Let us consider an economic sector characterized by the following data. The (inverse) demand function is p = 20 -2q with q the quantities produced by the firms in the sector and p the price. The total cost of production for any firm in the sector is: CT(q) = q* - 4q + 5 a) Firm 1 seeks to deter the entry of another firm, firm 2, into its market througha sustainable monopoly strategy. Calculate the equilibrium price, quantity and profit of firm 1 given this strategy Answer: Price set will be equal to marginal cost. P = MC MG = 29- 4 Putting this into demand function we get: 2q-4 = 20 - 2q q* = 6 (Quantity produced to deter competition) Putting this into demand function: p* = 8 (Price charged to deter competition) Firm 1 has failed to deter entry into its market. So, we now have a situation of non-cooperative oligopoly, with firm 2 having the same total cost as firm 1. The market quantity is such that q = q1 + q2 I) It is assumed that both firms 1 and 2 compete on quantity by making their decisions simultaneously. Identify the duopoly situation. Calculate the quantities offered by each firm, the equilibrium price, and their individual profits Answer: For firm 1: Optimal Condition: MR1 = MC1 TR1 = p°q1 = [20 - 2(q1 +q2)] *q1 MR1 = 20 - 4q1 - q2 MC1 = 2q1 - 4 Putting in optimal condition: 20 - 4q1 - q2 = 2q1 - 4 (24 - q2)/6 = q1• -) (Reaction function of firm1 for a given level of firm 2 output) For Firm 2: Optimal Condition: MR2 = MC2 TR1 = p*q1 = [20 - 2(q1 +q2)] *q2 MR1 = 20 - q1 - 4q2 MC1 = 2q2 - 4 q2* = (24 - q1)/6 -(ii) (Reaction function of firm2 for a given level of firm 1 output) Now we take value of q2* and put it into equation (i): 6q1 = 24 - (24 - q1)/6 36q1 = 120 + q1 q1* = 3.43 approx. Putting in equation (ii) we get: q2* = 3.43 Putting in demand function: P= 20 - 2*6.86 = 6.28 Profit of firm 1: => p*q1-q12 + 4q1 -5 Putting in optimal condition: q1 + q2 = q* = 6.86 = 23.52 - 11.76 + 13.72 -5 = 30.46 Similarly, Profit of firm 2 = 30.46 QUESTION (to answer): I) Represent the duopoly situation of question I) as a normal form game (matrix), with firm 1 in the row and firm 2 in the column. In addition to the equilibrium quantity strategies, you will choose as non-equilibrium quantity strategies q = 2 for both firm 1 and firm 2. By carefully explaining your reasoning, show that the equilibrium solution found in question I) is indeed a Nash equilibrium (For this question, please detail your profit calculations for out of equilibrium)

Let us consider an economic sector characterized by the following data. The (inverse) demand function is p = 20 -2q with q the quantities produced by the firms in the sector and p the price. The total cost of production for any firm in the sector is: CT(q) = q* - 4q + 5 a) Firm 1 seeks to deter the entry of another firm, firm 2, into its market througha sustainable monopoly strategy. Calculate the equilibrium price, quantity and profit of firm 1 given this strategy Answer: Price set will be equal to marginal cost. P = MC MG = 29- 4 Putting this into demand function we get: 2q-4 = 20 - 2q q* = 6 (Quantity produced to deter competition) Putting this into demand function: p* = 8 (Price charged to deter competition) Firm 1 has failed to deter entry into its market. So, we now have a situation of non-cooperative oligopoly, with firm 2 having the same total cost as firm 1. The market quantity is such that q = q1 + q2 I) It is assumed that both firms 1 and 2 compete on quantity by making their decisions simultaneously. Identify the duopoly situation. Calculate the quantities offered by each firm, the equilibrium price, and their individual profits Answer: For firm 1: Optimal Condition: MR1 = MC1 TR1 = p°q1 = [20 - 2(q1 +q2)] q1 MR1 = 20 - 4q1 - q2 MC1 = 2q1 - 4 Putting in optimal condition: 20 - 4q1 - q2 = 2q1 - 4 (24 - q2)/6 = q1• -) (Reaction function of firm1 for a given level of firm 2 output) For Firm 2: Optimal Condition: MR2 = MC2 TR1 = pq1 = [20 - 2(q1 +q2)] q2 MR1 = 20 - q1 - 4q2 MC1 = 2q2 - 4 q2 = (24 - q1)/6 -(ii) (Reaction function of firm2 for a given level of firm 1 output) Now we take value of q2* and put it into equation (i): 6q1 = 24 - (24 - q1)/6 36q1 = 120 + q1 q1* = 3.43 approx. Putting in equation (ii) we get: q2* = 3.43 Putting in demand function: P= 20 - 26.86 = 6.28 Profit of firm 1: => pq1-q12 + 4q1 -5 Putting in optimal condition: q1 + q2 = q* = 6.86 = 23.52 - 11.76 + 13.72 -5 = 30.46 Similarly, Profit of firm 2 = 30.46 QUESTION (to answer): I) Represent the duopoly situation of question I) as a normal form game (matrix), with firm 1 in the row and firm 2 in the column. In addition to the equilibrium quantity strategies, you will choose as non-equilibrium quantity strategies q = 2 for both firm 1 and firm 2. By carefully explaining your reasoning, show that the equilibrium solution found in question I) is indeed a Nash equilibrium (For this question, please detail your profit calculations for out of equilibrium)

Managerial Economics: A Problem Solving Approach

5th Edition

ISBN:9781337106665

Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Chapter9: Market Structure And Long-run Equilibrium

Section: Chapter Questions

Problem 2MC

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