Consider an industry with two firms (duopoly) the produce a homogenous product (
ID: 1207705 • Letter: C
Question
Consider an industry with two firms (duopoly) the produce a homogenous product (same product). The two firms strategically choose their price (price is the choice variable, not quantity). Note, this is a Bertrand duopoly setting. Firm 1 has a cost function c1(q1)=25q1. i.e. MC=25. and firm 2 has a cost function of C2(q2)=80+15q2 i.e. MC=15.The market demand is Q=100-p.
a. Use your knowledge of game theory to solve for the price charged in the market, the quantity produced by each firm, and the profit for each firm
Explanation / Answer
The firm 1 has a MC = 25 and firm 2 has a MC = 15.
The market demand is Q = 100 - p
Now the firm 2 can choose a price from 15 to maximum of 25. Firm 2 can never charge a price below 15 as itsit has incur losses in that case because it is unable to cover its MC. Firm 2 can never go above a price of 25 because then Firm 1 can charge a price between his MC(25) and the price charged by firm 2 and capture the whole market from firm 2.
Whereas, Firm 1 can never go below 25 as in that case it has to incur losses because he is unable to cover its MC.
Therefore, the firm 2 will charge a price of 25 due to which firm 1 will also charge 25. Hence, the market price is 25.
At a price of 25, Q = 100 - 25 = 75
=> Q = q1 + q2 = 75 .......................................................(1)
Now firm 1 is charging its MC only. So he is operating at MR = MC
Total Cost of firm 1, TR1 = p * q1 = [100 - (q1 + q2)]q1
Marginal Cost of firm 1, MR1 = 100 - 2q1 - q2
As firm 1 operates at MR1 = MC
=> 100 - 2q1 - q2 = 25
=> 2q1 + q2 = 75 ..................................................(2)
Adding (1) and (2), gives q1 = 0
=> q2 = 75 (from (1))
Now, Firm 1's profit = p * q1 - c1 * q1 = (25 * 0) - (25 * 0) = 0
Firm 2's profit = p * q2 - c2 * q2 = (25 * 75) - (15 * 75) = 1,875 - 1,125 = 750