Quantitative Demand Analysis
(as functions of output level)


Zebco management has expressed a desire to reduce its current inventory of fishing reels by 10%.
What price change is most likely to achieve this goal?


In August 2005, Hurricane Katrina basically shut down the production of oil in the southern Gulf coast area, which produces about 10% of the crude oil consumed in the
U.S. Transport Inc. is a trucking company that ships products all over the U.S. with a fleet of over 100 trucks. Immediately after
Katrina this company is trying to figure out the hurricane’s impact on its short-term and long-term fuel costs. What are reasonable projections?


The FCC has announced plans to auction off a license for the right to sell wireless communication products and services in a market with a population of 15 million. Tellcomm management is considering submitting a bid on the license.
Previous bids have averaged $80 million for markets averaging 10 million people. Tellcomm’s research department has also observed that previous bids have tended to increase by 1.4% for each 1% increase in population. What is your estimate of the minimum bid that will be required to acquire the new market’s license?


Reebock and Nike compete against each other in the athletic tennis shoe market.
Reebock has observed Nike’s decision to decrease its prices by roughly 4%. What is likely to happen to the quantity sales of
Reebock shoes if Reebock keeps its prices unchanged? How much will Reebock have to lower its price in order to maintain quantity sales at their previous level?


The marketing team of Global Concepts has observed total annual sales of $1.104 million for the company when a price of $24 was charged. More recently, the company has had total annual sales of
$1.320 million after it had raised its price to $33 for the year. The company’s statistician has just informed the marketing team that the firm’s demand curve has been linear and constant over this time period. If the marketing team would have had all of this information going into each of the previous years, what price should have been charged by the company in order to have maximized the dollar value of total company sales?


Jane is a huge Rod Stewart fan and recently attended one of his concerts. At the concert, she noticed there were a number of empty seats. She concluded the organizers of the concert could have sold more tickets and made more money if they had charged a lower price for the concert. Do you agree or disagree with
Jane?

1.
Graphical
Revenue
Concept
Output = q
2.
Mathematical
 Revenue Concept = f(q)


Example: P
P
1
2
= 10, Q
= 12, Q
1
2
= 20
= 18
Slope: measures the ‘unit’ or ‘absolute’ changes in Y associated with a one unit change in X

ΔP/ΔQ = +2/-2 = +1/-1
=> A 1 unit change in P is associated with a 1 unit change in Q in the opposite direction
Elasticity: measures the % change in Y associated with a
1% change in X
Example: %ΔQ/%ΔP = -10/+20 = -.5/1
=> A 1% change in P is associated with a .5% change in Q in the opposite direction


Case #1: P = a – bX
 ‘imperfect’ competition
* firm has some control over P (P maker)
 significant portion of mkt supply
 firm output influences mkt supply
* heterogeneous products
* difficult mkt entry (& exit)
* imperfect info


Case #2: P = a
 ‘perfect’ competition
* firm has no control over P (P taker)
 insignificant portion of mkt supply
 firm output does not impact mkt supply
* homogenous products
* easy mkt entry (& exit)
* perfect info

Concept/Definition
1. TR = Total Revenue
= total $ sales to firm
= gross income
= total $ cost to buyers
2. AR = Average Revenue
= revenue per unit of output
3. MR = Marginal Revenue
= additional revenue per unit of additional output
= slope of TR curve
If P = a – bx
= Px
= (a-bx)x
= ax-bx 2
= TR/x
= (ax-bx 2 )/x
= a – bx
= P
=  TR/  x
=  TR/  x
= a – 2bx
= Px
= ax
If P = a
= TR/x
= ax/x
= a
= P
=  TR/  x
=  TR/  x
= a

a
P
TR d=MR=AR=P=a
Q
TR=P

Q

P=a-bQ


P-Taking firm

No TR max as TR keeps increasing with Q

P-Setting firm

Max TR where MR = 0  P =a/2

Proof of Max TR (P-Setting Firm)
P = a – bQ (b>0)
TR = aQ – bQ 2
Slope of TR = a-2bQ = MR
Max TR => MR = 0
=> a-2bQ = 0 => Q =
Max TR => P = a-bQ
= a – b
= a – a
(
2
)
( a
)
( a
2 b
) a
2 b


If a firm wants to increase its dollar sales of a product, should it  P or  P?


