**
  - Because the market can absorb all at zero price, the entire 400 million gallons can be sold immediately.
  - **Answer:** **400 million gallons**

**Your statement: "perfect competition: 400 million gallons" is correct.**

---

## **Part 2: Monopoly, No Poodz, Year 1**

- **Assumption:**
  - The monopolist chooses how to allocate the 400 million gallons over four years to maximize total profit.
  - Since the interest rate is zero, the monopolist is indifferent to the timing of sales; they will spread sales evenly to maximize total revenue (or, equivalently, just split equally).

- **Calculation:**
  - Total supply: 400 million gallons
  - Equal distribution over 4 years: $400 / 4 = 100$ million gallons per year.
  
- **Price in Year 1:**
  - Demand at $P = 225 - Q$
  - $Q = 100$
  - $P = 225 - 100 = 125$

- **Answer:**
  - *Year 1 sales:* **100 million gallons**

**Your answer: "100 million gallons" matches this reasoning.**

---

## **Part 3: Introduction of Poodz (Years 3 and 4)**

### **A. Perfect competition with Poodz looming**

- The Poodz product will be available starting Year 3 at a marginal cost of $75.
- In Years 3 and 4, **Greaze cannot command a price above $75** because Poodz is a perfect substitute.
- **In Years 1 and 2:**
  - The price is zero (since no Poodz yet).
  - Sellers would want to sell as much as possible at $0.
  - The demand at $P=0$: $Q = 225 - 0 = 225$ million gallons per year.
  - But total supply is only 400 million gallons, and we need to allocate this over 4 years.

- **Strategy:**
  - Since in Years 3 and 4, Greaze can only command $75, and demand at that price is 150 million gallons per year.
  - To maximize total sales, sellers will want to sell as much as possible **before** the price drops to $75.
  - **In practice,** with zero interest, sellers will sell the entire 400 million gallons as soon as possible at $0 to avoid losing profit in later years where the price is capped at $75.

- **Distribution:**
  - **Years 1 and 2:** All 400 million gallons are sold at $0, unless there's an incentive to delay. But since there's no interest, selling earlier or later yields the same value.
  - **However,** demand at $0 is 225 million gallons per year, so in Year 1 and Year 2, the market could absorb up to 225 million gallons each year at $0.
  
- **But total supply is only 400 million**, so:
  - Year 1: 200 million gallons
  - Year 2: 200 million gallons
  - Years 3 and 4: 0 gallons (since all already sold)

- **Answer:**
  - **Year 1 sales:** **200 million gallons**

**Your answer: "50 million gallons" is incorrect. It should be 200 million gallons.**

---

### **B. Monopoly with Poodz looming**

- **The monopolist:**
  - Has 400 million gallons to allocate over 4 years.
  - Will want to maximize total profit.
  
- **Years 3 and 4:**
  - Price is fixed at $75 (due to Poodz).
  - Demand at $75: $Q = 225 - 75 = 150$ million gallons per year.
  - Total possible: **300 million gallons** over two years at $75.
  
- **Remaining supply:**
  - 400 total - 300 allocated to years 3 and 4 = **100 million gallons** to be sold in years 1 and 2.

- **Optimal strategy:**
  - The monopolist will prefer to sell as much as possible **before** the competition arrives, to maximize profits.
  - Given zero interest, the monopolist is indifferent about *when* the sales occur, but to maximize revenue, they would sell **all** remaining in Year 1 at the highest possible price *before* the price drops.

- **However:**
  - The demand in Year 1 is $Q = 225 - P$.
  - To sell all 100 million gallons in Year 1, the price must be:
    $$
    P = 225 - Q = 225 - 100 = 125
    $$
  - The revenue from selling 100 million gallons at $125 per gallon is higher than selling at $75 (which is the price in Years 3 and 4).

- **Conclusion:**
  - The monopolist will sell **all** 100 million gallons in Year 1 (at a price of $125).

- **Answer:**
  - **Year 1 sales:** **100 million gallons**

---

## **Summary of Corrected Answers:**

| Scenario | Year 1 Sales (million gallons) |
|-----------|------------------------------|
| Perfect competition (no Poodz) | **400** |
| Monopoly (no Poodz) | **100** |
| Perfect competition with Poodz looming | **200** |
| Mon

VIPSolutions · Answer

Let's carefully analyze each part of the problem step-by-step to ensure clarity and correctness.

---

### **Given Data Recap:**
- Total Greaze available: **400 million gallons** (fixed supply, perishable after 4 years)
- Demand in year t: $Q_t = 225 - P_t$
- No production beyond the initial stock
- Zero marginal cost
- Zero interest rate (no discounting)
- Years: 1 to 4
- **Additional info:** Starting Year 3, Poodz (a perfect substitute) can be produced at **$75 per million gallons**, and supply is unlimited.

