Using Impact Analysis to Calculate Arc Elasticity of Price

1. Using Impact Analysis to Calculate Arc Elasticity of Price Ted Mitchell

2. 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

3. 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

4. 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

5. 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

6. 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

7. Observation of Two Performances

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

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

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

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

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

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

14. Calculating Impact of Differences in Price and Quantity sold

15. 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 = Pmin(Q2-Q1) + Qmin(P2-P1) + J

16. ∆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 = Qmin(∆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 = Pmin (∆Q) = $4 x (2,500-3,000) = -$2,000

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

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

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

20. We have reviewed • To Price Elasticity

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

22. Revenue looks like R = aP - bP2 Revenue Price Elasticity -0.5 -0.75 -1 - 1.25 -1.5 -1.75 0 Price Optimal price, Pr = a/2b TJM

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

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

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

26. 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

27. 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 = Pmin(∆Q)/Qmin(∆P) • Remember the definition of elasticity! • Arc Elasticity of Price = (∆Q/Qmin)/(∆P/Pmin) • (∆Q/Qmin)/(∆P/Pmin) = (∆Q/Qmin) x (Pmin/∆P) = Pmin(∆Q)/Qmin(∆P)

28. Calculating Price Elasticity from Impact Analysis

29. 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 %∆Qmin / %∆Pmin • Arc Eqp=I∆Q / I∆P = %∆Qmin/ %∆Pmin

30. Any Questions?