Question

Consider a market with Stackelberg competition. The inverse demand curve is P = a−b Q, with a=13 and b=3. Firm 1 is the leader and produces at constant marginal costs equal to zero. Firm 2 is the follower and has the cost function: C(q) = cq^2, with c=5. (Note the square on q). What is the equilibrium quantity of firm 1?

Answer 1

Industry demand equation: P = 13-3Q

Where, Q=q1+q2

The marginal cost of Firm 1 (Leader) = MC 1 = 0

The marginal cost of Firm 2 (Follower) = MC2 = 10q

Profit function for Firm 2 = (13-3(q1+q2))q2-5(q2)^2 = 13(q2) - 3q1q2 - 3(q2)^2 - 5(q2)^2 = 13(q2) - 3q1q2 - 8(q2)^2

F.O.C

13 - 3(q1) - 16(q2) =0

3(q1) + 16(q2) = 13

q2 = (13 -3(q1))/16

Profit function for Firm 1 = (13-3(q1+q2))q1 - 0

= 13-3(q1 + (13 -3(q1))/16) (q1)

= 13(q1) - 3(q1)^2 - (39/16)(q1) + (9/16)(q1)^2

F.O.C

13-6(q1)-(39/16)+(18/16)(q1) = 0

q1 = 2.167

Hence, equilibrium quantity of Firm 1 = 2.167