William Company owns and operates a nationwide chain of movie theaters. The 500 properties in the
William chain vary from low-volume, small-town, single-screen theaters to high-volume, big-city,
multiscreen theaters.
The management is considering installing machines that will make popcorn on the premises.
These machines would allow the theaters to sell freshly popped popcorn rather than prepopped corn
that is currently purchased in large bags. This new feature would be advertised and is intended to
increase patronage at the company's theaters.
Annual rental costs and operating costs vary with the size of the machines. The machine capacities and costs are as follows: 

Economy Regular Super Annual capacity (boxes) 50,000 120,000 300,000 Costs Annual machine rental
$8,000 $11,000 $20,000 Popcorn cost per box 13c 13c 13c Other costs per box 22c 14c 05 Cost of each box
08c 080 08c

 Required:
a. Calculate the volume level in boxes at which the economy popper and regular popper
would earn the same profit (loss).
b. Management can estimate the number of boxes to be sold at each of its theaters. Present a
decision rule that would enable William's management to select the most profitable
machine without having to make a separate cost calculation for each theater.
c. Could management use the average number of boxes sold per seat for the entire chain and
the capacity of each theater to develop this decision rule? Explain your answer. 

a). There are two types of costs: Fixed costs and Variable costs

Variable costs per unit of Economy Popper is given by:

Total variable cost per unit = Popcorn Cost + Other costs + Box costs

                                               = $0.13 + $0.22 + $0.08

                                                = $0.43

Variable cost for the Regular popper is given by:

Total variable cost per unit = Popcorn Cost + Other costs + Box costs

                                               = $0.13 + $0.14 + $0.08

                                                 = $0.35

Number of boxes the company must produce for which both departments earn same profit is calculated as follows:

Number of boxes = Differential fixed cost/Differential variable cost

                               = $3,000/$0.08

                               = 37,500 boxes

Step-by-step explanation

a). There are two types of costs: Fixed costs and Variable costs

Variable costs per unit of Economy Popper is given by:

Total variable cost per unit = Popcorn Cost + Other costs + Box costs

                                               = $0.13 + $0.22 + $0.08

                                                = $0.43

Variable cost for the Regular popper is given by:

Total variable cost per unit = Popcorn Cost + Other costs + Box costs

                                               = $0.13 + $0.14 + $0.08

                                                 = $0.35

Number of boxes the company must produce for which both departments earn same profit is calculated as follows:

Number of boxes = Differential fixed cost/Differential variable cost

                               = $3,000/$0.08

                               = 37,500 boxes

b). To make a decision, the super model has too be included

Variable cost per unit for Super model is calculated as follows:

Total variable cost per unit = Popcorn Cost + Other costs + Box costs

                                               = $0.13 + $0.05 + $0.08

                                                 = $0.26

Fixed cost for Super popper is annual rental of $20,000

Differential analysis of the two models

Number of boxes the company must produce for which the two departments will earn the same profit is given by:

Number of boxes = Differential fixed cost/Differential variable cost

                                = $9,000/$0.09

                                = 100,000 boxes

The company can select the number of boxes to be sold as per the table below:

When less than 37,500 boxes are sold at the theater, the company must choose the Economy machine. when between 37,500 and 100,000 boxes are to be sold, the company should choose the Regular Machine and when over 100,000 boxes are to be sold, the company should choose the Super machine.

c). The management cannot make a decision based on the average number of boxes of popcorn sold in a theater. The total number of seats and the average number of customers attending the theater differ. Different people have different popcorn purchasing habits.