范课件by ccerch6需求比较静态分析yandx2p1 p2an

Chapter

SixDemand需求Properties

of

Demand

FunctionsComparative

statics

analysis

（比较静态分析）of

ordinary

demandfunctions

--

the

study

ofhowordinary

demands

x1*(p1,p2,y)

andx2*(p1,p2,y)

change

as

prices

p1,

p2and e

y

change.StructureOwn-price

changesPrice

offer

curve

（价格提供曲线）Ordinary

demand

curveInverse

demand

curve

（反需求函数）e

changese

offer

curve

（收入提供曲线）Engel

curve

（恩格尔曲线）Cross-price

effectsOwn-Price

ChangesHow

does

x1*(p1,p2,y)

change

as

p1changes,

holding

p2

and

y

constant?Suppose

only

p1

increases,

fromp1’to

p1’’

and

then

top1’’’.x1x2p1x1

+

p2x2

=

yp1

=

p1’Own-Price

ChangesFixed

p2

and

y.x2p1=

p1’’x1Own-Price

ChangesFixed

p2

and

y.p1x1

+

p2x2

=

yp1

=

p1’x1x2p1=

p1’’p1=p1’’’Own-Price

ChangesFixed

p2

and

y.p1x1

+

p2x2

=

yp1

=

p1’x1x1*(p1’)Own-Price

ChangesFixed

p2

and

y.x2p1

=

p1’x1x1*(p1’)p1x1*(p1’)p1’x1*Own-PriceChangesFixed

p2

and

y.x2p1

=

p1’x1x1*(p1’)x1*(p1’’)p1x1*(p1’)p1’x1*Own-PriceChangesFixed

p2

and

y.x2p1

=

p1’’x1x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’)x1*Own-PriceChangesFixed

p2

and

y.x2p1’’p1’x1x1*(p1’’’)

x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’)x1*Own-PriceChangesFixed

p2

and

y.x2p1

=

p1’’’p1’’p1’x2x1x1*(p1’’’)

x1*(p1’)x1*(p1’’)p1x1*(p1’’’)

x1*(p1’)x1*(p1’’)p1’’p1’p1’’’x1*Own-PriceChangesOrdinarydemand

curvefor

commodity

1Fixed

p2

and

y.x2x1x1*(p1’’’)

x1*(p1’)x1*(p1’’)p1x1*(p1’’’)

x1*(p1’)x1*(p1’’)p1’p1’’p1’’’x1*Own-PriceChangesOrdinarydemand

curvefor

commodity

1p1

priceoffercurveFixed

p2

and

y.Own-Price

ChangesThe

curve

containing

all

the

utility-maximizing

bundles

traced

out

as

p1changes,

with

p2

and

y

constant,isthe

p1-

price

offer

curve.The

plot

of

the

x1-coordinate

of

thep1-

price

offer

curve

against

p1

is

theordinary

demand

curve

forcommodity

1.Own-Price

ChangesWhat

does

a

p1

price-offer

curvelooklike

for

Cobb-Douglas

preferences?U(x1,

x2

)

=

xaxb.1

2Then

the

ordinary

demand

functionsfor

commodities

1

and

2

areTakeOwn-Price

Changesx*(p

,p

,

y)

=1

1

2a

·

yba

+

b

p1yb

p2*x2(p1,p2,

y)

=

a

+·

.andNotice

that

x2*

does

not

vary

with

p1

so

thep1

price

offer

curveis

flat

and

the

ordinarydemand

curve

for

commodity

1

is

arectangular

hyperbola.x1*(p1’’’)

x1*(p1’)x1*(p1’’)x1Own-PriceChangesFixed

p2

and

y.x22x*

=by(a

+

b)p2ay1*x1

=

(a

+

b)px1*(p1’’’)

x1*(p1’)x1*(p1’’)x2x1p1x1*Own-PriceChangesOrdinarydemand

curvefor

commodity

1isFixed

p2

and

y.by2x*

=(a

+

b)p2ay1*x1

=

(a

+

b)pay1x*

=(a

+

b)p1Own-Price

ChangesWhat

does

a

p1

price-offer

curvelooklike

for

a plements

utility

function?U(x1,

x2

)

=

min{x1,

x2}.Then

the

ordinary

demand

functionsfor

commodities

1

and

2

areOwn-Price

Changesy.x*(p

,p

,

y)

=

x*

(p

,p

,

y)

