Sample Quantitative Questions Chapter 3 Ted Mitchell

The Eventual Goal is To take two observations of marketing performance by replaying the same turn (changing only the Price, P, and thus the Quantity sold, Q) and To calculate the optimal selling price (Input)that will maximize the amount of gross profit generated (output).

Single Point of Observation with Calculated Conversion Rate and Forecasted Point H = Hours Open Quantity, Q x x x x x x x x x + x Q = r x H Q = (Q/H) x H Q = r x H Q = (Q/H) x H X

Two Points of Observation with Calculated Meta-Conversion Rate in Slope-Origin Form ∆H = change in Hours Open ∆Q = change in Quantity x x x x x x x x x x ∆Q = m x ∆H ∆Q = (∆Q/∆H) x ∆H ∆Q = m x ∆H ∆Q = (∆Q/∆H) x ∆H X

Meta-machine in Slope-Intercept Form with Conversion Rate, m, and Intercept, a H = Hours Open Quantity, Q x x x x x x x x x x x x Q = a + (m)(H) X a X

Multi-Point Estimation H = Hours Open Quantity, Q x x x x x x x x x x x x Q = kH h

Multi-point Meta-Conversion Rate H = Hours Open Quantity, Q x x x x x x x x x x x x Q = 6, cps(H) Q = kH h Q = r x H Q = (Q/H) x H Q = r x H Q = (Q/H) x H

Profits From Store Hours H = Hours Open Profit, Z

Reaching the Goal of Profit Analysis takes Four steps To take two observations of marketing performance by replaying the same turn (Price and Quantity sold) and 1) calculate the meta-conversion rate in the Slope-Origin Format Can you make a forecast of the change in output (demand, quantity sold) given a proposed change in input (Price) and a Meta- Marketing machine in Slope-Origin Format?

Reaching the Goal of Profit Analysis takes Four steps 2) Conceptualize the Slope-intercept Form of a Meta-Marketing Machine Can you convert the Slope-Origin Version into a Slope-Intercept Version by calculating the y- intercept? Can you use slope-intercept equation of the meta-demand machine to forecast the amount of output (demand, quantity sold) to be expected from a proposed amount of input (Price)?

Reaching the Goal takes Four steps 3) Conceptualize the demand model and convert it to a Revenue machine Can you Forecast the amount of revenue for any proposed level of price? Calculate the optimal amount of input (price) to produce the maximum amount of revenue (output)

Reaching the Goal takes Four steps 4) Conceptualize the Revenue model and convert it to a Profit machine Can you Forecast the amount of gross profit for any proposed level of price? Calculate the optimal amount of input (price) to produce the maximum amount of gross profit (output)

Today We are at Step #2 To take two observations of marketing performance by replaying the same turn (Price and Quantity sold) and 2) calculate the y-intercept and use slope- intercept equation of the meta-demand machine forecast the amount of output (demand, quantity sold) to be expected from a proposed amount of input (Price)

1) Calculate the meta-conversion rate from two observations You have observed two coffee shop performances Observation #1: open 100 hours and sold 10,000 cups of coffee at a rate of 100 cups per hour, cph Observation #2 open 120 hours and sold 10,800 cups of coffee at a rate of 80 cups per hour, cph What is the meta-conversion rate, m?

1) Calculate the meta-conversion rate from two observations Meta-coffee from hours marketing machine is Difference in output = (meta-conversion rate, m)x difference in input ∆Q = (meta-conversion rate, m) x ∆hours Measure Difference in output, ∆Q = Q2-Q1 ∆Q = 10,800 – 10,000 = 800 cups Measure Difference in Input, ∆H = H2-H1 ∆H = 120 hours -100 hours = 20 hours What is the Meta-conversion rate?

Single Point Meta-Marketing Machine Input Factor, ∆π Output, ∆ø 0, 0 ∆π ∆ø X Calibration point (∆π, ∆ø) Meta-Conversion rate m = rise/run m = ∆ø/∆π Meta-Conversion rate m = rise/run m = ∆ø/∆π Slope-Point Equation for Two-Factor Meta-machine Rise Run

1) Calculate the meta-conversion rate from two observations Observation 1Observation 2Meta-demand marketing machine Hours open, H Cups per hour, cps = Q/H m = ? Cups sold, Q 10,00010,800 What is the meta-conversion rate, m, for meta-demand marketing machine?

