Question

1. Consider a monopolist with unit cost c = 20, facing two separate markets with demand functions D1(p1) = 100 - p1 and D2(p2) = 60 - 2p2.

(a) Find the optimal prices (p1*, p2*) and quantities (q1*, q2*) with price discrimination.

(b) Find the optimal price p* and quantity q* without price discrimination. Compare them to the answers in (a)

(c) Compare total welfare with and without price discrimination. Explain your answer.

Answer 1

a)

First Market:

Q = 100 -P1

P1 = 100 -Q

TR = ( 100 -Q)*Q

= 100Q - Q^2

MR = 100 - 2Q

MC = 20

100 - 2Q = 20

2Q = 80

Q = 40

P = 100 - 40

= 60

Second market:

Q = 60 - 2P

P = 30 - 0.5Q

TR = 30Q - 0.5Q^2

MR = 30 -Q

MR = MC

30 -Q = 20

Q = 10

P = 30 - 0.5(10)

= 30 - 5

= 25

b)

Total demand = 100 -P +60 -2P

= 160 - 3P

P =53.33 - 0.33Q

TR = 53.33Q - 0.33Q^2

MR = 53.33 - 0.66Q

MR = MC

53.33 -0.66Q =20

33.33 = 0.66Q

Q = 50.50

P =53.33 -0.33(50.5)

=36.6

c)

No discrimination and total welfare:

= 0.5(53.33 -36.6)(50.5) +16.6*50.5

=1260.7325

With discrimination:

First market:

0.5(100-60)*40 + 40*40

= 2400

second market:

0.5(60-25)*10 + 10*5

=225

Price discrimination leads to rise in total welfare.