=1

1

2

2

1

2p1+

p2With

p2

and

y

fixed,

higher

p1

causessmaller

x1*

and

x2*.y

.p1

fi

0,

x*

=

x*

fi1

2Asp2As

p1

fi

¥

,

x*

=

x*

fi

0.1

2Own-PriceChangesFixed

p2

and

y.x1x2p1x1*Fixed

p2

and

y.y2x*

=y1x*

=Own-PriceChangesx1p1’y1x*

=p1’+

p2p1

=

p1’p1’+

p2p1’+

p2x2y/p2p1x1*Fixed

p2

and

y.*x2

=yy1x*

=Own-PriceChangesx1p1’p1’’p1

=

p1’’y1x*

=p1’+’p2p1’+’p2p1’+’p2x2y/p2p1x1*Fixed

p2

and

y.y2x*

=y1x*

=Own-PriceChangesx1p1’p1’’p1’’’y1x*

=p1

=

p1’’’p1’+’p’2p1’+’p’2p1’+’p’2x2y/p2p1Ordinarydemand

curvefor

commodity

1isy2x*

=p1

+

p2y1x*

=p1

+

p2y1

2*.x1

=

p+

pOwn-PriceChangesFixed

p2

and

y.x1x2p1’p1’’p1’’’y

x1*p2y/p2Own-Price

ChangesWhat

does

a

p1

price-offer

curvelooklike

for

a

perfect-substitutes

utilityfunction?U(x1,

x2

)

=

x1

+

x2.Then

the

ordinary

demand

functionsfor

commodities

1

and

2

areOwn-Price

Changes1

1

21,

if

p1

>

p2x*(p

,p

,

y)

=

0

y

/

p

2

1

22,

if

p1

<

p2,

if

p1

<

p2,

if

p1

>

p2.x*

(p

,p

,

y)

=

0

y

/

p

andOwn-PriceChangesx2x1Fixed

p2

and

y.2x*

=

0xyp11*=p1

=

p1’

<

p2’Own-PriceChangesx2x1p11x

*Fixed

p2

and

y.2x*

=

0xyp11*=p1’p1

=

p1’

<

p2’xy1*=p1’Own-PriceChangesx2x1p1x1*Fixed

p2

and

y.p1’p1

=

p1’’

=

p2Own-PriceChangesx2x1p1x1*Fixed

p2

and

y.p1’p1

=

p1’’

=

p2Own-PriceChangesx2x1p1x1*Fixed

p2

and

y.2y1x*

=p1’p1

=

p1’’

=

p2p1’’1x*

=

0

x*

=

0

2p2

x*

=

y

Own-PriceChangesx2x1p1x1*Fixed

p2

and

y.2y1p2x*

=p1’p =

p

’’

=

p1

1

21x*

=

0

x*

=

0

2p2

x*

=

y

1p20£

x*

£

y

p2

=

p1’’Own-PriceChangesx2x1p1x1*Fixed

p2

and

y.2p2x*

=

y1x*

=

0p1’p1’’’1x*

=

0p1

=

p1”’

>

p2p2

=

p1’’x2p1x1*Own-PriceChangesFixed

p2

and

y.1p

’p2

=

p1’’p1’’’1p1x*

=

y

2x1y0

£

x1

£

p*yp2p1

priceoffercurveOrdinarydemand

curvefor

commodity

1Own-Price

ChangesUsually

we

ask

“Given

the

price

forcommodity

1

what

is

the

quantitydemanded

of

commodity

1?”But

we

could

also

ask

the

inversequestion

“At

what

price

forcommodity

1

would

a

given

quantityof

commodity

1

be

demanded?”Own-Price

Changesp1x1*p1’Given

p1’,

what

quantity

isdemanded

of

commodity

1?Own-Price

Changesp1x1*p1’Given

p1’,

what

quantity

isdemanded

of

commodity

1?Answer: x1’

units.x1’Own-Price

Changesp1x1*x1’Given

p1’,

what

quantity

isdemanded

of

commodity

1?Answer: x1’

units.The

inverse

question

is:Given

x1’

units

aredemanded,

what

is

theprice

ofcommodity

1?Own-Price

Changesp11x

*p1’x1’Given

p1’,

what

quantity

isdemanded

of

commodity

1?Answer: x1’

units.The

inverse

question

is:Given

x1’

