finance 30210: managerial economics consumer demand analysis

Finance 30210: Managerial Economics

Consumer Demand Analysis

We can begin our representation of the consumer with a demand function…

csd PPIPDQ ,,,

Quantity demanded

Is a function of…

Price IncomePrices of Substitutes

Prices of Compliments

Quantity

10,40,100 cs PPID

Price

$20

40

10,40,100,2040 D

Given a demand function we can characterize the behavior of demand with elasticity..

Quantity

cs PPID ,,

Price

P

Qdp

%

%

$20

$15

40 60

%25% P

%50% dQ

225

50

p

Price elasticity will always be a negative number

Quantity

100ID

Price

$20

40 56

410

40I

I

QdI

%

%

Income elasticity will generally be a positive number


110ID

%40% dQ

%10% I

Quantity

40sPD

Price

$20

40 44

5.20

10

sP

s

dP P

Qs

%

%

Cross price elasticity will be a positive number for substitutes


48sPD

%10% dQ

%20% sP

Quantity

14cPD

Price

$20

32 40

5.40

20

I

c

dP P

Qc

%

%

Cross price elasticity will be a negative number for compliments


10sPD

%20% dQ

%40% sP

Note that if the demand relationship is linear, elasticity is not constant

csd ePdPcIbPaQ

Q

Pb

Q

P

P

Q

P

Q

p

p

dp

%

%

Q

Ic

Q

I

I

Q

I

Q

I

p

dp

%

%

Q

Pd s

Ps

Q

Pe c

Pc

For example….

IPQd 0025.260

Quantity

000,40ID

Price

$30

100

1100

000,400025.

6.100

302

I

p

$80

Let’s try something a little more complicated…a non-linear demand relationship

22

1

02.2100 IPQd

Quantity

20ID

Price

$400

29.

68

400 2

12

1

2

1

Q

P

Q

PP

Q

P

P

Qp

$2916

68

24.

68

2004.04.04.

22

Q

I

Q

II

Q

I

I

QI

Sometimes we use demand functions that are linear in logs…

Quantity

40ID

Price

$12

40

IPd eQ 012.15.150

LN(Quantity)

40ID

Price

$12

3.7

IPd eLNQLN 012.15.150

.15 .012150 P IdLN Q LN LN e

IPQLN d 01215.5

Quantity

40ID

Price

$12

40

IPd eQ 012.15.150

8.11215.15.150

150)15(.15.150

012.15.

012.15.012.15.

P

e

Pe

Q

Pe

Q

P

P

QIP

IPIP

p

48.40012.012.150

150)012(.012.150

012.15.

012.15.012.15.

I

e

Ie

Q

Ie

Q

I

P

QIP

IPIP

I

A little math trick…recall the derivative of the natural log

xx

x 1ln

x

xx

ln

This just says that the difference in logs is a percentage change

Therefore, if we start with elasticity

PP

Q

P

Q

P

Qp

ln

%

ln

%

%

LN(Quantity)

40ID

Price

$12

3.7

IPQLN d 01215.5

8.11215.15.ln

%

%

PPP

Q

P

Qp

48.40012.012.ln

%

%

III

Q

I

QI



Quantity

40ID

Price

$10

3

IQ ePe d 045.65.16

Quantity

40ID

LN(Price)

2.3

3

IQ ePLNeLN d 045.65.16

Id eLNPLNLNQ 045.65.16

IPLNQd 045.65.7.2


Quantity

40ID

LN(Price)

2.3

3

IPLNQd 045.65.7.2

22.3

65.165.

1

ln%

%

QQP

Q

P

Qp

6.3

40045.045.

%

%

Q

I

Q

I

I

Q

I

QI


Quantity

40ID

Price

$5

8.8

4.12.135. IPQd

LN(Quantity)

40ID

LN(Price)

1.6

2.2

4.12.135. IPLNQLN d

4.12.135. ILNPLNLNQLN d

ILNPLNQLN d 4.12.104.1


LN(Quantity)

40ID

LN(Price)

1.6

2.2

ILNPLNQLN d 4.12.104.1

2.1ln

ln

%

%

P

Q

P

Qp

4.1ln

ln

%

%

I

Q

I

QI

Log linear demand curves have constant elasticities!