---

### **Part 1: Perfect Competition, Year 1**

**Question:** How much Greaze is sold in Year 1 under perfect competition?

**Logic:**
- With zero marginal cost, firms are willing to supply at any price, but in competitive markets, the price drops to the marginal cost, which is zero.
- Since the entire stock is limited to 400 million gallons, and there's no reason to delay sales (interest rate = 0), all 400 million gallons will be sold immediately.

**Answer:**
$$
\boxed{\textbf{400 million gallons}}
$$

---

### **Part 2: Monopoly, Year 1**

**Question:** How much does a monopolist sell in Year 1?

**Reasoning:**
- The monopolist aims to maximize total discounted profit over 4 years.
- Since interest rate = 0, the timing doesn't matter; the monopolist will split the total stock evenly to maximize total revenue.
- **Total stock:** 400 million gallons
- **Number of years:** 4
- **Optimal per year:** $\frac{400}{4} = 100$ million gallons

**Price in Year 1:**
$$
P_1 = 225 - Q_1 = 225 - 100 = 125
$$

**Answer:**
$$
\boxed{\textbf{100 million gallons}}
$$

---

### **Part 3: Poodz enters in Year 3 with unlimited supply at $75**

---

#### **A. Perfect competition with Poodz looming (Years 1 & 2)**

**Question:** How much Greaze is sold in Year 1?

**Logic:**
- In Years 1 and 2, Poodz isn't available yet; the market price is determined solely by Greaze supply and demand.
- **Since there is no storage cost or interest**, sellers will want to sell as much as possible **before** the price drops in Year 3.
- In Year 3 and 4, the price can't exceed $75 because Poodz is a perfect substitute.

**Demand at $P=75$:**
$$
Q = 225 - 75 = 150 \text{ million gallons}
$$

- The maximum market demand per year at the Poodz price is 150 million gallons.
- The total stock is 400 million gallons.
- **To maximize profit**, sellers will sell as much as possible **before** Year 3, i.e., in Year 1 and Year 2.

**Distribution:**
- Year 1: 200 million gallons
- Year 2: 200 million gallons
- Total: 400 million gallons sold **before** Poodz arrives.

**But the question asks about Year 1 specifically:**

**Answer:**
$$
\boxed{\textbf{200 million gallons}}
$$

---

#### **B. Monopoly with Poodz looming**

**Question:** How much would a monopolist sell in Year 1?

**Logic:**
- The monopolist will want to sell **as much as possible at the highest price** before the Poodz price cap hits.
- Since there's no discounting, the monopolist prefers to sell **all** remaining stock **before** Year 3 to maximize profit.
- The **price** at which they sell in Year 1 will be the **reservation price** that maximizes revenue, considering that the demand at that price is $Q = 225 - P$.

- To sell the entire stock (or as much as possible) in Year 1 at the **highest feasible price**, the monopolist sets the price to optimize profit:

$$
\text{Set } MR = P
$$

- For a linear demand $Q = 225 - P$,
  
$$
\text{Total revenue } R = P \times Q = P (225 - P)
$$

- **Maximize** $ R $:
  
$$
\frac{dR}{dP} = 225 - 2P = 0 \implies P = 112.5
$$

- Corresponding demand:
  
$$
Q = 225 - 112.5 = 112.5
$$

- Since total stock is 400 million gallons, and demand at this price is 112.5 million gallons per year, the monopolist will sell **112.5 million gallons** in Year 1.

- But **note**: The monopolist might prefer to sell more in Year 1 at a *lower* price if it can be sold at a profit, but since the demand curve is linear and prices are flexible, setting the price at the revenue-maximizing point (112.5) is optimal.

**Alternatively**, because the monopolist controls the entire supply, they might choose to sell **the maximum possible** at the highest price *that consumers will pay*.

Given that, the monopolist will choose to sell **all** stock in Year 1 at the price where marginal revenue equals the opportunity cost (which is zero), but since the demand is linear and the marginal cost is zero, the monopolist prefers to sell **all** in Year 1 at the price $P = 112.5$, selling approximately:

$$
Q = 225 - 112.5 = 112.5
$$

**but** this is only **about 112.5 million gallons**.

**However**, the total supply is 400 million gallons, and the monopolist can allocate it to maximize total profit:

- **Best strategy**: sell the entire stock in Year 1 at $P = 112.5$, then sell the remaining in subsequent years.

**Answer:**
$$
\boxed{\textbf{Approximately 112.5 million gallons}}
$$

*(Note: The original hint suggests setting MR = $75 in Year 1, but in the absence o

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