Question

Suppose that the inverse demand function is described by, P = 60-4(q1 +q2), where q1 is the output of the incumbent firm and q2 is the output of the entrant. Let the labor cost per unit w = 6 and capital cost per unit be r = 6. In addition, each firm has a fixed setup cost of f. (a) Suppose that in stage one the incumbent invests in capacity K1. Find the incum- bentís and entrantís best response functions in stage two. (8 pts) (b) Suppose that incumbent commits to a production capacity of K1 = 6, what is the entrantís best response? Calculate their profits as a function of f. (8 pts) (c) For fixed setup cost f = 25, what is the equilibrium? (7 pts) (d) For fixed setup cost f = 64, what is the equilibrium? (7 pts)

Answer 1

(a) P = 60 - 4(q1 + q2)

Stage 1:

C1 = wL1 + rK1 = 6(L1 + K1) + f = C1(f)

C2 = wL2 = 6L2 + f = C2(f)

Stage 2:

R1 = P * q1 = 60q1 - 4q1² - 4q1q2

MR1 = 60 - 8q1 - 4q2

MC1 = C1^'(f)

MR1 = MC1

60 - 8q1 - 4q2 = C1'(f)

15 - 2q1 - q2 = C1'(f)

2q1 = 15 - q2 - C1'(f)

q1 = 7.5 - 0.5q2 - 0.5C1'(f)............................(1)

(1) gives the response function for the incumbent firm.

R2 = P * q2 = 60q2 - 4q1q2 - 4q2²

MR2 = 60 - 4q1 - 8q2

MC2 = C2'(f)

MR2 = MC2

60 - 4q1 - 8q2 = C2'(f)

15 - q1 - 2q2 = C2'(f)

2q2 = 15 - q1 - C2'(f)

q2 = 7.5 - 0.5q1 - 0.5C2'(f)....................(2)

(2) gives the response function for the entrant firm.

(b) C1 = 6L1 + 36 + f = C1(f)

Since, there is no change in the functional form of C1, hence, entrant firm's response function remains the same.

Simultaneously solving (1) and (2), we get:

q1 = 5 + 0.33C2'(f) - 0.67C1'(f) = q1(f)

q2 = 5 + 0.33C1'(f) - 0.67C2'(f) = q2(f)

R1 = Pq1(f) = R1(f)

R2 = Pq2(f) = R2(f)

1 = R1 - C1 = 60q1 - 4q1² - 4q1q2 - C1(f)

Substituting the values of q1 and q2, we get:

1 = R1(f) - C1(f) = 1(f)

Similarly,

2 = R2(f) - C2(f) = 2(f)

(c) f = 25

C1'(f) = 0

C2'(f) = 0

q1 = 5

q2 = 5

1 = 60 *5 - 4 * 25 - 4*5*5 - 25

= 300 -100 -100 - 25 = 75

2 = 60 * 5 - 4*5*5 - 4 * 25 - 25 = 75

(d) f = 64

Again, q1 = 5

q2 = 5

1 = 100 - 64 = 36

2 = 36