Suppose that you setting prices for US Air. You know that you face the following demand curve for the New York/Chicago Shuttle…

Quantity

Price

$200

400

PQ 41200

$80,000 If you wanted to increase revenues, should you increase or decrease your price?D

Quantity

Price

$199

400

PQ 41200

$80,396

D

PQTR

PPTR 41200

241200 PPTR

PP

TR81200

40020081200

P

TR

If you wanted to increase revenues, should you increase or decrease your price?

You should decrease price. A $1 price decrease will raise revenues by $400

$200

404

Why didn’t revenue go up by $400?

Quantity

Price

$150

400

PQ 41200

$90,000D

241200 PPTR

081200

PP

TR

150$P

Suppose that you wanted to maximize revenues?

$200

600

Now, let’s go about this a little differently…

Quantity

Price

$200

400

PQ 41200

$80,000

2400

2004

p

PQTR

QPTR %%%

PPTR p %%%

PTR p %1%

PTR %%

Every 1% drop in price will raise revenues by 1%

D

Using the point elasticity gives you the same answer…

Quantity

Price

$200

400

PQ 41200

$86,400

$180

480

%8100*000,80

000,80400,86

Why didn’t revenues go up by 10%?

5.1480

1804

p

PTR %5.%

D

We could also maximize revenues using elasticity…

Quantity

D

Price

$150

600

PQ 41200 PTR p %1%

When the elasticity hits -1, revenues stop growing when you lower your price.

14

Q

Pp

141200

4

P

P

PP 412004 150$P

$90,000

Quantity

D

Price

$150

600

PQ 41200

Quantity

D

Price

$200

600

$250

$140

$50

$90,000 2.0p

875.p

2p

5p

PDQ

Obviously, the more information you have, the better decisions you will make…

The best pricing decisions would come from a demand curve

However, knowing a few elasticities is quite helpful as well!

Revenue maximizing price Revenue maximizing price

Suppose that you setting prices for US Air. You know that you face the following demand curve for the New York/Chicago Shuttle…

IPQ 241000 Suppose that median income is equal to $50,000.

Income in thousands

Quantity

D

Price

$125

600

$75,000

Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?

40241000600 P

Quantity

D

Price

$125

600

$72,000


Now income is $40,000

IPQ 241000

120P

$120

20102 Q

2054 Q

Now, again with elasticity…

Suppose that median income is equal to $50,000.

Quantity

D

Price

$125

600

$75,000

83.600

1254

p

17.600

502

I


IPQ 241000

Quantity

D

Price

$125

600

4.32017.% Q

4.31.483.% Q

IPQ Ip %%%

2017.%83.0 P

$123

1.4% P

1.4% P


83.p

17.I

$73,800

Quantity

D

Price

$125

600

$75,000

Quantity

D

Price

$125

600

$75,000

83.p

17.I

IPQ 241000 IPDQ ,

Again, the more information you have, the better decisions you will make…

The best pricing decisions would come from a demand curve

However, knowing a few elasticities is quite helpful as well!

Now, suppose that you realized that you actually faced two types of customers : Recreational and business travelers. Could you do better?

Quantity

D

Price

$150

500Quantity

D

Price

$150

100

PQB 2800

Business

PQR 2400

Recreational

$400

$200

$75,000

$15,000

6.p

3p

Recall the aggregate demand curve we had previously….what we had was actually a piece of what aggregate demand really looked like

Quantity

D

Price

$200

600

PQB 2800

$400

$90,000

PQR 2400

At a price above $200, recreational travelers don’t fly. The only demand is coming from business travelers.

At a price below $200, we now have two types of demanders flying.

+

PQ 41200 $150

400

Could we do better?

PQ 2800

Suppose that we decided to ignore recreational travelers and cater to business clients…

Quantity

D

Price

$200

600

$400

$80,000PQ 41200

400

No…that’s not a good idea!

PQ 2800 22800 PPPQTR 04800

PP

TR

200$P

Why don’t we just charge them different prices?

Quantity

D

Price

$200

400Quantity

D

Price

$100

200

200$

4800

2800

28002

P

PP

TR

PPTR

PQB

Business Recreational

$400

$200

$80,000

$20,000

100$

4400

2400

24002

P

PP

TR

PPTR

PQR

1p1p

Quantity

Price

$200

$400

The real question is: Is this feasible to do empirically?

