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Using Impact Analysis to Calculate Arc Elasticity of Price Ted Mitchell

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Review Major Use of Impact Analysis To measure the individual impacts that the changes in two variables have on a third variable. – ∆Price and ∆Quantity each have an impact on the change in Revenue, ∆R – ∆Market Share and ∆Market Size each have an impact on the change in Quantity sold, ∆Q – ∆Advertising productivity and ∆Advertising Expense each have an impact on the change in Quantity sold, ∆Q

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Impact Analysis helps us explain 1) why revenue is at a maximum, when the price elasticity is equal to -1.0 2) why profit is at a maximum, when the elasticity of markup is equal to -1.0 3) why profit from promotional efforts, such as advertising, are at a maximum, when the elasticity of the Return on Advertising is equal to -1.0

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Impact Analysis is Related to 1) Price and Sales Variance Analysis for measuring Differences between Budgeted and Actual revenues in Managerial Accounting 2) Impact of Price and Quantity Changes on the Change in Revenue in Marketing Management 3) Ratio of Quantity Impact to the Price Impact is Arc Elasticity in Marketing, Economics

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We remember that There is a Two-Factor model of the marketing machine Output = (conversion rate, r) x Input Conversion rate, r = Output/Input Revenue, R =(conversion rate, r) x Price Tag, P Conversion rate, r = (Revenue, R)/(Price Tag, P) Mind bending observation: Quantity sold, Q= R/P Conversion rate, r = Quantity sold, Q

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Two-Factor Marketing Machine Revenue, R =(conversion rate, r) x Price Tag, P Conversion rate, r = (Revenue)/(Price Tag) Conversion rate, r = Quantity sold, Q Revenue, R = Quantity sold, Q x Price Tag, P R = Q(P) Review An Impact analysis of the Price and Quantity differences on a change in Revenue

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Observation of Two Performances Performance 1 Performance 2 Difference ∆P = P2-P1 Impact of Differences I∆, on change in Output Input: Price Tag, P $4$5 ∆P = $1I∆P = ∆P(Q min ) I∆P = $1 x $2,500 I∆P = $2,500 Conversion rate: Quantity, Q = R/P 3,0002,500 ∆Q= - 500I∆Q = ∆Q(P min ) I∆Q = -500($4) I∆Q = -$2,000 Output: Revenue, R = Q(P) $12,000$12,500 ∆R= $500 ∆R = I∆P+I∆Q ∆R = $2,500 - $2,000 ∆R = $500 Arc Price Elasticity = I∆Q/I∆P = -$2,000/$2,500 = -0.8

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Price per Unit Quantity Sold TJM X Q1 = 3,000 X P1 = $4 The starting point (Q1=3,000, P1 = $4) The revenue, R, is P x Q = $12,000

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The end point (Q2= 2,500, P1 = $5) The revenue is P x Q = $12,500

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the change in price on the change in revenue

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the change in price on the change in Revenue is I∆P = 2,500 x ($5-$4) I∆P = $2,500 The impact of the change in price on the change in Revenue is I∆P = 2,500 x ($5-$4) I∆P = $2,500

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the decrease in quantity on the change in Revenue

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the decrease in quantity on the change in Revenue I∆Q = $4 x (2,500 -3,000) I∆Q = -$2,000 The impact of the decrease in quantity on the change in Revenue I∆Q = $4 x (2,500 -3,000) I∆Q = -$2,000

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Calculating Impact of Differences in Price and Quantity sold Performance 1 Performance 2 Difference ∆P = P2-P1 Impact of Differences I∆, on change in Output Input: Price, P $4$5 ∆P = $1I∆P = ∆P(Q min ) I∆P = $1 x $2,500 I∆P = $2,500 Conversion rate: Quantity, Q = R/P 3,0002,500 ∆Q= - 500I∆Q = ∆Q(P min ) I∆Q = -500($4) I∆Q = -$2,000 Output: Revenue, R = Q(P) $12,000$12,500 ∆R= $500 ∆R = I∆P+I∆Q ∆R = $2,500 - $2,000 ∆R = $500 Arc Price Elasticity = I∆Q/I∆P = -$2,000/$2,500 = -0.8

