Question

A nightclub owner has both students and adult customers. The demand for drinks by a typical student is Qs= 30-3p The demand for a typical adult is Qa= 15-2p There are equal numbers of students and adults. The marginal cost of each drink is $3.

What price will the club owner set if she cannot discriminate between the two groups? What will her total profit be at this price?

If the club owner could separate the groups and practice third-degree price discrimination what price per drink would be charged to members of each group? What would be the club owner’s profit?

If the club owner can “card” patrons and determine who among them is a students and who is not, and in turn, can serve each group by offering a cover charge and a number of drink tokens to each group, what will the cover charge and the number of tokens given to students be? What will be the cover charge and number of tokens given to adults? What is the club owner’s profit under this regime?

Answer 1

If the club owner cannot discriminate between the two groups then the aggregate demand would be

Qd =Qs+Qa= 15-2p+ 30-3p
= 45- 5p

inverse demand is given by P= (45-Qd)/5

so total revenue would be Qd*P= (45Qd-Qd²)/5

so marginal revenue is given by MR= (45-2Qd)/5

now in order to get the equiliobrium price and quantity we equate the MC to MR

MC=MR

$3= (45-2Qd)/5

15 -45= -2Qd

=> Qd=30/2= 15

so aggregate equilibrium quantity is given by 15

putting value of Qd in inverse demand function

P= (45-Qd)/5 = (45-15)/5 = 30/5= $ 6

so $6 will be the equilibrium price in case of no price discrimination.

total profits = revenue - cost

= 6(15)- 3(15)

= 90- 45

= $45

if the owner practices price discrimination , then the MC would be equal to the marginal revenue of each type

so  Qs= 30-3p

P= (30-Qs)/3

total revenue is given by

P*Qs = (30Qs-Qs²)/3 =TR

so MRs= ( 30-2Qs)/3

MRs= MC => ( 30-2Qs)/3= $3  

Qs= 21/ 2= 10.5

Ps = (30-Qs)/3

= (30-10.5)/3

= 19.5/3 =$ 6.5

similarly we look at the demand function of adults

Qa= 15-2p

=> (15-Qa)/2 = P

TR = P*Qa = (15Qa-Qa²)/2

=> MRa= (15- 2Qa)/2

MC=MRa

$3= (15- 2Qa)/2

so Qa = 9/2 = 4.5

Pa= $ 5.25

total profits = total revenue is Qs*Ps+ Qa*Pa- total cost

= (6.5)*(10.5) +(5.25)(4.5)- (10.5)(3)-(4.5)(3)

=68.25 + 23.625- 31.5-13.5

= $ 91.875- 45

= $ 46.875

the owner practices two tariff pricing in this case

number of tokens would be the total quantity demanded at price $3 which would be the MC of the drink

putting the price as $3 in the demand functions of both students and adults

Qs= 30-3(3) = 30-9 = 21= number of tokens for students

Qa= 15-2(3) = 15-6 = 9= number of tokens for adults

total profits

consumer surplus of each would be equal to the cover charge for each party

cover charge for students = area above the price line and below the deamnd curve and Mc* no of tokens

= 1/2(10-3)(21)+3*21 [.[(10-3)is the price at 0 quantity minus the MC which is now the price assumed]

= 73.5 + 63

= 136.5

cover charge for adults = 1/2 * (7.5-3) * 9 +3(9) ......................[(7.5-3)is the price at 0 quantity minus the MC which is now the price assumed]

= 20.25 + 27

= 47.25

total profits = revenue - cost

=136.5 +47.25- 21(3)-9(3)....................cost = MCx no of units

= $93.75