EndogenousIncome

Theconsumption-leisuremodel

Modifying consumer’s problem

• Forthemoment,assumethereisnoadditionalexogenousincome

• Consumer’sincomeisthemarketvalueofherinitialendowment,

• Givenmarketpricespx andpy,theconsumer’sbudgetconstraintis

)y, x(

yxyx pypxypxp +£+

Budget constraint

x

y

x

y

y

x

ppxy +

x

y

ppy

x +

Changes in pricesAnincreaseinpx

y

x x

y

Changes in prices

Anincreaseinpy

y

x

y

x

Changes in prices

• Theimpactofpricechangesintheconsumer’sbudgetsetisnowmoresubtle:theincreaseofapricemaymaketheconsumer“relativelyricher”ifsheis“relativelyrich”inthatgood(i.e.,ifherendowmentofthisgoodislarge)

• Noticethat,independentlyofmarketprices,theendowmentisalwaysafeasiblebundle(consumeralwayscanavoidtradeandconsumeherownendowment)

Consumer Demand

• Assume derivableandthesystem

yieldsaninteriorsolutiontotheconsumer’sproblem,

• Thatis,

wherethefunctionsontheRHSaretheordinarydemands.

),(~ yx ppx ),(~ yx ppy

y

x

yxyx

ppyxMRS

pypxypxp

=

+=+

),(

),( yxu

),,(*),(~),,(*),(~

yxyxyx

yxyxyx

pypxppyppypypxppxppx

+=

+=

Consumer Demand: example

;

Calculateordinarydemands:

Hence,

yxyxu =),( )1,2(),( =yx

yyx

xyx

pIIppy

pIIppx

3),,(*

32),,(*

=

=

y

x

y

yxyx

x

y

x

yxyx

pp

ppp

ppy

pp

ppp

ppx

32

31

32

),(~

32

34

3)2(2

),(~

+=+

=

+=+

=

Income and substitution effects• Studytheeffectofanincreaseinpx forutilityfunction

andendowment),( yxu

),( yx

x

y

x

y

u1

u2A

B

C

Income and substitution effects

• Substitutioneffectisnegative

• Incomeeffectispositve

• Inthiscase,incomeeffectislargerthansubstitutioneffect,sothetotaleffectispositive:anincreaseinleadstheconsumertoincreaseitsconsumptionofbothgoods

xp

0<-= AB xxSE

0>-= BC xxIE

0>-=+= AC xxIESETE

Income and substitution effects

Supposeanincreaseinpx.

Thesubstitutioneffectisthesameasinthecasewithexogenousincome.

Theincomeeffectofapriceincreasenowinvolvestwocomponents:(1)ItmakestheconsumerpoorerasItbecomesmoreexpensive

buyingtheoptimalamountofthegood(ordinaryincomeeffect),and(2)Itmodifiesthevalueoftheconsumer’sendowment(endowment

effect).

Thesignofthe(total)incomeeffectdependsonwhethertheconsumerisanetseller(negative)oranetbuyer(positive)ofthegoodwhosepricehasincreased.

Income and substitution effects

Totaleffect=substitutioneffect+ordinaryincomeeffect+endowmenteffect

=substitutioneffect+(total)incomeeffect

Formally,

)(*|*

**|*~

xxIx

px

Ixx

Ixx

px

px

cteux

cteuxx

-¶¶

-¶¶

=

¶¶

+¶¶

-¶¶

=¶¶

=

=

Income and substitution effects

• Whenisthetotalincomeeffectpositive(theconsumerisricher)?Ifthegoodisnormalandtheconsumerisanetsellerofthegood

• Inanycase,noticethatpositiveincomeeffect(“richerconsumer”)doesnotmeanmoreconsumption:wehavetotakeintoaccountthesubstitutioneffect

0 and 0* 0)(*<->

¶¶

>-¶¶

- xxIxifxx

Ix

Income and substitution effects

Exogenousincome Endogenousincome

u2u1

x

y

B

A

C

u1

u0

y

x

A

B

C

The consumption-leisure model. Labor supply

• Twogoods:leisure(x-axis)andconsumption(y-axis)- Leisure,denotedbyhandmeasuredinhours.Thewageperhour(orpriceofleisure)isdenotedbyw.- Consumption,denotedbycandmeasuredineuros.Theprice

ofcisthereforepc =1).