1) Calculate the meta-conversion rate from two observations Observation 1Observation 2Meta-Demand Marketing Machine Hours open, H ∆H = 20 Cups per hour, cph = Q/H 100 Not Used 80 Not Used m = ∆Q/∆H m = 800/20 = m = 40 cph Cups sold, Q 10,00010,800∆Q = 800 Remember the single point r = cph is not used

1) Calculate the meta-conversion rate from two observed performances You don’t have to use the table format ∆Q = (meta-conversion rate, m) x ∆hours Measure Difference in output, ∆Q = Q2-Q1 ∆Q = 10,800 – 10,000 = 800 cups Measure Difference in Input, ∆H = H2-H1 ∆H = 120 hours -100 hours = 20 hours Calculate Meta-conversion rate, m = ∆O/∆H Meta-conversion rate, m = 800 cups/20 hrs = 40cph

#2 Forecast a change in output, ∆O, from a Meta-Marketing Machine You know the calibrate meta-machine that explains the differences in observed performances for coffee sales, ∆Q, from store hours, ∆H is represented as Output, ∆Q = 40cph x Input, ∆H The boss is proposing a decrease in the number of store of ∆H = -5 hours What is the forecasted change in the number of number of cups sold, ∆Q?

Single Point Meta-Marketing Machine Input Factor, ∆π Output, ∆ø 0, 0 ∆π = 20 ∆ø = 800 X Calibration point (800, 20) Meta-Conversion rate m = rise/run m = 40 cph Meta-Conversion rate m = rise/run m = 40 cph Slope-Point Equation for Two-Factor Meta-machine Rise Run

#2 Forecast a change in output, ∆O, from a Meta-Marketing Machine You know the calibrate meta-machine that explains the differences in observed performances for coffee sales, ∆Q, from store hours, ∆H is represented as Output, ∆Q = 40cph x Input, ∆H The boss is proposing a decrease in the number of store of ∆H = -5 hours What is the forecasted change in the number of number of cups sold, ∆Q? Answer ∆Q = 40cph x -5 hours = -200 cups

2) Forecast a Change in output, ∆O, from a Meta-Marketing Machine Observation 1Observation 2 Meta-demand marketing machine Forecasted change Hours open, H ∆H = 20∆H = -5 Cups per hour, cps = Q/H 10080m = ∆Q/∆H m = 800/20 = m = 40 cph Cups sold, Q 10,00010,800∆Q = 800 cups ∆Q = -200 cups

Forecasting the change in quantity sold due to a change in one of the marketing inputs Works in concert with the analysis of a breakeven quantity, BEQ, to calculate the minimum quantity that must be sold to cover the increase in the expenditure of marketing input In a future chapter we shall calculated breakeven quantity, BEQ

#3 Forecast an Actual Output In week 2 you are open for 120 hours a week and are selling 10,800 cups of coffee at at rate of 80 cups per hour. The Boss is proposing a reduction in store hours of 5 hours a week and you have forecasted a 200 cup reduction in sales in week 3. Three related questions: 1) What is the proposed number of hours in week 3? H3 = 120 hours – 5 hours = 115 hours 2) What is the forecasted sales volume in week 3? Q3 = 10,800 cups cups = 10,600 cups 3) What is the forecasted rate of sales per hour Conversion rate, r = Q/H = 10,600/115 = cph

Modification: to forecast an outcome from a current point of performance. Market research has provided you with a calibrated equation of the single point slope equation of a meta-marketing machine Change in Coffee sales, ∆Q = (40 cph) x Change in store hours, ∆H ∆Q = 40 cph x ∆H ∆Q= (40 cph) x -5 hours Convert the ∆Q and the ∆H to the proposed point and forecast minus the current point (input, output) (Q 3 – Q 2 ) = (40cph) x (H 3 - H 2 ) (Q 3 – 10,800) = (40cph) x (H 3 – 120) ∆H = H 3 – 120 hours -5 = H 3 – 120 hours H 3 = 115 hours (Q 3 – 10,800) = (40cph) x (115 – 120) Q 3 = 10,800 + (40cph) x -5 hours Q 3 = 10,800 – 200 = 10,600 cups

#3 Forecast a Specific Output and Conversion Rate Week 2 Forecasted change Forecasted week 3 Meta- marketing machine Hours open, H 120 hours ∆H = -5 hoursProposed Input 115 hours ∆H Cups per hour, cps = Q/H 80 cphm = 40 cph Forecasted r ≠ m m = ∆Q/∆H m = 40 cph Cups sold, Q10,800 cups ∆Q = -200 cups Forecasted Output = Q 3 = 10,600 cups ∆Q The forecasted conversion rate, r, is not the meta-conversion rate, m

