Question

Consider a more general Cournot model than the one presented in this chapter. Suppose there are \(n\) firms. The firms simultaneously and independently select quantities to bring to the market. Firm \(i\) 's quantity is denoted \(q_{i}\), which is constrained to be greater than or equal to zero. All of the units of the good are sold, but the prevailing market price depends on the total quantity in the industry, which is \(Q=\sum_{i=1}^{n} q_{i}\). Suppose the price is given by \(p=a-b Q\) and suppose each firm produces with marginal cost \(c\). There is no fixed cost for the firms. Assume \(a>c>0\) and \(b>0\). Note that firm \(i\) 's profit is given by \(u_{i}=p(Q) q_{i}-c q_{i}=(a-b Q) q_{i}-c q_{i}\). Defining \(Q_{-i}\) as the sum of the quantities produced by all firms except firm \(i\), we have \(u_{i}=\left(a-b q_{i}-b Q_{-i}\right) q_{i}-c q_{i}\). Each firm maximizes its own profit. (a) Represent this game in the normal form by describing the strategy spaces and payoff functions. (b) Find firm \(i\) 's best-response function as a function of \(Q_{-i}\). Graph this function. (c) Compute the Nash equilibrium of this game. Report the equilibrium quantities, price, and total output. (Hint: Summing the best-response functions over the different players will help.) What happens to the equilibrium price and the firm's profits as \(n\) becomes large? (d) Show that for the Cournot duopoly game \((n=2)\), the set of rationalizable strategies coincides with the Nash equilibrium.

Short answer

In equilibrium: \(q_i^* = \frac{a-c}{b(n+1)}\) per firm, total output \(Q = \frac{n(a-c)}{b(n+1)}\), price \(p \rightarrow c\) as \(n\) increases. For \(n=2\), Nash equilibrium matches rationalizable strategies.

Step-by-step solution

Define Strategy Spaces

In the Cournot model with \(n\) firms, each firm \(i\) chooses a quantity \(q_i \geq 0\) to produce. The strategy space for each firm is all non-negative quantities that can be produced. Let's denote firm \(i\)'s strategy space as \(S_i = \{ q_i \geq 0 \ \} \). The strategies for all firms together are represented by the set \(S = S_1 \times S_2 \times \cdots \times S_n \).

Define Payoff Functions

Firm \(i\)'s profit function given a production quantity \(q_i\) is \(u_i = (a-bQ)q_i - c q_i\), where total quantity \(Q = \sum_{j=1}^{n} q_j\). Simplifying, we get \(u_i = (a - bq_i - bQ_{-i})q_i - c q_i\), where \(Q_{-i} = \sum_{j eq i} q_j\). Each firm seeks to maximize its profit by selecting \(q_i\).

Derive Best-Response Function

Firm \(i\)'s profit \(u_i\) depends on \(q_i\). For maximizing profit, take the derivative of the profit function with respect to \(q_i\): \(\frac{\partial u_i}{\partial q_i} = a - 2bq_i - bQ_{-i} - c\). Setting it to zero gives the best-response function: \[ q_i^* = \frac{a - c - bQ_{-i}}{2b} \].

Graph the Best-Response Function

The best-response function \(q_i^* = \frac{a - c - bQ_{-i}}{2b}\) is a linear decreasing function in terms of \(Q_{-i}\). On a graph with \(q_i\) on the vertical axis and \(Q_{-i}\) on the horizontal axis, plot the line showing the relationship.

Compute Nash Equilibrium

To find the Nash equilibrium, substitute \(q_i^*\) in symmetrically for all firms: \(n q_i^* = Q = \sum_{i=1}^{n} q_i = \sum_{i=1}^{n} \left (\frac{a - c}{b(n+1)} \right ) = \frac{n(a - c)}{b(n+1)}\). Solving gives equilibrium quantity for each firm as \(q_i^* = \frac{a - c}{b(n+1)}\). The equilibrium price is \(p = a - bQ = a - b \left(\frac{n(a - c)}{b(n+1)}\right)\).

Analyze Effect of Increasing \(n\)

As \(n\) increases, equilibrium quantity for each firm \(q_i^* = \frac{a - c}{b(n+1)}\) decreases, and total output \(Q = \frac{n(a - c)}{b(n+1)}\) approaches \(a/b\). The market price approaches \(c\), reducing each firm's profit because the price converges with their marginal cost.

Cournot Duopoly and Rationalizable Strategies

For two firms (\(n=2\)), set \(q_1^* = \frac{a-c-bq_2}{2b}\) and \(q_2^* = \frac{a-c-bq_1}{2b}\). Solving this system gives \(q_1^* = q_2^* = \frac{a-c}{3b}\) and \(Q = \frac{2(a-c)}{3b}\), which is the Nash equilibrium. In this duopoly, these strategies are also the set of rationalizable strategies as no other strategy can yield a higher payoff assuming rationality.

Key concepts

Nash Equilibrium

The Nash Equilibrium is a concept in game theory where each player's strategy is optimal given the strategies of all other players. In the Cournot model, this means each firm selects a quantity to produce such that no firm can increase its profit by unilaterally changing its production quantity. In our example, the Nash Equilibrium can be calculated by finding the point where the best-response functions of all firms intersect. Here, each firm decides on the quantity to produce based on the quantities chosen by the other firms.

This results in an equilibrium where every firm is producing at a level that maximizes their profit, given the competition's output. If a firm produces more or less, it could decrease its profit. Thus, at Nash Equilibrium, the mutual best responses result in stable output quantities for each firm in the market.

Cournot Model

The Cournot Model is a foundational game theory model that describes an industry with firms competing by selecting quantities to produce. This model assumes that each firm makes its decision simultaneously and independently, with a focus on how much to produce rather than the price.

The key feature of the Cournot Model is that the market price is determined by the total output produced by all firms. The firms are aware of this and strategically choose their production levels to maximize their profits.

• Each firm's decision impacts the market price.
• The model assumes firms produce a homogeneous product.
• Firms base their production on the predicted behavior of competitors.

This model helps us understand how firms' output decisions affect market dynamics and how equilibrium is achieved in oligopolistic markets.

Strategy Space

In game theory, a strategy space refers to the set of all possible actions a player can choose from in a game. In the Cournot model, the strategy space for each firm consists of all non-negative quantities they can produce.

Each firm determines its strategy by considering potential quantities and identifying the one that maximizes its profit. The strategy space is essential because it defines the range of possible actions each firm can take.

• The strategy space is constrained, meaning firms cannot produce negative quantities.
• Every firm's strategy depends on expectations about rival firms' quantities.
• The collective strategy space of the firms shapes the competitive landscape.

This space interlinks with the payoff function, guiding firms to optimal choices in terms of production quantities.

Profit Maximization

Profit maximization is the process by which firms determine the level of output that will produce the highest possible profit. In the context of the Cournot model, each firm aims to choose a production quantity that maximizes its profit given the constraints of market pricing and competitor actions.

The profit function for a firm is based on the difference between total revenue and total cost. For the Cournot model, this involves calculating revenue based on the market price, which decreases as total market output increases. Firms adjust their output in response to competitors to ensure they are operating at their profit-maximizing level.

• Firms analyze the marginal cost and marginal revenue for optimal output.
• Profit maximization often requires firms to predict rivals’ behaviors accurately.
• Maximizing profit isn’t just about perfect output amounts; it’s about responsiveness to the entire market environment.

Ultimately, achieving profit maximization is a dynamic process that requires evaluating both current market conditions and competitor strategies.