### Pricing I: Linear Demand - Management By The Numbers

```Pricing I: Linear Demand
This module covers the relationships between price and
quantity, maximum willing to buy, maximum reservation
price, profit maximizing price, and price elasticity, assuming
a linear relationship between price and demand.
Authors: Paul Farris and Phil Pfeifer
Marketing Metrics Reference: Chapter 7
© 2010-14 Paul Farris, Phil Pfeifer and Management by the Numbers, Inc.
Definition
Linear demand functions are those in which the relationship between
quantity and price is linear. This means that any identical change in
price (no matter what the starting price) produces an identical change in
units demanded. The per unit change in Q caused by a change in P is
called the slope. With linear demand “curves”, the slope is constant (the
same for all prices). The demand “curve” is actually a demand “line”.
LINEAR PRICE-QUANTITY FUNCTIONS
Linear Price-Quantity Functions
This presentation covers the topics of maximum willing to buy (MWB),
maximum reservation price (MRP), profit maximizing price, and price
elasticity under the assumption of a linear relationship between price
and quantity.
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Increasing price from \$0.50 to \$1.50 causes a drop in
quantity from 9 to 7 units. Thus, the slope (delta Q / delta P)
is – 2 = (9 – 7) / (1.5 –.50).
10
9
8
7
Quantity 6
5
4
3
2
1
0
Increasing price from \$2 to \$3 causes
a drop in quantity from 6 to 4. Thus,
the slope (delta Q / delta P) is also -2,
for each \$1 change in price we see a
change on 2 unit of quantity sold in
the opposite direction.
0
\$1
\$2
Price
\$3
\$4
\$5
LINEAR PRICE-QUANTITY DEMAND
Linear Price-Quantity Demand
Insight
Price slope is
almost always
negative, but
often people
drop the sign.
Definition
Slope of demand = change in quantity / change in price.
3
MWB is where P = 0
MWB -- Maximum Willing to Buy
10
9
8
7
Quantity 6
5
4
3
2
1
0
Formulas for linear demand functions use the
following format (recall y = mx + b):
Quantity = Slope * Price + MWB
In this example, beta = - 2. One could then
solve for MWB and MRP using this
equation and a price quantity point.
MRP -- Maximum
Reservation Price
0
\$1
\$2
Price
\$3
\$4
LINEAR PRICE-QUANTITY DEMAND FORMULA
Linear Price-Quantity Demand Formula
\$5
MRP is where Q = 0.
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Definitions
= Quantity - Slope * Price
Conceptually, MWB is the quantity that would be sold (presuming linear demand)
if the price is free (price = 0).
Maximum Reservation Price (MRP)
= MWB / (- Slope)
Conceptually, MRP is the maximum price where some quantity would be sold
(again, presuming linear demand). This is actually where the linear demand line
crosses the axis or where quantity = 0.
LINEAR PRICE-QUANTITY DEMAND FORMULA
Linear Price-Quantity Demand Formula
We took a bit of a leap of faith there. For some of you, it may be helpful to
review the how these formulas were derived which we’ll do on the following
slide.
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You may recall from algebra that linear functions follow the format:
y = mx + b
Where b is the Y intercept (where the line crosses the y axis or where x = 0), m
is the slope of the line, and x and y are the coordinates of any point on the line.
First, let’s express the basic linear function using marketing terminology:
Quantity = Slope * Price + MWB
We can use this basic equation for price / quantity relationships as well as
bringing in some other important managerial concepts such as the maximum
willingness to buy (MWB) and maximum reservation price (MRP).
With a little substitution, the equation is, Quantity = MWB * [1 – Price / MRP]
LINEAR PRICE-QUANTITY DEMAND FORMULA
Linear Price-Quantity Demand Formula
From this, we can derive functions solving for MWB and MRP in the definitions.
Definitions
Maximum Reservation Price (MRP)
= Quantity - Slope * Price
= MWB / (- Slope)
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Now let’s turn to the question of
maximizing profit using the
following example.
MWB
10
9
8
7
Quantity 6
5
4
3
2
1
0
If you are told that the unit cost is
\$1, what is the optimal (profitmaximizing) price to charge?
