Question

Q5. You are a theater owner fortunate to book a summer box office hit into your single theater. You are now planning the length of its run. Your share of the film’s projected box office is

R = 10 W -0.25 (W)^2, where R is in thousands of dollars and W is the number of weeks that the movie runs. The average operating cost of your theater is AC =MC = $5 thousand per week.

1. To maximize your profit, how many weeks should the movie run? What is your profit?

You realize that your typical movie makes an average operating profit of $1.5 thousand per week. How does this fact affect your decision in part a above if at all?

Answer 1

Information given:

R = 10W - 0.25W2

AC = MC = $5000 per week

Taking the first differential of R with respect to W, we will get the marginal earnings per week

Let us call this MR

Now, profit will be maximized when MR = MC

Thus,

10 - 0.5W = 5

Here, we take MC as 5, because all units are in thousand $ per week. So we need to keep standard units.

5 = 0.5W

W = 10 weeks

Now, for us, profit is as under:

R = 10W - 0.25W2

R = (10 x 10) - 0.25 (100)

= 100 - 25

R = $75 thousand over the 10 weeks

Now, if a typical movie also has a per week operating profit of $1.5 thousand per week, it makes a difference to our per week operating costs.

The decision in a) will change, as under:

Take MR = MC

10 - 0.5W = 5 - 1.5

10 - 0.5W = 3.5

Here, we reduce MC by 1.5, due to the reduction in operating costs.

6.5 = 0.5W

W = 13 weeks

Now, for us, profit is as under:

R = 10W - 0.25W2

R = (10 x 13) - 0.25 (169)

= 130 - 42.25

R = $87.75 thousand over the 13 weeks

Thus, our movie will run longer and earn more profits.