Question

You are in charge of pricing tickets for an upcoming event at the outdoor amphitheater where all seating is by general admission(no assigned seats). Demand for the event can be segmented into students and non-students. Student demand is Qst=10,000-100P. Non- student demand can be expressed as Qns=5000-10P. Your marginal cost per attendee is $10 and is constant. The capacity of the amphitheater is 5000. Since you can check for a student ID, you can charge each group a different price. What price do you charge each group?

Answer 1

Each group can be charged a price such that MR=MC in each case to maximize profit.

Non Student's demand is given by

Q_ns=5000-10P

on rearranging we get

10P=5000-Q_ns

P=500-0.1Q_ns

TR=P*Q_ns=500Q_ns-0.1Q_ns²

Marginal Revenue=MR_ns=dTR/dQns=500-0.2Q_ns

Set MR_ns=MC for profit maximization.

500-0.2Q_ns=10

490=0.2Q_ns

Q_ns=2450

P=500-0.1Q_ns=500-0.1*2450=$255

Optimal price for non student is $255

Student's demand is given by

Q_st=10000-100P

on rearranging we get

100P=10000-Q_st

P=100-0.01Q_st

Total Revenue=P*Q_st=(100-0.01Q_st)*Q_st=100Q_st-0.01Q_st²

Marginal Revenue=MR_st=dTR/dQ_st=100-0.02Q_st

Set MR_st=MC for profit maximization

100-0.02Q_st=10

0.02Q_st=90

Q=4500

P=100-0.01Q_st=100-0.01*4500=$55

We see that optimal price is higher in case of non student group.

Since we have capacity constraint. Let us see how many seats are vacant after filling non student group.

Seats left for students=5000-2450=2550

We can see that demand is more than available seats.

Let us see what price student group can offer at this volume

P=100-0.01Q_st=100-0.01*2550=$74.50

So, student can be charged at $74.50 to and theater will be utilized at full capacity.

So, for profit maximization

Price for non student=$255

Price for non student=$74.50