Two Definitions of Price
and Calculation of the Average Price
Ted Mitchell
There are many perspectives
• On the nature of the price that sellers charge
and buyers pay for things in the market place
• The economist’s perspective
• The accountant’s perspective
• Consumer behaviorist’s perspective
• The marketing manager’s
Two Basic Definitions of Price
•
•
•
•
Create lots of confusion!
1) The Marketing Definition
2) The Accounting Definition
Marketing Managers Use Both
Two Basic Definitions of Price
• 1) The Marketing Definition is that selling
price is a price tag that signals the customer as
to the amount that must be given up to
acquire the product
• 2) The Accounting Definition is that selling
price is the (average) revenue generated per
unit sold.
The confusion starts with the basic
definition of Revenue
• Sales Revenue, R, is defined as the Selling Price, P,
times the Quantity or number of units sold, Q
• This means that are TWO Different ways to define
a Two-Factor Machine that produces Revenue as
the Machine’s Output
• 1) Revenue, R =
(Conversion rate, r) x (Size of the Price tag, Pt)
• 2) Revenue, R =
(Conversion rate, r) x (Number of Units Sold, Q)
First Definition of Price
• 1) The Two-Factor Marketing Machine that produces
Revenue as an Output and uses the Size of the Price
Tag as a Concrete Input, Pt
• Revenue, R=(conversion rate, r)x(Price Tag, Pt)
• The conversion rate is defined as
Conversion rate, r = Output/Input
• Conversion rate, r = (Revenue, R)/(Price tage, Pt)
• Conversion rate, r = R/Pt = the quantity sold, Q
• Quantity sold is defined as a conversion rate when
price is define as the price tag on each item sold
• Revenue, R = (dollars of revenue per dollar of price
tag) x Price Tag. Pt
• Revenue, R = (conversion rate, Q) x (Price Tag, Pt)
• R = Q x Pt
Second Definition of Price
• 2) Selling Price is a conversion rate, r of a Two-Factor
machine which produces Revenue from a concrete
input measured as the number of units sold
• Revenue, R = conversion rate, r) x (Quantity sold, Q)
• Conversion rate, r = Output/Input
• Conversion rate, r = (Revenue, R)/(Quantity Sold)
• Conversion rate, r = R/Q = average revenue per unit
sold
• Conversion rate, r = Price per unit sold. P
• Revenue, R = (Conversion rate, P) x (Quantity Sold, Q)
• Price is now defined as the abstract conversion rate,
average revenue per unit sold
• Revenue, R = (P, revenue per unit sold) x (Units sold, Q)
• R=PxQ
Basic Definitions of Price
• 1) The Marketing Definition holds that the selling
price is a price tag and that it is an input and the
conversion rate, r = R/P = Q is same value as the
quantity sold
• 2) The Accounting Definition is that selling price
is the (average) revenue generated per unit sold.
• 3) The Economist’s Definition is that the Quantity
sold is an Input and conversion rate, r = R/Q = P, is
the average revenue per unit sold and has the
same value as the Price Tag
Confusion Stems From
• 1) The quantity sold, Q, does NOT sound like
an abstract conversion rate which would be
described as the Revenue per Dollar of Price
Tag
• 2) The average revenue per unit sold sounds
like a conversion rate and sounds too much
like a price tag.
• 3) In accounting the average revenue per unit
is seldom equal to the price tag
Confusion due to Different Definitions
of Price in the Revenue Machine
Price Tag
Total Cups
Total Revenue
Large Coffee $4.00 per cup
1,000 cups
$4,000
Medium
Coffee
2,000 cups
$7,200
3,000 cups
$11,200
3.60 per cup
Definitions
of Price
What is the Average Price?
Confusion due to Different Definitions
of Price in the Revenue Machine
Price Tag
Total Cups
Total Revenue
Large Coffee $4.00 per cup
1,000 cups
$4,000
Medium
Coffee
2,000 cups
$7,200
3,000 cups
$11,200
3.60 per cup
Definitions
of Price
Definition of Prices
1) Average Revenue per Unit sold
2) Price Tag
Confusion due to Different Definitions
of Price in the Revenue Machine
Price Tag
Total Cups
Total Revenue
Large Coffee $4.00 per cup
1,000 cups
$4,000
Medium
Coffee
2,000 cups
$7,200
3,000 cups
$11,200
Definitions
of Price
3.60 per cup
Average Price Tag
($4 + $3.60)/2 =
$7.60/2 = $3.80
Average Revenue per Cup
$11,200/3,000 cups = $3.73
Avoid the Mistake
A good Exam question
• It is legitimate to calculate the average price tag
as the mean of the selling prices but it is wrong to
calculate the mean of the price tags as the
average revenue per unit sold.
• True or False
• The correct answer is True
• Output, R = (R/P) x (Input, Price Tag)
• Output, R = (R/Q) x (Input, Quantity sold, Q
Output, R = (conversion rate, P) x Q
Any Questions about
• The many perspectives on price
• The two definitions of price?
• The calculation of an average price?
Price and Quantity in the Analysis of
Sales Revenue Variance
Ted Mitchell
The Simple Two-Factor Revenue
Machines
• Are most useful for diagnostic purposes when
comparing performances between two machines
• 1) Revenue, R = (R/π) x (number of servers, π)
• 2) Revenue, R = (R/P) x (price, tag, P)
Revenue, R = Q x (price, tag, P)
Remember: Quantity, Q, is considered a
conversion rate, R/P, when Price Tag is a Input.