#3 Forecast a Specific Output and Conversion Rate Week 2 Forecasted change Forecasted week 3 Meta- marketing machine Hours open, H 120 hours ∆H = -5 hoursProposed Input 115 hours ∆H Cups per hour, cps = Q/H 80 cphm = 40 cphForecasted conversion rate, r=10,600/115 r = cph m = ∆Q/∆H m = 40 cph Cups sold, Q10,800 cups ∆Q = -200 cups Forecasted Output = 10,600 cups ∆Q The forecasted conversion rate, r, is not the meta-conversion rate, m

Meta-Conversion Rate, m H = Hours Open Quantity, Q x x x x x x x x x x x x Q = a + (m cph)(H) + Q = r x H Q = (Q/H) x H Q = r x H Q = (Q/H) x H O X

When you have to do this calculation many times 1) Forecasting a specific outcome is awkward when using the single point Slope-Origin equation of the meta-marketing ‘coffee sales from hours’ machine Does not help us easily identify optimal levels of store hours for maximum profit

We want a slope-intercept equation H = Hours Open Quantity, Q x x x x x x x x x x x x Linear Meta-Machine is a secant that approximates the Quantity sold as a function of hours open Q = a + m(H) Q = 6,000 + (40cph)(H) Q = a + m(H) Q = 6,000 + (40cph)(H) a = 6,000 cups

5) Construct a slope-intercept equation of a meta-marketing machine Market research has provided you with a calibrated equation of the single point slope equation of a meta-marketing machine Change in Coffee sales, ∆Q = (meta-conversion rate, 40 cph) x Change in store hours, ∆H ∆Q = 40 cph x ∆H (Q 3 -10,800 cups) = 40cph x (H 3 – 120 hours) What is the y-intercept of the slope-intercept equation of the meta- marketing machine? Set H3 = zero and Q3 = value of y-intercept, a a -10,800 cups = 40 cph x (0-120 hours) a = 10,800 cups – 4,800 cups a = 6,000 cups sold when the store is closed and only the drive through is open What is the slope-intercept equation of the meta-marketing machine? Forecasted Cups sold, Q = 6,000 cups + (40 cps)(Hours open, H)

Linear versus Q = kH h H = Hours Open Quantity, Q x x x x x x x x x x x x Linear Meta-Machine is a secant that approximates the Quantity sold as a function of hours open Q = 6,000 + (40cph)(H) Q = kH h

6) Forecast an output from the slope-intercept equation given a proposed level of input Market research department has provided an estimate of the relationship between hours the store is open and the number of cups that is sold as a slope-intercept equation Q = a + m(H) Forecasted Cups sold, Q = 6,000 cups + (40 cph)(Hours open, H) It is proposed to stay open for 105 hours a week. What is the forecasted sales volume for staying open that many hours? Answer: Cups sold, Q = 6,000 cups + (40 cph)(105 hours) Cups sold, Q = 6,000 cups + 4,200 cups = 10,200 cups

We need to forecast demand, quantity sold, Q From changes in any one of the 4 P’s as input Promotion (advertising, radio spots, servers) Product quality ( coffee, servers) Place ( ambience, location, hours open and P rice Tag

slope-intercept equations for representing meta-marketing machines from two observations are always constructed in the same way to provide: Demand, Q = a + m(positive, π) Demand, Q = a – m(negative, P)

7) Forecast a Quantity sold from a proposed Price tag Market Research has provided you with the slope- intercept equation which they are calling a demand curve given the price tag on each cup, P, is an input Q = a – m(P) Quantity Demanded, Q = 6,000 cups -900 cpP x Price tag, P When the price is $4.00 for a cup what is the forecasted quantity of cups sold, Q? Quantity, Q = 6000 cups – 900 cp$ x ($4) Quantity, Q = 6000 cups – 3,600 cups = 2,400 cups

Lower Price Sells More Units Price per Cup $3.90$4.00 Quantity Sold 2,400 Demand Equation Q = 6,000 – 900(P) Revenue = 2,400 x $4.00 Revenue = $9,600 TJM

The Demand Equation Q = a – m(P) Often called demand equation or price response function The Slope-Intercept equation of the meta- marketing machine that produces a demand or quantity sold, Q, using the price tag, P, as an input

Any Questions?