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Impact Analysis The $500 change in Revenue has to be equal to the impact of the change in price and the impact of the change in quantity ∆R = R2 – R1 = $12,500 – $12,000 = $500 ∆R = I∆Q + I∆P + Joint $500 = I∆Q + I∆P + J $500 = P min (Q2-Q1) + Q min (P2-P1) + J

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∆R = I∆Q + I∆P + J The net of two impacts equals the change in Revenue = $500 Since ∆P is positive and ∆Q is negative the Joint Impact, J = 0 The impact on the change in Revenue by the increase in the price is calculated as I∆P = Q min (∆P) = 2,500 x ($5-$4) = $2,500 The impact on the change in Revenue by the decrease in Quantity is calculated as I∆Q = P min (∆Q) = $4 x (2,500-3,000) = -$2,000

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the change in price on the change in Revenue = I∆P = 2,500 The impact of the decrease in quantity on the change in Revenue = I∆Q = -$2,000 The impact of the decrease in quantity on the change in Revenue = I∆Q = -$2,000

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the change in price on the change in Revenue = I∆P = 2,500 The impact of the decrease in quantity on the change in Revenue = I∆Q = -$2,000 The impact of the decrease in quantity on the change in Revenue = I∆Q = -$2,000 Joint Impact, J = 0

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 The impact of the change in price on the change in Revenue = I∆P = 2,500 The impact of the decrease in quantity on the change in Revenue = I∆Q = -$2,000 The impact of the decrease in quantity on the change in Revenue = I∆Q = -$2,000 Net Impact is a I∆Q + I∆P + J = $500 increase in Revenue Net Impact is a I∆Q + I∆P + J = $500 increase in Revenue

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We have reviewed To Price Elasticity

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Price Elasticity = -1 Price per Unit a/2b Quantity Sold a/2 TJM -0.5 -0.75 -1 -1.25 -1.5 -1.75

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Revenue looks like R = aP - bP 2 Revenue Price0 TJM -0.5 -0.75 -1 - 1.25 -1.5 -1.75 Price Elasticity Optimal price, Pr = a/2b

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Start with a low price As it grows larger, then the sizes of the two impacts become more equal to each other

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4

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Price per Unit P2 = $5 Quantity Sold Q2 = 2,500 TJM X Q1 = 3,000 X P1 = $4 Q3 = 2,000 P3 =$6 Larger impact due to ∆Q Smaller Impact due to ∆P

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The Concept You have to Know When the impacts of the two changes are equal the revenue is at a maximum and ratio of the two impacts is equal to -1 Arc Price Elasticity = I∆Q/I∆P = -1 Arc Eqp = Impact of the difference in Quantity divided by the Impact of the difference in Price Tag

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Linkage The ratio of the impact due to the changing quantity and the impact due to the changing price is the Arc Elasticity of Price. Arc Elasticity of Price = I∆Q/I∆P Arc Elasticity of Price = P min (∆Q)/Q min (∆P) Remember the definition of elasticity! Arc Elasticity of Price = (∆Q/Q min )/(∆P/P min ) (∆Q/Q min )/(∆P/P min ) = (∆Q/Q min ) x (P min /∆P) = P min (∆Q)/Q min (∆P)

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Calculating Price Elasticity from Impact Analysis Performance 1 Performance 2 Difference ∆P = P2-P1 Impact of Differences I∆, on change in Output Input : Price, P $4$5 ∆P = $1I∆P = ∆P(Q min ) I∆P = $1 x $2,500 I∆P = $2,500 Conversion rate: Quantity, Q = R/P 3,0002,500 ∆Q= - 500I∆Q = ∆Q(P min ) I∆Q = -500($4) I∆Q = -$2,000 Output: Revenue, R = Q(P) $12,000$12,500 ∆R= $500 ∆R = I∆P+I∆Q ∆R = $2,500 - $2,000 ∆R = $500 Arc Price Elasticity = I∆Q/I∆P = -$2,000/$2,500 = -0.8

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What did we learn? Arc Elasticity of Price, Eqp, is equal to the ratio of the impact of the change in quantity, I∆Q, on the change in revenue, to the ratio of the impact of the change in price, I∆P, on the change in revenue and the %∆Q min / %∆P min Arc Eqp = I∆Q / I∆P = %∆Q min / %∆P min

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