OPTIMAL (PROFIT MAXIMIZING) PRICE
Optimal (Profit Maximizing) Price
MRP
0
\$1
\$2
Price
\$3
\$4
\$5
Unit
Cost
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As you might expect, the profit maximizing price
is always more than cost and less than the MRP.
MWB
10
9
8
7
6
Quantity
5
4
3
2
1
0
But, it is nice, and maybe a little
surprising, that the profit-maximizing
price is ALWAYS exactly half-way
between unit costs and MRP.
Profit = Q * (P-C)
= 4 * (\$3-\$1)
= \$8
0
\$1
\$2
Price
\$3
OPTIMAL (PROFIT MAXIMIZING) PRICE
Optimal (Profit Maximizing) Price
MRP
\$4
\$5
C = Unit Cost, P = Price, Q = Quantity
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So, for linear demand functions, we only need two pieces of information
to calculate the profit maximizing price:
1. Unit Variable Cost and
2. Maximum Reservation Price (MRP is where Q is equal to 0)
BIG CONCLUSION!
Big Conclusion!
Definition
Profit Maximizing Price = ½ (Unit Cost + MRP)
In addition, since linear demand functions have a constant slope, with
any two points (price quantity combinations) you can calculate the slope,
the MWB, and the MRP.
Now let’s apply these definitions in a couple of examples.
Question 1: For a product, we observe that at a price of \$5, a quantity of 8
is sold and that at \$4 we sell 12 units of a product. The cost of the product is
\$3. What is the MWB, MRP and Profit Maximizing Price?
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First, we have two data points, necessary to calculate the slope which
we’ll need for solving for MWB.
Slope = delta Q / delta P = (8 – 12) / (5 – 4) = – 4
Using the MWB = Quantity - Slope * Price formula, substitute one set
of price & quantity values from above along with the slope.
MWB = 12 – (-4 * 4)
MWB = 12 – (-16) = 12 + 16 = 28
Verify with the other point: MWB = 8 – (-4 * 5) = 8 + 20 = 28 (Matches)
Now, it is easy to find MRP by solving for the price at which we sell
zero units. 0 = 28 – 4 * price = \$7 or by using MRP = MWB / (- Slope)
MRP = 28 / - (-4) = 28 / 4 = \$7
EXAMPLE OF HOW TO CALCULATE MWB AND MRP
Example - How to Calculate MWB & MRP
Profit Maximizing Price = 1/2 (Cost + MRP) = ½ (3 + 7) = \$5.
MBW = 28, MRP = \$7, Profit Maximizing Price = \$5
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Question 2: Suppose we apply regression to sales data, and find the
following demand function where slope = – 4 and MWB = 100:
q = slope * p + MWB = – 4 * p + 100
If the unit cost equals \$5, what is the optimal price?
First, find MRP by dividing MWB by – slope (e.g. a + value),
100 / 4 = 25, so MRP = 25
FINDING OPTIMAL PRICE WITH REGRESSION
Finding Optimal Price with Regression
Then add to the cost ½ the difference between cost and MRP.
Cost + ½ (MRP – Cost) = \$5 + ½ (\$25 – \$5) = \$15
So, the profit-maximizing price is \$15.
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The slope is the unit change in quantity for a small unit change in
price. For linear demand curves, the slope is constant (the same at all
prices).
Another measure of how “much” quantity reacts to changing prices is
ELASTICITY. Whereas slope is a unit per unit change rate, elasticity is
a percentage per percentage change rate. It is the slope times (P/Q),
and is often thought of as the percentage change in Q for a small
percentage change in P.
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SLOPE VERSUS ELASTICITY
Slope versus Elasticity
12
For a linear demand curve, the slope is a constant. This means that the
elasticity will NOT be a constant but will depend on the initial price. For
any linear demand curve, the elasticity is larger for higher prices.
This makes sense because if a unit change in price produces a constant
unit change in Q, the unit increase in price is a smaller percentage of P
if P is high and the unit decrease in Q is a larger percentage of Q when
Q is low (which it is if P is high).
