Question

a homogenous products duopoly faces a market demand function given by P=a-Q where Q=q1+q2 and a>300. both firms have constant marginal cost MC=100. there are no fixed cost.

a) what is firm 1's optimal quantity given that firm 2 produces an output of 50 units per year?

Answer 1

Profit of Firm 1:

$Π_1 = (P-MC)q_1 \\ = (a-q_1-q_2-100)q_1 \\ \frac{∂Π_1}{∂q_1} = (a-q_1-q_2-100)(1) + q_1(-1) = 0 \\ a – q_1-q_2-100-q_1 = 0 \\ 2q_1 = a-q_2-100 \\ q_1 = \frac{a-q-2-100}{2} \\ q_2=50 \\ q_1 = \frac{a-50-100}{2} \\ = \frac{a-150}{2}$