unit 4. quantitative demand analysis (as functions of output level)

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

QQ

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

QQ

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.