Question

1. Consider two firms, Firm A and Firm B, who compete as duopolists. Each firm produces an identical product. The total inverse demand curve for the industry is P=250-QA+QB. Firm A has a total cost curve CAQA=100+QA2. Firm B has a total cost curve CBQB=100+2QB.
   1. Suppose for now, only Firm A exists (QB=0). What is the Monopoly equilibrium quantity and price? What is Firm A’s profit?
   2. Find the Nash Cournot equilibrium price and output level. What are the firms’ profits?
   3. Find the equilibrium price and output level in the market if firm A acts as a Stackelberg leader. What are the firms’ profits?
   4. Suppose that the two firms are able to form a cartel. Derive the output each firm will produce, the market price, and the total profit under the cartel solution.
   5. Compare the Cournot, Stackelberg, and Cartel outcomes to the monopoly outcome you calculated in part a.

Answer 1

Monopoly Equilibrium

We have the following information

Demand equation: P = 250 – Q_A

Total Cost (TC) equation for Firm A: C_A = 100 + Q_A²

The equilibrium is at the point where the marginal cost (MC) is equal to the marginal revenue (MR)

MC = ∂TC/∂Q_A = 2Q_A

Total Revenue (TR) = Price × Quantity

TR = (250 – Q_A)Q_A

TR = 250Q_A – Q_A²

MR = ∂TR/∂Q_A = 250 – 2Q_A

MR = MC

250 – 2Q_A = 2Q_A

4Q_A = 250

Q_A = 62.5

P = 250 – QA

P = 187.50

Profit (Π) = TR – TC

TR = 187.50 × 62.5 = 11,718.75

TC = 100 + Q_A²

TC = 100 + (62.5)²

TC = 4,006.25

Profit (Π) = 11,718.75 – 4,006.25

Profit (Π) = 7,712.50

Cournot Equilibrium

We have the following information

Demand equation: P = 250 – (Q_A + Q_B)

Assuming Q = Q_A + Q_B

Total Cost (TC) equation for Firm A: C_A = 100 + Q_A²

Total Cost (TC) equation for Firm B: C_B = 100 + 2Q_B

The profits of the duopolists are

Π_A = PQ_A – C_A = [250 – (Q_A + Q_B)]Q_A – 100 – Q_A²

Π_A = 250Q_A – Q_A² – Q_AQ_B – 100 – Q_A²

Π_A = 250Q_A – 2Q_A² – Q_AQ_B – 100

Π_B = PQ_B – C_B = [250 – (Q_A + Q_B)]Q_B – 100 – 2Q_B

Π_B = 250Q_B – Q_AQ_B – Q_B²– 100 – 2Q_B

Π_B = 248Q_B – Q_AQ_B – Q_B² – 100

For profit maximization under the Cournot assumption we have

∂Π_A/∂Q_A = 0 = 250 – 4Q_A – Q_B

∂Π_B/∂Q_B = 0 = 248 – 2Q_B – Q_A

The reaction functions are

Q_A = 62.5 – 0.25Q_B

Q_B = 124 – 0.5Q_A

Replacing Q_B into the Q_A reaction function we get

Q_A = 62.5 – 0.25(124 – 0.5Q_A)

Q_A = 62.5 – 31 + 0.125Q_A

0.875Q_A = 31.5

Q_A = 36

And

Q_B = 124 – (0.5 × 36)

Q_B = 106

Thus, the total output in the market is

Q = Q_A + Q_B = 36 + 106 = 142

And the market price

P = 250 – (Q_A + Q_B)

P = 250 – 142

P = 108

Total Revenue (TR) = Price × Quantity

Π_A = PQ_A – C_A

Π_A = (108 × 36) – 100 – (36)²

Π_A = 3888 – 1396

Π_A = 2498

And

Π_B = PQ_B – C_B

Π_B = (108 × 106) – 100 – (2 × 106)

Π_B = 11448 – 214

Π_B = 11,234

Stackelberg Equilibrium

The reaction functions are found by taking the partial derivatives of the duopolists' profit functions and equating them to zero:

