Nash equilibrium in duopoly with differentiated products

Suppose Ajax and Bayern operate in the same market, but produce differentiated products, goods A and B, respectively. They sell these goods in the market at prices Pa and PB,

AEB 5188: Problem Set #6 - Managerial decisions under oligopoly

respectively. The linear demands for the two goods are, respectively,

QA = 100 - 4PA + 1.5PB

QB = 120 - 2B + 0.5 PA

Production costs are constant, but not equal:

ACA = MCA = $2

1. Assuming that firms choose their prices (Pa and Ps, respectively) to maximize profits given the price they expect their rival to set, derive the equations for each firm's best response functions.
2. Sketch a graph of the best response curves, being sure to clearly label each curve.
3. If Ajax expects Bayern to set its price at $20, what is Ajax's best response? If Bayern expects Ajax will price good A at $36, what is Bayern's best response?
4. What is the Nash equilibrium pricing strategy in this market?
5. How much profit does each firm earn at the Nash equilibrium?
6. If Ajax and Bayern set prices at $22 and $35, respectively, how much profit does each firm earn? Why don't they choose these prices then?