Please help with two questions from managerial economics... (explanations would
ID: 1188762 • Letter: P
Question
Please help with two questions from managerial economics... (explanations would be greatly appreciated)
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 (@s = qr + q2). The cost function for Firm 1 and Firm 2 are Cx - 10and C2 = 10q2 respectively. What is the profit maximizing price Ps and quantity @sin 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