Chapter 12: Problem 1

Explain Cournot's Duopoly model. What are reaction curves?

Short Answer

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Question: Explain the importance of reaction curves in the Cournot Duopoly model and their role in determining each firm's output levels to maximize their profits.

Step by step solution

Understand the Cournot Model

In the Cournot duopoly model, there are two firms (firm 1 and firm 2) that produce homogeneous goods in a market with no other competitors. They simultaneously make decisions about their production levels (q1 and q2) to maximize their own profits. The total market quantity is the sum of their individual quantities (Q = q1 + q2). They face a market demand curve (P(Q)) that determines the price at which they can sell their goods.

Derive the profit function for each firm

To analyze each firm's decision-making process, we need to derive their profit functions. Each firm wants to maximize its profit (π), which is the difference between its total revenue (TR) and total cost (TC). The profit function for each firm can be expressed as: Firm 1: π1(q1,q2) = P(Q) × q1 - C1(q1) Firm 2: π2(q1,q2) = P(Q) × q2 - C2(q2) Where C1(q1) and C2(q2) are the total cost functions for firm 1 and firm 2, respectively.

Calculate the best response functions

To find the best output level for each firm, we calculate their best response (or reaction) functions. These functions express the optimal output level for each firm (q1 or q2) in response to the output level chosen by the other firm. This is done by taking the first-order partial derivative of profits for each firm (π1 and π2) with respect to their respective output levels (q1 and q2) and setting them equal to zero: Firm 1: (∂π1/∂q1) = 0 Firm 2: (∂π2/∂q2) = 0 By solving these equations simultaneously, we obtain reaction functions for both firms: Firm 1: q1 = R1(q2) Firm 2: q2 = R2(q1)

Find the Cournot equilibrium

Next, we determine the Cournot equilibrium by finding the output levels q1 and q2 at which both firms' reaction functions are equal. In other words, we identify the point where both firms have no incentive to change their output levels, given the output chosen by their competitor. This can be done by solving the simultaneous equations: R1(q2) = q1 R2(q1) = q2 These solutions (q1* and q2*) represent the Cournot equilibrium output levels for each firm.

Understand the role of reaction curves

Reaction curves play a critical role in the Cournot duopoly model as they represent the strategic behavior of each firm in response to their competitor's decisions. By plotting the reaction curves for both firms on a graph with q1 on the horizontal axis and q2 on the vertical axis, we can visually determine the Cournot equilibrium at the intersection point of the two reaction curves. Similarly, any changes in demand or cost conditions can be analyzed by observing shifts in the reaction curves and finding the new Cournot equilibrium point. This illustrates how firms can strategically adjust their output levels to maximize profits in a duopolistic market.