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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.

Prepared for SSAC by Gary Franchy – Davenport University

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007

SSAC2007:HF5415.GTF1.3

Core Quantitative IssueOptimization

Supporting Quantitative ConceptsQuadratic FunctionsLinear ModelingGraphing

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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.

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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.

Overview of Module

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.

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Recreate this spreadsheet.

= Cell with a number in it

= Cell with a formula in it

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?

Each Discount Taken:1. Subtracts $0.25 from the price2. Adds 25 to the sales

You must use “Discount” and “Discount Number” in computing “Price” and “Sales”.

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.

Initial Condition:1. Original ticket price: $102. Original sales: 500

B C D E2 Original Price $10.003 Discount $0.254 Original Sales 5005 Sales Gain per Discount 2567 Discount Number Price Sales Revenue8 0 $10.00 500 $5,0009 1 $9.75 525 $5,11910 2 $9.50 550 $5,225

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At what ticket price will the theater owner maximize his revenue?

Expand the table until it matches the one below.

Question 1 (cont.)

To “Copy Drag” additional rows1. 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”.

B C D E2 Original Price $10.003 Discount $0.254 Original Sales 5005 Sales Gain per Discount 2567 Discount Number Price Sales Revenue8 0 $10.00 500 $5,0009 1 $9.75 525 $5,11910 2 $9.50 550 $5,22511 3 $9.25 575 $5,31912 4 $9.00 600 $5,40013 5 $8.75 625 $5,46914 6 $8.50 650 $5,52515 7 $8.25 675 $5,56916 8 $8.00 700 $5,60017 9 $7.75 725 $5,61918 10 $7.50 750 $5,62519 11 $7.25 775 $5,61920 12 $7.00 800 $5,60021 13 $6.75 825 $5,56922 14 $6.50 850 $5,52523 15 $6.25 875 $5,46924 16 $6.00 900 $5,40025 17 $5.75 925 $5,31926 18 $5.50 950 $5,22527 19 $5.25 975 $5,11928 20 $5.00 1000 $5,000

B C D E2 Original Price $10.003 Discount $0.254 Original Sales 5005 Sales Gain per Discount 2567 Discount Number Price Sales Revenue8 0 $10.00 500 $5,0009 1 $9.75 525 $5,11910 2 $9.50 550 $5,22511 3 $9.25 575 $5,31912 4 $9.00 600 $5,40013 5 $8.75 625 $5,46914 6 $8.50 650 $5,52515 7 $8.25 675 $5,56916 8 $8.00 700 $5,60017 9 $7.75 725 $5,61918 10 $7.50 750 $5,62519 11 $7.25 775 $5,61920 12 $7.00 800 $5,60021 13 $6.75 825 $5,56922 14 $6.50 850 $5,52523 15 $6.25 875 $5,46924 16 $6.00 900 $5,40025 17 $5.75 925 $5,31926 18 $5.50 950 $5,22527 19 $5.25 975 $5,11928 20 $5.00 1000 $5,000 6

At what ticket price will the theater owner maximize his revenue?

Question 1 (cont.)

Find the largest revenue value and its corresponding ticket price.

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).

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At what ticket price will the theater owner maximize his revenue?

How would you describe the shape of the graph?

Question 1 (cont.)

Notice that the y-axis of the graph has been rescaled.

$4.00 $5.00 $6.00 $7.00 $8.00 $9.00 $10.00 $11.00$4,600

$4,800

$5,000

$5,200

$5,400

$5,600

$5,800

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Recreate this spreadsheet on a new worksheet.

= Cell with a number in it

= Cell with a formula in it

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?

Each Discount Taken:1. Subtracts $1 from the price2. Adds 2 to the sales

You must use “Discount” and “Discount Number” in computing “Sales” and “Price”.

Initial Condition:1. Original price: $302. Original sales: 203. Per-unit cost: $10

Will maximizing revenue equate to maximizing profit like it did in the fixed-cost model?

B C D E F2 Original Price 303 Cost 104 Discount 15 Original Quantity Sold 206 Sales Gain per Discount 2789 Discount Number Sales Price Revenue Profit10 0 20 30 600 40011 1 22 29 638 41812 2 24 28 672 432

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At what price will the store owner maximize his profit?

Question 2 (cont.)

Expand the table until it matches the one to the left.

To “Copy Drag” additional rows1. Highlight the bottom two rows of

the table2. 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

B C D E F2 Original Price 303 Cost 104 Discount 15 Original Quantity Sold 206 Sales Gain per Discount 2789 Discount Number Sales Price Revenue Profit10 0 20 30 600 40011 1 22 29 638 41812 2 24 28 672 43213 3 26 27 702 44214 4 28 26 728 44815 5 30 25 750 45016 6 32 24 768 44817 7 34 23 782 44218 8 36 22 792 43219 9 38 21 798 41820 10 40 20 800 40021 11 42 19 798 37822 12 44 18 792 35223 13 46 17 782 32224 14 48 16 768 28825 15 50 15 750 250

B C D E F2 Original Price 303 Cost 104 Discount 15 Original Quantity Sold 206 Sales Gain per Discount 2789 Discount Number Sales Price Revenue Profit10 0 20 30 600 40011 1 22 29 638 41812 2 24 28 672 43213 3 26 27 702 44214 4 28 26 728 44815 5 30 25 750 45016 6 32 24 768 44817 7 34 23 782 44218 8 36 22 792 43219 9 38 21 798 41820 10 40 20 800 40021 11 42 19 798 37822 12 44 18 792 35223 13 46 17 782 32224 14 48 16 768 28825 15 50 15 750 250

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At what price will the store owner maximize his profit?

Question 2 (cont.)

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).

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At what price will the store owner maximize his profit?

Question 2 (cont.)

Next, create scatter plots with

Revenue (y-axis) and Discount Number (x-axis)

andProfit (y-axis) and Discount Number

(x-axis)

to see what, if any, patterns emerges.

B C D E F2 Original Price 303 Cost 104 Discount 15 Original Quantity Sold 206 Sales Gain per Discount 2789 Discount Number Sales Price Revenue Profit10 0 20 30 600 40011 1 22 29 638 41812 2 24 28 672 43213 3 26 27 702 44214 4 28 26 728 44815 5 30 25 750 45016 6 32 24 768 44817 7 34 23 782 44218 8 36 22 792 43219 9 38 21 798 41820 10 40 20 800 40021 11 42 19 798 37822 12 44 18 792 35223 13 46 17 782 32224 14 48 16 768 28825 15 50 15 750 250

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Question 2 (cont.)

At what price will the store owner maximize his profit?

0 2 4 6 8 10 12 14 160

100

200

300

400

500

600

700

800

900

Revenue

Profit

How would you describe the shape of each graph?

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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?

End of Module Questions

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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: $25Original 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: $45Original quantity sold: 30 Sales gain per discount: 5

End of Module Questions