This is “the Truth”. Two types of individuals making purchasing decisions. Individual decisions added up across all individuals create an aggregate demand curve.

PQ 2400 PQ 2800

$200 ,41200

200$ ,2800

PP

PPQ

Quantity

Price

$200

$400

If you do a linear estimation, what you end up with something like this…not really what we want

bPaQ

$200 ,41200

200$ ,2800

PP

PPQ

bPaQ

However, if you put a dummy variable in the regression for business/recreational traveler, you could get at the truth…

Quantity

Price

$200

$400

PQ 2400 PQ 2800

cDbPaQ bPaQ

cDbPaQ

Business ,1

Recreation ,0D

YES! We can do this!

Now, lets change it up a little….

QuantityD

Price

$150

450Quantity

D

Price

$150

150

PQB 600

Business

PQR 3600

Recreational

$600

$200

$67,500

$22,500

33.p

3p

Now, if we charge different prices….

QuantityD

Price

$300

300Quantity

D

Price

$100

300

$600

$200

$90,000

$30,000

300$

2600

600

6002

P

PP

TR

PPTR

PQB

Business

1p

100$

6600

3600

36002

P

PP

TR

PPTR

PQB

Recreational

1p

Quantity

Price

$200

$600

Again, we have to ask: Is this feasible to do empirically?

This is “the Truth”. Two types of individuals making purchasing decisions. Individual decisions added up across all individuals create an aggregate demand curve.

PQR 3600

PQB 600

PQ 41200

$300

PQB 600

PQR 3600 +

Quantity

Price

$200

$600

PQR 3600

PQB 600

bPaQ

If you do a linear estimation, what you end up with something like this…

bPaQ

Quantity

Price

$200

$600

PQR 3600

PQB 600

cDbPaQ

What if we tried the dummy variable trick…

cDbPaQ

Business ,1

Recreation ,0D

bPaQ

We’re Screwed! Time to try something else.

Here’s the process that takes place in the economy…

Individual consumers have preferences over a variety of goods…they have limited incomes and face market prices. Consumers make choices on what to buy

P

QD

P

QD

P

QD

P

QD

Individual decisions can be represented by individual demand curves

P

QD

The market aggregates those decisions into a market demand curve

Here is the problem we face…

P

QD

We can estimate a market demand curve…

P

QD

P

QD

P

QD

P

QD

The problem is that the market demand often tells us very little about what is happening in the background…

So, here is how we attack the problem…

We see if we can come up with a numerical representation of preferences…

P

QD

P

QD

P

QD

P

D

We then derive the resulting demand curves that come from the consumer choice problem…

P

QD

We then aggregate the individual demand curves to get a market demand. This we can compare to aggregate data to see if we are correct

How do we get an understanding of consumer preferences? We observe what they do!

Television ACost = $500

Television BCost = $2500

Suppose you walk into the store with a choice between two TVs.

Suppose that you choose Television A

Either you prefer Television A or you can’t afford Television B

Suppose that you choose Television B You must prefer Television B

Suppose that you observed the following consumer behavior

P(Bananas) = $4/lb.

P(Apples) = $2/Lb.

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

P(Bananas) = $3/lb.

P(Apples) = $3/Lb.

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

What can you say about this consumer?

Is strictly preferred to

Choice A

Choice B

Choice B Choice A

How do we know this?

Consumers reveal their preferences through their observed choices!

P(Bananas) = $4/lb.

P(Apples) = $2/Lb.

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

P(Bananas) = $3/lb.

P(Apples) = $3/Lb.

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

Cost = $80 Cost = $90

Cost = $90 Cost = $90

B Was chosen even though A was the same price!

Choice A Choice B

What about this choice?

P(Bananas) = $2/lb.

P(Apples) = $4/Lb.

Q(Bananas) = 25lbs

Q(Apples) = 10lbs

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

Cost = $90

Q(Bananas) = 15lbs

Q(Apples) = 15lbsCost = $90

Cost = $100

Is strictly preferred to Choice C Choice B

Choice C

Is choice C preferred to choice A?

