Question

You’ve been assigned to set the price at the local movie theater. You know that market demand can be broken into two groups, students and non-students. Those functions are: Students: p = 12 − 1/4q Non-Students: p = 22 − q (if you aggregated those functions, you would obtain p = 22 − q if p > 12 and p = 14 −1/5q if p ≤ 12). You’ve also been given the total cost and marginal cost for the theater. TC = 6 + 2q + q^2, MC = 2 + 2q

a) Plot the inverse demand curves for students and non-students. Make sure to also include both marginal revenue curves and supply curves, as well as labeling correctly.

b) Now, let’s assume that you can only charge one price for tickets. What is the equilibrium price, quantity and profits if you choose to act as a monopolist?

c) You’ve decided to try price discrimination for each group. What is the equilibrium price and quantity for students and non-students? What is your new total profit? Round answers to two decimal places.

d) Do the prices in part (c) make sense? Why would the price for one group be higher than the price for the other group?

e) What kind of price discrimination is this? Is there any way you can prevent arbitrage in this case? If so, how? Why is it important to prevent arbitrage in price discrimination?

Answer 1

a) The graph is attached as a image below

b) In case of acting as a monopolist, the price will be charged where aggregate marginal revenue is equal to marginal cost. It is evident from the graph that the quantity produces will fall on the lower portion of the demand curve.

b) TR = 14q - 1/5q²

MR= 14-2/5q = 2+2q (MC)

On solving, we get, q=5

On equating the value of q in equation p = 14 −1/5q, we get p=13

Profit= TR-TC

=(14q - 1/5q²) - (6 + 2q + q^2)

Equating the value of q i.e. 5 in above, we get

Profit= (70-5) - (6+10+25)

Profit =65-41 = 24

c) In case of price discrimination, we would sell where the marginal revenue of student is equal to marginal cost from the aggregate market

For student, p= 12 − 1/4q. So, TR= 12q-1/4q^2

MR=MC

12-q/2=2+2q

24-q = 4+4q

Equibrium quantity of students, q=4

p=12-1/4(4)

Equilibrium price of students, p=11

Profit = TR-TC

=(12q-1/4q^2) - (6 + 2q + q^2)

=14

In case of price discrimination, we would sell where the marginal revenue of non student is equal to marginal cost from the aggregate market

p=22-q. So, TR=P*Q= 22q-q^2

MR=MC

22-2q=2+2q

Equibrium quantity of non students, q=5

p=22-5 = 17

Equilibrium price of non students, p=17

Profit= (22q-q^2)- (6 + 2q + q^2)

=44

So, total profit = 14+44 = 58

d) The price for one group is higher than the price for the other group as the profits earned by non-students is much higher than earned by students. Also, the profit that is earned as a result of separating the markets is greater than the profit that is earned as a result of keeping the markets combined.

Also, the prices charged from students and non-students is higher than the prices charged if there is no price discrimination.