Finance 510: Microeconomic Analysis Consumer Demand Analysis

Finance 510: Microeconomic Analysis

Consumer Demand Analysis

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!

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) = 15lbs

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 consumer goods. Its usually convenient to represent these preferences with a utility function

BAU :

A BU

Set of possible 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 only require a couple restrictions on Utility functions

For any two choices (X and Y), either U(X) > (Y), U(Y) > U(X), or U(X) = U(Y) (i.e. any two choices can be compared)

For choices X, Y, and Z, if U(X) > U(Y), and U(Y) > U(Z), then U(X) > U(Z) (i.e., the is a definitive ranking of choices)

However, we usually add a couple additional restrictions to insure “nice” results

If X > Y, then U(X) > U(Y) (More is always better)

If U(X) = U(Y) then any combination of X and Y is preferred to either X or Y (People prefer moderation to extremes)

Suppose we have the following utility function

),( yxUU

Imagine taking a “cross section” at some utility level.

U = 20

The “cross section” is called an indifference curve (various combinations of X and Y that provide the same level of utility)

20),( yxUx

y

A

B

C

20)()( BUAU

)()()()( BUCUAUCU

Any two choices can be compared

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

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

kyxU ),(

),(),( **** yxUyxxU ),(),( *** yxUyyxU

x

y

+ = 0

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


x

y

*y

*x

kyxU ),(

xx

yxUyxxU

),(),( ****

yy

yxUyyxU

),(),( ***

x

y

+ = 0

Now, let the change in X become arbitrarily small


x

y

*y

*x

kyxU ),(

0),(),( **** dyyxUdxyxU yx

Marginal Utility of YMarginal Utility of X

),(

),(**

**

yxU

yxU

dx

dyMRS

y

x


x

y

*y

*x

kyxU ),(

)','(),( ** yxMRSyxMRS

'y

'x

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

An Example

yxyxU ),(

yxyxU x1),(

1),( yxyxU y

x

y

yx

yx

yxU

yxU

y

x

1

1

**

**

),(

),(

The elasticity of substitution measures the curvature of the indifference curve

x

y'

x

y

x

y

MRSxy

%

%

MRSdxy

d

xy

MRS

An Example yxyxU ),(

yxyxU x1),(

1),( yxyxU y

x

y

yx

yx

yxU

yxU

y

x

1

1

**

**

),(

),(

MRSdxy

d 1

xy

xy

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

),(max0,0

I ypx ptosubject

yxU

yx

yx

As before, set up the lagrangian…

)(),(),,( ypxpIyxUyx yx

)(),(),,( ypxpIyxUyx yx

First Order Necessary Conditions

0),(),,( xxx pyxUyx

0),,( ypxpIyx yx

0),(),,( yy pyxUyyx y

x

y

x

P

P

yxU

yxU

),(

),(

Iypxp yx x

x

y

y

p

yxU

p

yxU ),(),(

y

x

),(max0,0

I ypx ptosubject

yxU

yx

yx

xp

I

yp

I

*y

*x

),,(* Ippxx yx),,(* Ippxy yx

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

Iypxp yx

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

IxP

PPxp

y

xyx

Iypxp yx

y

xp

Ix

2

yp

Iy

2

y

x

xp

I

yp

I

*y

*x

Suppose that we raise the price of X

Can we be sure that demand for x will fall?

y

x

xp

I

yp

I

*y

*x

Suppose that we raise the price of X, but at the same time, increase your income just enough so that your utility is unchanged

y

x

y

x

P

P

yxU

yxU

),(

),(Substitution effect

y

x

xp

I

yp

I

*y

*x

Now, take that extra income away…

Income effectIypxp yx

y

x*x

Demand Curves present the same information in a different format

x

xp

'x 'x *x

xp

xp'

D

y

x

Demand Curves present the same information in a different format

x

xp

*x

xp

*x

MRSxy

%

% x

x p

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

(Almost)

x

xp

*x

xp

x

p

dp

dx

p

x x

xxx

%

%

xp

Ix

2

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

22 xx p

I

dp

dx

1

2

2 2

x

x

xx

pI

p

p

I

y

x

xp

I

yp

I

*y

*x

Suppose that we raise the price of Y…

y

x

y

x

P

P

yxU

yxU

),(

),(Substitution effect (+)

Income effect (-) Iypxp yx

Net Effect = ????

