using impact analysis to calculate arc elasticity of price ted mitchell
TRANSCRIPT
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
Price per UnitP2 = $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
Price per UnitP2 = $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
Price per UnitP2 = $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
Price per UnitP2 = $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
Calculating Impact of Differences in Price and Quantity sold
Performance 1
Performance 2
Difference∆P = P2-P1
Impact of DifferencesI∆, on change in Output
Input: Price, 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
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
Price per UnitP2 = $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
Price per UnitP2 = $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
Joint Impact, J = 0
Price per UnitP2 = $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
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
Price per UnitP2 = $5
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
Impact of DifferencesI∆, on change in Output
Input: Price, 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
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?