• Not very useful for forecasting or optimization
purposes
For Diagnostic Purposes
• You want to explore the differences between
two performances
∆Revenue due to ∆P and ∆Q
• 1) a machine and an benchmark performance
• 2) a machine and a standard performance
• 3) a machine and an average performance
• 4) a machine and its previous performance
Two-Factor Revenue Performance
Café #1
Quantity of Cups Sold, Q
Factor #1
Q1 = 2,000
Selling price per Cup, P
Factor #2
P1 = $4.00
Sales Revenue, R =P x Q
$8,000
Do NOT Forget: If you know 2 of the 3 elements,
you can calculate the third element of the TwoFactor Machine
Compare the Revenue performance to
another typical machine
Café #1
Café #2
Difference #2-#1
Quantity of Cups Sold, Q
Q1 = 2,000
Q2 = 2,200
Selling price per Cup, P
P1 = $4.00
P2 = $3.90
∆Q = 200 cups
∆P = -$0.10
Sales Revenue, R =P x Q
$8,000
$8,580
∆R = $580
You can see the differences in the two performances
Identify the impact the ∆P and the impact ∆Q had on the ∆R
∆R = Impact of ∆Q + Impact of ∆P
R=PxQ
Quantity Factor
Observed point
($3.90, 2,200)
2,200 cups
Observed Output =
$3.90 x 2,200 =
$8,580 revenue
Observed point
($4.00, 2,000)
∆Q
2,000 cups
Observed Output =
$4.00 x 2,000 =
$8,000 Revenue
0, 0
$4.00 per cup
$3.90 per cup
Price Factor
∆P
∆R = I∆P + I∆Q = $780 -$200 = $580
Quantity Factor
Observed point
($3.90, 2,200)
2,200 cups
Impact of ∆Q=
$3.90 x 200 =
$780 revenue
Observed point
($4.00, 2,000)
∆Q
2,000 cups
Impact of ∆P =
-$0.10 x 2,000 =
-$200 Revenue
0, 0
$4.00 per cup
$3.90 per cup
Price Factor
∆P
The Simple Two-Factor Model for Diagnostics
• Positive impact due to increase in quantity,
Impact∆Q=$780
• Negative Impact due to decrease in price tag
• Impact∆P = -$200
• Net Impact = ∆R = $780 + (-$200) = $580
• The impact due to change in quantity more
than off-sets the net impact of the price
reduction
∆Revenue due to ∆P and ∆Q
Café #1
Café #2
Impact of Changes
Quantity of Cups Sold, Q
Q1 = 2,000
Q2 = 2,200
Selling price tag per Cup,
P
P1 = $4.00
P2 = $3.90
Sales Revenue, R =P x Q
$8,000
$8,580
∆Q = 200 cups
Impact = $780
in revenue
∆P = -$0.10
Impact = $200 in
revenue
∆R = $580
Also use this for calculating Price Elasticity
Elasticity of Price =( ∆Q/Q1) / (∆P/P1)
Elasticity of Price = (200/2,000) /(-$0.10)/$4) = 0.1/-0.025 = -4
After Aggregation
• Is there any chance for decomposition?
• Yes!
Decomposition for
More Diagnostic Detail
• The problem with using high levels of
aggregation such as
• Total Promotion Budget rather than radio
budget and print budget
• Total Revenue rather than revenue from
pastry, large cups, small cups, etc.
• is you lose too much information detail
Example You find your total budget too
aggregated for you analysis
• Revenue, R =
(conversion rate, R/π) x (total promotion, π)
• Decompose the Two Factors into Three
Factors
• Revenue, R = (Revenue returned by cost of
radio spots, R/S) x (Ratio of Radio cost to Total
cost of Promotion, S/π) x Total promotion, π)
• R = (R/S) x (S/π) x π
Decompose The Aggregated Input
Into A Multi-factor machine
You need to have recorded total promotion, π,
total revenue, R, and total radio spot expense, S
Decomposing the Revenue in the
conversion rate
• Total revenue has been aggregated into
revenue from pastry sales and from coffee
sales
• You find that total revenue is too aggregated
for your analysis
Transform from a Two-Factor to a
Three-Factor Machine
• Revenue, R =(conversion r, R/Q) x Cups sold, Q
• You need to know the number of pastries sold,
T, to expand the analysis
• Decompose to Three Factors
• Revenue, R =
(Sales Revenue per pastry, R/T) x (Pastry per
cup sold, T/Q) x Number of cups sold, Q
• R = (R/T) x (T/Q) x Q
Decomposing the Conversion Factor
into a Multi-Factor Model
You need to know that 600 pastries were sold in
café #1 and 900 pastries were sold in café #2
• Any Questions?
You may have to Calibrate the Revenue
Producing Meta-π Machine
• Using the basic 7 steps for calibrating the
• Slope-Intercept Equation
of the Meta-Marketing Machine
• Output = a – b(Input)
• Where
a = calibrated value of the y-intercept
b = calibrated value of the slope. ∆ø/∆I
Review the 7 Calibration Steps
•
•
•
•
•
1) Observe two inputs to the machine, ∆π = π2-π1.
2) Observe two outputs of the machine, ∆ø = ø2-ø2.
3) Establish the Meta-Machine, ∆ø = m x ∆π
4) Determine the meta-conversion rate, m = b = ∆ø/∆π.
5) Set Slope-Proposed Point Equation, (ø – ø2)/(π-π2) = m
where the input is set at π=0 and the output is the y-intercept, ø=a.
• 6) Use observed values of ø2 =y, π2=x, and the calculated value
of conversion rate, m = b, to calculate the value of the yintercept, ø=a,
(a-y)/(0-x) = b
a = b(-x) + y
• 7) Establish the Slope-Intercept equation of the metamarketing model as
Output = a + b(Input)