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weshouldconsiderarealincomeas:
pi
l
>oi.i12.Whenthepricesofgoodsandthe
consumer’snominalincomechangeproportionally(theyincreasewhenλ>1ordecrease
whenλE(0,1))therealincomedoesnotchange.Thus,thevalueofthedemandfunctionfor
bothgoodsandthevalueoftheindirectutilityfunctioncannotchange.
(
|
|
k
0
,
X
X
p
I
x
2
(
2
0
N
|
|
)
,
0
)
x
(
|
|
k
X
X
u
p
I
1
(x
,
0
)
N
|
|
)
±
x
u
1
Figure1.1.Homogeneityofdegree0oftheconsumerdemandfunctionandofthe
indirectutilityfunction
Df.1.5Aderivativeoftheindirectutilityfunctionwithrespecttotheconsumer’sincomeis
calledamarginalutilityofincomeandgivenas:
(77)
IV(plip2il)
Il
il.m
∆lą0
V(plip2il+∆l)żV(plip2il)
∆l
,
whichdeterminesbyhowmuchtheutilityoftheoptimalconsumptionbundlewill
approximatelychange(generallyincrease)whentheconsumer’sincomeincreasesby
a(notional)unit.
Th.1.4IftheutilityfunctionuiR+
2ąRlisincreasing,twicedifferentiable,andstrictly
concave,thenVplip2iI>o:
1.theconsumerdemandfunctionfli.ntR+
3ąR+
2isdifferentiableinitsdomain,
2.theindirectutilityfunctionVi.ntR+
3ą.ntR+
lisdifferentiableinitsdomain
3.
IV(plip2il)
Il
±o
ż
δV(plip2il)
fli(plip2iI)i1
δV(plip2il)
δpi
i.i1i2ł
δl
(Itisso-calledRoy’sidentity,whichallowstodeterminetheconsumer’sdemand
functionforthei-thproductusinganalyticalformoftheindirectutilityfunction.)
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