Question

In: Economics

The market demand curve for mineral water is P=15-Q. Suppose that there are two firms that...

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.

Solutions

Expert Solution

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


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