Decomposing Two Factor ModelsCups per Hour

Ted Mitchell

Two Factor Model of Hours as an input in Biz Cafe

• If you keep Biz-Café open longer hours you sell more cups of coffee

• Output = Conversion Rate x Input• Cups Sold, Q = Cups per Hour x Hours open, H• Q = Qph x H• You have other related models• Revenue, R = Revenue per hour x Hours open, H• R = Rph x H• Gross Profit = Gross return per hour x Hours open, H• G = Gph x H

The hours are an important determinant of the gross profit

• Gross Profit = Gross return per hour x Hours open, H

• G = Gph x H• Sample Problem• The gross profit, G was higher this week than

the week before. • Can we identify and explain how much of the

change in gross profit was due to a change in the hours open, H, or a change in the Gph

Comparing the change in gross profitfrom week to week

Week 13 Week 14 What is causing the difference in the two outputs?

Input: Hours Open 112 hr. 95 hr -17 hour

Conversion RateGross Profit per Hour

$86.43 per hr $95.05 per hr + $8.62 per hr

Output: Gross Profit $9,680 $9,030 ∆G = -$650

The hours are an important determinant of the gross profit

• Gross Profit = Gross return per hour x Hours open, H

• G = Gph x H• G = (G/H) x H• But we would like to have other important

variables to be included in the explanation of changes in gross profit, ∆G

• A variable such as Sales Revenue, R• A variable such as number of cups sold, Q

To include Revenue, R and Quantity sold

• We use a process called expand, aggregate and decompose.

• Sometimes just called the Decomposition Process

• To Decompose the Two-Factor Model into a richer Four-Factor Model that includes Revenue, R, and Quantity Sold, Q.

To include Revenue, R and Quantity sold, Q• G = (G/H) x H• The expansion of G = (G/H) x 1 x 1 x H• Changes nothing• the fact that 1 = R/R and 1 = Q/Q• G = (G/H) x (R/R) x (Q/Q) x H• Changes nothing• Combine The conversion ratios into an Aggregated Conversion Factor• G = (G x R x Q)/(R x Q x H) x H• Decompose the Aggregated Conversion Factor• G = (G/R) x (R/Q) x (Q/H) x H• Interpret the three new conversion factors• G = Gross Return on Sales x Selling Price x Cups per hour x Hours• Gross Profit = GROS x P x cph x Hours

Now explain the change in Gross profit

• Gross Profit = GROS x P x cph x Hours• Any change in the Gross Profit =• any change in the gross return on sales x• any change in selling price x• any change the selling rate of cph• any change in the number of hours

Decomposition process allows

• The decomposition of simple 2-factor marketing models into richer explanations with more factors being made explicit

• Revenue and Price are always at work in the conversion of Hours into Gross Profits

• But the Expansion, Aggregation and Decomposition made them explicit!

Decomposition process allows

• A move From• Gross Profit = Gross profit per Hour x Hours• To a 4-Factor Explanation• Gross profit = GROS x Price x cph x Hours

In Future Lectures

• Show how to measure how much of the change in Gross Profit, ∆G, was due to

• 1) the changes in gross return on sales, I∆GROS• 2) the changes in the selling price, I∆P• 3) the changes in the sales rate per hour, I∆cph• 4) the change in the hours of operation I∆H• ∆G = I∆GROS + I∆P + I∆cph + I∆H

• Any Questions