Diewert Essays In Index Number Theory Problems

On By In 1

178EssaysinIndexNumberTheory7.TheEconomicTheory179

2.PriceIndexesandtheKon¨usCostofLivingIndex

WeassumethataconsumerismaximizingautilityfunctionF(x)subjecttothe

expenditureconstraintp

T

x≡

P

N

i=1

p

i

x

i

≤ywherex≡(x

1

,...,x

N

)

T

≥0

N

isanonnegativevectorofcommodityrentals,p≡(p

1

,...,p

N

)

T

0

N

isa

positivevectorofcommodityprices

1

andy>0isexpenditureontheNcom-

modities.Wecouldalsoassumethataproducerismaximizingaproduction

functionF(x)subjecttotheexpenditureconstraintp

T

x≤ywherex≥0

N

is

nowaninputvector,p0

N

isaninputpricevectorandy>0isexpendi-

tureontheinputs.Inordertocoverboththeconsumerandproducertheory

applications,weshallcalltheutilityorproductionfunctionFanaggregator

functioninwhatfollows.

Theconsumer’s(orproducer’s)aggregatormaximizationproblemcanbe

decomposedintotwostages:inthefirststage,theconsumer(orproducer)

attemptstominimizethecostofachievingagivenutility(oroutput)level,

and,inthesecondstage,hechoosesthemaximalutility(oroutput)levelthat

isjustconsistentwithhisbudgetconstraint.

Thesolutiontothefirststageproblemdefinestheconsumer’s(orpro-

ducer’s)costfunctionC:

(1)C(u,p)≡min

x

{p

T

x:F(x)≥u,x≥0

N

}

ThecostfunctionCturnsouttoplayapivotalroleintheeconomic

approachtoindexnumbertheory.

Throughoutmuchofthischapter,weshallassumethattheaggregator

functionFsatisfiesthefollowingconditionsI:FisarealvaluedfunctionofN

variablesdefinedoverthenonnegativeorthantΩ≡{x:x≥0

N

}whichhasthe

threepropertiesof(i)continuity,(ii)increasingness

2

and(iii)quasiconcavity.

3

LetUbetherangeofF.FromI(i)and(ii),itcanbeseenthatU≡{u:

u≤u≤ou}whereu≡F(0

N

)<ou.Notethattheleastupperboundou

couldbeafinitenumberor+∞.Inthecontextofproductiontheory,typically

u=0andou=+∞,but,forconsumertheoryapplications,thereisnoreason

torestricttherangeoftheutilityfunctionFinthismanner.

1

Notation:x≥0

N

meanseachcomponentofthecolumnvectorxisnonneg-

ative,x0

N

meanseachcomponentispositive,x>0

N

meansx≥0

N

but

x6=0

N

where0

N

isanNdimensionalvectorofzeros,andx

T

denotesthe

transp oseofx.

2

Ifx

00

x

0

≥0

N

,thenF(x

00

)>F(x

0

).

3

Foreveryu∈rangeF,theupperlevelsetL(u)≡{x:F(x)≥u}isaconvex

set.AsetSisconvexiffx

0

∈S,x

00

∈S,0≤λ≤1impliesλx

0

+(1−λ)x

00

∈S:

i.e.thelinesegmentjoininganytwopointsbelongingtoSalsobelongstoS.

Definethesetofpositivepricesp≡{p:p0

N

}.Itcanbeshown

that(seeDiewert[1978c])ifFsatisfiesconditionsI,thenthecostfunctionC

definedby(1)satisfiesthefollowingconditionsII:

(i)C( u,p)isarealvaluedfunctionofN+1variablesdefinedoverU×P

andisjointlycontinuousin(u,p)overthisdomain.

(ii)C(

u,p)=0foreveryp∈P.

(iii)C( u,p)isincreasinginuforeveryp∈P;i.e.,ifp∈P,u

0

,u

00

∈U,

withu

0

<u

00

,thenC(u

0

,p)<C(u

00

,p).

(iv)C(

ou,p)=+∞foreveryp∈P;i.e.,ifp∈P,u

n

∈U,lim

n

u

n

=u,

thenlim

n

C(u

n

,p)=+∞.

(v)C( u,p)is(positively)linearlyhomogenousinpforeveryu∈U;i.e.,

u∈U,λ>0,p∈PimpliesC(u,λp)=λC(u,p).

(vi)C( u,p)isconcaveinpforeveryu∈U;i.e.,ifp

0

0

N

,p

00

0

N

,0≤

λ≤1,u∈U,thenC(u,λp

0

+(1−λ)p

00

)≥λC(u,p

0

)+(1− λ) C(u,p

00

).

(vii)C( u,p)isincreasinginpforu>

uandu∈U.

(viii)CissuchthatthefunctionF

(x)≡max

u

{u:p

T

x≥C(u,p)for

everyp∈P,u∈U}iscontinuousforx≥0

N

.

Forsomeofthetheoremstobepresentedinthischapter,wecanweaken

theregularityconditionsontheaggregatorfunctionFtojustcontinuityfrom

above.

4

UnderthisweakenedhypothesisonF,thecostfunctionCdefinedby

(1)willstillsatisfymanyofthepropertiesinconditionsIIabove.

5

Finally,someofthetheoremsbelowmakeuseofthefollowing(stronger)

regularityconditionsontheaggregatorfunction:wesaythatFisaneoclassical

aggregatorfunctionifitisdefinedoverthepositiveorthant{x:x0

N

}and

is(i)positive,i.e.F(x)>0forx0

N

,(ii)(positively)linearlyhomogeneous,

and(iii)concaveover{x:x0

N

}.Undertheseconditions(letuscall

themconditionsIII)FcanbeextendedtothenonnegativeorthantΩ,andthe

extendedFwillbenonnegative,linearlyhomogeneous,concave,increasingand

continuousoverΩ(seeDiewert[1978c]).Moreover,ifFisneoclassical,then

F’scostfunctionCfactorsinto

(2)C(u,p)≡uC(1,p)≡uc(p)

4

Fiscontinuousfromaboveoverx≥0

N

iffforeveryu∈rangeF,L(u)≡

{x:F(x)≥u}isaclosedset.

5

Specifically,Diewert[1978c]showsthatCwillsatisfythefollowingcondi-

tionsII

00

:(i)C( u,p)isarealvaluedfunctionofN+1variablesdefinedover

U×Pandiscontinuousinpforfixeduandcontinuousfromb elowinufor

fixedp(thesetUisnowtheconvexhulloftherangeofF),(ii)C(u,p)≥0

foreveryu∈Uandp∈P,(iii)C(u,p)isnondecreasinginuforfixedp,(iv)

C(u,p)isnondecreasinginpforfixedu,andproperties(v)and(vi)arethe

sameas(v)and(vi)ofconditionsI I.

Вцепившись в левую створку, он тянул ее на себя, Сьюзан толкала правую створку в противоположном направлении.

Через некоторое время им с огромным трудом удалось расширить щель до одного фута. - Не отпускай, - сказал Стратмор, стараясь изо всех сил.

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