Cournot Model

Assumptions

• Homogenous product, so firms maximize profit by choosing how much to produce
• Competitor’s output will not change
• No collusion

Derivatives

Slope of the tangent line of a point in a function

Useful derivatives for this model

An example situation

A market for gas has 2 competitors, Company A and Company B. The Demand curve for the market is represented by the equation P = 120 - 2Q, where P is the price and Q is the total output by the firms. Company A has a total cost of 30Q and Company B has a total cost of 20Q. At what price and quantity would each of these firms produce at in a cournot equilibrium?

Demand Curve for market

P = 120 - 2Q

Solving

Notice that the equation P = 120 - 2Q can be rewritten as P = 120 - 2(Q_A + Q_B), where Q_A and Q_B are the quantity produced by Company A and B respectively.

Find “Reaction” Function for Company A

Π_A = (120 - 2(Q_A + Q_B))Q_A - 30Q_A = 90Q_A - 2Q²_A - 2Q_AQ_B

(dΠ_A / dQ_A ) = 90 - 4Q_A - 2Q_B = 0

4Q_A = 90 - 2Q_B

Q_A = 22.5 - .5Q_B

Find “Reaction” Function for Company B

Π_B = (120 - 2(Q_A + Q_B))Q_B - 20Q_B = 100Q_B - 2Q²_B - 2Q_AQ_B

(dΠ_B / dQ_B ) = 100 - 4Q_B - 2Q_A = 0

4Q_B = 100 - 2Q_A

Q_B = 25 - .5Q_A

Visualizing the Reaction Functions

Q_A

Q_B

Finding the equilibrium

Q_A = 22.5 - .5(25 - .5Q_A) = 10 - .25Q_A

Q_A = 8

Q_B = 25 - .5(8) = 21

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P = 120 - 2(21 + 8) = 62

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Adding more firms