Problem 15 Two farmers, Tito and Helen, sup... [FREE SOLUTION]

Chapter 11: Problem 15

Two farmers, Tito and Helen, supply a chain of islands with kale. The inverse demand for kale in the islands is given by $P=60-0.5 Q$ where $Q$ is the combined output of Tito $\left(q_{T}\right)$ and Helen $\left(q_{H}\right)$. measured in 10 -pound bunches. Tito grows kale at a constant marginal and average cost of $\$ 12$ per bunch; Helen grows kale at a constant marginal and average cost of $\$ 10$ per bunch. a. Suppose this market is a Stackelberg oligopoly and Tito is the first-mover. How much will he and Helen produce? What will the market price of kale be? How much profit will each farmer earn? b. Now suppose that Helen is the first-mover in this Stackelberg oligopoly. How much will each farmer produce? What will the market price of kale be? How much profit will each farmer earn? c. Quantify the value to Helen of being the first-mover in this Stackelberg game.

Short Answer

Expert verified

a) Tito: 28.67, Helen: 42.83, Price: $24.25, Tito's profit: $349.63, Helen's profit: $615.64. b) Helen: 60, Tito: 18, Price: $21, Helen's profit: $660, Tito's profit: $162. c) Helen gains $44.36 more as the first mover.

Step by step solution

Understanding Stackelberg Model

In the Stackelberg model, one firm is the leader and the other is the follower. The leader commits to a quantity first, and the follower reacts to this quantity.

Tito as First Mover (Finding Reaction Function)

Since Tito is the first mover, he chooses his quantity $q_T$ considering Helen's reaction. The reaction function for Helen is derived by maximizing her profit $\pi_H = (60 - 0.5(q_T + q_H))q_H - 10q_H$. Taking the derivative and setting it to zero gives her reaction function: $q_H = 50 - 0.25q_T$.

Tito's Profit Maximization

Substitute Helen's reaction $q_H = 50 - 0.25q_T$ into the inverse demand function $P = 60 - 0.5(q_T + q_H)$ to express price in terms of $q_T$. Tito maximizes his profit $\pi_T = (60 - 0.5(q_T + (50 - 0.25q_T)))q_T - 12q_T$. Simplifying gives $\pi_T = (33.5 - 0.375q_T)q_T - 12q_T$, then solve $\frac{d\pi_T}{dq_T} = 0$ to find $q_T$.

Solving for Tito's Optimal Quantity

Solving $\frac{d((21.5 - 0.375q_T)q_T)}{dq_T} = 0$ results in $q_T = 28.67$. Substitute $q_T$ into $q_H = 50 - 0.25q_T$ to find $q_H$.

Calculate Helen's Output, Market Price, and Profits

With $q_T = 28.67$, substituting into $q_H = 50 - 0.25q_T$ gives $q_H = 42.83$. Total output $Q = q_T + q_H = 71.5$. The price is $P = 60 - 0.5 \times 71.5 = 24.25$. Tito's profit: $\pi_T = (24.25 - 12) \times 28.67 = 349.63$. Helen's profit: $\pi_H = (24.25 - 10) \times 42.83 = 615.64$.

Helen as First Mover (Finding Reaction Function)

Now Helen is the first mover. Tito's reaction function is derived by maximizing his profit $\pi_T = (60 - 0.5(q_T + q_H))q_T - 12q_T$. Solving yields $q_T = 48 - 0.5q_H$.

Helen's Profit Maximization

Insert Tito's reaction $q_T = 48 - 0.5q_H$ into the demand $P = 60 - 0.5(q_H + (48 - 0.5q_H))$, and maximize Helen's profit $\pi_H = (60 - 0.5(q_H + (48 - 0.5q_H)))q_H - 10q_H$. Solving the derivative, we find $q_H$.

Solving for Helen's Optimal Quantity

The solution of $\frac{d((36 - 0.25q_H)q_H)}{dq_H} = 0$ leads to $q_H = 60$. Substitute $q_H$ into $q_T = 48 - 0.5q_H$ to determine $q_T$.

Calculate Tito's Output, Market Price, and Profits

With $q_H = 60$, $q_T = 48 - 0.5 \times 60 = 18$. Total output $Q = q_T + q_H = 78$. The price is $P = 60 - 0.5 \times 78 = 21$. Helen's profit: $\pi_H = (21 - 10) \times 60 = 660$. Tito's profit: $\pi_T = (21 - 12) \times 18 = 162$.

Value of Helen Being First Mover

The value of Helen being the first mover is the difference in her profit: $660 - 615.64 = 44.36$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Demand

Inverse demand is a concept that defines the relationship between the price of a good and its quantity demanded, but in a reverse framework. The formula given as an example in our exercise is:

$ P = 60 - 0.5Q $, where $ P $ is the market price, and $ Q $ is the total quantity produced by both farmers.

This setup implies that as more kale is produced, the price consumers are willing to pay decreases. This negative relationship is an essential feature of most demand curves. Here鈥檚 how it works in our exercise:

When the total quantity $ Q $ is low, the price $ P $ is closer to 60, indicating high consumer willingness to pay more for lower availability.
Conversely, when $ Q $ is high, the price $ P $ approaches 0, showing that with increased availability, price must decrease for the quantity to be sold.

Understanding inverse demand is vital in determining the market strategies that Tito and Helen use in their production decisions.

Marginal Cost

Marginal cost (MC) is a critical metric used to determine the cost of producing one additional unit of a product. In the case of our two kale producers:

Tito has a marginal and average cost of $ \\(12 $ per bunch.
Helen, on the other hand, has a slightly lower marginal and average cost at $ \$10 \) per bunch.

The marginal cost influences how much each farmer decides to produce. Since Helen has a lower marginal cost, she has a potential competitive advantage as she can profit from lower prices. This advantages her in situations where market prices fall, as she continues to make a profit where Tito might not.

Profit Maximization

Profit maximization occurs when a firm chooses the level of output where it can make the most profit possible. For both farmers in this Stackelberg game, the goal is to determine

a quantity of kale that exceeds their cost of production significantly while taking market dynamics into account.

The basic formula used throughout involves calculating profit

$ \pi = PQ - C(Q) $, where $ \pi $ is profit, $ PQ $ is total revenue, and $ C(Q) $ is total cost.

For Stackelberg leaders and followers:

The leader determines their production level first, anticipating the follower's response鈥攃alculated from the follower鈥檚 reaction function.
Using these calculations, each farmer can determine the profit-maximizing level of output and adapt to trends or changes in market conditions accordingly.

Through sequential decision-making, focusing on the reaction functions allows players to maximize their respective profits carefully.

Market Price

In the context of Stackelberg oligopoly, the market price of a product is contingent upon the combined output of the firms and the inverse demand function. To find the market price, use the formula:

$ P = 60 - 0.5Q $, which involves substituting the total output $ Q = q_T + q_H $.

This means:

As Tito and Helen adjust their production bases strategically upon their roles as first movers or followers, they influence the total market supply $ Q $.
The resulting market price $ P $ adjusts based on this new total quantity, indicating what consumers would pay for the available kale.

Additionally, the Stackelberg model emphasizes the first-mover advantage in setting a favorable output level that can positively shape market prices to a firm's benefit. Consequently, the skilled integration of market conditions enables Tito or Helen to lead with market prices that optimize their profits efficiently.

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