economics 102 lecture 6 demand rev

7/28/2019 Economics 102 Lecture 6 Demand Rev

1/54

8/3/200

Lecture 6: Theory of the Consumer:

Demand

Changes in income

Income changes and offer curves and Engel

curves

Changes in prices

Price offer curve and demand curves

Substitutes and complements

Inverse demand function


2/54

8/3/200

Consumers demand functions give the

optimal amounts of each of the goods as

a function of the prices and income

faced by the consumer.

),,(

),,(

2122

2111

yppxx

yppxx

Change the consumers income, holding prices as

fixed

Increase in income shifts the budget line outward

in a parallel fashion

Effects on quantity demanded:

the quantity demanded increases as income increases

the quantity demanded decreases as income increases


3/54

8/3/200

Income offer curve- set of bundles of goods that are

demanded at different levels of income.

Derived by connecting together the bundles that are

obtained as the budget line is shifted outwards.

Income offer curve is also known as the income

expansion path

If both goods are normal goods, then the income

expansion path will have a positive slope.

Engel curve graph of one of goods as a

function of income, with all prices being held

constant.

Derived by holding prices of good 1 and 2 fixed,

and then changing income, much like thederivation of the income expansion path.

We then plot the quantity demanded of one good

as a function of income


4/54

8/3/200

x2

x1

Income ChangesFixed p

1

and p2

.

y < y < y

x2

x1

Income ChangesFixed p1 and p2.

y < y < y

x1x1

x1

x2x2x2


5/54

8/3/200

x2

x1

Income ChangesFixed p

1

and p2

.

y < y < y

x1x1

x1

x2x2x

2

Income

offer curve

x2

x1

Income ChangesFixed p1 and p2.

y < y < y

x1x1

x1

x2x2x2

Income

offer curve

x1*

x2*

y

y

x1x1

x1

x2x2

x2

yy

y

yyy

Engel

curve;

good 2

Engel

curve;

good 1


6/54

8/3/200

An example of computing the equations of

Engel curves; the Cobb-Douglas case.

The ordinary demand equations are

U x x x xa b( , ) .1 2 1 2

x aya b p

x bya b p

11

22

* *

( );

( ).

xay

a b px

by

a b p1

12

2

* *

( );

( ).

Rearranged to isolate y, these are:

ya b p

a x

ya b p

bx

( )

( )

*

*

1 1

22

Engel curve for good 1



7/54

8/3/200

y

yx1*

x2*

y a b pa

x( ) *1 1Engel curve

for good 1

ya b p

b

x( ) *2

2

Engel curvefor good 2

Perfect complements

income offer curve- straight line through thequantities demanded since the consumer will alwaysconsume the same amount of each good no matterwhat.

The Engel curve is therefore a straight line with slope:

This is just derived from the demand for good 1:

21 pp

)/( 211 ppyx


8/54

8/3/200

Rearranged to isolate y, these are:

y p p x

y p p x

( )

( )

*

*

1 2 1

1 2 2


x x yp p

1 21 2

* * .


Income Changes

x1

x2y < y < y

x1x1

x2x2

x2

x1 x1*

x2*

y

y x2x2

x2

yy

y

yyy

Engel

curve;

good 2

Engel

curve;

good 1

x1x1

x1

Fixed p1 and p2.


9/54

8/3/200

Income Changes

x1*

x2*

y

y x2x2

x2

yyy

y

y

y

x1x1

x1

*

221 )( xppy

Engel

curve;

good 2

Engel

curve;

good 1

Fixed p1

and p2

.

*

121 )( xppy

Perfect substitutes

If the consumer is specializing in

the consumption of good 1. If his incomeincreases, so will his consumption of good 1.

Thus the income offer is the horizontal axis.

The Engel curve will be a straight line with a

slope of p1.

21 pp


10/54

8/3/200

1

The perfect-substitution case.

The ordinary demand equations are

U x x x x( , ) .1 2 1 2

x p p yif p p

y p if p p1 1 2

1 2

1 1 2

0*( , , )

,

/ ,

x p p yif p p

y p if p p2 1 2

1 2

2 1 2

0*( , , )

,

/ , .

