unit 4. quantitative demand analysis (as functions of output level)
TRANSCRIPT
Unit 4.Quantitative Demand Analysis (as functions of output level)
Inventory Sale
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?
Katrina Impact?
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?
Expected Bid Price 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’s Response to Nike
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?
Let’s Maximize $ Sales 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?
Empty Seats, Lost Revenue?
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?
Revenue Concepts and Output Relationships
1. Graphical
RevenueConcept
Output = q
2. Mathematical Revenue Concept = f(q)
‘Unit’ vs % Marginal Analysis
Example: P1 = 10, Q1 = 20 P2 = 12, Q2 = 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 directionElasticity: measures the % change in Y associated
with a 1% change in XExample: %ΔQ/%ΔP = -10/+20 = -.5/1=> A 1% change in P is associated with a .5% change
in Q in the opposite direction
D Curves Facing Individual Firms
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
D Curves Facing Individual Firms
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
Revenue Concepts
Concept/Definition If P = a – bx If P = a
1. TR = Total Revenue = total $ sales to firm = gross income = total $ cost to buyers
= Px= (a-bx)x= ax-bx2
= Px= ax
2. AR = Average Revenue = revenue per unit of output
= TR/x= (ax-bx2)/x= a – bx= P
= TR/x= ax/x= a= P
3. MR = Marginal Revenue = additional revenue per unit of additional output = slope of TR curve
= TR/x= TR/x= a – 2bx
= TR/x= TR/x= a
Market & Firm D (Perfect Competition)
Revenue ConceptsP = a
a
P
Q
d=MR=AR=P=a
TR
Q
TR=PQ=aQ
Revenue ConceptsP=a-bQ
TR Max
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 – bQ2
Slope of TR = a-2bQ = MRMax TR => MR = 0
=> a-2bQ = 0 => Q = Max TR => P = a-bQ
= a – b= a –= = mid pt of D curve
b
a
2
)2
(b
a
)2
(a
)2
(a
Question
If a firm wants to increase its dollar sales of a product, should it P or P?
Quote of the Day
“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 Income1 40,0002 42,000
% Change:= (income change/orig income) x 100= (+2,0000/40,000) x 100= (.05) x 100= 5%
Elasticity of D Definition (Meaning)
= A measure of responsiveness of D to changes in a factor that influences D
Two components1. Magnitude of change (number)2. 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
Elasticities of Demand
EQ,F = %Qdx/% F = %Q/%F
where,Qdx = the quantity demanded of X
F = a factor that affects Qdx
Notes:
sign > 0 Qdx & F, ‘directly’ related
sign < 0 Qdx & F, ‘indirectly’ related
number > 1 %Qdx>%F
Elasticity Calculation%
%
Q
F
x
FF
x
QQFF
Q
QxF
F
Q
FxF
Q
Q
F
F
Q
d
1 0 0
1 0 0
Types of Elasticities ( )re Q x
d
Type F
E0 = own P PX
EC = cross P PY
EI = Income I
EA = advertising A
Elasticity Value Meanings (e.g.)
E0 = -2 for each 1% Px,Qd for X will by 2% in opposite direction
EC = +1/2 for each 1% PY,Qd for X will by 1/2% in same direction
EI = +.1 for each 1% I,Qd for X will by .1% in same direction
Summary of demand elasticity values
E0
always < 0ignoring sign:
< 1 => inelastic= 1 => unitary> 1 => elastic
Ec
> 0 => substitutes< 0 => complements
E1
> 0 => normal good< 0 => inferior good
Own Price Elasticity of Demand
Negative according to the ‘law of demand’
EQ
PQ Pxd
xx x, ,
%
%
E lastic E
Inelastic E
Unitary E
Q P
Q P
Q P
x x
x x
x x
:
:
:
,
,
,
1
1
1
Perfectly Elastic & Inelastic Demand
Price Price
Quantity
DD
D
Quantity
Perfectly Elastic Perfectly Inelastic
Elasticity Calculation Overview
= %ΔQ / %ΔF
= (∂Q / ∂F) (F/Q)
= (slope of Q wrt F) (given values of F&Q)
E0 Calculation
E0 = EX,Px
%
%
/
X
P
x
P
P
X
slope o f D curve
P
X
P X
P
X
X
P
P
X
x
x
x
x
x
x
x
x
1
1
E0 and Linear D (P = a – bx)
EX
P
P
X
b
P
X
x
x
x
0
1
a
Px
a/2
a/2b a/b x
Px E0
a/2
> a/2
< a/2
Qd = 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
Example of Linear Demand
Factors Affecting Own Price Elasticity
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.