“Students of Economics need to be taught, in business, sometimes you should raise your price, and sometimes you should lower your price.”
CEO of Casey’s

Business managers often want to know:

If a D factor affecting sales of their product changes by a given %, what will be the corresponding % impact on
Q sold of their product.
= “Elasticity of Demand”

Calculate the % change in income below
Yr Income
1 40,000
2 42,000
% Change:
= (income change/orig income) x 100
= (+2,0000/40,000) x 100
= (.05) x 100
= 5%

(Meaning)




= A measure of responsiveness of D to changes in a factor that influences D
Two components
1.
2.
=
Magnitude of change (number)
Direction of change (sign)
The number shows the magnitude of how much
D will change due to a 1% change in a D factor
The sign shows whether the D factor and D are changing in the same or opposite directions
+  same direction
 opposite direction


 E
Q,F
= %  Qd x
/%  F = %  Q/%  F where,
Notes:
Qd x
= the quantity demanded of X
F = a factor that affects Qd x sign > 0  Qd x
& F, ‘directly’ related sign < 0  Qd x
& F, ‘indirectly’ related number > 1  %  Qd x
>%  F

%
%

Q d

F


Q x 100
Q

F x 100
F


Q
Q

F
F


Q x
Q
F

F


Q

F
F x
Q



Q
F

F
Q

( re Q x d
)
Type
E
0
E
C
= own P
= cross P
F
P
X
P
Y
= Income I E
I
E
A
= advertising A


E
0
= -2  for each 1%  P x
,Q d for X will
 by 2% in opposite direction

E
C
= +1/2  for each 1%  P
Y
,Q d for X will
 by 1/2% in same direction

E
I
= +.1  for each 1%  I,Q d for X will  by .1% in same direction

Summary of demand elasticity values
E
0 always < 0 ignoring sign:
< 1 => inelastic
= 1 => unitary
> 1 => elastic
E c
> 0 => substitutes
< 0 => complements
E
1
> 0 => normal good
< 0 => inferior good

Own Price Elasticity of Demand
E
Q P x
,
, x

%
%

Q x d

P x

Negative according to the ‘law of demand’

1
, x x

1
, x x

1
, x x

Price
Quantity
Perfectly Elastic
Price
D
Quantity
Perfectly Inelastic

= % ΔQ / %ΔF
= (∂Q / ∂F) (F/Q)
= (slope of Q wrt F) (given values of F&Q)

E
0
= E
X,Px
0

%

X
%

P x

 x

P x

P x
X

1 slopeof Dcurve

P x
X

1

P x
/

X

P x
X



X
P x

P x
X

0
E
0



X
P x

P x
X
P x a a/2


1
 b
P x
X a/2b a/b x
P x a/2
> a/2
< a/2
E
0





Q d = 10 – 2P
Own-Price Elasticity: (-2)P/Q
If P=1, Q=8 (since 10 – 2 = 8)
Own price elasticity at P=1, Q=8:
(-2)(1)/8 = -0.25




Available Substitutes

The more substitutes available for the good, the more elastic the demand.
Time


Demand tends to be more inelastic in the short term than in the long term.
Time allows consumers to seek out available substitutes.
Expenditure Share

Goods that comprise a small share of consumer’s budgets tend to be more inelastic than goods for which consumers spend a large portion of their incomes .

0
E
0
= % ΔQ / %ΔP
 Can use this 3-variable equation to solve for one variable given the value of the two other variables
1) project %ΔQ due to given %ΔP & E
0
=> %ΔQ = E
0 x %ΔP
2) project %ΔP to be associated with given %ΔQ, given E
0
=> %ΔP = %ΔQ / E
0

Example 1: Pricing and Cash Flows



According to an FTC Report by Michael
Wad, AT&T’s own price elasticity of demand for long distance services is –
8.64.
AT&T needs to boost revenues in order to meet it’s marketing goals.
To accomplish this goal, should AT&T raise or lower it’s price?

Example 2: Quantifying the Change

If AT&T lowered price by 3 percent, what would happen to the volume of long distance telephone calls routed through AT&T?


Calls would increase by 25.92 percent!
E
Q P x
, x
 
8 64

%
%

Q x d

P x

8 64

%

Q x d

3%

3% x (

.
)

%

Q x d
%

Q x d 




Elastic

Increase (a decrease) in price leads to a decrease
(an increase) in total revenue.
Inelastic

Increase (a decrease) in price leads to an increase
(a decrease) in total revenue.
Unitary

Total revenue is maximized at the point where demand is unitary elastic.

