Question

2. Suppose that a monopoly faces two markets for its product. D: Q1 = 100 - P; and D2 : Q2 = 80-P The monopoly can verify consumers to decide which market they belong to so that it charge different prices in the two markets. The cost of production is CQ) = 100 where Q- Q. +Q2. a) Please write out the total profit function for the monopoly as functions of Qı and Q2 b) Find out the profit maximizing Q1 and Q2, and prices Pi and P2 charged by the monopoly in each of these two markets, figure out the total profit. c) Suppose the monopoly must charge the same price in both market, please calculate the price and quantity for the monopoly, what would be the profit for the monopoly.

Answer 1

(a)

In market 1: P1 = 100 - Q1

In market 2: P2 = 80 - Q2

Profit (Z) = TR1 + TR2 - C(Q) = (P1 x Q1) + (P2 x Q2) - 10 x (Q1 + Q2)

Z = 100Q1 - Q12 + 80Q2 - Q22 - 10Q1 - 10Q2

Z = 90Q1 - Q12 + 70Q2 - Q22

(b)

Profit is maximized when $\partial$ Z/$\partial$Q1 = 0 and $\partial$ Z/$\partial$Q2 = 0.

$\partial$Z/$\partial$Q1 = 90 - 2Q1 = 0

2Q1 = 90

Q1 = 45

P1 = 100 - 45 = 55

$\partial$Z/$\partial$Q2 = 70 - 2Q2 = 0

2Q2 = 70

Q2 = 35

P2 = 80 - 35 = 45

Profit = (55 x 45) + (45 x 35) - [10 x (45 + 35)]

= 2475 + 1575 - (10 x 80)

= 4050 - 800

= 3250

(c)

In this case, P1 = P2 = P

Market demand (Q) = Q1 + Q2 = 100 - P + 80 - P

Q = 180 - 2P

P = (180 - Q)/2 = 90 - 0.5Q

TR = PQ = 90Q - 0.5Q2

MR = dTR/dQ = 90 - Q

MC = dC(Q)/dQ = 10

Setting MR = MC,

90 - Q = 10

Q = 80

P = 90 - 0.5 x 80 = 90 - 40 = 50

Profit = Q x (P - MC) = 80 x (50 - 10) = 80 x 40 = 3200