Managerial Uses of E0
E0 = %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 & E0
=> %ΔQ = E0 x %ΔP2) project %ΔP to be associated with given %ΔQ, given E0
=> %ΔP = %ΔQ / E0
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?
Answer
Calls would increase by 25.92 percent!
EQ
P
Q
x Q
Q
Q Pxd
x
xd
xd
xd
x x, .%
%
.%
( . ) %
% .
8 6 4
8 6 43 %
3 % 8 6 4
2 5 9 2 %
Own-Price Elasticity and Total Revenue
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.
Change in TR (math) (If ↓P)
TR1 = P1Q1
TR2 = P2Q2
= (P1+P)(Q1+Q)
= P1Q1+PQ1+QP1+PQ
TR = TR2 – TR1
= PQ1+QP1+PQ
= PQ1+Q (P1+ P)
= PQ1+QP2
= lost TR + added TR
Change in TR Due to Q (i.e. MR)
TR PQ QP
TR
QP
P
MR PP
Q
Q
PP
MR PE
MR PE
E E
MR PE
E
[ ]
[ ]
[ ]
11
1
1
NOTE:
MR = 0 if E is unitary
> 0 if E is elastic
< 0 if E is inelastic
Change in TR and E0
TR P Q Q P
PQQ
QPQ
P
P
TRQ
Q
P
P
TRQ
Q
P
P
P
P
P
P
P
P
TR EP
P
[ ]
[ ]
[ ]1 0
Quantifying the Change inTR
= ($100 mil) (1 – 8.64) (-.03)
= (100 mil) (-7.64) (-.03)
= $ + 22.92 mil.
Cross Price Elasticity of Demand
+Substitutes
- Complements
EQ
PQ Pxd
Yx Y,
%
%
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?
Answer
AT&T’s demand would fall by 36.24 percent!
EQ
P
Q
x Q
Q
Q Pxd
Y
xd
xd
xd
x Y, .%
%
.%
. %
% .
9 0 6
9 0 64 %
4 % 9 0 6
3 6 2 4 %
Income Elasticity
+ Normal Good
- Inferior Good
EQ
MQ Mxd
x,
%
%
Demand Functions
Mathematical representations of demand curves
Example:
Q P P Mxd
x Y 1 0 2 3 2
X and Y are substitutes (coefficient of PY is positive)
X is an inferior good (coefficient of M is negative)
Elasticity Calculation
x
F
F
X
ownX
P
P
X
crossX
P
P
X
Incom eX
I
I
X
x
x
Y
Y
Specific Demand Functions
Linear Demand
Own Price Cross Price IncomeElasticity Elasticity Elasticity
Q a a P a P a M a Hxd
x x Y Y M H 0
E aP
QQ P xx
xx x, E a
P
QQ P YY
xx Y, E a
M
QQ M Mx
x ,
EX,Px Calculation Given D Function Equation
X = 10 – 2Px + 3PY – 2M
= 10 – 2Px + 3(4) – 2(1)
X = 20 – 2PX
Px = 10 - .5X
EX,Px at PX = 4 ?
X
P
P
Xx
x
( )( )
/ .
24
1 22 3 6 7
EX,I Calculation Given D Equation
X = 10 – 2PX + 3PY – 2I
= 10 – 2(1) + 3(4) – 2I
X = 20 – 2I
EX,I at I = 2 ?
X
I
I
X
( )( )
/ .
22
1 61 4 2 5
Log-Linear Demand
Own Price Elasticity: X
Cross Price Elasticity: Y
Income Elasticity: M
lo g lo g lo g lo g lo gQ P P M Hxd
x x Y Y M H 0
E0 & P volatility
If E0 is inelastic (=> %ΔQ / %ΔP < 1),=> %ΔQ < %ΔP=> %ΔP > %ΔQ=> small changes in Q can result in big changes in P
e.g. if E0 = -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% ↓PX if E0 = -.4and
2) 10% ↑I if EI = +.2
%ΔQ due to:
10% ↓ PX = +4%10% ↑ I = + 2%=> combined = +6%
Summary
Elasticities are tools you can use to quantify the impact of changes in prices, income, and advertising on sales and revenues.
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.