units

aredemanded,

what

is

theprice

ofcommodity

1?1Answer: p

’Own-Price

ChangesTaking

quantity

demanded

as

givenand

then

asking

what

must

be

pricedescribes

the

inverse

demandfunction

of

a

commodity.Own-Price

ChangesA

Cobb-Douglas

example:ay1x*

=(a

+

b)p1is

the

ordinary

demand

function

anday1p1

=(a

+

b)x*is

the

inverse

demand

function.Own-Price

ChangesA plements

example:y1x*

=p1

+

p2is

the

ordinary

demand

function

and1p1

=

y

-

p2x*is

the

inverse

demand

function.Meaning

of

the

Inverse

DemandFunctionAt

optimal

choiceOrIf

taking

good

2

as

money

on

other

goods,then

p2=1

and

p1=MRS.This

is

the

marginal

willingness

to

pay.MRS

=

P1P2P1

=

P2

MRSe

ChangesHow

does

the

value

of

x1*(p1,p2,y)change

as

y

changes,

holding

bothp1

and

p2

constant?x1e

ChangesFixed

p1

and

p2.x2y’

<

y’’

<

y’’’x1e

ChangesFixed

p1

and

p2.x2y’

<

y’’

<

y’’’x1e

ChangesFixed

p1

and

p2.x2y’

<

y’’

<

y’’’x1’

x1’’’x1’’x2’’’x2’’x2’x1e

Changesx1’

x1’’’x1’’Fixed

p1

and

p2.x2y’

<

y’’

<

y’’’eoffer

curvex2’’’x2’’x2’e

ChangesA

plot

of

quantity

demanded

againste

is

called

an

Engel

curve.x1e

Changesx1’

x1’’’x1’’Fixed

p1

and

p2.x2y’

<

y’’

<

y’’’eoffer

curvex2’’’x2’’x2’e

ChangesFixed

p1

and

p2.x2x1’

x1’’’x1’’x2’’’x2’’x2’y’

<

y’’

<

y’’’eoffer

curve1x

*x1’’x1’

x1’’’yy’’’y’’y’x1Engelcurve;good

1x2x1e

ChangesFixed

p1

and

p2.x1’

x1’’’x1’’x

’’’2x2’’x2’y’

<

y’’

<

y’’’eoffer

curvex2*yx2’

x2’’’x2’’y’’’y’’y’Engelcurve;good

2x2e

ChangesFixed

p1

and

p2.x1’

x1’’’x1’’x

’’’2x2’’x2’y’

<

y’’

<

y’’’eoffer

curve1x

*x2*yx1’’x1’

x1’’’x2’

x2’’’x2’’y’’’y’’y’yy’’’y’’y’x1Engelcurve;good

2Engelcurve;good

1e

Changes

and

Cobb-DouglasPreferencesAn

example

of

computing

theequations

of

Engel

curves;

the

Cobb-Douglas

case.U(x1,

x2

)

=

xaxb.1

2The

ordinary

demand

equations

areay

by12x*

=;

x*

=

.(a

+

b)p1

(a

+

b)p2e

Changes

and

Cobb-DouglasPreferencesay

by12.x*

=;

x*

=(a

+

b)p1

(a

+

b)p2Rearranged

to

isolate

y,

these

are:aby

=

(a

+

b)p1

x*1y

=

(a

+

b)p2

x*2Engel

curve

for

good

1Engel

curve

for

good

2e

Changes

and

Cobb-DouglasPreferencesyyx1*x2*ay

=

(a

+

b)p1

x*1Engel

curvefor

good

1by

=

(a

+

b)p2

x*2Engel

curvefor

good

2e

Changes

and

Perfectly-Complementary

Preferencesy.x*

=

x*

=1

2p1

+

p2Another

example

of

computing

theequations

of

Engel

curves;

theplementary

case.U(x1,

x2

)

=

min{x1,

x2}.The

ordinary

demand

equations

aree

Changes

and

Perfectly-Complementary

PreferencesRearranged

to

isolate

y,

these

are:y

=

(p1

+

p2

)x*1y

=

(p1

+

p2

)x*2Engel

curve

for

good

1y.x*

=

x*

=1

2p1

+

p2Engel

curve

for

good

2e

Changesx1Fixed

p1

and

p2.x2e

Changesx1Fixed

p1

and

p2.x2y’

<

y’’

<

y’’’e

Changesx1Fixed

p1

and

p2.x2y’

<

y’’

<

y’’’e

Changesx1x1’’x2’’’x2’’x2’x1’

x1’’’Fixed

p1

and

p2.x2y’

<

y’’

<

y’’’e

Changesx1x1’’x2’’’x2’’x2’x1’

x1’’’x1*y’’’y’’y’Engelcurve;good

1x1’

x1’’’x1’’Fixed

p1

and

p2.x2y’

<

y’’

