Suppose that a movie theater owner faces the demand curve given by P=10-.001Q where P is the price of a ticket in dollars and Q is the number of tickets sold per week. Suppose that all the cost associated with the theater other than film costs do not vary with the number of tickets sold.
4a. What price will the theater owner charge if she must pay a fee of $12,000 per week to rent the movie?
4b. What price will she charge if she must pay $4 per ticket to the film distributor?
4c. What price will she charge if she must pay 80% of her revenues to the distributor?
4d. How much revenue will she get ($12,000 per week rental) if she can perfectly price discriminate?
Can you please tell me how to break each problem down.
Sure! Let's break down each problem step by step:
4a. What price will the theater owner charge if she must pay a fee of $12,000 per week to rent the movie?
To find the price, we need to determine the ticket price (P) that maximizes the theater owner's profit. Profit is calculated as revenue minus costs. In this case, the only cost is the rental fee of $12,000 per week.
The revenue earned by the theater owner is the product of the ticket price (P) and the number of tickets sold (Q). We can express revenue as P * Q.
The profit function is then given by:
Profit = Revenue - Cost = P * Q - $12,000
Now, to find the price that maximizes the profit, we need to find the value of P at which the derivative of the profit function with respect to P equals zero. This is because the maximum or minimum point of a function occurs when the derivative is zero.
Taking the derivative of the profit function with respect to P, we get:
d(Profit)/dP = d(P * Q - $12,000)/dP = Q - 0 = Q
Setting Q (which is the number of tickets sold) to zero will give us the maximum profit, as the derivative is equal to zero when there is no change in the number of tickets sold. Hence, we can calculate the price at this point as:
10 - 0.001 * 0 = 10
Therefore, the theater owner will charge a price of $10 if she must pay a fee of $12,000 per week to rent the movie.
4b. What price will she charge if she must pay $4 per ticket to the film distributor?
In this case, the cost per ticket is $4 to the film distributor. The profit function remains the same, but now the cost term changes.
Profit = Revenue - Cost = P * Q - $4 * Q
Taking the derivative with respect to P, we get:
d(Profit)/dP = d(P * Q - $4 * Q)/dP = Q - 0 = Q
Setting Q to zero, we get:
10 - 0.001 * 0 = 10
Therefore, the theater owner will still charge a price of $10 if she must pay $4 per ticket to the film distributor.
4c. What price will she charge if she must pay 80% of her revenues to the distributor?
Now, the cost is based on a percentage of the revenue. This means the cost per ticket will depend on the ticket price (P). Let's denote the cost per ticket as C.
C = 0.80 * P
The profit function will be:
Profit = Revenue - Cost = P * Q - C * Q
Taking the derivative with respect to P, we get:
d(Profit)/dP = d(P * Q - C * Q)/dP = Q - 0.80 * Q = 0.2 * Q
Setting this derivative to zero, we get:
0.2 * Q = 0
Q = 0
In this case, the number of tickets sold is zero, which means the revenue and profit will also be zero. Therefore, the theater owner will not charge any price if she must pay 80% of her revenues to the distributor.
4d. How much revenue will she get ($12,000 per week rental) if she can perfectly price discriminate?
Perfect price discrimination means that the theater owner can charge different prices to different customers based on their willingness to pay. In this case, there is no demand curve equation provided, so we can assume that the theater owner can charge each customer a price equal to their individual maximum willingness to pay.
Since we don't have a demand equation or any other information about customers' willingness to pay, it is not possible to calculate the exact revenue the theater owner will receive in this scenario.
Hopefully, this breakdown helps you understand how to approach each problem.