Question

1. Consider two companies A and B sharing a market by producing identical goods (or highly substitutable goods). Company A’s marginal cost is MC=20 and company B’s marginal cost is MC=10. Market demand is known to be P=100-0.001Q.

1. Find profit maximizing level of Q_A and Q_B under oligopoly setting.
2. Determine the market price.
3. Determine the revenue of company A and B.
4. Determine the profit of company A and B.
5. Find collusive level of profit maximizing output for A and B (Under collusion A and B share the same MC=10 and share the market equally).
6. Using a simple game theory method, show that the collusive outcome is not sustainable. Be sure to construct a 2x2 matrix with correct payoffs.

Answer 1

a. The market demand function is given as : P = 100 - 0.001Q

Let Q = QA + QB       {QA : Firm A’s Output ; QB : Firm B’s Output}

Thus, P = 100 – 0.001QA – 0.001QB

Given MC(A) = 20 and MC(B) =10

For Firm A

Total Revenue of Firm A: TR(A): P*QA= (100 – 0.001QA – 0.001QB) *QA

= 100QA – 0.001QA²- 0.001QAQB

Marginal Revenue of Firm A: MR(A) = 100 – 0.002QA –0.001QB

At profit maximizing levels : MR(A) = MC(A)

100 – 0.002QA –0.001QB = 20

100 – 20 – 0.001QB = 0.002QA

80 – 0.001QB = 0.002QA

40000 – 0.5QB=QA     {Best Response Function of Firm A}

For Firm B

Total Revenue of Firm B: TR(B): P*QB= (100 – 0.001QA – 0.001QB) *QB

= 100QB – 0.001QB²- 0.001QAQB

Marginal Revenue of Firm B: MR(B) = 100 – 0.002QB –0.001QA

At profit maximizing levels: MR(B) = MC(B)

100 – 0.002QB –0.001QA = 10

100 – 10 - 0.001QA = 0.002QB

90 – 0.001QA = 0.002QB

45000 – 0.5QA = QB     {Best Response Function of Firm B}

Using the value of QB in the best response function of Firm A we get QA. Then using the value of QA in the best response function of Firm B we get QB.

QA = 40000 – 0.5QB

QA = 40000 – 0.5 (45000 – 0.5QA)

QA = 40000 – 22500 + 0.25QA

0.75QA = 17500

QA = 17500/0.75

QA = 23333.33

Since QB = 45000 – 0.5QA

QB = 45000 – 0.5(23333.33)

QB = 45000 – 11666.66

QB= 33333.33

b. The Market price be : P = 100 – 0.001QA – 0.001QB

P = 100 – 0.001(23333.33) – 0.001(33333.33)

P = 100 – 23.33 – 33.33

P = 43.34

c. Total Revenue for Firm A and B :

Total Revenue: TR(A) = 100QA – 0.001QA²- 0.001QAQB

= 100(23333.33) – 0.001(23333.33)² – 0.001 * 23333.33 *33333.33

= 2333333 – 544444.28 – 777777.58

= 1011111.14

Total Revenue: TR(B) = 100QB – 0.001QB²- 0.001QAQB

= 100(33333.33) – 0.001(33333.33)² – 0.001*23333.33*33333.33

= 3333333 – 1111110.8 – 777777.58

= 1444444.62

d. Total Cost of Firm A : TC(A) = 20QA = 20 * 23333.33 = 466666.6

Total Cost of Firm B : TC(B) = 10QB = 10 *33333.33 = 333333.3

Profits for Firm A = TR(A) – TC(A) = 1011111.14 - 466666.6 = 544444.54

Profits for Firm B = TR(B) – TC(B) = 1444444.62 - 333333.3 = 1111111.32

e. If the firms collude then market demand be : P = 100 – 0.001Q

TR = P*Q = (100 – 0.001Q)Q = 100Q – 0.001Q²

MR = 100 – 0.002Q

Given MC = 10, at the profit maximizing level: MR = MC

100 – 0.002Q = 10

90 =0.002Q

90/0.002 = Q

45000 = Q

Thus each firm produces Q/2 = 45000/2 = 22500 units