Choice B

Choice A

Is strictly preferred to Choice B Choice A

Is strictly preferred to Choice C Choice B

Is strictly preferred to Choice C Choice A

Rational preferences exhibit transitivity

C > B > A

Consumer theory begins with the assumption that every consumer has preferences over various combinations of consumer goods. Its usually convenient to represent these preferences with a utility function

BAU :

A BU

Set of possible consumption choices “Utility Value”

Q(Bananas) = 25lbs

Q(Apples) = 10lbs

Q(Bananas) = 10lbs

Q(Apples) = 20lbs

Q(Bananas) = 15lbs

Q(Apples) = 15lbs

Choice C

Choice A

Choice B

Using the previous example (Recall, C > B > A)

)20,10()15,15()10,25( UUU

We require that utility functions satisfy a few basic properties

20),( yxUx

y

A

B

C

BUCU

BUAU

AUCU

)()(

There is a definite ranking of all choices (i.e. transitivity)

25),( yxU

This tells us that indifference curves can’t cross on another

20),( yxU

x

y

A

B

C BUCU

BUAU

AUCU

)()(

We run into a problem!

25),( yxU

Suppose that indifference curves did cross…

)()( CUAU

20),( yxUx

y

A

B

C

More is always better!

)()( AUCU


20),( yxUx

y

A

B

C

People Prefer Moderation!

)()( AUCU

15

5 15

5

10

10

25),( yxU


Note: This is a result of diminishing marginal utility…this guarantees that demand curves slope down!

20),( yxUx

yA

BC

People prefer extremes!

)()( AUCU

25),( yxU

What if we didn’t have decreasing marginal utility?

Increasing marginal utility produces some weird decisions!!

x

y

*y

*x

20),( yxU

We can characterize preferences with a few statistics. First, how does a consumer prefer one good relative to another.

x

yMarginal Utility of Y

Marginal Utility of X

),(

),(**

**

yxU

yxUMRS

x

y

y

x

The marginal rate of substitution (MRS) measures the amount of Y you need to be get in order to give up a little of X

The marginal rate of substitution (MRS) measures the amount of Y you require to give up a little of X

x

y

*y

*x

20),( yxU

)','(),( ** yxMRSyxMRS

'y

'x

If you have a lot of X relative to Y, then X is much less valuable than Y MRS is low!

The elasticity of substitution measures the curvature of the indifference curve

x

y'

x

y

x

y

MRSxy

%

%

Elasticity of substitution measures the degree to which your valuation of X depends on your holdings of X

y

x

small is

y

x

large is

The elasticity of substitution measures the curvature of the indifference curve

If the elasticity of substitution is small, then small changes in x and y cause large changes in the MRS

If the elasticity of substitution is large, then large changes in x and y cause small changes in the MRS

MRSxy

%

%

X

Y

X

Y

Elasticity of Substitution measures the degree in which you can alter the mix of goods. Consider a couple extreme cases:

Perfect substitutes can always be can always be traded off in a constant ratio

Perfect compliments have no substitutability and must me used in fixed ratios

Elasticity is 0Elasticity is Infinite

Consumers solve a constrained maximization – maximize utility subject to an income constraint.

),(max,

I ypx ptosubject

yxU

yx

yx

As before, set up the lagrangian…

)(),(),( ypxpIyxUyx yx

)(),(),( ypxpIyxUyx yx

First Order Necessary Conditions

0),( xx pyxU

ypxpI yx

0),( yy pyxU

y

x

y

x

P

P

yxU

yxU

),(

),(

x

x

y

y

p

yxU

p

yxU ),(),(

y

x

),(max0,0

I ypx ptosubject

yxU

yx

yx

xp

I

yp

I

*y

*x

y

x

y

x

P

P

yxU

yxU

),(

),(

ypxpI yx

y

x

Individual demand Curves represent a summary of the individual consumer choice problem

x

xp

*x

xp

*x

ypID ,

yp

I

xp

I

y

x x

y

x x

small is MRS

large is MRS

xp

xp

Willingness to pay is low

Willingness to pay is high

The marginal rate of substitution controls the height of the demand curve

ypID ,

ypID ,

y

x*x

The elasticity of substitution will effect the price elasticity of the demand curve

x

xp

'x *x

xp

xp'