Suppose p1 < p2. Then

xy

p1

1

* x2

0* and

x2 0*

.y p x1 1*

and


11/54

8/3/200

x2 0*

.y p x1 1

*

y y

x1* x2*0Engel curve

for good 1

Engel curve

for good 2

Homothetic preferences result in straight lineEngel curves or where the demand for the good goesup by the same proportion as income.

Homothetic preferences consumers preferencesdepend on the ratio of good 1 to good 2.

Given:

Then for any positive value of t:

),(),(s.t.),,(),,( 21212121 yyxxyyxx

),(),( 2121 tytytxtx


12/54

8/3/200

1

Homothetic preferences

Three examples are perfect complements, perfectsubstitutes and Cobb-Douglas

If the consumer has homothetic preferences, thenthe income offer curves are all straight linesthrough the origin.

If preferences are homothetic, scaling income upor down by twill scale the quantity demanded bythe same amount.

The consumers MRS is the same anywhere on astraight line drawn from the origin.

Engel curves do not have to be straight lines. In

general, when income goes up, the demand for a

good goes up by a greater proportion than

income.

Luxury good when demand for the good

increases by a greater proportion than theincrease in income

Necessary good when the demand for that

good increases by a lesser proportion than the

increase in income.


13/54

8/3/200

1

Quasilinear preferences Example of non-homothetic preferences

The utility function for these preferences take the form:

If an indifference curve is tangent to the budget line at

then another indifference curve must also be tangent at

for any constant k.

21)( xxvu

),( *2*

1 xx

),( *2*1 kxx

Quasilinear preferences Example of non-homotheticpreferences

Increasing income doesnt change the demand forgood 1 at all, and all the extra income goes entirely tothe consumption of good 2.

With quasilinear preferences, there is a zero incomeeffect for good 1.

Engel curve for good 1 is a vertical line, as youchange income, the demand for good 1 remainsconstant.

This kind of income offer curve is relevant for someitems that do not form a large amount of theconsumers budget.


14/54

8/3/200

1

Quasilinear preferences are not homothetic.

For example,

U x x f x x( , ) ( ) .1 2 1 2

U x x x x( , ) .1 2 1 2

x2

x1

Each curve is a vertically shifted

copy of the others.

Each curve intersects

both axes.


15/54

8/3/200

1

Income Changes; Quasilinear

Utilityx2

x1x1~

x1*

x2*y

y

x1~

Engel

curvefor

good 2

Engel

curve

for

good 1

Normal good- a good is a normal good when

the demand for the good increases as income

increases, and decreases when income

decreases.

the quantity demanded always changes the

same way as income changes:

A normal goods Engel curve is positively

sloping

0/1

mx


16/54

8/3/200

1

Inferior good a good whose quantity

demanded decreases as income

increases.

examples are low quality goods

normality or inferiority of a good depends on

the income range that we are considering

An income inferior goods Engel curve is

negatively sloped.

x2

x1

Income Changes; Goods1 & 2 Normal

x1x1

x1

x2x2x2

Income

offer curve

x1*

x2*

y

y

x1x1

x1

x2x2

x2

yy

y

yyy

Engel

curve;

good 2

Engel

curve;

good 1


17/54

8/3/200

1

Income Changes; Good 2 Is Normal,

Good 1 Becomes Income Inferior

x2

x1

Income Changes; Good 2 Is Normal,Good 1 Becomes Income Inferior

x2

x1


18/54

8/3/200

1



x2

x1


x2

x1


19/54

8/3/200

1



x2

x1


x2

x1

Income

offer curve


20/54

8/3/200

2



x2

x1 x1*

yEngel curve

for good 1


x2

x1 x1*

x2*

y

y

Engel curve

for good 2

Engel curvefor good 1


21/54

8/3/200

2

Decrease the price of good 1 holding the price

of good 2 and income constant.

Depict changes via price offer curve and

ordinary demand curve

Price offer curve depicts the optimal choicesas the price of good 1 changes.

