Question

3 Price Discrimination ( Show all work)

Suppose the demand for ticket sales is given by the following function:

P = 315 − 2Q

Further suppose that marginal cost is 3Q and total cost is 3/2Qsquare2

a) Find the profit maximizing price and quantity.

b) What is the maximum profit? Suppose now that the ticket seller can price discriminate by checking IDs. There are two demands in the market:

Adult Demand: PA = 315 − 3Q

Student Demand: PK = 315 − 6Q

Again, suppose that marginal cost is 3Q and total cost is 3/2Q square2

c) What is the profit maximizing price (PA) that will be charged to the adults?

d) What is the profit maximizing price (PK) that will be charged to the kids?

e) What is the maximum profit achieved by profit discrimination (add the profits from selling to the adult and kid markets)?

Answer 1

(a)

Demand function is as follows -

P = 315 - 2Q

Calculate the Total revenue -

TR = P * Q = (315 - 2Q) * Q = 315Q - 2Q²

Calculate the marginal revenue -

MR = dTR/dQ = d(315Q - 2Q²)/dQ = 315 - 4Q

MC = 3Q

A monopolist maximize the profit when it produce that level of output corresponding to which MR equals MC.

MR = MC

315 - 4Q = 3Q

7Q = 315

Q = 315/7

Q = 45

P = 315 - 2Q = 315 - (2*45) = 315 - 90 = 225

Thus,

The profit-maximzing price is $225 per ticket.

The profit-maximizing quantity is 45 tickets.

(b)

Calculate the maximum profit -

Profit = Total revenue - Total cost

Profit = (P * Q) - [(3/2)(Q)²]

Profit = ($225 * 45) - [(3/2) * (45)²]

Profit = $7,087.5

The maximum profit is $7,087.5

(c)

Market for Adults -

Demand is as follows -

P = 315 - 3Q

Calculate the Total revenue -

TR = P * Q = (315 - 3Q) * Q = 315Q - 3Q²

Calculate the marginal revenue -

MR = dTR/dQ = d(315Q - 3Q²)/dQ = 315 - 6Q

MC = 3Q

A monopolist maximize the profit when it produce that level of output corresponding to which MR equals MC.

MR = MC

315 - 6Q = 3Q

9Q = 315

Q = 315/9

Q = 35

P = 315 - 3Q = 315 - (3*35) = 315 - 105 = 210

Thus, in adult market,

The profit-maximzing price is $210 per ticket.

The profit-maximizing quantity is 35 tickets.

(d)

Market for Kids -

Demand is as follows -

P = 315 - 6Q

Calculate the Total revenue -

TR = P * Q = (315 - 6Q) * Q = 315Q - 6Q²

Calculate the marginal revenue -

MR = dTR/dQ = d(315Q - 6Q²)/dQ = 315 - 12Q

MC = 3Q

A monopolist maximize the profit when it produce that level of output corresponding to which MR equals MC.

MR = MC

315 - 12Q = 3Q

15Q = 315

Q = 315/15

Q = 21

P = 315 - 6Q = 315 - (6*21) = 315 - 126 = 189

Thus, in kids market

The profit-maximzing price is $189 per ticket.

The profit-maximizing quantity is 21 tickets.

(e)

Calculate the maximum profit achieved by price discrimination -

Profit = (PA * QA) + (PK * QK) - [3/2(QA +QK)²]

Profit = (210 * 35) + (189 * 21) - [3/2(35+21)²]

Profit = 7,350 + 3,969 - 4,704

Profit = 6,615

The maximum profit is $6,615.