Suppose that 1,000 people are interested in attending ElvisLand.Once a person arrives at ElvisLand, his or her demand for rides is given by x = max{6 - p, 0} , where p is the price per ride.There is a constant marginal cost of $3 for providing a ride at ElvisLand.If ElvisLand charges a profit-maximizing two-part tariff, with one price for admission to ElvisLand and another price per ride for those who get in.How much should it charge per ride and how much for admission?
A)$3 per ride and $6 for admission
B)$3 per ride and $4.50 for admission
C)$0 per ride and $3 for admission
D)$0 per ride and $7.50 for admission
E)$6 per ride and $6 for admission

Question

Quizplus · Accepted Answer

To find the profit-maximizing two-part tariff, we need to find the price that maximizes the profit of ElvisLand.

First, let's find the demand for admission only. We know that the demand for rides will depend on the price of the rides, so we need to find the demand for rides at different prices:

- If p is $0, then x = 106. This is the maximum number of rides that anyone will take, since if p = $0, then everyone will take the maximum number of rides possible (106 rides) to maximize their utility. 
- If p is $1, then x = 105. 
- If p is $2, then x = 104. 
- And so on, until... 
- If p is $103, then x = 3. 
- If p is $104 or higher, then x = 0, since no one will pay more for a ride than the amount they value it (i.e. their willingness to pay).

Now we can find the demand for admission at different prices. To do this, we need to subtract the marginal cost of providing one ride ($3) from the price per ride, multiply that by the number of rides demanded at that price, and add the admission price:

- If p is $0, then the price per ride is $0 - $3 = -$3. This doesn't make sense, since we can't have a negative price, so we'll skip this one. 
- If p is $1, then the price per ride is $1 - $3 = -$2, so no one will buy any rides. The demand for admission is simply the total number of people interested in attending, which is 1,000. Therefore, the total revenue is $1,000 x $1 = $1,000. 
- If p is $2, then the price per ride is $2 - $3 = -$1, so again no one will buy any rides. The demand for admission is still 1,000, so the total revenue is $1,000 x $2 = $2,000. 
- And so on, until... 
- If p is $103, then the price per ride is $103 - $3 = $100, and the demand for rides is 3. The demand for admission is 1,000, so the total revenue is (1,000 x $100) + (3 x $103) = $100,300. 
- If p is $104, then the price per ride is $104 - $3 = $101, but the demand for rides is 0, so the total revenue is simply 1,000 x $104 = $104,000. 
- If p is $105 or higher, then the demand for rides is 0 and the revenue is just 1,000 x the admission price.

To find the profit-maximizing prices, we need to choose the prices that maximize revenue.

For the admission price, it's clear that the higher we go, the less revenue we make. Therefore, we should set the admission price as low as possible (i.e. $3).

For the price per ride, we want to choose the price that leads to the highest total revenue. From our calculations above, we can see that the revenue first increases as the price per ride increases, but then it starts to decrease once the price per ride exceeds $100. Therefore, the profit-maximizing price per ride is the highest price before the revenue starts to decrease, which is $3.

Therefore, the profit-maximizing two-part tariff is a price of $3 per ride and an admission price of $4.50 ($3 for the rides + $1.50 for the admission).

Suppose That 1,000 People Are Interested in Attending ElvisLand