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SLOPE VERSUS ELASTICITY
Slope versus Elasticity
13
Increasing price from \$0.50 to \$1.50 causes a drop in quantity
from 9 to 7 units. Thus, the slope (delta Q / delta P) is 2 and
the elasticity (– 2 / 9) / (\$1 / \$0.5) = –.11
10
9
8
7
Q 6
5
4
3
2
1
0
Increasing price from \$2 to \$3 causes a
drop in quantity from 6 to 4. Thus, the slope
(delta Q/delta P) is 2, but the elasticity is
(– 2 / 6) / (\$1 / \$2) = –.67
0
\$1
\$2
\$3
Note that if we calculate the
elasticity for the same interval
using a decrease in price from
\$3 to \$2, we get
(2 / 4) / (– \$1 / \$3) = –1.5,
a different value.
\$4
\$5
LINEAR PRICE-QUANTITY DEMAND
Linear Price-Quantity Demand
Price
Insight
For linear demand functions, elasticity changes at each point on the
price-quantity demand function.
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Example: Using the formula, q = – 4 * p + 100, calculate the price
elasticity at the point where p = \$15, the profit-maximizing price.
• First, calculate q = – 4 * 15 + 100 = 40
• Next, the price elasticity is
equal to the slope * p / q = – 4 * (15 / 40) = – 1.5
• Now calculate the percentage margin on selling price at profitmaximizing price…(\$15 - \$5) / \$15 = 66.7%
• Divide the margin into 1 = 1 / .667 = 1.5
• At the profit maximizing price, the elasticity is equal to the reciprocal
of the margin and vice versa. The minus sign is ignored for these
purposes. This is a very powerful and important result that always
holds, regardless of the nature of the form of demand.
PRICE ELASTICITY AND OPTIMAL MARGIN
Price Elasticity and Optimal Margin
Definition
At the Profit Maximizing Price:
Elasticity = 1 / Margin%, and Margin% = 1 / Elasticity.
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When selling through resellers, we still need to calculate the MRP in
terms of retail price, but now we have the added complication of the
channel margins impacting the margin calculation. Since, retailers often
require a percentage* margin, it is no longer a constant dollar variable
cost as price changes. One way to handle this is to use the retail margin
to convert Retail MRP to the Marketer MRP. Let’s use the linear demand
function below to illustrate.
SELLING THROUGH RESELLERS
Selling Through Resellers
Example: Q = – 4 * Retail Price + 100 and assume retailers earn a 40% margin
and that unit costs are \$5.
First, calculate MRP = MWB/4 = 100/4 = \$25 (Retail)
Since retailers take 40% margins, our marketer only receives 60% of retail price.
For her profit calculations, the
MRP (Marketer) = \$25 * 60% = \$15.
Maximum profit is earned at the Marketer price that is halfway between \$15 and
\$5, or \$10. This corresponds to a Retailer Price of \$16.66 (\$10/.6).
* If retailers use constant dollar margins, we could just add that dollar margin to our unit variable cost.
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Although price elasticity is not very useful if the demand function is
linear, elasticity is the key to finding the optimal price for nonlinear
demand curves. As we just observed, at the optimal price, the elasticity
is always equal to the reciprocal of the margin (or vice versa) if the sign
is dropped. We can easily verify this for our linear example.
MORE ON PRICE ELASTICITY
More on Price Elasticity
In Pricing II – Constant Elasticity, we will examine demand curves that
are not linear. For non-linear demand curves, it will not be as easy to
find the optimal price. However, what we can do is compare the
elasticity (if we know it at the current price) to the reciprocal of our
current margin. If they are not equal, we know which direction to adjust
our price to improve profit. Eventually, these adjustments will lead to the
optimal price.
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Marketing Metrics by Farris, Bendle, Pfeifer and
Reibstein, 2nd edition, chapter 7.
- And the following MBTN modules -
FURTHER REFERENCE
Further Reference
Pricing II: Constant Elasticity - This module explores
pricing under the assumption of demand curves that are
not linear.
Profit Dynamics - This module is a more basic module