1 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 SSAC2007:HF5415.GTF1.3 Core Quantitative Issue Optimization Supporting Quantitative Concepts Quadratic Functions Linear Modeling Graphing

2 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.

3 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 have you create a scatter plot and observe the results. Slides give the assignment to hand in.

4 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 price 2.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: $10 2.Original sales: 500

5 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 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”.

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).

7 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.

8 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 price 2.Adds 2 to the sales You must use “Discount” and “Discount Number” in computing “Sales” and “Price”. Initial Condition: 1.Original price: $30 2.Original sales: 20 3.Per-unit cost: $10 Will maximizing revenue equate to maximizing profit like it did in the fixed-cost model?

9 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 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

10 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).

11 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) and Profit (y-axis) and Discount Number (x-axis) to see what, if any, patterns emerges.

12 Question 2 (cont.) At what price will the store owner maximize his profit? How would you describe the shape of each graph?

13 Save your completed Excel file and 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

14 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: $50Discount amount: $2Unit cost: $25 Original quantity sold: 30Sales 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: $50Discount amount: $2Unit cost: $45 Original quantity sold: 30Sales gain per discount: 5 End of Module Questions