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Math 267
Professor Luke Froeb
Copyright 2002, Froeb
Owen Graduate School of Management
Vanderbilt University
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Pricing is an extent
decision

Profit=Revenue-Cost

Definition: Demand curves
are functions that relate the
price of a product to the
quantity demanded by
consumers.

Demand Curves help us
make decisions to increase
profits by modeling revenue
» Particularly MR
» Should I sell another unit?
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Aggregate Demand


Aggregate Demand: each consumer
wants one unit.
To construct demand, sort by value.
Aggregate Demand
14
12
10
Price
Pric
Revenu Marginal
e Quantity
e Revenue
12
1
12
12
11
2
22
10
10
3
30
8
9
4
36
6
8
5
40
4
7
6
42
2
6
7
42
0
5
8
40
-2
4
9
9
-7
8
6
4
2
0
0
5
Quantity

Discussion: Why do aggregate
demand curves slope downward?
» Role of heterogeneity?
» How to estimate?
10
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Pricing Tradeoff

Lower pricesell more, but
earn less on each unit sold

Higher pricesell less, but
earn more on each unit sold

Tradeoff created by
downward sloping demand
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Marginal Analysis

Marginal analysis finds the right
solution to the pricing tradeoff.
» Also requires less information.

Definition: The marginal
revenue (MR) is the change in
total revenue with an extra unit.

Proposition: If MR>0, then total
revenue will increase if you sell
one more unit.

Proposition: If MR>MC, then
total profits will increase if you
sell one more unit.

Proposition: Profits are max.
when MR=MC
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Elasticity of Demand

Motivation: price elasticity is used
to do marginal analysis.

Definition: price elas.=(%change
in quantity demanded) 
(%change in price)
» If |e| is less than one, demand is
said to be inelastic.
» If |e| is greater than one, demand is
said to be elastic.
» If |e|=1, demand is said to be
unitary elastic.
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Other Elasticities

Definition: income
elasticity=(%change in quantity
demanded)  (%change in income)
» Inferior (neg.) vs. normal (pos).

Definition: cross-price elasticity of
good one with respect to the price of
good two = (%change in quantity of
good one)  (%change in price of
good two)
» Substitute (pos.) vs. complement
(neg.).

Definition: advertising
elasticity=(%change in quantity) 
(%change in advertising) .
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Describing demand
with price elasticity

First law of demand: e<0
(price goes up, quantity goes
down).
» Discussion: Do all demand
curves slope downward?

Second law of demand: in
the long run, |e| increases.
» Discussion: Give an example
of the second law of demand.
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Describing demand
(cont.)

Third law of demand: as
price increases, demand
curves become more price
elastic, |e| increases.
» Discussion: Give an example
of the third law of demand.
HFCS
Price
Sugar Price
HFCS Demand
HFCS Quantity
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Estimating Elasticities

Definition: Arc (price)
elasticity=[(q1-q2)/(q1+q2)]
 [(p1-p2)/(p1+p2)].
» Discussion: price changes
from $10 to $8, quantity
changes from 1 to 2.

Discussion: On a promotion
week for Vlasic, the price of
the Vlasic pickles drops by
25% and quantity increases
by 300%.
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Estimating Elasticities
(cont.)

3-Liter Coke Promotion
» Instituted to meet Wal-Mart
Promotion
Product
Price/bottleQ 3-liter
P of 3-liter
Initial
210
$1.79
Final % change
420
66.67%
$1.50 -17.63%
elas.
-3.78
Price/bottleQ 2-liter
P of 3-liter
120
$1.79
48
$1.50
-85.71%
-17.63%
4.86
Price/litre Q liters
P liters
870
$0.52
1356
$0.46
10.92%
-3.12%
-3.50
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Quick and Dirty
Estimators

Linear Demand Curve Formula,
e=p/(pmax-p)

Discussion: How high would the
price of the brand have to go
before you would switch to
another brand of running
shoes?

Discussion: How high would the
price of all running shoes have to
go before you should switch to a
different type of shoe?
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Market Share Formula

Proposition: The individual brand
demand elasticity is
approximately equal to the
industry elasticity divided by the
brand share.
» Discussion: Suppose that the
elasticity of demand for running
shoes is –0.4 and the market share
of a Saucony brand running shoe is
20%. What is the price elasticity of
demand for Saucony running
shoes?

Proposition: Demand for
aggregate categories is lesselastic than demand for the
individual brands in aggregate.
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Using Elasticities for
Prediction

Discussion: The income elasticity of
demand for WSJ is 0.50. Real income
grew by 3.5% in the United States.
» Estimate WSJ demand

Discussion: The 1998 real per-capita
median income in Arizona income in
Arizona is $30,863; and in Colorado,
$40,706
» Estimate difference between per capita
consumption in Colorado and in Arizona.
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Elasticity and
Revenue

Approximate relationship
» %Rev.= %P + %Q
» =%P(1+ %Q / %P)
» =%P(1+ e)
» =%Price(1- |e|)

Discussion: In 1980, Marion
Barry, mayor of the District of
Columbia, raised the sales tax on
gasoline sold in the District by
6%.
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Elasticity and MR

Proposition: MR=P(1-1/|e|)
» If |e|>1, MR>0.
» If |e|<1, MR<0.

Discussion: If demand for
Nike sneakers is inelastic,
should Nike raise or lower
price?

Discussion: If demand for
Nike sneakers is elastic,
should Nike raise or lower
price?
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Elasticity and Pricing

MR>MC is equivalent to
» P(1-1/|e|)>MC
» P>MC/(1-1/|e|)
» (P-MC)/P>1/|e|
 Discussion: elas= –2, p=$10
mc= $8, should you raise
price?

Discussion: mark-up of 3-liter
Coke is 2.7%. Should you
raise price?
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Elasticity and pricing
(cont.)

Discussion: Sales people
MR>0. vs. marketing
MR>MC.

Discussion: The Kentucky
legislature allows only one
race track to be open at a
time.