<

y’’’yx1x1’’x

’’’2x2’’x2’x1’

x1’’’x2*x2’

x2’’’x2’’yy’’’y’’y’Engelcurve;good

2e

ChangesFixed

p1

and

p2.x2y’

<

y’’

<

y’’’x1x1’’x

’’’2x2’’x2’x1’

x1’’’x1*x2*x2’

x2’’’x2’’y’’y’yy’’’yy’’’y’’y’Engelcurve;good

2Engelcurve;good

1x1’

x1’’’x1’’e

ChangesFixed

p1

and

p2.x2y’

<

y’’

<

y’’’x1*x2*x2’

x2’’’x2’’yy’’’y’’y’yy’’’y’’y’x1’

x1’’’x1’’y

=

(p1

+

p2

)x*2*y

=

(p1

+

p2

)x1Engelcurve;good

2Engelcurve;good

1e

ChangesFixed

p1

and

p2.e

Changes

and

Perfectly-Substitutable

PreferencesAnother

example

of

computing

theequations

of

Engel

curves;

theperfectly-substitution

case.U(x1,

x2

)

=

x1

+

x2.The

ordinary

demand

equations

aree

Changes

and

Perfectly-Substitutable

Preferences1

1

21,

if

p1

>

p2x*(p

,p

,

y)

=

0

y

/

p

2

1

22,

if

p1

<

p2,

if

p1

<

p2,

if

p1

>

p2.x*

(p

,p

,

y)

=

0

y

/

p

1

21p12Suppose

p <

p

.

Then

x*

=

y

and

x*

=

021

1y

=

p

x*

and

x*

=

0.e

Changes

and

Perfectly-Substitutable

Preferences2x*

=

0.y

=

p1x1*yyx2*0x1*Engel

curvefor

good

1Engel

curvefor

good

2e

ChangesIn

every

example

so

far

the

Engelcurves

have

all

been

straight

lines?Q:

Is

this

true

in

general?A:

No. Engel

curves

are

straightlines

if

the

consumer’s

preferencesare

homothetic.Homotheticity

（位似偏好）A

consumer’s

preferences

arehomothetic

if

and

onlyifThat

is,

the

consumer’s

MRS

isthesame

anywhere

on

a

straight

linedrawn

from

the

origin.(ky1,ky2)(x1,x2)

(y1,y2)

(kx1,kx2)

for

every

k

>

0.e

Effects

--

A

NonhomotheticExampleQuasilinear

preferences

are

nothomothetic.1

21

2U

(x

,

x

)

=

v(x

)

+

x

.For

example,U(x1,

x2

)

=

x1

+

x2

.Optimal

interior

consumption:12.ppv

'(x1*)

=Quasi-linear

Indifference

Curvesx2x1Each

curve

is

a

vertically

shiftedcopy

of

the

others.Each

curve

intersectsboth

axes.e

Changes;

Quasilinear

Utilityx2x1x1~e

Changes;

Quasilinear

Utilityx2x11x~y1x~Engelcurveforgood

1x1*e

Changes;

Quasilinear

Utilityx2x1x1~yEngelcurveforgood

2x2*e

Changes;

Quasilinear

Utilityx2x11x~yy1x~Engelcurveforgood

2x2*Engelcurveforgood

1x1*e

EffectsA

good

for

which

quantity

demandedrises

with e

is

called

normal

（正常品）.Therefore

a

normal

good’s

Engelcurve

is

positively

sloped.e

EffectsA

good

for

which

quantity

demandedfalls

as e

increases

is

called

einferior

（劣质品）.Therefore

an e

inferior

good’s

Engelcurve

is

negatively

sloped.x2e

Changes;

Goods1

&

2

Normalx1’

x1’’’x1’’x

’’’2x2’’x2’eoffer

curve1x

*x2*yx1’’x1’

x1’’’x2’

x2’’’x2’’y’’’y’’y’yy’’’y’’y’x1Engelcurve;good

2Engelcurve;good

1e

Changes;

Good

2

Is

Normal,Good

1

es e

Inferiorx1x2eoffer

curvee

Changes;

Good

2

Is

Normal,Good

1

es e

Inferiorx2x1x1*yEngel

curvefor

good

1e

Changes;

Good

2

Is

Normal,Good

1

es e

Inferiorx2x1x1*yyEngel

curvefor

good

2x2*Engel

curvefor

good

1Ordinary

Goods

（一般商品）A

good

is

called

ordinary

if

thequantity

demanded

of

it

alwaysincreases

as

its

own

price

decreases.Ordinary

Goo