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8/3/2019 Micro Consumption
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8/3/2019 Micro Consumption
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Contentsofthelecture
neoclassicalmicroeconomictheoryofconsump2on
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Neoclassicaleconomy
Founda2ons:perfectra;onality:peoplemaketheirchoicessoastomaximizetheiru2lityunderabudgetconstraint
perfectinforma;on:alltheinforma2ontheyneedinordertotaketheirdecisionisavailablewithoutcosts
u2lityismeasurablethankstoascoringsystemormonetaryunits-problem:theu2lityofoneunitofmoneyisnotabsolutebutdependsontheconsumerswealth
independenceofUifromqj:U=U1(q1)+U(q)++Un(qn)marginalu2lityposi2veanddecreasing:
)q(UU iii =
0q
U
,0q
U2i
i
2
i
i
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Cardinalu;lity
iq
iU
0
iq
i
i
q
U
00
iq 'qi0
iq 'qi
tg
tg
0q
U,0
q
U2
i
i
2
i
i
eachaddi2onalunitconsumedincreasesthe
levelofsa2sfac2on,butatdecreasingratesasthesatura2onlevelisreached
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Axioma;ctheoryofpreferences
2q
1q
0
1q
2q
two-goodsbasket:q=(q 1>0,q>0)
subsistencebasket
collec2onXofall
availablebaskets
undertheperfectra2onalityand
perfectinforma2onassump2ons,a
ra2onalsubjectisabletoexpresshis
preferences.Forexample,hecouldsay:
aq
bq
cq
ab qq cb qq
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Axioma;ctheoryofpreferences
Axiomsfororderingpreferences:
1.
reflexivity:. completeness:foreachcoupleofbaskets,weeitherhave
3. transi2vity:
4. nonsatura2on:themoreofagood,thebeRer:
5. strictconvexity
:Xq,q,q,q cba
caqq
~
ba qq
~
cb qq
~
ab qq ~
ba qq ~
or
if and then
( ) ( ) b2a2b1a1b2b1ba2a1a qqorqqifq,qqq,qq >>==
babacba qq,10forqqqq)1(q +
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Axioma;ctheoryofpreferences
ThesetXcollec2ngallbaskets(eachbasketbeingassociatedtoapreferencelevel)is:
1. closed:itincludesitsboundary. convex:
3. strictlyconvex10withXb)1(a +
10withIntXb)1(a +
A B
NONCONVEX
A
B
CONVEX
A
B
STICTLYCONVEX
Thereexixtsatleastonepairof
thesetwhoselinearcombina2onmayfalloutsidetheset
Anylinearcombina2onofanytwo
elementsofthesetfallseitherinsideorontheboundaryoftheset
Anylinearcombina2onofany
twoelementsofthesetfallsstrictlyinsidetheset
C
D
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Indifferencecurve
Indifferencecurve:curveconnec2ngallindifferentbaskets
Features:
1. decreasing(fromnonsatura2onaxiom). atdecreasingrates(fromstrictconvexityaxiom)3. notintersec2ngcartesianaxis(qi>0)4. nonintersec2ngotherindifferencecurves(transi2vityaxiom):ifthey
intersected,theintersec2onbasketwouldbeassociatedtotwodifferentlevelsofu2lity
2q
1q
0
1q
2q
2q
1q
0
1qq
'q
cq
aq
bq
qcannotbeonthesameindifferencecurveofqa,qborqc,becauseqa,qb
andqchavemoreofbothgoods,soforthenonsatura2onaxiomtheymust
bepreferred.Forabundletobeonthesameindifferencecurveofq,itmusthavelessofonegoodbutmoreoftheother,soastocompensate;qcould
meetthisre uirement.Thus indifferencecurvearedecreasin .
1. .
q
'q
cq
aq
bq
2q
theincrementalquan2tyneededtotradeanabundant
goodhastobehigherthantheincrementalquan2ty
neededtotradeascarcegood:togiveuponeunitofascarcegoodIwantalargequan2tyoftheothergood.
This ivestheindifferencecurveastrictl convexsha e
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Mapofindifferencecurves
2q0
2q
1q
1q
q
aq
cq
bq
bc qq ac
qq ab qq
qqa
qqqq acb
thehigheristheindifference
curve,thehigheristheu2lityprovided:theu2lityprovidedby
basketqb(orequivalentlybyqc)is
higherthantheu2lityprovidedby
qa,whichinturnispreferredtoq
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Marginalrateofsubs;tu;on
MRS:themarginalrateofsubs2tu2onisthefirstderiva2veoftheindifferencecurve.
