Question

Smith Distributing sells videocassettes in two separable markets. The marginal cost of each
cassette is $2. For the first market, demand is given by
Q1 = 20 - 5P1
The demand equation for the second market is
Q2 = 20 – 2P2i)
If the firm uses third-degree price discrimination, what will be the profit-maximizing
price and quantity in each market? How much economic profit will the firm earn?
ii) If the firm charges the same price in both markets, what will be the profit-maximizing
price and total quantity? How much economic profit will the firm earn?

Answer 1

I.

Under third degree price discrimination sellers charge different prices to difference consumer groups

MC = 2

TC = 2Q

Market One

Q$_{1}$ = 20 - 5P$_{1}$

$P_{1} = ​$ 4 - $\frac{Q_{1}}{5} ​$

TR = 4$Q_{1} ​$ - $\frac{Q_{1}^{2}}{5}$

Differentiating the TR with respect to Q we get MR

MR = 4 - $\frac{2Q_{1}}{5}$

mc = 2

Equating MR = MC

$4 - \frac{2Q_{1}}{5} = 2$ solving this we get

$Q_{1} = 5 ​$

substituting $Q_{1} = 5$ into $P_{1} ​ =​ 4 - \frac{Q_{1}}{5} ​$

we get $P_{1}$ = 4

substituting quantity we get actual TR

TR = $4Q_{1} -$$\frac{Q_{1}^{2}}{5}$

TR = 4 $\times$ 5 - $\frac{5\times 5}{5}$

TR= 15

TC = 2Q

TC = 2 $\times$ 5

TC = 10

Hence economic profit = 15 - 10 = 5

market two

$Q_{2} = 20 - 2P_{2} ​$

$P_{2} =​ ​10 -\frac{Q_{2}}{2} ​$

$TR = 10Q_{2} - \frac{Q_{2}^{2}}{2}$

Differentiating the TR with respect to Q we get MR

$MR = 10 - Q_{2}$

MR = MC

$10−Q_{2} = 2$

$Q_{2} = 8$

$P_{2} =​ ​10 -\frac{Q_{2}}{2} ​ ​ ​ ​$

$P_{2} =​ ​10 -\frac{8}{2} ​$

$P_{2} =​ 6$

Economic profit

$TR = 10Q_{2} - \frac{Q_{2}^{2}}{2}$ but $Q_{2} = 8$

Therefore TR = 48

TC = 2Q but $Q_{2} = 8$

TC = 16

Economic profit = 48 - 16 = 32

ii.

$Q_{1} = 20 - 5P_{1}$

$Q_{2} = 20 - 2P_{2} ​$

Price is the same

Quantity to be used will be the same

Average quantity in both of the markets

Q = $20 - 5P_{1}$ + $20 - 2P_{2}$

Q = $\frac{(20 - 5P + 20−5P ) }{2} ​$

Q = 40 - 3.5P

P = 11.4 - 0.3Q

TR = 11.4Q - 0.3$Q^{2}$

MR = 11.4 - 0.6Q

MC = 2

MR = MC

11.4 - 0.6Q =2

Q = 15.7 units

P = 11.4 - 0.3Q but q = 15.7

P = 11.4 - 0.3$\times 15.7$

P = 6.7

Economic profit

TR = 11.4Q - 0.3$Q^{2}$ but q = 15.7

TR = 105

TC = 2 Q

TC = 31.4

ECONOMIC PROFIT = 105 - 31.4

= 73.6