Question

ou are given the market demand function Upper Q equals 1000 minus 1000 p​, and that each duopoly​ firm's marginal cost is ​$0.28 per​ unit, which implies the cost​ function: Upper C (q Subscript i Baseline )equals 0.28 q Subscript i​, assuming no fixed costs for i​ = 1, 2. The Cournot equilibriumLOADING... quantities are q 1equals 240 and q 2equals 240 ​(enter your responses as whole numbers​). The Cournot equilibrium price is ​$ 0.52 ​(round to the nearest penny​). Calculate the Cournot​ profits: firm 1 ​$ 57.6 and firm 2 ​$ 57.6 ​(round both responses to the neare

Answer 1

a. The Cournot equilibrium quantities are q_1 equals 240 and q_2 equals 240.

b. p = 0.52 which is the Cournot equilibrium price.

c. The Cournot​ profits: for firm 1 = ​$57.6, and for firm 2 = ​$57.6.

Step-by-step explanation:

Q = 1000 - 1000p .............................................. (1)

From equation (1), we have:

p = (1000 - Q)/1000

p = 1 - 0.001Q ..................................................... (2)

MC_i = $0.28

C = 0.28q_i

Q = q_1 + q_2, we have:

C = 0.28Q

p = 1 - 0.001(q_1 + q_2) = 1 - 0.001q_1 - 0.001q_2

For Firm 1:

TR_1 = p * q_1 = (1 - 0.001q_1 - 0.001q_2)q_1 = q_1 - 0.001q_1^2 - 0.001q_2q_1

MR_1 = dTR_1/dq_1 = 1 - 0.002q_1 - 0.001q_2

Since at the optimum MR_1 = MC, we have:

1 - 0.002q_1 - 0.001q_2 = 0.28

q_1 = (1 - 0.28 - 0.001q_2) / 0.002

q_1 = 360 - 0.5q_2 .......................................... (3)

For Firm 2:

TR_2 = p * q_2 = (1 - 0.001q_1 - 0.001q_2)q_2 = q_2 - 0.001q_1q_2 - 0.001q_2^2

MR_2 = dTR_2/dq_2 = 1 - 0.001q_1 - 0.002q_2

Since at the optimum MR_2 = MC, we have:

1 - 0.001q_1 - 0.002q_2 = 0.28

q_2 = (1 - 0.28 - 0.001q_1) / 0.002

q_2 = 360 - 0.5q_1 .......................................... (4)

a. Calculation of Cournot equilibrium quantities

Substituting equation (4) for q_2 into equation (3), we have:

q_1 = 360 - 0.5(360 - 0.5q_1)

q_1 = 360 - 180 + 0.25q_1

q_1 - 0.25q_1 = 180

0.75q_1 = 180

q_1 = 180 / 0.75

q_1 = 240 <------------- Cournot equilibrium quantity for firm 1

Substitute for q_1 in equation (4), we have:

q_2 = 360 - 0.5(240)

q_2 = 360 - 120

q_2 = 240 <------------- Cournot equilibrium quantity for firm 2

Therefore, the Cournot equilibrium quantities are q_1 equals 240 and q_2 equals 240.

b. Calculation of Cournot equilibrium price

Since p = 1 - 0.001q_1 - 0.001q_2, we substitue for q_1 and q_2 as follows:

p = 1 - 0.001(240) - 0.001(240)

p = 1 - 0.24 - 0.24

p = 0.52 <------------- Cournot equilibrium price.

c. Calculate the Cournot​ profits

For Firm 1:

TR_1 = q_1 - 0.001q_1^2 - 0.001q_2q_1

TR_1 = 240 - (0.001 * 240^2) - (0.001 * 240 * 240)

TR_1 = $124.80

C_1 = 0.28q_1

C_1 = 0.28 * 240

C_1 = $67.20

Profit_1 = $124.80 - $67.20

Profit_1 = $57.60 <------------- Cournot profit for firm 1.

For Firm 2:

TR_2 = q_2 - 0.001q_1q_2 - 0.001q_2^2

TR_2 = 240 - (0.001 * 240 * 240) - (0.001 * 240^2)

TR_2 = $124.80

C_2 = 0.28q_2

C_2 = 0.28 * 240

C_2 = $67.20

Profit_2 = $124.80 - $67.20

Profit_2 = $57.60 <------------- Cournot profit for firm 2.