Characteris2csofMS:
1. nega2ve(goodsaresubs2tutes:thedisposalofoneunitofonegoodrequirestoaddupsomeunitsoftheothergoodsinordertoleavethelevelof
u2lityunchanged). increasing(ordecreasinginabsolutevalue):goodsareimperfect
subs2tutes
2q
1q
0
.aq
bq
2q
1q
0
1.aq
bq
0tgqU
qU
dq
dqMRS
1
2
2
1
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Ordinalu;lity
thecardinaltheoryhasoneimportantflaw:itassumesitispossibletoquan2fytheu2lityassociatedtoonebasketinabsoluteterms,butthisisnotrealis2candmoreovertheu2lityof
oneunitofmoneyisnotabsolutebutdependsontheconsumerswealth.So,wehavetogive
uptheideaofassocia2ngeachbasketacardinalamountofu2lity,andresignourselvestojust
putallthebasketsinaordinalranking.Thatstheideaoftheordinalu2litytheory.
u2lityisnotmeasurableinabsolutetermsdependenceofUfromqj:U=U(q1,q,qn):u2litybecomesaconven2onallabelbuiltinsuchawaythatitreflectstheorderedpreferencesonthesetXofallbaskets.
)q,,q,q(UU n21 =
)q(U)q(Uqqif baba >
)q(U)q(Uqqif baba =
)q(U)q(Uqqif baba
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Isou;litycurves
2q0
2q
1q
1q
q
aq
bq
0dqq
Udq
q
UdU 2
2
1
1
=
+
=
thehigheristheindifference
curve,thehigheristheu2lityprovided:theu2lityprovidedby
basketqb(orequivalentlybyqa)is
higherthantheu2lityprovidedby
q
0U
0U'U >
Isou;litycurve:curveconnec2ngallthosebasketsprovidingthesamelevelofu2lity.
0qU
qU
dq
dqMRS
1
2
2
1firstordercondi2ons
ofthemaximiza2onproblemarenotverified
(theslopeofthebudgetlinewillneverbe
equaltotheslopeofisou2litycurves):notangentinternalsolu2on.
onecornersolu;on
2M21 pCq,0q ==
solu2on:
=>= tgpptgdqdqif 1221
ofcoursesolu2oncouldalsostandontheothercornerif
isou2litycurveswereflaRerthanthebudgetline
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Specialcases
3.NONCONVEXITYINONEINTERVAL
betweenqaandqbthereisadiscon2nuity;
firstordercondi2onsofthemaximiza2on
problemcannotbewriRen(theisou2lity
curvesarenotdifferen2able)
mul;plesolu;ons2q0
1q
1
M
p
C
2
M
p
C
aq
bq
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individualdemand
Eachconsumerhandlesthemaximiza2onproblemandgetsasolu2onwhichcons2tutesherindividualdemand.
Thesolu2onofthei thconsumercancons2tute:
ademandfunc;on(whengoodsareimperfectsubs2tutes,thatistheisou2litycurveisstrictlyconvex):givenCM,p1,p,oneandonlyonebundlecons2tutesthe
op2malsolu2on:
ademandcorrespondence(whengoodsareperfectsubs2tutes,thatistheisou2litycurveisconvexbutnotstrictly):giventheternCM,p1,pmorethanone
bundlecons2tutestheop2malsolu2on
adiscon;nuousdemand(whenisou2litycurveisnonconvex):inthiscaseanequilibriummayNOTexist!
rememberweareworkingwithinthe
neoclassicalseng,whichmeansthat
consumerisra2onalandperfectlyinformed
( ) 2,1iforp,p,Cqq 21Mid
i ==
Thedemandcurveishomogeneousofzerodegreeinpricesand
income:ifallpricesandtheincomechangeinthesamepropor2on,qddoesnotchange:absenceofmonetaryillusion
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Marshalliandemand
Inordertoretrievethemarshalliandemandweneedtoworkundertheassump2onofcardinalu2lity
( ) ( )2211M21 qpqpCq,qUZ +=
0pqUqZ 111 ==
0pqUqZ 222 ==
0qpqpCZ 2211M ==
firstorder
condi2on
M
M
2
2
1
1
C
U
UZ
CZ
p
qU
p
qU
=
==
=
( )iiii
i qppq
U1p =
=
lagrangian
0q
Ufor0
dq
dp2
i
2
i
i
ifCM
increases,thebudgetline
shisup.
Ifthegoodisnormal,increasing
theincomecausesanincreaseinits
consump2on.
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Incomeeffect
2q0
1q
2
M
p
C
'U
'q
U
q
2
M
p
'C
1M p'C
1M pC
INFERIORGOODS
0C
q
M
d
1subs2tu2oneffect
. thedecreaseinphasincreasedtheresourcesoftheconsumer,soshecanbuymoreofboth
goods->incomeeffect
U
Then,onepricechangesfromp top.Assumeforexample:
22 p'p
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Price-consump;oncurves
Thankstoesandec,wecanclassifygoodsin:
2q0
1q
inferior
good
normal
good
Giffen
good
0e,0e Cs