SSAC2007:HF5415.GTF1.3
The Price is Right - Or is it?
How can we find the right sale price to actually increase profits?
You see them on TV, you hear them
on the radio, and you read them in the
paper: stores advertising big sales.
Maybe it’s a holiday special, maybe
they’re overstocked, maybe they just
want to get you in the front door. How
can they make money at these prices?
Well, sometimes less is really more.
Core Quantitative Issue
Optimization
Supporting Quantitative
Concepts
Quadratic Functions
Linear Modeling
Graphing
Prepared for SSAC by
Gary Franchy – Davenport University
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007
1
Overview of Module
Business will have “sales” for a variety of reasons. Sometimes stores
will price an item below cost, known as a loss leader, with the goal of
getting people into the store with the hope they will make additional
purchases. Examples of this include Day-After-Thanksgiving
(a.k.a.“Black Friday”) specials and grocery stores’ milk prices.
Sometimes the goal is simply to get rid of inventory before a product
spoils. For example, in Michigan and other northern states you can
find many plants and flowers at “giveaway” prices at the end of
summer. The stores would rather sell it below cost than have to throw
it away and get nothing.
Most of the time, however, the goal is to increase the profit made
from an item. This can be accomplished if the increase in sales more
than offsets the decrease in price.
2
Overview of Module
We will be examining two cases: The first is a movie theater owner
who, having paid a fixed amount to secure the movie, is trying to
maximize ticket revenue. The second involves a store owner trying to
maximize profit on an item that also has a per-unit cost to consider.
In both cases there will be two items changing: the retail price and the
quantity sold. For simplicity, both will change at a constant rate (i.e.,
linear) with an increase in sales corresponding to each drop in price.
Slides 2-3 provide an overview of module.
Slides 4-5 ask you to set up your worksheet and format the cells.
Slides 6-7 have you create a scatter plot and observe the results.
Slides 8-9 ask you to set up your worksheet and format the cells.
Slides 10-12 have you create a scatter plot and observe the results.
Slides 13-14 give the assignment to hand in.
3
Question 1
If the movie theater gets an additional 25 customers for every 25-cent drop
in price, at what ticket price will the theater owner maximize his revenue?
You must use “Discount” and “Discount Number”
in computing “Price” and “Sales”.
Recreate this spreadsheet.
= Cell with a number in it
Initial Condition:
1. Original ticket price: $10
2. Original sales: 500
= Cell with a formula in it
B
C
D
E
2
Original Price
$10.00
3
Discount
$0.25
4
Original Sales
500
5
Sales Gain per Discount
25
7
Discount Number
Price
Sales
Revenue
8
0
$10.00
500
$5,000
9
1
$9.75
525
$5,119
10
2
$9.50
550
$5,225
Each Discount Taken:
1. Subtracts $0.25 from the price
2. Adds 25 to the sales
6
In order to be able to “cut and paste”
additional rows into the table, use
absolute cell references in building your
Price, Sales, and Revenue formulas.
4
Question 1 (cont.)
At what ticket price will the theater owner maximize his revenue?
Expand the table until it matches the one below.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
B
Original Price
Discount
Original Sales
Sales Gain per Discount
C
$10.00
$0.25
500
25
D
E
Discount Number
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Price
$10.00
$9.75
$9.50
$9.25
$9.00
$8.75
$8.50
$8.25
$8.00
$7.75
$7.50
$7.25
$7.00
$6.75
$6.50
$6.25
$6.00
$5.75
$5.50
$5.25
$5.00
Sales
500
525
550
575
600
625
650
675
700
725
750
775
800
825
850
875
900
925
950
975
1000
Revenue
$5,000
$5,119
$5,225
$5,319
$5,400
$5,469
$5,525
$5,569
$5,600
$5,619
$5,625
$5,619
$5,600
$5,569
$5,525
$5,469
$5,400
$5,319
$5,225
$5,119
$5,000
To “Copy Drag” additional rows
1. Highlight the bottom two rows of the
table.
2. Move cursor to bottom right of
highlighted area until cursor looks like
“+”.
3. Hold the left mouse button and roll the
mouse down until the number 20
appears.
4. Release the left mouse button.
To format cell(s) as dollars:
1.Highlight cell(s).
2.Right-click mouse.
3.Choose “Format Cells”.
4.Click on “Number” tab.
5.Choose “Currency”.
6.Press “OK”.
5
Question 1 (cont.)
At what ticket price will the theater owner maximize his revenue?
Find the largest revenue value and its corresponding ticket price.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
B
Original Price
Discount
Original Sales
Sales Gain per Discount
C
$10.00
$0.25
500
25
D
E
Discount Number
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Price
$10.00
$9.75
$9.50
$9.25
$9.00
$8.75
$8.50
$8.25
$8.00
$7.75
$7.50
$7.25
$7.00
$6.75
$6.50
$6.25
$6.00
$5.75
$5.50
$5.25
$5.00
Sales
500
525
550
575
600
625
650
675
700
725
750
775
800
825
850
875
900
925
950
975
1000
Revenue
$5,000
$5,119
$5,225
$5,319
$5,400
$5,469
$5,525
$5,569
$5,600
$5,619
$5,625
$5,619
$5,600
$5,569
$5,525
$5,469
$5,400
$5,319
$5,225
$5,119
$5,000
Next, create a scatter plot
of Price (x-axis) and
Revenue (y-axis) to see
what, if any, pattern
emerges.
This row contains the
largest revenue ($5625).
It occurs when the ticket
price is reduced to $7.50
(i.e., after 10 discounts).
6
Question 1 (cont.)
At what ticket price will the theater owner maximize his revenue?
How would you describe the shape of the graph?
