using impact analysis to calculate arc elasticity of price ted mitchell

Using Impact Analysis to Calculate Arc Elasticity of Price

Ted Mitchell

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

Observation of Two PerformancesPerformance

1Performance

2Difference∆P = P2-P1

Impact of DifferencesI∆, on change in Output

Input: Price Tag, P

$4 $5 ∆P = $1 I∆P = ∆P(Qmin)I∆P = $1 x $2,500I∆P = $2,500

Conversion rate: Quantity, Q = R/P

3,000 2,500 ∆Q= -500 I∆Q = ∆Q(Pmin)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

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

Price per UnitP2 = $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


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


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


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


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

Calculating Impact of Differences in Price and Quantity sold

Performance 1

Performance 2

Difference∆P = P2-P1


Input: Price, P

$4 $5 ∆P = $1 I∆P = ∆P(Qmin)I∆P = $1 x $2,500I∆P = $2,500


3,000 2,500 ∆Q= -500 I∆Q = ∆Q(Pmin)I∆Q = -500($4)I∆Q = -$2,000


$12,000 $12,500 ∆R= $500 ∆R = I∆P+I∆Q∆R = $2,500 - $2,000∆R = $500


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


Quantity

Sold

Q2 = 2,500

TJM

X

Q1 = 3,000 X

P1 = $4


Revenue =I∆P = 2,500

The impact of the decrease in quantity on the change in Revenue =

I∆Q = -$2,000


Quantity

Sold

Q2 = 2,500

TJM

X

Q1 = 3,000 X

P1 = $4




I∆Q = -$2,000

Joint Impact, J = 0


Quantity

Sold

Q2 = 2,500

TJM

X

Q1 = 3,000 X

P1 = $4




I∆Q = -$2,000

Net Impact is aI∆Q + I∆P + J = $500 increase in Revenue

We have reviewed

• To Price Elasticity

Price Elasticity = -1

Price per Unita/2b

Quantity

Sold

a/2

TJM

Maximum Revenue

-0.5 -0.75 -1 -1.25 -1.5 -1.75

Revenue looks like R = aP - bP2

Revenue

Price0

TJM

-0.5 -0.75 -1 - 1.25 -1.5 -1.75Price Elasticity

Optimal price, Pr = a/2b

Start with a low price

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


Quantity

Sold

Q2 = 2,500

TJM

X

Q1 = 3,000 X

P1 = $4

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

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)

Calculating Price Elasticity from Impact Analysis

Performance 1

Performance 2

Difference∆P = P2-P1


Input: Price, P

$4 $5 ∆P = $1 I∆P = ∆P(Qmin)I∆P = $1 x $2,500I∆P = $2,500


3,000 2,500 ∆Q= -500 I∆Q = ∆Q(Pmin)I∆Q = -500($4)I∆Q = -$2,000


$12,000 $12,500 ∆R= $500 ∆R = I∆P+I∆Q∆R = $2,500 - $2,000∆R = $500


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

• Any Questions?

using impact analysis to calculate arc elasticity of price ted mitchell

Documents