Question

There are two firms (Firm A and Firm B) competing in a Cournot game. The market demand is P=350-2Qa-2Qb. The marginal cost for both firms is constant is equal to $50. a) Derive the reaction functions for these firms. b) Draw the reaction functions on the graph similar to what we did in class. Be sure to label the axes and make it clear who each curve belongs to. c) Solve for the Cournot equilibrium quantities for each firm and the market price. d) Draw a market graph that shows where the Cournot equilibrium lies relative to the perfectly competitive market equilibrium. e) Identify on your graph and calculate the value of the deadweight loss associated with this outcome relative to the market equilibrium.

Answer 1

The Cournot duopoly model reaction functions for Firm A and Firm B are derived based on maximizing their respective profits given the market demand and constant marginal cost. The equilibrium quantities for both firms are found to be 50 units each, with a market price of $250. Graphically, the equilibrium and deadweight loss can be illustrated by comparing the Cournot outcome with the perfectly competitive market equilibrium.

Step-by-step explanation:

To derive the reaction functions for Firm A and Firm B in the Cournot game with market demand P=350-2Qa-2Qb and marginal cost of $50, we first write down the profit maximization condition for each firm. Profit for Firm A is πA = (P - MC)QA, where P is the price received by Firm A for its output QA and MC is the marginal cost. Similarly, for Firm B, profit is πB = (P - MC)QB. To find the reaction function, we set the derivative of the profit with respect to QA and QB equal to zero (i.e., ∂πA/∂QA = 0 and ∂πB/∂QB = 0) and solve for QA and QB respectively.

For Firm A, the reaction function is QA = 75 - 0.5QB and for Firm B, it is QB = 75 - 0.5QA. The reaction functions graph will display Firm A's output on the x-axis and Firm B's output on the y-axis, with each firm's reaction curve labeled correspondingly.

To find the Cournot equilibrium quantities, we substitute one reaction function into the other and solve the system of equations or intersect the two reaction curves graphically. The equilibrium values are QA = QB = 50. The market price at equilibrium can be calculated by substituting these quantities into the demand function, giving us P = 250.

For part d and e), we would typically draw the respective curves to show the Cournot equilibrium and the perfectly competitive market equilibrium to identify the deadweight loss in a graph which is the loss of economic efficiency when the competitive equilibrium is not achieved.