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Using Impact Analysis to Calculate Arc Elasticity of Price

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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

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

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

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

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

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

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

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

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

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

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

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

∆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

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

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

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

We have reviewed

- To Price Elasticity

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

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

Start with a low price

- As it grows larger, then the sizes of the two impacts become more equal to each other

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

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

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)

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

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