Question

Consider a market with two firms, 1 and 2 producing a homogeneous good. The market demand is \(P=130-2\left(Q_{1}-Q_{2}\right)\), 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 \(T C_{1}=\) \(10 Q_{1}\), while the total cost for firm 2 is \(T C_{2}=10 Q_{2}\). Each firm chooses the quantity to best maximize profits. (a) From the condition \(M R_{1}=M C_{1}\), find the reaction function of firm 1 , and from \(M R_{2}=M C_{2}\), find the reaction function of firm \(2 .\) (b) Find the equilibrium quantity produced by each firm by solving the system of the two reaction functions you found in (a). Sketch your solution graphically. (c) Find the equilibrium price and then find the profit of each firm.

Short answer

Equilibrium quantities: \(Q_1 = 20\), \(Q_2 = 20\); Price: 50; Profits each: 800.

Step-by-step solution

Determine Marginal Cost and Revenue for Firm 1

To maximize profits, we first need to find the marginal revenue (MR) for Firm 1. Starting with the inverse demand function:\[ P = 130 - 2(Q_1 + Q_2) \]We can find the total revenue (TR) for Firm 1 as:\[ TR_1 = P \times Q_1 = (130 - 2(Q_1 + Q_2))Q_1 \]Simplifying, we get:\[ TR_1 = 130Q_1 - 2Q_1^2 - 2Q_1Q_2 \]The marginal revenue is then:\[ MR_1 = \frac{d(TR_1)}{dQ_1} = 130 - 4Q_1 - 2Q_2 \]Given that total cost (TC) for Firm 1 is \( TC_1 = 10Q_1 \), the marginal cost (MC) is:\[ MC_1 = \frac{d(TC_1)}{dQ_1} = 10 \]

Define Firm 1's Reaction Function

Set the marginal revenue equal to the marginal cost to find Firm 1's reaction function:\[ 130 - 4Q_1 - 2Q_2 = 10 \]Solving for \( Q_1 \):\[ 4Q_1 = 120 - 2Q_2 \]\[ Q_1 = 30 - 0.5Q_2 \]This is the reaction function for Firm 1.

Determine Marginal Cost and Revenue for Firm 2

For Firm 2, follow a similar process. The total revenue (TR) and marginal revenue (MR) are:\[ TR_2 = P \times Q_2 = (130 - 2(Q_1 + Q_2))Q_2 \]Thus we have:\[ TR_2 = 130Q_2 - 2Q_2^2 - 2Q_1Q_2 \]The marginal revenue is:\[ MR_2 = \frac{d(TR_2)}{dQ_2} = 130 - 4Q_2 - 2Q_1 \]Given Firm 2's total cost is \( TC_2 = 10Q_2 \), the marginal cost (MC) is:\[ MC_2 = \frac{d(TC_2)}{dQ_2} = 10 \]

Define Firm 2's Reaction Function

Set the marginal revenue equal to the marginal cost to find Firm 2's reaction function:\[ 130 - 4Q_2 - 2Q_1 = 10 \]Solving for \( Q_2 \):\[ 4Q_2 = 120 - 2Q_1 \]\[ Q_2 = 30 - 0.5Q_1 \]This is the reaction function for Firm 2.

Calculate Equilibrium Quantities

Substitute \( Q_1 = 30 - 0.5Q_2 \) into Firm 2's reaction function \( Q_2 = 30 - 0.5Q_1 \).We have:\[ Q_2 = 30 - 0.5(30 - 0.5Q_2) \]\[ Q_2 = 30 - 15 + 0.25Q_2 \]\[ 0.75Q_2 = 15 \]\[ Q_2 = 20 \]Using \( Q_2 = 20 \) back in \( Q_1 = 30 - 0.5Q_2 \):\[ Q_1 = 30 - 0.5 \times 20 = 20 \]So, \( Q_1 = 20 \) and \( Q_2 = 20 \).

