91Ó°ÊÓ

The platypus is a shy and secretive animal that does not breed well in captivity. But two breeders, Sydney and Adelaide, have discovered the secret to platypus fertility and have effectively cornered the market. Zoos across the globe come to them to purchase their output; the world inverse demand for baby platypuses is given by \(P=1,000-2 Q,\) where \(Q\) is the combined output of Sydney \(\left(q_{S}\right)\) and Adelaide \(\left(q_{A}\right)\). a. Sydney wishes to produce the profitmaximizing quantity of baby platypus. Given Adelaide's choice of output, \(q_{A}\), write an equation for the residual demand faced by Sydney. b. Derive Sydney's residual marginal revenue curve. c. Assume that the marginal and average total cost of raising a baby platypus to an age at which it can be sold is \(\$ 200\). Derive Sydney's reaction function. d. Repeat steps (a), (b), and (c) to find Adelaide's reaction function to Sydney's output choice. e. Substitute Sydney's reaction function into Adelaide's to solve for Adelaide's profitmaximizing level of output. Then use your answer to find Sydney's profit-maximizing level of output. f. Determine industry output, the price of platypus, and the profits of both Sydney and Adelaide. g. If Adelaide were hit by a bus on her way home from work, and Sydney were to become a monopolist, what would happen to industry quantity, price, and profit?

Step by step solution

01

Sydney's Residual Demand

The world inverse demand for baby platypuses is given by \( P = 1000 - 2Q \). Sydney's residual demand is influenced by Adelaide's output \( q_A \). Therefore, Sydney's demand is \( P = 1000 - 2(q_S + q_A) \), which simplifies to \( P = 1000 - 2q_S - 2q_A \).

02

Sydney's Residual Marginal Revenue

Sydney's marginal revenue can be derived from the demand function \( P = 1000 - 2q_S - 2q_A \). To find this, express total revenue as \( TR = P imes q_S = (1000 - 2q_S - 2q_A) imes q_S \). Differentiate to find MR: \( MR = \frac{d(TR)}{d(q_S)} = 1000 - 4q_S - 2q_A \).

03

Sydney's Reaction Function

With a marginal and average total cost of raising a platypus being \( \$ 200 \), set \( MR = MC \). Thus, \( 1000 - 4q_S - 2q_A = 200 \). Solve for \( q_S \) in terms of \( q_A \): \( q_S = 200 - 0.5q_A \).

04

Adelaide's Residual Demand

Similarly to Sydney, Adelaide faces a residual demand \( P = 1000 - 2(q_S + q_A) \). Therefore, Adelaide's demand simplifies to \( P = 1000 - 2q_A - 2q_S \).

05

Adelaide's Residual Marginal Revenue

Adelaide's marginal revenue is derived from \( P = 1000 - 2q_A - 2q_S \). Total revenue for Adelaide is \( TR = P \times q_A = (1000 - 2q_A - 2q_S) \times q_A \). Differentiate to get MR: \( MR = \frac{d(TR)}{d(q_A)} = 1000 - 4q_A - 2q_S \).

06

Adelaide's Reaction Function

Set \( MR = MC \) for Adelaide where \( MC = \$ 200 \), yielding \( 1000 - 4q_A - 2q_S = 200 \). Solve to get \( q_A = 200 - 0.5q_S \).

07

Finding Profit-Maximizing Quantities

Substitute Sydney's reaction function \( q_S = 200 - 0.5q_A \) into Adelaide's, \( q_A = 200 - 0.5q_S \). Simplify to find \( q_A = 133.33 \) and substitute back to find \( q_S = 133.33 \).

08

Determine Industry Output and Price

The combined output \( Q = q_S + q_A = 266.67 \). Substitute into the inverse demand \( P = 1000 - 2(266.67) = 466.67 \).

09

Profits for Sydney and Adelaide

Profit for each equals total revenue minus total cost. Profit for Sydney is \( (466.67 \times 133.33) - (200 \times 133.33) = 35,555.56 \), and similarly for Adelaide. Both earn \( 35,555.56 \) in profits.

10

Monopoly Scenario

If Sydney is a monopolist, \( Q = q_S \) and demand is \( P = 1000 - 2Q \). With \( MR = MC \), solve \( 1000 - 4Q = 200 \), resulting in \( Q = 200 \), \( P = 600 \), and profit \( (600 \times 200) - (200 \times 200) = 80,000 \).

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Key Concepts

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

Residual Demand

In an oligopolistic market, understanding residual demand is crucial for strategic decision-making. This concept refers to the portion of market demand that is not met by competitors, which a specific firm undertakes to serve. Given that Sydney and Adelaide follow the inverse demand curve defined by \( P = 1000 - 2Q \), Sydney’s residual demand needs to subtract Adelaide's output \( q_A \) from the total. Thus, Sydney faces a residual demand expressed as \( P = 1000 - 2(q_S + q_A) \). Simplifying, we find Sydney's portion: \( P = 1000 - 2q_S - 2q_A \).
This formula reveals directly how the actions of Adelaide affect Sydney's market potential. When Adelaide increases her output, Sydney's available share of the market diminishes, demonstrating the interconnected nature of decision-making in an oligopoly.

Marginal Revenue

Marginal revenue (MR) is the additional revenue that a company earns by selling one more unit of a product. In an oligopolistic setting like that of Sydney's platypus business, marginal revenue is influenced by the output of both parties. To derive Sydney's marginal revenue, we start with the total revenue function: \( TR = (1000 - 2q_S - 2q_A) \times q_S \). Differentiating \( TR \) with respect to \( q_S \) provides the marginal revenue: \( MR = \frac{d(TR)}{d(q_S)} = 1000 - 4q_S - 2q_A \).
This equation shows that Sydney's marginal revenue decreases not only as she increases her output, but also as Adelaide increases hers. This relationship is key to understanding strategic interactions in oligopolies, as each firm's decision impacts market dynamics and potential revenue.

Reaction Function

A reaction function in an oligopoly helps delineate a firm's optimal response to the output levels of a competitor. For Sydney, her reaction to Adelaide's output \( q_A \) can be calculated using her marginal revenue and cost considerations. Given that Sydney's marginal and average total cost for raising a platypus is \( \$200 \), and by setting \( MR = MC \), we solve \( 1000 - 4q_S - 2q_A = 200 \). This yields the reaction function: \( q_S = 200 - 0.5q_A \).
The reaction function essentially maps the quantity of output Sydney will choose in response to various levels of production by Adelaide. It reflects how Sydney's profit-maximizing output decreases with an increase in Adelaide's output, confirming the strategic interdependence characteristic of oligopolies.

Profit Maximization

The goal in any market structure, including an oligopoly, is profit maximization, which involves determining the quantity of output that maximizes the difference between total revenue and total cost. For both Sydney and Adelaide, setting \( MR = MC \) is pivotal in achieving this. By substituting Sydney’s reaction function \( q_S = 200 - 0.5q_A \) into Adelaide’s equivalent function \( q_A = 200 - 0.5q_S \), we can solve for their respective outputs: \( q_A = 133.33 \) and \( q_S = 133.33 \).
These outputs combine to yield an industry quantity \( Q = 266.67 \), with each firm capturing a market share that reflects their strategic interactions. Consequently, the market price is set at \( P = 1000 - 2 \times 266.67 = 466.67 \). Profit for each is calculated as revenue minus cost resulting in \( 35,555.56 \) for both Sydney and Adelaide, demonstrating the balancing act of maintaining optimal production levels in competitive environments.