Consumers live uniformly in a “linear-1-mile city”. There are two firms, located at x = 0 and x = 1, which each produce the same physical good at marginal cost of c > 0. Consumers have transportation cost t per unit of distance. Firms are competing for customers by selecting their prices p1 ≥ 0 and p2 ≥ 0. It is assumed that each consumer will buy exactly one unit of the product.
In class, we have shown that the location of the “indifferent” consumer x′ can be found by solving:
p1 +tx′ =p2 +t(1−x′)
x′ = 1 + (p2−p1)/2t
Individual Demands of Firm 1 and 2 are then given by:
q1(p1,p2)=x′ =1/2 +(p2−p1)/2t
q2(p1,p2)=1−x′ =1/2 +(p1−p2)/2t
Finally, the payoffs are given by:
π1(p1, p2) = [p1 − c] ∗ q1(p1, p2)
π2(p1, p2) = [p2 − c] ∗ q2(p1, p2)
(a) Find best response functions for both firms and plot them on the same graph.
(b) Find the Nash equilibrium.
(c) Find market price, firms’ margins and firms’ profits at equilibrium. Hint: Follow the same steps we took in class to solve the “Partnership Game”.