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Chapter 13: Problem 7

A market has an inverse demand curve \(p=340-2 Q\) and five firms, each of which has a constant marginal cost of \(M C=20 .\) If the firms form a profit- maximizing cartel and agree to operate subject to the constraint that each firm will produce the same output level, how much does each firm produce? (Hint: See Chapter 11's treatment of monopoly.) A

Short Answer

Expert verified
Each firm produces 16 units.

Step by step solution

01

Understand the Cartel Scenario

When firms form a cartel, they act like a single monopolist to maximize total profits. In this case, the five firms will decide together on the total quantity to produce in the market and then split that quantity equally.
02

Identify the Marginal Revenue

For the inverse demand curve given as \( p = 340 - 2Q \), the corresponding total revenue \( TR \) is given by \( TR = p \times Q = (340-2Q) \times Q = 340Q - 2Q^2 \). The marginal revenue \( MR \) is the derivative of the total revenue with respect to \( Q \), so \( MR = \frac{d(340Q - 2Q^2)}{dQ} = 340 - 4Q \).
03

Equate Marginal Revenue to Marginal Cost

To find the profit-maximizing quantity for the cartel, we set the marginal revenue (\( MR \)) equal to the marginal cost (\( MC \)), which is constant at 20. So, we have \( 340 - 4Q = 20 \).
04

Solve for the Total Cartel Output, Q

Rearrange the equation from the previous step: \( 340 - 4Q = 20 \) becomes \( 320 = 4Q \). Solving for \( Q \), we divide both sides by 4: \( Q = \frac{320}{4} = 80 \).
05

Divide the Output Equally Among Firms

Since there are five firms in the cartel, and it's agreed to share the production equally, each firm will produce \( \frac{80}{5} = 16 \) units.

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Key Concepts

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

Inverse Demand Curve
An inverse demand curve is a key concept in economics that establishes a relationship between the price of a product and the quantity demanded by consumers. In simple terms, it expresses price as a function of quantity. In this exercise, the inverse demand curve is given by the equation \( p = 340 - 2Q \). This equation means that as the quantity \( Q \) increases, the price \( p \) decreases.
  • "Inverse" because it expresses price rather than quantity as a function.
  • Helps determine the maximum price consumers are willing to pay for each unit of output.
  • Crucial for firms in deciding how much to produce to maximize revenue.
The inverse demand curve allows firms to understand how price adjustments might affect demand levels. This understanding is especially important for firms operating within a cartel, like in this exercise, as they collectively decide on output levels to maximize profits.
Marginal Cost
Marginal cost (MC) refers to the additional cost of producing one more unit of a product. In the exercise, each firm in the cartel has a constant marginal cost of \( MC = 20 \). Constant marginal costs mean that producing additional units always costs the same amount.
  • Crucial for profit maximization because it represents the cost of increasing production.
  • A constant MC simplifies calculations and decision-making.
  • In a competitive market, firms produce until MC equals marginal revenue (MR).
In our scenario, the firms operate as a cartel, setting MC equal to marginal revenue to determine the optimal production level. By understanding and minimizing marginal costs, firms ensure efficient production and maximize potential profits.
Marginal Revenue
Marginal revenue (MR) is the increase in total revenue from selling one more unit of a product. To find the marginal revenue, we derive it from the total revenue equation, which is often related to the inverse demand curve. For this problem, the total revenue \( TR \) was calculated as \( TR = 340Q - 2Q^2 \), and its derivative gives the marginal revenue \( MR = 340 - 4Q \).
  • Helps determine how output changes affect total revenue.
  • Essential for understanding how to scale production efficiently.
  • In monopoly and cartel situations, firms equate MR to MC for profit maximization.
Marginal revenue considerations help the cartel decide the level of output to collectively produce and sell, balancing against marginal costs to achieve optimal profitability.
Monopoly Behavior
Monopoly behavior refers to how a single firm or a group acting as a monopoly, like a cartel, controls the market to maximize profits. Such entities focus on adjusting output to influence prices, contrary to firms in a competitive market that are price takers.
  • Monopolies and cartels set prices higher and output lower than in competitive markets.
  • They equate MR and MC for profit maximization, just like in this exercise where the cartel sets \( MR = MC \).
  • Monopoly power allows them to adjust quantities to affect prices directly.
In our example, five firms form a cartel that behaves similarly to a monopoly, deciding total output collectively. By treating the group as a single entity, they can manipulate market conditions to their advantage, highlighting the strategic benefits of monopoly behavior.

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Most popular questions from this chapter

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