Question

Suppose that we have two firms that face a linear demand curve \(p(Y)=\) \(a-b Y\) and have constant marginal costs, \(c,\) for each firm. Solve for the Cournot equilibrium output.

Short answer

In Cournot equilibrium, each firm produces \( \frac{a - c}{3b} \).

Step-by-step solution

Define the market demand

The market demand function is given by the linear equation \( p(Y) = a - bY \), where \( Y \) is the total output in the market, \( p \) is the price, \( a \) is the intercept, and \( b \) is the slope of the demand curve.

Determine individual firm's output

Since there are two firms, let \( q_1 \) and \( q_2 \) be the output by Firm 1 and Firm 2, respectively. The total market output \( Y \) is then \( Y = q_1 + q_2 \).

Write profit function for each firm

The profit for each firm is given by profit \( \pi_i = (p - c)q_i \), where \( p \) is the price determined by the demand function. Substituting the demand function, the profit for Firm 1 becomes \( \pi_1 = (a - b(q_1 + q_2) - c)q_1 \). Similarly, for Firm 2, \( \pi_2 = (a - b(q_1 + q_2) - c)q_2 \).

Calculate the reaction function for Firm 1

To find Firm 1's reaction function, take the derivative of its profit function with respect to \( q_1 \) and set it equal to zero for maximization: \[ \frac{d\pi_1}{dq_1} = a - 2bq_1 - bq_2 - c = 0 \]. This gives Firm 1's reaction function: \( q_1 = \frac{a - c}{2b} - \frac{q_2}{2} \).

Calculate the reaction function for Firm 2

Similarly, differentiate Firm 2's profit function with respect to \( q_2 \) and set it equal to zero: \[ \frac{d\pi_2}{dq_2} = a - 2bq_2 - bq_1 - c = 0 \]. This gives Firm 2's reaction function: \( q_2 = \frac{a - c}{2b} - \frac{q_1}{2} \).

Solve the system of reaction functions

Substitute Firm 2's reaction function into Firm 1's reaction function: \( q_1 = \frac{a - c}{2b} - \frac{1}{2} \left( \frac{a - c}{2b} - \frac{q_1}{2} \right) \). Simplify and solve for \( q_1 \): \( q_1 = \frac{a - c}{3b} \).

Determine equilibrium quantities

Substitute \( q_1 = \frac{a - c}{3b} \) into Firm 2's reaction function: \( q_2 = \frac{a - c}{3b} \). Thus, \( q_1 = q_2 = \frac{a - c}{3b} \) is the Cournot equilibrium output for each firm.

Key concepts

Linear Demand Curve

In economics, a linear demand curve is a straightforward representation of consumer demand where price and quantity have a direct relationship. For our exercise, the demand curve is expressed as \( p(Y) = a - bY \). Here, \( p \) is the price, \( Y \) signifies total quantity in the market, \( a \) is the maximum price consumers are willing to pay when no items are available, and \( b \) is the slope indicating how price changes with quantity.
Understanding the slope \( b \) is pivotal because it tells us how sensitive the price is to changes in the total supply. A steeper slope means prices fall quickly as quantity increases, signaling high sensitivity. Thus, firms must carefully consider their output decisions, knowing that increasing supply may drop prices substantially due to the linear nature of the demand curve.

• \( a \): maximum willingness to pay
• \( b \): rate of price decline with supply increase
• \( p(Y) \): market price at total output \( Y \)

Reaction Function

A reaction function outlines how a firm adjusts its output based on its competitor's choices to maximize profits. In the Cournot model, where firms choose quantities simultaneously, these functions are crucial.
For Firm 1, its reaction to Firm 2's output \( q_2 \) is expressed mathematically by first taking the derivative of its profit function concerning its output \( q_1 \). Setting this derivative to zero reveals Firm 1's best response: \( q_1 = \frac{a - c}{2b} - \frac{q_2}{2} \).
This calculation shows that Firm 1's optimal output depends not only on the market parameters \( a \), \( b \), and \( c \), but also on how much Firm 2 produces.

• Firm 1's reaction: \( q_1 = \frac{a - c}{2b} - \frac{q_2}{2} \)
• Firm 2's mimicry: \( q_2 = \frac{a - c}{2b} - \frac{q_1}{2} \)

Their outputs are intertwined, reflecting strategic dependencies—a hallmark of Cournot competition.

Profit Maximization

Profit maximization is the primary aim of firms in the Cournot model. Think of it as the company trying to get the most peanuts from the peanut butter jar without spilling any! For each firm, that means choosing the level of output that maximizes their profit given by \( \pi_i = (p - c)q_i \), which accounts for both the revenue and costs. Here, \( p \) reflects the demand curve, and \( c \) is the marginal cost.
To maximize profit, each firm's task is to adjust their output until the marginal profit—the extra profit from selling one more unit—is zero. By differentiating their profit function and setting it to zero, they find this optimal point.
This leads us to the reaction functions, which signal how much one firm should produce, considering the other firm's output. These interactions make the Cournot equilibrium a fine balancing act: each firm must constantly guess the other’s next move, tweaking their production level to maximize profit while avoiding oversupply.

Marginal Cost

Marginal cost is a critical concept in economics, particularly in understanding firms' decision-making with production. It refers to the additional cost of producing one more unit of output.
In Cournot competition, where firms vie for market dominance, marginal cost is pivotal to setting output levels. In the formula \( \pi_i = (p - c)q_i \), \( c \) is the constant marginal cost for each firm.
Constant marginal cost simplifies calculations, as each additional unit costs the same to produce. This means firms will only continue to produce extra units that allow them to achieve a profit greater than this cost. Therefore, equilibrium is reached when each firm's production levels are such that the market price \( p \) exactly covers the marginal cost \( c \), without any resulting surplus.

• Key relationship: Production ceases when \( p = c \)
• Constant \( c \): enables straightforward profit maximization calculations

If \( c \) were higher, firms would need to reduce output to maintain profitability, showcasing the strategic importance of managing costs.