Question

The builder of a new movie theater complex is trying to decide how many screens she wants. Below are her estimates of the number of patrons the complex will attract each year, depending on the number of screens available.

Number of screens	Total number of patrons
1	40,000
2	65,000
3	85,000
4	100,000
5	110,000

After paying the movie distributors and meeting all other noninterest expenses, the owner expects to net $2.5 per ticket sold. Construction costs are $1,000,000 per screen.

Instructions: Enter your responses as whole numbers.

a. Make a table showing the value of marginal product for each screen from the first through the fifth.

Number of screens	Value of marginal product
1	$
2	$
3	$
4	$
5	$

What property is illustrated by the behavior of marginal products?

• Diminishing returns to capital

• Increasing returns to capital

• Negative returns to capital

b. How many screens will be built if the real interest rate is 5.5 percent?

screen(s)


c. How many screens will be built if the real interest rate is 7.5 percent?

screen(s)


d. How many screens will be built if the real interest rate is 10 percent?

screen(s)

e. If the real interest rate is 5.5 percent, what is the highest construction cost per screen that would make a five-screen complex profitable?

$

Answer 1

a)

No. of Screens	Total no. of patrons	Additional patrons	Price (in $)	Value of Marginal Product
1	40,000	40,000	2.5	100,000
2	65,000	25,000	2.5	62,500
3	85,000	20,000	2.5	50,000
4	100,000	15,000	2.5	37,500
5	110,000	10,000	2.5	25,000

( NOTE: The additional patron is calculated by subtracting the total no. of patrons in previous year from total no. of patrons in the year for which it is to be calculated.

For eg, Additional patrons

For year 1 = 40,000-0 = 40,000

For year 2 = 65,000-40,000 = 25,000 and so on. )

The behaviour of marginal product shows the diminishing returns as total no. of patrons increases but with a diminishing rate i.e. the marginal product falls and becomes less than the average product.

b) The interesr rate = 5.5%

Thus, The interest cost of each screen = Interest rate*construction cost

= 5.5% * $1,000,000

= $55,000

Since, there are no other costs mentioned, So, the value of marginal product exceeds $55,000 for 2 screens.

Thus, 3 screens should be built.

c) (This will be solved as same as the second solution)

The interesr rate = 7.5%

Thus, The interest cost of each screen = Interest rate*construction cost

= 7.5% * $1,000,000

= $75,000

Since, there are no other costs mentioned, So, the value of marginal product exceeds $75,000 for 1 screen.

Thus, 1 screens should be built.

d) (This will also be solved as same as the second and the third solution)

The interesr rate = 10%

Thus, The interest cost of each screen = Interest rate*construction cost

= 10% * $1,000,000

= $100,000

Since, there are no other costs mentioned, So, the value of marginal product equals $100,000 for 1 screen.

Thus, 1 screens should be built.

e) The value of marginal product for the fifth screen is $25,000.

At an interest rate of 5.5% building a five screen complex is profitable only if 5.5% times the pre-screen contsruction cost is no greater than $25,000.

Financial cost per screen = real rate of interesr * Construction cost per screen

= $25,000 / 5.5%

= $454,545.45 (approx.)