Question

Two identical firms, 1 and 2, compete on quantities. The reaction function of firm 1 is \(\mathrm{Q}_{1}=15-1 / 2 \mathrm{Q}_{2}\), while for firm 2 we have \(\mathrm{Q}_{2}=15-1 / 2 \mathrm{Q}_{1}\), In the table below we have the total quantity produced in the market: $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline Q 1+Q 2 & 2 & 6 & 10 & 14 & 18 & 22 & 26 & 30 & 34 \\ \hline \end{array} $$ Using the fact that both firms must produce the same quantity, plot the reaction functions of the two firms in a graph. How is the equilibrium quantity determined?

Short answer

The equilibrium quantity for each firm is 10, and the total market quantity is 20.

Step-by-step solution

Understand the Reaction Functions

First, let's clarify that the reaction function describes how each firm's output depends on its competitor's output. For Firm 1, the reaction function is \( Q_1 = 15 - \frac{1}{2} Q_2 \) and for Firm 2, it is \( Q_2 = 15 - \frac{1}{2} Q_1 \).

Set Both Quantities Equal

In equilibrium, both firms produce the same quantity (\( Q_1 = Q_2 \)). Hence, we set the two reaction functions equal: \( 15 - \frac{1}{2} Q = Q \), where \( Q \) represents the equilibrium quantity produced by each firm.

Solve for Equilibrium Quantity

Substitute \( Q \) for \( Q_1 \) and \( Q_2 \): \( Q = 15 - \frac{1}{2} Q \). To solve, add \( \frac{1}{2} Q \) to both sides: \( Q + \frac{1}{2} Q = 15 \). This simplifies to \( \frac{3}{2} Q = 15 \), hence \( Q = 10 \).

Determine Total Market Quantity

Once we have \( Q = 10 \) for each firm at equilibrium, the total quantity in the market, \( Q_1 + Q_2 \), is \( 10 + 10 = 20 \).

Verify Equilibrium and Plot Reaction Functions

The equilibrium quantity \( Q = 10 \) should satisfy both original reaction functions, and indeed: \( 15 - \frac{1}{2}(10) = 10 \). On the graph of the reaction functions, plot \( Q_1 = 15 - \frac{1}{2} Q_2 \) and \( Q_2 = 15 - \frac{1}{2} Q_1 \) and observe the intersection at point \((10, 10)\).

Key concepts

Reaction Functions

In a Cournot Duopoly, reaction functions play a crucial role in understanding how firms determine their output. Essentially, a reaction function shows the optimal quantity a firm should produce, given the quantity produced by its rival. Here, Firm 1's reaction function is expressed as \( Q_1 = 15 - \frac{1}{2} Q_2 \). This means that Firm 1’s optimal output decreases as Firm 2 increases its output. Similarly, Firm 2's reaction function \( Q_2 = 15 - \frac{1}{2} Q_1 \) has the same dynamic.

This interdependence highlights the strategic nature of competition in duopolies. Each firm adjusts its quantity based on the other's actions, striving to maximize profits amidst the uncertainties of its competitor's responses.

Equilibrium Quantity

The equilibrium quantity is the amount each firm produces when both firms perfectly anticipate each other's output decisions. In the Cournot model, equilibrium is reached when each firm's output choice is optimal given the other firm's choice. To find this equilibrium, we set \( Q_1 \) equal to \( Q_2 \), leading to the equation \( Q = 15 - \frac{1}{2} Q \).

Solving this results in \( Q = 10 \), meaning both Firm 1 and Firm 2 produce 10 units. This balance is critical because it defines a stable state where neither firm benefits from altering its output, assuming the other keeps its output constant, thereby demonstrating a key aspect of strategic decision-making in competitive markets.

Market Competition

Market competition is the driving force behind the outputs in a Cournot Duopoly. Here, firms compete by choosing quantities rather than prices, focusing on maximizing their individual profits through their production decisions. The competition is inherently strategic, as each firm's profits depend on its own output and on the output chosen by the competitor.

This model mirrors many real-world scenarios where oligopolistic firms must continuously adjust to their rivals' output levels. The reaction functions illustrate how market competition shapes business strategies, with each firm aiming to outmaneuver the other by predicting and reacting to its competitor's moves—a classic hallmark of duopolistic market structures.

Game Theory

Game theory provides the foundation for analyzing strategic interactions, like those seen in a Cournot Duopoly. Within this framework, firms are considered players in a game, each seeking to maximize their payoff through optimal decision-making.

Game theory’s application here demonstrates the importance of anticipating competitors’ actions and responses. The reaction functions serve as strategies that each firm employs, showcasing their understanding of the strategic environment. Analyzing such models through game theory helps explain why firms might settle at a particular equilibrium, highlighting the balanced, yet competitive nature of their interactions.

• The Nash Equilibrium in this context is pivotal, as it ensures that no firm has anything to gain by changing its production unilaterally, providing insight into both firm behavior in competitive markets and broader strategic considerations in business.