$5,700
$5,600
$5,500
$5,400
$5,300
$5,200
$5,100
$5,000
$4,900
$0.00
$2.00
$4.00
$6.00
$8.00
$10.00
$12.00
Notice that the y-axis of the graph has been rescaled.
7
Question 2
If the store owner gets an additional two sales for every $1 drop in price, at
what price will the store owner maximize his profit?
You must use “Discount” and
“Discount Number” in computing
“Sales” and “Price”.
Recreate this spreadsheet
on a new worksheet.
= Cell with a number in it
Initial Condition:
1. Original price: $30
2. Original sales: 20
= Cell with a formula in it
B
C
2
Original Price
30
3
Cost
10
4
Discount
1
5
Original Quantity Sold
20
6
Sales Gain per Discount
2
D
E
F
Each Discount Taken:
1. Subtracts $1 from the price
2. Adds 2 to the sales
7
8
9
Discount Number
Sales
Price
Revenue
Profit
10
0
20
30
600
400
11
1
22
29
638
418
12
2
24
28
672
432
Will maximizing revenue equate
to maximizing profit like it did in
the fixed-cost model?
8
Question 2 (cont.)
At what price will the store owner maximize his profit?
B
C
Original Price
30
3
Cost
10
4
Discount
1
5
Original Quantity Sold
20
6
Sales Gain per Discount
2
9
Discount Number
Sales
Price
Revenue
Profit
10
0
20
30
600
400
11
1
22
29
638
418
12
2
24
28
672
432
13
3
26
27
702
442
14
4
28
26
728
448
15
5
30
25
750
450
16
6
32
24
768
448
17
7
34
23
782
442
18
8
36
22
792
432
19
9
38
21
798
418
20
10
40
20
800
400
21
11
42
19
798
378
22
12
44
18
792
352
23
13
46
17
782
322
24
14
48
16
768
288
25
15
50
15
750
250
2
D
E
F
Expand the table until it
matches the one to the left.
7
8
To “Copy Drag” additional rows
1. Highlight the bottom two rows of
the table
2. Move cursor to bottom right of
highlighted area until cursor looks
like “+”
3. Hold the left mouse button and roll
the mouse down until the number
15 appears
4. Release the left mouse button
9
Question 2 (cont.)
At what price will the store owner maximize his profit?
B
C
2
Original Price
30
D
E
F
3
Cost
10
4
Discount
1
5
Original Quantity Sold
20
6
Sales Gain per Discount
2
9
Discount Number
Sales
Price
Revenue
Profit
10
0
20
30
600
400
11
1
22
29
638
418
12
2
24
28
672
432
13
3
26
27
702
442
14
4
28
26
728
448
15
5
30
25
750
450
16
6
32
24
768
448
17
7
34
23
782
442
18
8
36
22
792
432
19
9
38
21
798
418
20
10
40
20
800
400
21
11
42
19
798
378
22
12
44
18
792
352
23
13
46
17
782
322
24
14
48
16
768
288
25
15
50
15
750
250
7
8
This row contains the
largest profit ($450).
It occurs when the sale
price is reduced to $25
(i.e., after five discounts).
This row contains the
largest revenue ($800).
It occurs when the ticket
price is reduced to $20
(i.e., after ten discounts).
10
Question 2 (cont.)
At what price will the store owner maximize his profit?
B
C
2
Original Price
30
3
Cost
10
4
Discount
1
5
Original Quantity Sold
20
6
Sales Gain per Discount
2
D
E
F
Next, create scatter plots with
7
8
9
Discount Number
Sales
Price
Revenue
Profit
10
0
20
30
600
400
11
1
22
29
638
418
12
2
24
28
672
432
13
3
26
27
702
442
14
4
28
26
728
448
15
5
30
25
750
450
16
6
32
24
768
448
17
7
34
23
782
442
18
8
36
22
792
432
19
9
38
21
798
418
20
10
40
20
800
400
21
11
42
19
798
378
22
12
44
18
792
352
23
13
46
17
782
322
24
14
48
16
768
288
25
15
50
15
750
250
Revenue (y-axis) and Discount
Number (x-axis)
and
Profit (y-axis) and Discount Number
(x-axis)
to see what, if any, patterns
emerges.
11
Question 2 (cont.)
At what price will the store owner maximize his profit?
900
800
700
600
500
Revenue
Profit
400
300
200
100
0
0
5
10
15
20
How would you describe the shape of each graph?
12
End of Module Questions
Save your completed Excel file and e-mail it to your instructor.
Looking back at Case #1
1. Create the revenue equation by multiplying the price formula by the
sales formula. For each formula, let Discount Number be the only variable
(i.e., Let discount number be x and use the actual values for original price,
original sales, discount, and sales gain per discount).
2. At what other price do we get revenue the same as our original price?
3. After seeing the table created from case #1, the theater owners decide to
price tickets at $5 each and knowingly forgoe the extra $625 in ticket
revenue. Give a likely reason for such a strategy.
4. Change the Sales Gain per Discount to 20. What is the maximum
revenue and at what price (or prices) do we attain it?
13
End of Module Questions
Looking back at Case #2
5. Was maximizing revenue the same as maximizing profit?
6. What is the lowest price you could sell the item for and still make a
profit?
7. What sale price would yield the maximum profit in the following
scenario:
Original price: $50
Discount amount: $2
Unit cost: $25
Original quantity sold: 30
Sales gain per discount: 5
8. Sometimes a company would like to raise prices (Hint: use negative
numbers for both discount amount and sales gain per discount)
What sale price would yield the maximum profit in the following scenario:
Original price: $50
Discount amount: $2
Unit cost: $45
Original quantity sold: 30
Sales gain per discount: 5
14