In: Economics
The market demand curve for mineral water is P=15-Q. Suppose that there are two firms that produce mineral water, each with a constant marginal cost of 3 dollars per unit. Suppose that both firms make their production decisions simultaneously. How much each firm should produce to maximize its profit? Calculate the market price.
The quantity produced by firm 1 is denoted by Q1
The quantity produced by firm 2 is denoted by Q2.
The total quantity produced in the market is denoted by Q.
The market price is denoted by P.
Each firm’s marginal cost function is MC = 3 and the market demand function is P = 15 – Q
Hence demand is P = 15 – (q1 + q2) where Q is the sum of each firm’s output q1 and q2.
Find the best response functions for both firms:
Revenue for firm 1
R1 = P*q1 = (15 – (q1 + q2))*q1 = 15q1 – q12 – q1q2.
Firm 1 has the following marginal revenue and marginal cost functions:
MR1 = 15 – 2q1 – q2
MC1 = 3
Profit maximization implies:
MR1 = MC1
15 – 2q1 – q2 = 3
12 – q2 = 2q1
which gives the best response function:
q1 = 6 - 0.5q2.
By symmetry, Firm 2’s best response function is:
q2 = 6 - 0.5q1.
Cournot equilibrium is determined at the intersection of these two best response functions:
q1 = 6 - 0.5(6 - 0.5q1)
q1 = 3 + 0.25q1
This gives q1 = q2 = 4 units
Market quantity = q1 + q2 = 8 units
Price is (15 – (4 + 4) = $7 per unit