x

xp

*x

xp

x

p

dp

dx

p

x y

yyy

%

%

Cross Price Elasticity

x%

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xp

Ix

2

yp

Iy

2

0%

%

x

p

dp

dx

p

x y

yyy

Income and Substitution effects cancel each other out!!

y

x

xp

I

yp

I

*y

*x

Suppose that we raise Income

y

x

y

x

P

P

yxU

yxU

),(

),(Substitution effect = 0

Income effect (-) Iypxp yx

'x

x

xp

*x

xp

x

I

dI

dx

I

xI

%

%

Income Elasticity

x%

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

xp

Ix

2

yp

Iy

2

x

I

dI

dx

I

xI

%

%

1

2

2

1

x

x

pI

I

p

Q

P

Willingness to pay

Suppose that we have the following demand curve

100

$50

D

$100

PQ 2200

A demand curve tells you the maximum a consumer was willing to pay for every quantity purchased.

For the 100th sale of this product, the maximum anyone was willing to pay was $50

Q

P

Willingness to pay

Suppose that we have the following demand curve

100

$50

D

$100

PQ 2200


$75

50

Q

P

Consumer Surplus

100

$50

D

$100

PQ 2200


$75

50

Consumer surplus measures the difference between willingness to pay and actual price paid

Whoever purchased the 50th unit of this product earned a consumer surplus of $25

Q

P

Consumer Surplus

100

$50

D

$100

PQ 2200

Consumer surplus measures the difference between willingness to pay and actual price paid

If we add up that surplus over all consumers, we get:

CS = (1/2)($100-$50)(100-0)=$2500$2500

$5000Total Willingness to Pay ($7500)

- Actual Amount Paid ($5000)

Consumer Surplus ($2500)

A useful tool…

In economics, we are often interested in elasticity as a measure of responsiveness (price, income, etc.)

xx p

x

%

%

xdx

dxx ln%

xx

xx pd

p

dpp ln%

)(ln

)(ln

xx pd

xd

Estimating demand curves

Given our model of demand as a function of income, and prices, we could specify a demand curve as follows:

yxd paIapaax 3210

x

pa

x

p

dp

dx

p

x xx

xxx 1%

%

yxd paIapaax 3210

x

pa

x

p

dp

dx

p

x xx

xxx 1%

%

x

xp

High Elasticity

Low Elasticity

Linear demand has a constant slope, but a changing elasticity!!


We could, instead, use a semi-log equation:

yxd paIapaax lnlnln 3210

x

a

xpd

dx

p

x

xxx

11

ln%

%


We could, instead, use a semi-log equation:

yxd paIapaax 3210ln

xxxx

x papdp

xd

p

x1

ln

%

%


The most common is a log-linear demand curve:

yxd paIapaax lnlnlnln 3210

1ln

ln

%

%a

pd

xd

p

x

xxx

Log linear demand curves are not straight lines, but have constant elasticities!

max 5.5.

0,0

I ypx ptosubject

yx

yx

yx

If we assumed that this was the maximization problem underlying a demand curve, what form would we use to estimate it?

yxd paIapaax lnlnlnln 3210

0 1 1: 3210 aaaH


x

xp

Suppose you observed the following data points. Could you estimate the demand curve?

D


x

xp

Market prices are the result of the interaction between demand and supply!!

dxd Iapaax 210

A bigger problem with estimating demand curves is the simultaneity problem.

D

S

xp

sd xx


x

xp

Case #1: Both supply and demand shifts!!

D’’

S’S

S’’

D’D

x

xp

D

S’S

S’’

Case #2: All the points are due to supply shifts

An example…

dxd Iapaax 210

sxs pbbx 10Supply

Demand

Equilibrium ds xx

Suppose you get a random shock to demand

The shock effects quantity demanded which (due to the equilibrium condition influences price!

Therefore, price and the error term are correlated! A big problem !!

sxdx pbbIapaa 10210

Suppose we solved for price and quantity by using the equilibrium condition

ds xx

11

11

11

22

1111

2

ab

abI

ab

abx

abI

ab

ap

sd

sdx

We could estimate the following equations

22

11

Ix

Ipx

11

222

11

21

ab

ab

ab

a

The original parameters are related as follows:

1

22

b

We can solve for the supply parameter, but not demand. Why?

dxd Iapaax 210

sxs pbbx 10

x

xpS

D

D

D

By including a demand shifter (Income), we are able to identify demand shifts and, hence, trace out the supply curve!!

Documents

Finance 510: Microeconomic Analysis Consumer Demand Analysis