the choice of consumption and leisure, labor supply
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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
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