D

MRSxy

%

%

xx p

x

%

%

yp

I

xp

I

xp

I

''x

y

x x

xp

small is small is x

y

x x

xp

large is large is x

Elasticity of Substitution vs. Price Elasticity

y

x x

xp

0 0x

y

x x

xp

x

Perfect Complements vs. Perfect Substitutes

An Example: Cobb-Douglas Utility

yxyxU ),(

yxyxU x1),(

1),( yxyxU y

x

y

yx

yx

yxU

yxU

y

x

1

1

),(

),(

),(

),(**

**

yxU

yxUMRS

y

x Recall, Marginal Rate of Substitution measures the relative value of x. This will determine the intercept of the demand curve

1

x

yMRS

xy

MRS

MRSxy

MRSxy

*%

% 1

xy

xy

xy

MRS

MRSxy

%

%

Recall, elasticity of substitution is measuring the degree of flexibility in the consumption of X and Y and will determine the slope of the demand curve

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

)(),( 5.5. ypxpIyxyx yx

y

x

y

x

p

p

yx

yx

yxU

yxU

5.5.

5.5.

5.

5.

),(

),(x

p

py

y

x


Marginal Rate of Substitution

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

Ixp

ppxp

y

xyx

Iypxp yx

y

xp

Ix

2

yp

Iy

2


x

xp

*x

xp

x

p

p

x

p

x x

xxx

%

%

xp

Ix

2

, 5.5. yxyxU

22 xx p

I

p

x

1

2

2 2

x

x

xx

pI

p

p

I

Constant price elasticity

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xp

Ix

2 0

%

%

x

p

p

x

p

x y

yyy

Zero cross price elasticity

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xp

Ix

2

x

I

I

x

I

xI

%

%

1

2

2

1

x

x

pI

I

p

Constant income elasticity

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xp

Ix

2

yp

Iy

2

Ipx x lnln2

1lnln

Ipy y lnln2

1lnln

Log-linear demand curves…

Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers


, 95.05. yAyAU

, 98.02. yAyAU

Airline Travel Everything Else

95.

05.

y

AMRS

1

98.

02.

y

AMRS

1

Different intercepts but the same slope…



, 95.05. yAyAU

, 98.02. yAyAU

Airline Travel Everything Else

IpA lnln05.lnln

p

IA

05.

IpA lnln02.lnln

p

IA

02.

LN(Quantity)

LN(Price)

IpAB lnln05.lnln

IpAR lnln02.lnln +

IpA ln2ln202.ln05.lnln

We end up with something like this…

BARA A

Business

Recreation

Aggregate

We could estimate a log linear demand curve with a dummy variable to check this

dDIcpbaA lnlnln

Suppose that we wanted different slopes…

1,1

yxyxU

The Cobb-Douglas utility function is a special case of the CES (Constant elasticity of substitution) utility functions…

1

1

x

yMRS

1

1

The parameter alpha will govern the marginal rate of substitution which influences the intercept of the demand curve

The parameter rho will govern the elasticity of substitution which influences the slope of the demand curve

Suppose that we wanted different slopes…Suppose that we wanted different slopes…

The Cobb-Douglas utility function is a special case of the CES (Constant elasticity of substitution) utility functions…

The resulting demand curve looks like this…

PIpx x lnlnlnlnln

A price index comprised of both good’s prices

1,1

yxyxU



95.

05.8.

y

AMRS

25.1

98.

02.2.

y

AMRS

5

95.05., 2.

12.2. yxyxU

98.02., 8.

18.8. yxyxU

Different intercepts and different slopes…


PIpA lnlnln2.05.ln2.ln


95.05., 2.

12.2. yxyxU

98.02., 8.

18.8. yxyxU

PIpA lnlnln8.02.ln8.ln

Quantity

Price

Now, we have something like this…

PIpAB lnlnln2.05.ln2.ln

PIpAR lnlnln8.02.ln8.ln +

PIpA ln2ln2ln02.ln8.05.ln2.ln

Business

Recreation

Aggregate

We could estimate a log linear demand curve to check this

PdIcpbaA lnlnlnln

RABA

A

finance 30210: managerial economics consumer demand analysis

Documents

price slide

elasticity quantity

logs quantity price

answer quantity price

like quantity price

thousands quantity price

lnquantity price

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