Suppose the demand for Pepsi is qp = 50 - 2pp + 1pc. The firm faces a constant marginal cost of m, and denotes the price of Coke
Assuming Bertrand behavior, derive Pepsi's best-response function and explain how the firm changes price in response to changes in its own marginal cost and changes in Coke's price.
For Pepsi, profit maximization means δπ/δpp = 50 + 2m - 4pp + pc = 0. Solving for pp yields Pepsi's best-response function: pp = 12.5 + (1/2 )m + (1/4)pc. A $1 increase in marginal cost yields a $0.50 increase in price. A $1 increase in Coke's price yields a $0.25 increase in price.
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