Question

Suppose demand in a market is given by x = 250 - Px. Marginal cost of...

Suppose demand in a market is given by x = 250 - Px. Marginal cost of any of two firms operating is MC = $4.

What is the Cournot duopoly equlibrium price and quantity?

Homework Answers

Answer #1

Each firm’s marginal cost function is MC= 4 and the market demand function is Px = 250 – x

Where x is the sum of each firm’s output x1 and x2.

Find the best response functions for both firms:

Revenue for firm 1

R1 = P*x1 = (250 – (x1 + x2))*x1 = 250x1 – x12 – x1x2.

Firm 1 has the following marginal revenue and marginal cost functions:

MR1 = 250 – 2x1 – x2

MC1 = 4

Profit maximization implies:

MR1 = MC1

250 – 2x1 – x2 = 4

which gives the best response function:

x1 = 123 - 0.5x2.

By symmetry, Firm 2’s best response function is:

x2 = 123 - 0.5(123 - 0.5x2)

= 123 - 61.5 + 0.25x2

x2 = 82 and so x1 is also 82. Total quantity is 82 + 82 = 164 and price = 250 - 164 = $86

Cournot exuilibrium is determined at the intersection of these two best response functions:

x2 = 123 - 0.5x1.

x1 = x2 = 173.33

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