Question

(i) Find the simultaneous-move Nash equilibrium in the Cournot model, where two duopolists choose their outputs in anticipation of the market-clearing price. Each duopolist's total cost function is given by 4Q_i, where Q_i represents the output of duopolist i. The market demand is represented as P = 288 - 8Q, where P is the price and Q is the total output. Provide the output levels chosen by each firm, the market-clearing price, and each firm's profits.

(ii) Create a graph of the best-response functions for the duopolists based on the Nash equilibrium found in part (i). Identify the Nash equilibrium on the graph, and be sure to label the axes and the best-response functions, specifying which one corresponds to duopolist 1 and which to duopolist 2.

(iii) Now, consider the Stackelberg model where duopolist 1 first selects its output, and duopolist 2 observes this choice before making its own. Afterward, the market-clearing price is determined, and profits are calculated. Find the Subgame Perfect Nash Equilibrium (SPNE) in this scenario, indicating the output levels and profits for each duopolist. Discuss whether this problem exhibits a first-mover advantage, a second-mover advantage, or neither.

Answer 1

(i) The simultaneous-move Nash equilibrium in the Cournot model is reached when both duopolists produce an output of 18 units each. The market-clearing price is $144, and each duopolist earns a profit of $1,296.

(ii) The graph of the best-response functions illustrates the Nash equilibrium at the intersection of the reaction functions for duopolist 1 and duopolist 2. The horizontal axis represents the output of duopolist 1, the vertical axis represents the output of duopolist 2, and the labeled curves depict each duopolist's best-response function.

(iii) In the Stackelberg model, duopolist 1, as the leader, chooses an output of 24 units, and duopolist 2, the follower, produces 12 units. The market-clearing price is $144, and profits for duopolist 1 and duopolist 2 are $2,304 and $864, respectively. This scenario exhibits a first-mover advantage for duopolist 1, as it can influence the market and secure higher profits.

Step-by-step explanation:

(i) The Cournot model involves two firms simultaneously determining their output levels to maximize profits. In this case, both duopolists choose to produce 18 units each, leading to a total market output of 36 units. The market-clearing price is calculated using the demand function P = 288 - 8Q, where Q is the total output. Substituting Q = 36 into the demand function gives P = $144. Each duopolist's profit is then computed as profit = (P - MC) * Q, resulting in a profit of $1,296 for each.

(ii) The graph of the best-response functions showcases the reaction functions of both duopolists. The equilibrium is found at the intersection, indicating the output levels chosen by each in response to the other's output. The labeled axes and curves help visualize the strategic interaction between the duopolists.

(iii) In the Stackelberg model, duopolist 1 acts as the leader, selecting its output first. Duopolist 2 observes this choice and responds optimally. Duopolist 1 produces 24 units, influencing the market-clearing price to be $144. Duopolist 2, as the follower, produces 12 units. The profits are calculated based on the Cournot profit function. Duopolist 1, with a higher output, enjoys a first-mover advantage, obtaining higher profits compared to duopolist 2.