Demand curve shows for each level of p1 theoptimal level of consumption of good 1. The

demand curve is a plot of the demand function:

holding p2 and m constant.

),,( 2111 yppxx


22/54

8/3/200

2

Two alternative effects:

the consumption of good 1 increases

the consumption of good 1 decreases

Ordinary good when the quantity demandedof the good increases as the price of the good isdecreased

Giffen good a good such that when its pricedeclines, the quantity demanded of the gooddeclines as well.

the reduction in the price of good 1 has freed upsome extra money that can be spent on otherthings, so much so that consumer decides toconsume more of the other good and reduce yourconsumption of this good

The price change is to some extent like an incomechange. Even though money income remains thesame, the change in the price of one good changespurchasing power and thus changes demand.


23/54

8/3/200

2

Fixed p2

and y.

x1

x2

Fixed p2 and y.

x1

x2p1 price

offer

curve


24/54

8/3/200

2

Fixed p2

and y.

x1

x2p1 price

offer

curve

x1*

Downward-sloping

demand curve

Good 1 is

ordinary

p1

Fixed p2 and y.

x1

x2


25/54

8/3/200

2

Fixed p2

and y.

x1

x2 p1 price offercurve

Fixed p2 and y.

x1

x2 p1 price offercurve

x1*

Demand curve has

a positively

sloped part

Good 1 is

Giffen

p1


26/54

8/3/200

2

Ordinarily, when the price of a good decreases,the demand for the good will increase and viceversa. Thus the demand curve has a negativeslope since price and consumption move inopposite directions.

Therefore:

01

1

p

x

x1

x2

p1 = p1

Fixed p2 and y.

p1x1 + p2x2 = y

Own-Price Changes


27/54

8/3/200

2

Own-Price Changes

x1

x2

p1= p

1

p1 = p1

Fixed p2

and y.

p1x1 + p2x2 = y

Own-Price Changes

x1

x2

p1= p1p1=

p1

Fixed p2 and y.

p1 = p1

p1x1 + p2x2 = y


28/54

8/3/200

2

x2

x1

p1 = p1

Own-Price ChangesFixed p

2

and y.

x2

x1x1*(p1)

Own-Price Changes

p1 = p1

Fixed p2 and y.


29/54

8/3/200

2

x2

x1x1*(p1)

p1

x1*(p1)

p1

x1*


2

and y.

p1 = p1

x2

x1x1*(p1)

p1

x1*(p1)

p1

p1 = p1

x1*

Own-Price ChangesFixed p2 and y.


30/54

8/3/200

3

x2

x1x1*(p1)

x1*(p1)

p1

x1*(p1)

p1

p1 = p1

x1*


2

and y.

x2

x1x1*(p1)

x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

x1*



31/54

8/3/200

3

x2

x1x1*(p1)

x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

p1 = p1

x1*


2

and y.

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

p1 = p1

x1*



32/54

8/3/200

3

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*


2

and y.

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

Own-Price Changes Ordinarydemand curve

for commodity 1Fixed p2 and y.


33/54

8/3/200

3

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*


for commodity 1Fixed p

2

and y.

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*


for commodity 1

p1 price

offer

curve

Fixed p2 and y.


34/54

8/3/200

3

For Cobb-Douglas preferences

Take

Then the ordinary demand functions for

commodities 1 and 2 are

U x x x xa b( , ) .1 2 1 2

x p p ya

a b

y

p1 1 2

1

*( , , )

x p p yb

a b

y

p2 1 2

2

*( , , ) .

and


35/54

8/3/200

3

x p p y aa b

yp

1 1 21

*( , , )

x p p yb

a b

y

p2 1 2

2

*( , , ) .

and

Notice that x2* does not vary with p1 so the p1price offer curve is flat and the ordinary demand

curve for commodity 1 is a rectangular hyperbola.

x1*(p1) x1*(p1)

x1*(p1)

x2

x1

p1

x1*


for commodity 1

is

Fixed p2 and y.

x

bya b p

2

2

*

( )

xay

a b p1

1

*

( )

xay

a b p1

1

*

( )


36/54

8/3/200

3

For a perfect-complements utility function

U x x x x( , ) min , .1 2 1 2Then the ordinary demand functions

for commodities 1 and 2 are

x p p y x p p yy

p p1 1 2 2 1 2

1 2

* *( , , ) ( , , ) .

