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Three-Factor Model: A Guide to Decomposition Analysis

Learn how to create a three-factor model by expanding, aggregating, and decomposing variables such as cups sold, sales revenue, and gross profit in a marketing context.

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Three-Factor Model: A Guide to Decomposition Analysis

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  1. Intro to Decomposition:Creating a Three Factor Model(Cups/Server) (Servers/Hour) (Hours) Ted Mitchell

  2. We have seen that both • Number of Servers as an Inputand • Number of Café Hours as an Input • Have meaningful impacts on the Outputs of • 1) Number of Cups Sold, Q • 2) Dollar of Sales Revenue, R • 3) Dollars of Gross Profit, G

  3. When we consider the • Explicit impact of hours of operation, H, we leave the number servers implicit and constant • When we consider the number of servers, S, we leave the number of hours implicit and constant

  4. We wish to have both made Explicit • Number of Servers, S, and the Number of Store Hours, H, made explicit in the same analysis

  5. Creating a Three Factor Model • Requires • A process of expansion • A process of aggregation • A process of decomposition

  6. Step 1 • Identify the Two Factor Model you wish to expand into a Three Factor Model and which previously implicit variable is to be made explicit • Cups sold, Q = cups per hour x hours, H Cups Sold, Q = (Q/H) x H • Expand the cups per hour machine to make the number of servers explicit, S

  7. Step 2 • Introduce the variable to be made explicit as unity into the Two Factor model • (Number of Servers, S) /(Number of Servers, S) = Unity • S/S = 1 • Cups Sold, Q = (Q/H) x 1 x Hours, H • Cups Sold, Q = (Q/H) x (S/S) x Hours, H

  8. Step 3 Aggregate the Expansion Factors • Cups sold, Q = (Q/H) x (S/S) x Hours open, H • Cups Sold, Q = [(QxS) / (HxS)] x Hours open, H • r = [(QxS) / (HxS)] is an ugly and large conversion factor

  9. Step 4 Decompose the Ugly Aggregated Conversion Factor • Into two conversion rates • [(QxS) / (HxS)] = (Q/S) x (S/H) • Cups Sold, Q = (Q/S) x (S/H) x Hours, H • where • (Q/S) = Conversion Factor #1 = (cups sold per server) • (S/H) = Conversion Factor #2 = (number of servers per hour) • H = Input Factor = (Number of Hours, H

  10. Three Factor Marketing Machine • Model that makes the number of servers, and the number of operating hours explicit elements in the Marketing Machine

  11. Three Factor Machine Note: The original rate of Cups per Hour is lost and has been replaced by two new rates: Servers per Hour and Cups per Server.

  12. Many people think of this as decomposing the original rate • The Rate of (Cups per Hour) into • (Cups sold per Server) x (Servers per Hour) • But this is inaccurate • Reorganize the 3 Factor Machine as an identity of rates • Q = (Q/S) x (S/H) x H • Divide both sides by H • (Q/H) = (Q/S) x (S/H)

  13. The other Outputs of the Two-Factor Models can be expended as well • 1) Output: Dollars of Sales RevenueRevenue, R = (dollar sales per server) x (servers per hour) x number of hoursR = (R/S) x (S/R) x H • 2) Output: Dollars of Gross ProfitGross Profit, G = (gross profit per server) x (servers per hour) x (number of hours, H)G = (G/S) x (S/H) x H

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