You are the manager of a new amusement park that has monopoly power over the service it offers: roller coaster rides. You have figured that the park will attract 1,000 people per day, and each person will take x = 50 − 50p rides, where p is the price of a ride. The marginal cost of a ride is essentially zero.

 i. If admission to the park were free and you had to choose the profit maximizing price of a ride, p, what would it be? 

 ii. Suppose now that you can use a two-part tariff, i.e., you can set a price for admission to the park, and another one for each roller coaster ride. Which admission fee and price of a ride would you choose? 

Answer:

(i) The profit-maximizing price of a ride would be $0.5

(ii) The admission fee would be $25 and the price of a ride would be $0.5

Step-by-step explanation

(i)

If admission to the park is free, then the profit-maximizing price will be determined by putting MR=MC.

Given, MC = 0 and the demand function is:

Q = 50 - 50P

So, P = 1- Q/50

TR = P.Q

So, ?TR=Q−Q^2/50?

MR is derived by taking the first-order derivative of TR.

So, ?MR=1−Q/25?

On putting MR=MC and solving for Q, we get:

1-Q/25 = 0

So, Q= 25

Plugging the value of Q in the demand function and solving for P, we get:

P = 1 - 25/50 = 1- 0.5 = 0.5

Therefore, the profit-maximizing price and quantity are $0.5 and 25 respectively.

Hence, the profit-maximizing price of a ride would be $0.5

(ii)

A monopolist uses a two-part tariff two capture consumer surplus.

The consumers receive the maximum surplus when P=MC.

Given MC =0

Putting P = MC= 0, we get:

Q = 50 - 50P

So, Q = 50

So, at P=0, consumers will take 50 rides.

In the case of two-part tariff, the admission fee charged is equal to the consumer surplus.

As shown in the above diagram, the consumer surplus is represented by the yellow shaded triangle.

Here, consumer surplus = 1/2 * (1-0) * (50-0) = $25

Thus, the admission fee charged will be $25.

The price for each roller coaster ride will be determined by MR=MC. So, as seen above, the price for each rise will be $0.5

Therefore, the admission fee would be $25 and the price of a ride would be $0.5

PFA