Question

Consider a market with two firms, 1 and 2 , producing a homogeneous good. The market demand is \(P=100-3\left(Q_{1}+Q_{2}\right)\), where \(Q_{1}\) is the quantity produced by firm 1 and \(Q_{2}\) is the quantity produced by firm 2 . The total cost for firm 1 is \(T C_{1}=\) \(40 Q_{1}\), while the total cost for firm 2 is \(T C_{2}=40 Q_{2}\). Each firm behaves like a competitive firm. a) What is the equilibrium quantity in the market? b) Suppose both firms exhibit Cournot behaviour. Given that their reaction functions are \(Q_{1}=20-2 Q_{2}\) and \(Q_{2}=20-2 Q_{1}\), how would their output change compared to \((\mathrm{a}) ?\)

Short answer

Equilibrium in (a): Total output = 20. In (b): Firm 1 = 8, Firm 2 = 4; Total = 12.

Step-by-step solution

Understand the Market Demand

The market demand function is given by \( P = 100 - 3(Q_1 + Q_2) \). This expresses the price (\( P \)) as a function of the total quantity (\( Q = Q_1 + Q_2 \)) produced by both firms.

Calculate Equilibrium in Perfect Competition

In a competitive market, each firm maximizes profit by producing where price equals marginal cost. Here, the marginal cost for each firm is the derivative of their total cost function, which is 40 (since \( TC_1 = 40Q_1 \) and \( TC_2 = 40Q_2 \)). Set price equal to marginal cost: \( 100 - 3(Q_1 + Q_2) = 40 \). Simplifying gives \( Q_1 + Q_2 = 20 \).

Cournot Equilibrium Analysis

Given the reaction functions for Cournot competition: \( Q_1 = 20 - 2Q_2 \) and \( Q_2 = 20 - 2Q_1 \), solve these simultaneously. Substitute \( Q_2 = 20 - 2Q_1 \) into the equation for \( Q_1 \), leading to \( Q_1 = 20 - 2(20 - 2Q_1) \), which simplifies to \( Q_1 = 8 \). Similarly, substituting \( Q_1 = 8 \) into the equation for \( Q_2 \) gives \( Q_2 = 20 - 2(8) = 4 \).

Compare Outputs from A and B

In (a), the total output is \( Q_1 + Q_2 = 20 \). For (b), the total output is \( Q_1 + Q_2 = 8 + 4 = 12 \), indicating a decrease in total market output under Cournot behavior.

Key concepts

Cournot competition

In the world of oligopolies, where just a few firms dominate the market, Cournot competition offers a fascinating glimpse into strategic decision-making. Cournot competition is named after the French economist Augustin Cournot. He introduced this model in 1838, focusing on the behaviour of firms that decide on the quantity to produce simultaneously. What makes Cournot competition unique is that each firm considers the output level of its competitors when deciding its own output.

To dive a bit deeper:

• Firms in a Cournot model do not set prices directly; instead, they determine the quantity to produce.
• Each firm assumes its competitor's output is fixed when deciding their own production level, leading to strategic interdependence.
• Reaction functions, like those given in the exercise as \( Q_1 = 20 - 2Q_2 \) and \( Q_2 = 20 - 2Q_1 \), are crucial in determining each firm's best response to the expected quantities produced by competitors.
• The equilibrium reached, known as the Cournot equilibrium, is where neither firm has an incentive to change its output, given the competitor's production decision.

This model is particularly relevant in industries where firms can act strategically over their production quantities, influencing the total market quantity and, subsequently, the price.

Market demand

Understanding market demand is critical when analyzing how firms operate in an oligopoly. The market demand curve shows the relationship between the price customers are willing to pay and the quantity of goods available. In this exercise, the demand function provided is \( P = 100 - 3(Q_1 + Q_2) \). This equation tells us that as total production from both firms increases, the market price decreases.

Key elements of a demand curve:

• The demand curve is typically downward sloping, illustrating the inverse relationship between price and quantity demanded.
• Parameters like "100" indicate the maximum price consumers would pay if no goods were available.
• The "3" in the equation represents the sensitivity of price changes to changes in quantity (the slope of the demand curve).

In the context of oligopolies such as this, understanding the demand curve helps firms predict how changes in their output, as well as that of their competitors, can impact the overall market price. It's crucial for identifying the potential profit-maximizing levels of output while considering market constraints.

Equilibrium quantity

The concept of equilibrium quantity is central in economic models involving oligopolies like Cournot competition. In our context, the equilibrium quantity refers to the point where the quantities each firm chooses lead to a stable outcome — neither firm has the desire to deviate from its chosen production level given the other firm's choice.

Calculating the equilibrium quantity involves:

• Understanding firms' cost structures, which influence their production decisions. Here, both firms have a similar total cost of 40 times the quantity produced.
• Using the market demand function to set prices equal to marginal costs. Initially, this led to a total quantity of 20 in perfect competition, where the sum of \( Q_1 + Q_2 \) matches production to meet zero economic profit.
• Using Cournot reaction functions to find where both firms' best responses intersect, resulting in the Cournot equilibrium. Solving the given reaction functions \( Q_1 = 8 \) and \( Q_2 = 4 \) gives us a total equilibrium quantity of 12, showing a decrease compared to a perfectly competitive scenario.

Through these calculations, we see how different competitive behaviours can lead to variations in equilibrium quantities. This impacts consumer prices and market dynamics, which are vital considerations for firms in oligopolistic markets.