• Initialendowmentis(M,H),where:-M:initialexogenouswealth(ornon-laborincome).- H:numberofhoursavailableforleisureandwork.

The consumption-leisure model. Labor supply

• Budgetset(recall)

:expenditureonleisure:monetaryvalueofinitialendowment

1=cp

MwHhwc +£+

h

c

H

M

MwH +

wslope -=hwMwH +

The consumption-leisure model. Labor supply

• Solvetheproblemasusual,butwatchoutforadditionalconstraints

hcMax , ),( hcu

st MwHhwc +£+

00³

££c

Hh

The consumption-leisure model. Labor supply. Example

Interior solution requires

And

hcMax , hc ln2+

st 416 +=+ wwhc

0160

³££

ch

wwhw

hwchMRS 2)(2),( =Þ=Û=

81162)(

002)(

³Û£=

>"Û³=

ww

wh

ww

wh

The consumption-leisure model. Labor supply. Example

Therefore,

=)(wh8/1 16 <wif

8/1 2³wif

w

=)(wc8/1 4 <wif

The consumption-leisure model. Labor supply. Example

And labor supply

=-= )()( whHwl

8/1 0 <wif

8/1 216 ³- wifw

1/8

The consumption-leisure model. Labor supply. Effect of changes in wages

• Assume

• Forinteriorsolutions,theconsumerisanetsupplierofleisure

• Totalincomeeffect:if leisureisanormalgood,,TIEispositive,leadingtheconsumertodemandlessleisure(orsupplymorelabor)

• Substitutioneffect:isalwaysnon-positive.Sinceleisureischeaper,thiseffectleadstheconsumertodemandmoreleisure(orsupplylesslabor)

• Totaleffectisambiguous(itdependsontheshapeoftheutilityfunction)

ww <'

0/ >¶¶ Ih

0)( >-¶¶

- HlIh

>0 <0

The consumption-leisure model. Labor supply. Effect of changes in wages

c

h

M

H

MwH +

MHw +' AB

C

The consumption-leisure model. Labor supply. Effect of changes in wages

c

MwH +

MHw +'

M

H h

B

C

A

The consumption-leisure model. Labor supply. Effect of changes in wages

For,SEdominates(leisureismoreexpensiveandconsumeroffersmorelabor)

For,TIEdominates(consumerisricheranddoesnotneedtoworkasmuchasbefore)

)10,0(Îw

)20,10(Îw

Application: a tax on labor income

• Impose• Thenewbudgetconstraintis

• Thetaxisequivalenttoareductionofwage:itsimpactonleisureconsumption(orlaborsupply)isambiguous

• Itsimpactonwelfareisunambiguous.Ofcourse,taxpolicieshaveotherobjectiveswearenotconsideringhere.

]1,0[Ît

MwHtwhtc +-£-+ )1()1(

Application: a tax on labor income

c

h

M

H

MwH +

MwHt +- )1(

*h

*c

Application: a tax on labor income

• Alternative:anon-laborincome tax T.• The newbudget constraint is

• If both goods arenormal,then the introduction ofT reducestheir demands (increases laborsupply,inparticular)

)( TMwHwhc -+£+

Application: a tax on labor income

MwH +

)( TMwH -+

M

TM - h

c

H

Application: a tax on labor income

• Exercise:what if T=tw(H-h*)?

• Hint:is (c*,h*)optimal for T?

h*h

*c

cMwH +

MwHt +- )1(

)( TMwH -+

MTM -

A

B