1. Review and Examples of Percentages and Percentage Changes Ted Mitchell

2. In-Class problem You have one coupon that provides a 20% discount You have a second coupon that provides a 25% discount You can use both coupons at the same time on a single purchase. What is the total percentage discount? Your answer is __________?

3. Do NOT Treat Rates as Whole Numbers Discount Percentages are Rates You have one coupon that provides a 20% discount You have a second coupon that provides a 25% discount You can NOT add the two numbers together! The answer is NOT 45%

4. Brute Force Solution You have one coupon that provides a 20% discount You have a second coupon that provides a 25% discount Assume a price tag of $100 First discount is 20% of $100 = $20 price after first discount = $100-$20 = $80 Second discount is 25% of $80 = $20 Total dollar discount is $20 +$20 = $40 Total Percentage discount = $40/$100 = 40%

5. The vast majority of Consumers Are sufficiently innumerate to think that you add 20% + 25% to get a total discount of 45% You are to be marketing managers You are NOT consumer behaviorists You have to choose the coupon rates!

6. The Need to Choose Rates Last week you sent out a coupon worth 20% to your customers This week your competitor just sent the customers in your market a coupon worth 40% off You want to sent out a second coupon that will match your competitor’s discount and give your customers a total discount of 40% using two coupons What should be the size of the discount on the second coupon?

7. Solution Requires You To understand percentages and percentage changes The general solution for the total discount using two discounts (D 1 and D 2 ) is Total Discount = D 1 +D 2 –(D 1 x D 2 ) The total discount has to be 40% and D1 = 20% 40% = 20% +D 2 –(20%xD 2 ) (get rid of the %) 0.4 =0.2 +D 2 –.2D 2 0.4-0.2 = (1-.2)D 2 0.2=0.8D 2 D 2 = 0.2/0.8 =.25 or make it a percent, 25% Discount

8. Review Percentage Rates and Percentage Rates of Change Ted Mitchell

9. Examples of Three Types of Percentage Calculations that are possible given the basic Two-Factor Model of an Initial Price, Input = current price, P, and a Final Price Point, F The General Two-Factor Model is Output, O = O/I x Input, I The percentage conversion rate, r = O/I = %i Output, O = %i x Input I

10. Consider Percentage Conversion rate Size of Output, O = O/I x Size of Input, I The conversion rate, r =O/I = %i O = %i x I Final Price, F = F/P x Current price, P The conversion rate, r = F/P = %p F = %p x P

11. 1) Calculate Output or Final State, F Biz-Café announces a decrease in the current price of its coffee. The current price is P=$4 a cup. The new price will be %p = 90% of the current price. What is the new price, F? Output, F = (Conversion rate, %p) x Input, P New Price, F = %p x Current Price, P New Price, F = 90% x $4 New Price, F = 0.90 x $4 = $3.60 The New Price, F, is $3.60

12. 2) Calculate the Percentage Rate or Ratio of Conversion, %i Biz-Café is preparing to announce its new price as a percentage of its old price for a cup of coffee. The new selling price will be F = $4 a cup. The current selling price is P = $5 a cup. What percentage is the new price of the current price? Output, O = (conversion rate, %i) x Input, I Conversion rate from I, %i= O/I New Price, F = %p x Old Price, P %p = New Price, F / Old Price, P %p= $4 / $5 = 0.80 = 80% The new price is 80% of the old price New price, F = 80% x Old Price, P

13. 3) Calculate the Price or Initial state, I Biz-Café reviews last week’s decision to reduce the selling price of its coffee. The newly determined selling price is F=$3.50 a cup. The new price is 70% of the old price. What was the old price, P? Output, F = (conversion rate, %i) x Input, I New Price, F = %p x Old Price, P Old Price, P = (New Price, F) / %p Old Price, P = $3.50/0.70 = $5.00 The Old Price, P, was $5.00 a cup

14. Percentage Changes in a Price Ted Mitchell

15. Examples of Percentage Change Calculations that are possible given the basic Two-Factor Model where the Size of Change from the Input is now the Output, Output, Size of Change from the Input, (O–I) = ∆i Percentage Change in size is the Conversion Rate conversion rate = (O-I)/I = %∆i conversion rate = ∆i/I = %∆i The Two-Factor Model is written as Output, ∆i = %∆i x Input I

16. General Model of Percentage Change Calculations that are possible given the basic Two-Factor Model of an Initial Price, Input = current price, P, and the amount of the change between the Current Price, P, and the Final Price Point, F Size of Change from the Current Price is now the Output, Output is the Size of Change from the Input, P Output = (Final price) – (current Price) Output = (F–P) = ∆p The conversion rate is Output/Input (F-P)/P = ∆p/P ∆p/P = %∆p As a Two-Factor Model Output, change from current price=(%∆p) x (Input, Current Price, P)Change from current price, ∆p = (∆p/P) x (Current Price, P) ∆p = %∆p x P

17. Example #1 Calculating Percentage Changes in Prices Biz-café gives customers a coupon that reduces the current selling price of P = $4.00 a cup to a new lower price of F =$3.20. What is the percentage size of the discount? Remember a coupon is a price discount or a reduction in price, ∆p a percentage discount is a negative percentage change in the price, %∆p %∆p = ∆p/P %∆p = (F-P)/P %∆p = ($3.20-$4.00)/$4.00 = $0.80/$4.00 = 20%

18. Example #2 Calculating Percentage Changes in Prices Biz-café is providing a coupon that gives customers a 20% discount on the current selling price of P = $4.00 a cup. How much money will the customer save, ∆p? By how much will the current price be reduced, ∆p? How much money will be subtracted from the current price, –∆p? Remember a coupon is a price discount or a reduction in price a percentage discount is a negative percentage change in the price, %∆p = –20% ∆p = %∆p x P ∆p= –0.20 x $4 = –$0.80 80 cents off the current price

19. Example #3 Calculating Percentage Changes in Prices Biz-café is providing a coupon that gives customers a %∆p = 20% discount on the current selling price of P = $4.00 a cup. What is the amount of the final price, F, that customers actually pay? ∆p = %∆p x P Remember The output is the difference from the current price! (F-P) = %∆p x P (F-$4) = –0.20 x $4 (F-$4) = –$0.80 F = $4–$0.80 = $3.20

20. The first place we used %∆P

21. First place we %∆P was Elasticity, e by Popular definition Percentage change in quantity sold divided by the percentage change in the price e = %∆Q/%∆P e = (∆Q/Q) / (∆P/P) e = (Q 2 -Q 1 )/P 1 )) / ((P 2 -P 1 )/P 1 )