Consider a market for a homgeneous product with demand given by Q = 37.5 - .25P.
ID: 1253844 • Letter: C
Question
Consider a market for a homgeneous product with demand given by Q = 37.5 - .25P. There are two firms, each with a constant marginal cost equal to 40.a. Determine the output and price under a cournot equilibrium.
b. Determine the output and price under a Bertrand equilibrium.
c. Show graphically the e¢ ciency gain of Cournot and Bertrand outcomes relative to monopoly.
Explanation / Answer
Monopolist maximizes profit with MC = MR -> MC = 40, MR = 150 - 8Q Therefore Qm = 110/8 or 13.75 and Pm = 150 - 4(13.75) = 95 Cournot: (Profit = Pi) (P-c)q1 = Pi = (150 - 4(q1+q2) - 40)q1 = (110-4q2)q1 - 4q1^2 Max Pi at First order condition with respective to q1-> 110 - 4q2 = 8q1 q1 = (110/8) - 0.5q2 By symmetry q2 = (110/8) - 0.5q1 Sub q2 into q1 -> q1 = (110/8) -0.5([110/8]-0.5q1) q1 = 55/6 = 9.17 Do the same for q2 -> q2 = 55/6 = 9.17 Qc = q1+q2 = 110/6 = 18.33 Pc = 150 - 4Qc = 230/3 = 76.67 Bertrand: Since firms can undercut each other when price is above MC to capture full market share and would make negative profit when price is below MC, both firms would charge MC as their respective prices. Pb = P1 = P2 = 40 Qb = 37.5 - .025(40) = 27.5 To show efficiency gain graphically, draw the demand function on a graph with P on the y-axis and Q on the x-axis. Identify Qm, Qc, and Qb on the demand function line and compare.
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