Question

Question #1 (this is a traditional perfect competition, monopoly, and duopoly question) Suppose that the (inverse) demand curve for Ginseng is given by P = 124 − 6Q and TC =10 + $4Q + $3Q2

What is equilibrium Price and Quantity and Profit if the market is competitive? 4 Points

What is equilibrium Price and Quantity and Profit if there are two firms in the market (note Q = q1 + q2)?

What is equilibrium Price and Quantity and Profit if there are monopoly in the market (note Q = Q)? 5 Points

If there were 3 firms, where do you estimate the output and the price would be—this does not require a mathematical calculation—it is based on the expectations created by the prior three answers (1-b, 1-c, and 1-d). 2 Points

Answer 1

MC = dTC/dQ = 4 + 6Q

(1) In competitive market, P = MC.

124 - 6Q = 4 + 6Q

12Q = 120

Q = 10

P = 4 + 6 x 10 = 4 + 60 = 64

Total revenue (TR) = P x Q = 64 x 10 = 640

Total cost (TC) = 10 + 4 x 10 + 3 x 10 x 10 = 10 + 40 + 300 = 350

Profit = TR - TC = 640 - 350 = 290

(2) When there are 2 firms, P = 124 - 6q1 - 6q2 [since Q = q1 + q2]

For firm 1,

TR1 = P x q1 = 124q1 - 6q12 - 6q1q2

MR1 = TR1/q1 = 124 - 12q1 - 6q2

MC1 = 4 + 6q1 + 6q2 [as Q = q1 + q2]

Equating MR1 and MC1,

124 - 12q1 - 6q2 = 4 + 6q1 + 6q2

18q1 + 12q2 = 120

3q1 + 2q2 = 20...........(1) [best response function, firm 1]

For firm 2,

TR2 = P x q2 = 124q2 - 6q1q2 - 6q22

MR2 = TR2/q2 = 124 - 6q1 - 12q2

MC2 = 4 + 6q1 + 6q2 [since Q = q1 + q2]

Equating MR2 and MC2,

124 - 6q1 - 12q2 = 4 + 6q1 + 6q2

12q1 + 18q2 = 120

2q1 + 3q2 = 20...........(2) [best response function, firm 2]

(1) x 2 results: 6q1 + 4q2 = 40.........(3)

(2) x 3 results: 6q1 + 9q2 = 60.........(4)

(4) - (3) results: 5q2 = 20

q2 = 4

q1 = (20 - 3q2)/2 [from (2)] = [20 - (3 x 4)]/2 = (20 - 12)/2 = 8/2 = 4

Q = 4 + 4 = 8

P = 124 - (6 x 8) = 124 - 48 = 76

TR = P x Q = 76 x 8 = 608

TC = 10 + 4 x 8 + 3 x 8 x 8 = 10 + 32 + 192 = 234

Profit = 608 - 234 = 374

(3) With monopoly, MR = MC.

TR = P x Q = 124Q - 6Q2

MR = dTR/dQ = 124 - 12Q

So,

124 - 12Q = 4 + 6Q

18Q = 120

Q = 6.67

P = 124 - (6 x 6.67) = 124 - 40 = 84

TR = 84 x 6.67 = 560.28

TC = 10 + 4 x 6.67 + 3 x 6.67 x 6.67 = 10 + 26.68 + 133.47 = 170.15

Profit = 560.28 - 170.15 = 390.13

(4) The higher the number of firms, the lower the price and the higher the (total) output. So, if P* and Q* be price and quantity with 3 firms, then

76 < P* < 64 and

8 < Q* < 10