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# Using Impact Analysis to Calculate Arc Elasticity of Price - PowerPoint PPT Presentation

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

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### 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

• 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

• 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

• 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

• 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

• 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

• To Price Elasticity

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

Quantity

Sold

Maximum Revenue

a/2

Price per Unit

a/2b

TJM

Revenue

Price Elasticity

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

0

Price

Optimal price, Pr = a/2b

TJM

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

Sold

Q1 = 3,000

X

Q2 = 2,500

X

P1 = \$4

Price per Unit

P2 = \$5

TJM

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

• 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

• 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

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