TR
TR
1
2
=
=
P
1
Q
1
P
2
Q
2
= (P
1
+  P)(Q
1
= P
1
Q
1
+  Q)
+  QP
1
+  P  Q
 TR = TR
2
=  PQ
1
=  PQ
1
=  PQ
1
+  PQ
1
– TR
1
+  QP
+  QP
1
+  Q (P
2
+
1
 P
+


Q
P)
= lost TR + added TR

Change in TR Due to  Q (i.e. MR)

TR
 
PQ
 
QP

MR


TR

Q
P

P

Q
Q
P
P

P

Q
Q
MR

P [ 1

1
E
]
MR

P [
MR

P [
E

1
E E
E

1
]
E
]
NOTE:
MR = 0 if E is unitary
> 0 if E is elastic
< 0 if E is inelastic

0

TR
   

PQ

Q
Q

PQ

P
P

TR [

Q


Q P
P
]

TR [

Q
Q
P

P


TR [ 1

E
0
]

P
P

P
P
P

P
]

P
P


= ($100 mil) (1 – 8.64) (-.03)

= (100 mil) (-7.64) (-.03)

= $ + 22.92 mil.

E
, x Y

%
%

Q x d

P
Y
+Substitutes
- Complements

Example 3: Impact of a change in a competitor’s price

According to an FTC Report by Michael
Ward, AT&T’s cross price elasticity of demand for long distance services is
9.06.

If MCI and other competitors reduced their prices by 4%, what would happen to the demand for AT&T services?


AT&T’s demand would fall by 36.24 percent!
%
E
 

Q x d
, x Y %

P
Y


%

Q x d
%


Q x d
4%
.

%

Q x d
 

E
Q M x
,

%
%

Q x d

M
+ Normal Good
Inferior Good



Mathematical representations of demand curves
Example:
Q x d 
10

2 P x

3 P
Y

2 M

X and Y are substitutes (coefficient of P
Y is positive)

X is an inferior good (coefficient of M is negative)

Elasticity Calculation
 x

F

F own

X
X

P x
 cross

Income
X

P
Y

X

I

P x
X
P
Y

X
I
X


Linear Demand
Q d x
 a
0
 a P x
 a P
Y
 a M
 a H
E
, x x
 a x
P
Q x x
Own Price
Elasticity
E
, x Y
 a
Y
P
Y
Q x
Cross Price
Elasticity
E
Q M x
,
 a
M
M
Q x
Income
Elasticity

X,Px

X = 10 – 2P x
= 10 – 2P x
+ 3P
Y
– 2M
+ 3(4) – 2(1)
 X = 20 – 2P
X
 P x
= 10 - .5X


X,Px
X

X

P x

P x
X

(

2 )(
4
)
12
 
/
 
.
67

E
X,I
Calculation Given D Equation

X = 10 – 2P
X
+ 3P
Y
– 2I
= 10 – 2(1) + 3(4) – 2I
 X = 20 – 2I

X,I


X

I

I
X

(

2 )(
2
)
16
 
/
 
.
25

log Q x d  
0
  x log P x
 
Y log P
Y
 
M log M
 
H log H

Own Price Elasticity: 
X

Cross Price Elasticity: 
Y

Income Elasticity: 
M

0
If E
0 is inelastic (=> %
=> %ΔQ < %ΔP
ΔQ / %ΔP < 1),
=> %ΔP > %ΔQ
=> small changes in Q can result in big changes in P e.g. if E
0
= -0.2
=> .2% ΔQ => 1% ΔP
=> 1% ΔQ => 5% ΔP
(=> %ΔP is 5 x %ΔQ)
=> 5% ΔQ => 25% ΔP

When Two or More D Factors
Change
Combined impact of:
1) 10% ↓P
X if E
0 and
2) 10% ↑I if E
I
= -.4
= +.2
%ΔQ due to:
10% ↓ P
X
10% ↑ I
= +4%
= + 2%
=> combined = +6%




Elasticities are tools you can use to advertising on sales and revenues.
quantify the impact of changes in prices, income, and
Given market or survey data, regression analysis can be used to estimate:


Demand functions
Elasticities

A host of other things, including cost functions
Managers can quantify the impact of changes in prices, income, advertising, etc.