Suppose the demand for pizza in a small town where there are only two firms, A a

Question

Suppose the demand for pizza in a small town where there are only two firms, A and B, is p = 10 - Q, with Q = q_a + q_b middot where q_i is the quantity produced by firms i=A, B Each firm has a cost function TC = 2 + q_i. Describe the behavior of the firms if they act as Cournot duopolists and determine the Courtnot Equilibrium. Illustrate your answer by a clearly labeled graph. Describe the behavior of the firms if they act as Stackelberg duopolists and determine the Stackelberg equilibrium if Firm A is the Stackelberg leader. Describe the behavior of the firms if they collude and determine the collusion equilibrium if both firms share the market equally. Compute the equilibrium price for each of the three models. Compute the profit for each of the firms in the three models.

Explanation / Answer

1. P = 10 - Q

TC = 2 + qi

In case of cournot duopoly,

the profit for firm A will be

Profit A = TR - TC = P*qA - TCA

= (10 - qA -qB)qA - 2 - qA

dProfitA/dqA = 10 - 2qA -qB - 1

Putting   dProfitA/dqA = 0

qA = (9-qB)/2

This is the reaction function for firm A,

Similiarly reaction function for firm B

qB = (9-qA)/2

Solving these two reaction function for qA and qB

qA = (9-((9-qA)/2)/2

4qA = 9 + qA

qA = 9/3 = 3

qB = 3

Q = qA + qB = 6

P = 10 - Q = 10 - 6 = 4

Profit of firm A = 4*3 - 2 - 3 =7

  Profit of firm B = 4*3 - 2 - 3 =7

For the diagram draw the two reaction curve equations.

In case of Stackelberg

If firm A is the stackelberg leader, then

Reaction function of firm B is  qB = (9-qA)/2 (same as cournot)

Profit maximization equation for firm A

Profit =  (10 - qA -qB)qA - 2 - qA

Putting reaction function of firm 2

    ProfitA = (10 - qA - (9-qA)/2)qA - 2 - qA

ProfitA = (5.5 - 0.5qA)qA - 2 - qA

dProfitA/dqA = 5.5 - qA - 1

Putting   dProfitA/dqA = 0

qA = 4.5

qB = (9-qA)/2 = 2.25

Q = qA + qB = 6.75

P = 10 - Q = 3.25

Profit of firm A = 3.25*4.5 - 2 - 4.5 = 8.125

Profit of firm B = 3.25*2.25 - 2 - 2.25 = 3.0625

c. In case, if they collude than they will produce the monopoly output.

Profit maximization condition for monopoly , MR = MC

TR = P*Q = (10 - Q)*Q

MR = dTR/dTC = 10 - 2Q

MC = dTC/dQ = 1

Equating MR = MC

10 - 2Q = 1

Q = 9/2 = 4.5

hence qA = qB = Q/2 = 4.5/2 = 2.25

P = 10 - Q = 10 - 4.5 = 5.5

Profit of firm A = 5.5*2.25 - 2 - 2.25 = 8.125

  Profit of firm B = 5.5*2.25 - 2 - 2.25 = 8.125

If you don't understand anything then comment, I'ill revert back on the same. :)

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.