Π_A = PQ_A – C_A = [250 – (Q_A + Q_B)]Q_A – 100 – Q_A²

Π_A = 250Q_A – Q_A² – Q_AQ_B – 100 – Q_A²

Π_A = 250Q_A – 2Q_A² – Q_AQ_B – 100

Π_B = PQ_B – C_B = [250 – (Q_A + Q_B)]Q_B – 100 – 2Q_B

Π_B = 250Q_B – Q_AQ_B – Q_B²– 100 – 2Q_B

Π_B = 248Q_B – Q_AQ_B – Q_B² – 100

For profit maximization under the Cournot assumption we have

∂Π_A/∂Q_A = 0 = 250 – 4Q_A – Q_B

∂Π_B/∂Q_B = 0 = 248 – 2Q_B – Q_A

The reaction functions are

Q_A = 62.5 – 0.25Q_B ----------------- A’s Reaction Curve

Q_B = 124 – 0.5Q_A ------------------- B’s Reaction Curve

Stackelberg's solution with A being the sophisticated leader

Firm A will substitute B's reaction function in its own profit equation, which it will then maximise as if it were a monopolist:

Π_A = 250Q_A – 2Q_A² – Q_AQ_B – 100

Substitute, Q_B = 124 – 0.5Q_A

Π_A = 250Q_A – 2Q_A² – Q_A(124 – 0.5Q_A) – 100

Maximise:                                        Π_A = 126Q_A – 1.5Q_A² – 100

First-order condition: ∂Π_A/∂Q_A = 126 – 3Q_A = 0

This yields output:                 Q_A = 42

Π_A = 126Q_A – 1.5Q_A² – 100

Π_A = (126 × 42) – 1.5(42)² – 100

Π_A = 2546

Second order condition: ∂²Π_A/∂Q²_A = – 3 < 0

So, the second-order condition for profit maximisation is fulfilled

Firm B would be the follower. It would assume that A would produce 42 units; thus B substitutes this amount in its reaction function

Q_B = 124 – 0.5Q_A

Q_B = 124 – (0.5 × 42)

Q_B = 103

Π_B = 248Q_B – Q_AQ_B – Q_B² – 100

Π_B = (248 × 103) – (42 × 103) – (103)² – 100

Π_B = 25544 – 4326 – 10609 – 100

Π_B = 10,509

And the market price

P = 250 – (42 + 103)

P = 250 – 145

P = 105

Cartel Equilibrium

The main aim of the central agency running the cartel is to maximize the total profit of the cartel

Π_T = Π_A + Π_B

Where

Π_A = TR_A – TC_A and Π_B = TR_B – TC_B

TR = Total Revenue

TC = Total Cost

Thus

Π_T = (TR_A + TR_B) – TC_A – TC_B

Π_T = P(Q_A + Q_B) – 100 – Q_A² – 100 – 2Q_B

Π_T = (250 – Q_A – Q_B)(Q_A + Q_B) – 100 – Q_A² – 100 – 2Q_B

Π_T = 250Q_A – Q_A² – Q_AQ_B + 250Q_B – Q_AQ_B – Q_B² – Q_A² – 200 – 2Q_B

Π_T = 250Q_A + 248Q_B – 2Q_AQ_B – Q_B² – 2Q_A² – 200

Setting the partial derivatives equal to zero we obtain

∂Π_A/∂Q_A = 250 – 4Q_A – 2Q_B = 0

∂Π_B/∂Q_B = 248 – 2Q_B – 2Q_A = 0

Solving for Q_A and Q_B we obtain

Q_A = 1

Q_B = 123

P = 250 – (Q_A + Q_B)

P = 250 – (1 + 123)

P = 250 – 124

P = 126

Total Profit

Π_T = 250Q_A + 248Q_B – 2Q_AQ_B – Q_B² – 2Q_A² – 200

Π_T = (250 × 1) + (248 × 123) – (2 × 1 × 123) – (123)² – (2 × 1)² – 200

Π_T = 250 + 30504 – 246 – 15129 – 2 – 200

Π_T = 30754 – 15577

Π_T = 15,177