With p2 and y fixed, higher p1causes smaller x1* and x2*.

p x x yp

1 1 22

0 , .* *Asp x x1 1 2 0 , .* *As


37/54

8/3/200

3

Fixed p2

and y.

x1

x2

p1

x1*

Fixed p2 and y.

x

yp p

2

1 2

*

xy

p p1

1 2

* x1

x2

p1

xy

p p1

1 2

*

p1 = p1

y/p2


38/54

8/3/200

3

p1

x1*

Fixed p2

and y.

x

y

p p

2

1 2

*

xy

p p1

1 2

* x1

x2

p1

p1

p1 = p1

x

y

p p1

1 2

*

y/p2

p1

x1*

Fixed p2 and y.

x

y

p p

2

1 2

*

xy

p p1

1 2

* x1

x2

p1

p1

p1

xy

p p1

1 2

*

p1 = p1

y/p2


39/54

8/3/200

3

p1

x1*

Ordinary

demand curve

for commodity 1is

Fixed p2

and y.

x

y

p p

2

1 2

*

xy

p p1

1 2

*

xy

p p1

1 2

*.

x1

x2

p1

p1

p1

y

p2

y/p2

For a perfect-substitutes utility

function

U x x x x( , ) .1 2 1 2Then the ordinary demand functions

for commodities 1 and 2 are


40/54

8/3/200

4

x p p yif p p

y p if p p1 1 2

1 2

1 1 2

0*( , , )

,

/ ,

x p p yif p p

y p if p p2 1 2

1 2

2 1 2

0*( , , )

,

/ , .

and

Fixed p2 and y.

x2

x1

p1

x1*

Fixed p2 and y.

x2 0*

xy

p1

1

*

p1

p1 = p1 < p2

xy

p1

1

*


41/54

8/3/200

4

Fixed p2

and y.

x2

x1

p1

x1*

Fixed p2

and y.

x2 0*

xy

p1

2

*

p1

p1 = p1 = p2

x1 0*

xy

p2

2

*

0 1

2

x yp

*

p2 = p1

Fixed p2 and y.

x2

x1

p1

x1*

Fixed p2 and y.

xy

p

2

2

*

x1 0*

p1

p1

x1 0*

p2 = p1


42/54

8/3/200

4

Fixed p2

and y.

x2

x1

p1

x1*

Fixed p2

and y.

p1

p2 = p1

p1

xy

p1

1

*

0 1

2

x yp

*

y

p2

p1 price

offer

curve

Ordinary

demand curve

for commodity 1

Discrete good - Suppose that good 1 is a

discrete good.

At some high price, consumption will bezero

At some low price, consumption will beone unit

At some price r1, the consumer will beindifferent between consuming good 1or not consuming it.

r1 is called the reservation price.


43/54

8/3/200

4

Demand behavior can be described by asequence of reservation prices at whichthe consumer is just willing to purchaseanother unit of the good.

at a price r1, he is just willing topurchase one unit,

at a price r2, he is just willing topurchase another unit.


44/54

8/3/200

4

Reservation prices can be described in

terms of the original utility function.

r1 satisfies the condition:

r2 satisfies the condition:

),1(),( 1rmumou

)2,2(),1( 22 rmurmu

For quasilinear utilities - the formulas

describing the reservation prices are simpler:

)1(

,rforsolving

,)1()0(:asequationfirstthewritecanwe,0)0(and

2)(),(

1

1

1

121

vr

rmvmmv

v

xxvxxu


45/54

8/3/200

4

Similarly, we can write the second equation as:

Proceeding in this manner, the reservation price for

the third unit is just:

)1()2(

:grearranginandtermscanceling

,2)2()1(

2

22

vvr

rmvrmv

)2()3(3 vvr

In each case, the reservation price measures theincrement in utility necessary to induce theconsumer to choose an additional unit of the good.

