Two firms, an incumbent (I) and an entrant (E), can produce an identical good at a cost of [tex]c > 0[/tex] per unit. The incumbent chooses its quantity [tex]q \geq 0[/tex] first. The entrant chooses its quantity [tex]q_e \geq 0[/tex] after having observed [tex]q[/tex]. The price is given by the inverse demand function [tex]p = a - q - q_e[/tex], with [tex]a > c[/tex]. The entrant faces an entry fee [tex]f \geq 0[/tex] if it produces a positive quantity; the entrant's cost of producing is 0 if [tex]q_e = 0[/tex] and [tex]f + cq_e[/tex] if [tex]q_e > 0[/tex]. The incumbent faces no such fee, so its cost of producing is [tex]cq[/tex]. Both firms are profit maximizers, and for simplicity, assume that if a firm is indifferent among multiple quantities, it chooses the smallest one.

(a) (15 points) Fixing an incumbent quantity [tex]q[/tex], find the optimal entrant quantity.

(b) (30 points) Calculate the equilibrium quantities set by the two firms.

(c) (15 points) Briefly provide intuition for your result in (b). [max: 100 words]