Finance 30210: Managerial Economics

Finance 30210: Managerial Economics

Consumer Demand Analysis

Suppose that you observed the following consumer behavior

P(Bananas) = $4/lb.P(Apples) = $2/Lb.

Q(Bananas) = 10lbsQ(Apples) = 20lbs



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!





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?




Cost = $90


Cost = $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”




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

)()()()( BUCUAUCU

There is a definite ranking of all choices

25),( yxU

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


x

y

*y

*x

20),( yxU

Suppose you are given a little extra of good X. How much Y is needed to return to the original indifference curve?

1x

?yMarginal Utility of Y

Marginal Utility of X

),(),(

**

**

yxUyxUMRS

xy

y

x

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more 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'

xy

xy

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

20),( yxU

40),( yxU30),( yxU

xy

We often assume that the marginal rate of substitution is dependant only on the ratio of X and Y – i.e. preferences are homogeneous

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

PP

yxUyxU

),(),(

x

x

y

y

pyxU

pyxU ),(),(

y

x

),(max0,0

I ypx ptosubject

yxU

yx

yx

xpI

ypI

*y

*x

y

x

y

x

PP

yxUyxU

),(),(

ypxpI yx

y

x

Demand Curves present the same information in a different format – therefore, all the properties of preferences are present in the demand curve

x

xp

*x

xp

*x

ypID ,

Demand relationships are based off of the theory of consumer choice. We can characterize the average consumer by their utility function.

HLU ,

“Utility” is a function of lemonade and hot dogs

Consumers make choices on what to buy that satisfy two criteria:

L

L

H

H

PMU

PMU

ILPHP LH

Their decision on what to buy generates maximum utility

Their decision on what to buy generates is affordable

IPPDQ LHH ,,These decisions can be represented by a demand curve

Example: Suppose that you have $10 to spend. Hot Dogs cost $4 apiece and glasses of lemonade cost $2 apiece.

# Hot Dogs MU (Hot Dogs)

# Lemonade MU (Lemonade)

1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

24

48

ILPHP LH

210,2,4 DQH

L

L

H

H

PMU

PMU

101224

This point satisfies both conditions and, hence, is one point of the demand curve

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

$10

$2

ypID ,

ypID ,

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend)



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

24

68

ILPHP LH L

L

H

H

PMU

PMU

101226

Your decision at the margin has been affected. You need to buy less hot dogs and more lemonade (Substitution effect)

You can’t afford what you used to be able to afford – you need to buy less of something! (Income effect)

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend)



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

23

69

ILPHP LH L

L

H

H

PMU

PMU

102216

110,2,6 DQH


Demand curves slope downwards – this reflects the negative relationship between price and quantity. Elasticity of Demand measures this effect quantitatively

Quantity

Price

$4.00

2

10$ID

$6.00

1

%50100*2

21

%50100*4

46

15050

%%

PQ

D

y

x*x

The elasticity of substitution will control the slope of the demand curve

x

xp

'x *x

xp

xp'

D

MRSxy

%

%

xx p

x

%%

y

x x

xpsmall is small is x

y

x x

xp

large is large is x

Elasticity of Substitution vs. Price Elasticity

y

x x

xp0 0x

y

x x

xp

x

Perfect Complements vs. Perfect Substitutes

(Almost)

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend.



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

24

48

ILPHP LH L

L

H

H

PMU

PMU

201224

Your decision at the margin is unaffected, but you have some income left over (this is a pure income effect)

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend.



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

23

46

ILPHP LH L

L

H

H

PMU

PMU

202244

420,2,4 DQH


For any fixed price, demand (typically) responds positively to increases in income. Income Elasticity measures this effect quantitatively

Quantity

Price

$4.00

2

10$ID

4

%100100*2

24%

Q

%100100*10

1020%

I

1100100

%%

IQ

D

20$ID

y

x x

xp

*x

xp x%

Ix

I

%%

Income elasticity measures the response of consumers to changes in income holding prices constant – the homogeneity of preferences will effect this

*x

Cross price elasticity refers to the impact on demand of another price changing

Quantity

Price

$4.00

2

2$LPD

6

%200100*2

26%

Q

%100100*2

24%

LP

2100200

%%

L

HL P

Q

4$LPD

Note: These numbers aren’t coming from the previous example!!

A positive cross price elasticity refers to a substitute while a negative cross price elasticity refers to a compliment

y

x*x

Cross price elasticity measures consumer response to changes in other prices – this is influenced by both homogeneity and elasticity of substitution

x*x

xp

xpy

y px

%%

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

)(),( 5.5. ypxpIyxyx yx

y

x

y

x

pp

yxyx

yxUyxU

5.5.

5.5.

5.5.

),(),(

xppy

y

x

An Example: Cobb-Douglas Utility

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

Ixpppxp

y

xyx

Iypxp yx

y

xpIx

2

ypIy

2



yxyxU ),(

yxyxU x1),(

1),( yxyxU y

xy

yxyx

yxUyxU

y

x

1

1

**

**

),(),(

With Cobb-Douglas Utility functions, your MRS is directly proportional to your relative consumption of the two goods.

xyMRS

MRSdxyd 1

xy

xy

Cobb-Douglas Utility functions have constant elasticity of substitution

MRSxy

An Example: Cobb-Douglas Utility yxyxU ),(

x

xp

*x

xp

xp

dpdx

px x

xxx

%%

xpIx

2

, 5.5. yxyxU

22 xx pI

dpdx

1

22 2

x

x

xx

pIp

pI

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xpIx

2

ypIy

2

0%%

xp

dpdx

px y

yyy

Cobb-Douglas demands are independent of other prices!

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xpIx

2

ypIy

2

xI

dIdx

Ix

I

%%

1

22

1

x

x

pII

p

Finance 30210: Managerial Economics

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