Assumption of convex preferences mean that thesequence of reservation prices must decrease:

Given any price p, we just find where it falls in thelist of reservation prices.

321 rrr


46/54

8/3/200

4

Suppose p is between r6 and r7.

r6 > p means that the consumer is willingto give up p pesos per unit bought to get 6units of good 1

p > r7 means that the consumer is not

willing to give up p pesos to get the 7thunit of good 1.

Substitutes If the demand for good 1 goesup when the price of good 2 goes up, then wesay that good 1 is a substitute for good 2:

Complements If the demand for good 1goes down as the price of good 2 increases,we say that good 1 is a complementto good2:

.0

2

1

p

x

.02

1

p

x


47/54

8/3/200

4

A perfect-complements example:

xy

p p1

1 2

*

x

p

y

p p

1

2 1 22

0*

. so

Therefore commodity 2 is a gross complement

for commodity 1.

p1

x1*

p1

p1

p1

y

p2

Increase the price of

good 2 from p2 to p2

and


48/54

8/3/200

4

p1

x1*

p1

p1

p1

y

p2

Increase the price of

good 2 from p2 to p2

and the demand curve

for good 1 shifts inwards

-- good 2 is a

complement for good 1.

A Cobb- Douglas example:

xby

a b p2

2

*

( )

so


49/54

8/3/200

4

A Cobb- Douglas example:

xby

a b p2

2

*

( )

x

p2

1

0*

.

so

Therefore commodity 1 is neither a grosscomplement nor a gross substitute for

commodity 2.

As long as we have a downward sloping

demand curve, it is meaningful to speak of the

inverse demand function.

Inverse demand function is the demandfunction viewing price as a function of

quantity.

Therefore, for each quantity, what would be the

level of prices such that the consumer would

choose that level of consumption.


50/54

8/3/200

5

p1

x1*

p1

Given p1, what quantity isdemanded of commodity 1?

p1

x1*

p1

Given p1, what quantity is

demanded of commodity 1?

Answer: x1 units.

x1


51/54

8/3/200

5

p1

x1*x1

Given p1, what quantity isdemanded of commodity 1?

Answer: x1 units.

The inverse question is:

Given x1 units are

demanded, what is the

price of

commodity 1?

p1

x1*

p1

x1

Given p1, what quantity is

demanded of commodity 1?

Answer: x1 units.

The inverse question is:

Given x1 units are

demanded, what is the

price ofcommodity 1?

Answer: p1


52/54

8/3/200

5

A Cobb-Douglas example:

xay

a b p1

1

*

( )

is the ordinary demand function and

pay

a b x

1

1

( ) *

is the inverse demand function.

A perfect-complements example:

xy

p p1

1 2

* is the ordinary demand function and

py

xp1

1

2*is the inverse demand function.


53/54

8/3/200

5

An economic interpretation

For well-behaved preferences, the optimal choice

must satisfy the condition that the absolute value of

the MRS equals the price ratio:

This says that at the optimal level of demand for

good 1, we must have:

At the optimal level of good 1, the price of good 1 is

proportional to the absolute value of the MRS

between good 1 and good 2.

2

1

p

pMRS

MRSpp 21

Suppose that p2 =1, the equation states that at theoptimal level of demand, the price of good 1measures how much the consumer is willing to giveup of good 2 in order to gain a little of good 1.

If we think of good 2 as the amount of money tospend on other goods, then we can think of the MRSas being how many pesos the individual would bewilling to give up in order to have more of good 1.


54/54

8/3/200

MRS is therefore the marginal willingness topay . the marginal willingness to pay is just the price of

the good.

At each quantity x1, the inverse demand functionmeasures how many pesos a consumer is willingto give up for a little more of good 1.

Marginal willingness to pay, i.e., in the senseof the marginal willingness to sacrifice good 2for good 1 is decreasing as we increase theconsumption of good 1.

economics 102 lecture 6 demand rev

Documents