Question

Assume that Netflix can charge European users a different price from US users, but that the costs of providing programming to each user increases as the total number of users increases. Specifically, the U.S. has demand:

??? = 22 − (3/2) ???

And Europe has demand:

?? = 12 − ??

And Netflix’s cost function is: ? = 30 + 5(??? + ?? ) + (1/2) (??? + ??) 2

Resulting in marginal cost: ?? = 5 + (??? + ??) Identify the optimal price(s), quantities of customers, and resulting profits for this third-degree price discriminating firm.

Answer 1

P_US = 22 - (3/2)QUS If demand function P = a - bQ then   

MR_US = 22 - 2(3/2)Q_US MR = a - 2bQ

= 22 - 3Q_US  

P_E = 12 - Q_E

MR_E = 12 - 2Q_E  

MC =  5 + Q_US + Q_E

Profit maximizing condition

MR_US = MR_E = MC

So  MR_US  = MC  

   22 - 3Q_US =   5 + Q_US + Q_E

22 - 5 = Q_US + 3Q_US + Q_E

17 = 4Q_US + Q_E (i)

Now    MR_E = MC  

12 - 2Q_E =    5 + Q_US + Q_E  

12 - 5 = Q_E + 2Q_E + Q_US

7 = 3Q_E  +    Q_US (ii)

from (i) and (ii)  

17 = 4Q_US + Q_E

   7 =   Q_US + 3Q_E

Multiply equ.(i) by 3 on both sides

51 = 12Q_US + 3Q_E

   7 =   Q_US + 3Q_E

subtracting these two equations and we get

44 = 11Q_US

Q_US = 44/11

= 4

put Q_US = 4 in equ.(i)

17 = 4Q_US + Q_E

17 = 4(4) + Q_E   

17 = 16 + Q_E   

Q_E = 17 - 16

= 1

P_US = 22 - (3/2)Q _US

= 22 - (3/2)(4)

= 22 - 6

= 18

P_E = 12 - Q_E

= 12 - 1

= 11

Profit   = 111 + 184 - 30 - 5(4+1) - 1/2(4+1)²

= 11 + 72 - 30 - 25 - 25/2

= 83 - 55 - 12.5

= 83 - 67.5

= 15.5