Determine Equilibrium Price and Profits

Substitute \( Q_1 = 20 \) and \( Q_2 = 20 \) into the demand equation to find the equilibrium price:\[ P = 130 - 2(20 + 20) = 130 - 80 = 50 \]Now, calculate the profit for each firm using \( \text{Profit} = \text{Total Revenue} - \text{Total Cost} \):For Firm 1:\[ \text{Profit}_1 = 50 \times 20 - 10 \times 20 = 1000 - 200 = 800 \]For Firm 2:\[ \text{Profit}_2 = 50 \times 20 - 10 \times 20 = 1000 - 200 = 800 \]

Graphical Representation

To sketch the solution, plot both reaction functions with \( Q_1 = 30 - 0.5Q_2 \) and \( Q_2 = 30 - 0.5Q_1 \) on a coordinate plane. The equilibrium occurs at the intersection of these lines, which we found as \( Q_1 = 20 \) and \( Q_2 = 20 \). This point reflects the equilibrium quantities for both firms in the market.

Key concepts

Reaction Functions

In Cournot competition, firms determine their output based on the quantity they believe their competitor will produce. This strategic decision-making is captured in what we call reaction functions. Each firm calculates its optimal response, or its best quantity to produce, given the production level of the other firm.

The reaction function for a firm is derived from setting its marginal revenue equal to its marginal cost. For example, Firm 1's reaction function in the market is established from the equation \(MR_1 = MC_1\). By rearranging the demand function and calculating the marginal revenue for Firm 1, we find:

\(Q_1 = 30 - 0.5Q_2\),

where \(Q_1\) depends on the quantity \(Q_2\) produced by Firm 2. Similarly, the reaction function for Firm 2 is:

\(Q_2 = 30 - 0.5Q_1\).

The intersection of these functions determines the equilibrium in the Cournot model. The reaction functions provide insights into how each firm's output decision is a response to the expected output of its rival. Understanding them is crucial for determining competitive equilibrium in oligopoly markets.

Equilibrium Quantities

The equilibrium quantities in a Cournot competition are found where the reaction functions of both firms intersect. This intersection point represents the optimal quantity for each firm such that neither wants to change its output, given the output of the other.

To find the equilibrium quantities in our example, we substitute the expression for \(Q_1\) from Firm 1's reaction function into Firm 2's reaction function and solve:

• Start with \(Q_1 = 30 - 0.5Q_2\)
• Insert into Firm 2's function: \(Q_2 = 30 - 0.5Q_1\)
• Solve to find \(Q_2 = 20\) and \(Q_1 = 20\)

The equilibrium quantities, \(Q_1 = 20\) and \(Q_2 = 20\), suggest that each firm opts to produce 20 units. This decision ensures mutual profit maximization, with neither firm having an incentive to alter its production while the other maintains its quantity. The equilibrium reflects a balance where competitive tensions are neutralized by strategic output choices.

Marginal Cost and Marginal Revenue

In Cournot competition, understanding marginal cost (MC) and marginal revenue (MR) is essential for determining each firm's reaction function. Marginal cost is the additional cost incurred by producing one more unit of output, while marginal revenue is the additional revenue generated from selling one more unit.

For profit maximization, a firm adjusts its production until its marginal revenue equals its marginal cost. In our example, both firms have a marginal cost expressed simply as \(MC = 10\).

The marginal revenue is found from differentiating the total revenue with respect to quantity. For Firm 1, this led to a marginal revenue expression of \(MR_1 = 130 - 4Q_1 - 2Q_2\). For Firm 2, the corresponding calculation produced \(MR_2 = 130 - 4Q_2 - 2Q_1\).

Equating marginal revenue with marginal cost derived each firm’s reaction function, demonstrating how closely tied marginal figures are with strategic production decisions. A thorough grasp of these concepts is key in understanding the broader mechanics of market competition and equilibrium.

Profit Maximization

Profit maximization is the primary goal for firms in Cournot competition. It entails making output decisions that maximize the difference between total revenue and total cost.

The steps to achieve profit maximization start with understanding the relationship between output, cost, and revenue. Each firm calculates its total cost and total revenue to find its profit.

In the exercise, after determining the equilibrium quantities \(Q_1 = 20\) and \(Q_2 = 20\):

• Substitute these quantities into the demand equation to find the equilibrium price: \(P = 50\).
• Calculate profit for each firm using the formula: \(\text{Profit} = \text{Revenue} - \text{Cost}\).
• For both firms, the profit is \(800\).

By setting marginal cost equal to marginal revenue, each firm identifies its optimal output level. Saturating the market at this point ensures maximum profit without initiating price wars that could erode profits. This delicate balancing act forms the crux of Cournot firms' strategic output determination.