# Calculate optimal profits of each firm.

Two firms produce differentiated products and set prices to maximize their individual profits. Demand functions for the firms are given by

Q1 =64 -4P1 +2P2

Q2 =50 -5P2+ P1

where P1, P2, Q1, Q2, refer to prices and outputs of firms 1 and 2 respectively. Firm 1â??s marginal cost is $5 while firm 2â??s marginal cost is $4. Each firm has a fixed cost of $50.

Assuming that the two firms decide on prices independently and simultaneously, calculate the best response function of each firm in terms of prices. Calculate the resulting equilibrium price quantity combination for each firm. Illustrate your answer with a suitable graph. Also calculate optimal profits of each firm.

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#### Solution Preview

Q1 = 64 - 4P1 + 2P2 => P1 = 16 - Q1/4 + P2/2

Q2 = 50 - 5P2 + P1 => P2 = 10 - Q2/5 + P1/5

To make it easier to read, I use a = Q1, b = P1, c = Q2, d = P2,

then

a = -4c -18d + 264 (best response function of 1)

b = 16 - a/4 + d/2

c = b - 5d + 50 (best response function of 2)

d = 10 - ...

#### Solution Summary

Calculate optimal profits of each firm in an oligopoly market.