Notation: Q = Aggregate Quantity and q = individual firm quantity Suppose we hav

Question

Notation: Q = Aggregate Quantity and q = individual firm quantity Suppose we have 2 firms: firm 1 is the leader and firm 2 is the follower. The demand function that is faced by both firms are Ps = 100 - 2Qs, where Qs is the sum of quantity that is produced by both Firm 1 and Firm 2 or (Q_s = q_1+ q_2). The cost function for Firm 1 and Firm 2 are C_1 = 10q_1 and C_2 = 10q_2 respectively. What is the profit maximizing price Ps and quantity Q_s in this stackelberg equilibrium? Suppose we have a monopoly market. P* and Q* are the price and quantity that are produced by monopoly. Qc is the quantity that will be produced if we have a competitive market and 65 is the maximum price that consumers are willing to pay. Based on the figure below, calculate producer surplus, consumer surplus, and deadweight loss associated with monopoly.

Explanation / Answer

(1)

P = 100 – 2Q where Q = q1 + q2

P = 100 – 2q1 – 2q2

So,

Total revenue of firm 1, TR1 = P x q1 = 100q1 – 2q12 – 2q1q2

Total revenue of firm 2, TR2 = P x q2 = 100q2 – 2q1q2 – 2q22

So, Marginal revenue of firm 2, MR2 = dTR2 / dq2 = 100 – 2q1 – 4q2

C2 = 10q2, so MC2 = dTC2 / dq2 = 10

Equating MR2 = TC2,

100 – 2q1 – 4q2 = 10

Or,

2q1 + 4q2 = 90

Dividing both sides by 2:

q1 + 2q2 = 45

Or,   q2 = (45 – q1) /2 = 22.5 – 0.5q1    ..... (1)

This is firm 2’s response function. Substituting (1) in TR1:

TR1 = 100q1 - 2q12 – 2q1q2 = 100q1 – 2q12 – 2q1 (22.5 – 0.5q1)

= 100q1 – 2q12 – 45q1 – q12

= 55q1 – 3q12

So, MR1 = dTR1 / dq1 = 55 – 6q1

Equating MR1 = MC1 [Where MC1 = dC1 / dq1 = 10]

55 – 6q1 = 10

6q1 = 45

q1 = 45/6 = 7.5

therefore, q2 = 22.5 – 0.5q1    [From (1)]

= 22.5 – 0.5 x 7.5 = 18.75

Q = q1 + q2 = 7.5 + 18.75 = 26.25

P = 100 – 2Q = 100 – (2 x 26.25) = 100 – 52.5

P = 47.5

(2)

Consumer surplus = area between demand curve and price

= (1/2) x (65 – 55) x 30 = 150

Producer surplus = Area between supply curve (= MC) and price

= [(55 – 10) x 30] + [(1/2) x (50 – 30) x (55 – 10)]

= 1350 + 450 = 1800

Deadweight loss = (1/2) x (55 – 10) x (50 – 30) = 450

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

---

---

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