垄断竞争DS模型的详细推导(2).pdf

垄断竞争DS模型的详细推导(2).pdf

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IncreasingReturnstoScale（IRS）andMonopolisticCompetition（MC）:

Dixit-StiglitzModel1-PreferencesU=Y=?

Zn0y?

1?

idi?

1?

1

（1）Notice:

equation

（1）impliesloveforvariety.Toseethis,guessconstantpriceandoutput（wewillverifytheseguesseslateron）:

?

yi=y=Enppi=pwhereEdenotesexpenditure.Then:

Y=?

y?

1?

Zn0di?

1=?

ny?

1?

1=?

Enp?

n?

1=?

Ep?

n1?

1

（2）Clearly,equation

（2）impliesthatthehigherthenumberofvarietiesn,thehigherthelevelofutilityU（hence,loveforvariety）.?

UtilityMaximizationProblem（UMP）:

MaxYyi?

0s:

t:

Rn0piyi?

ETheLagrangianforthisproblemtakestheform:

L=?

Zn0y?

1?

idi?

1?

Zn0piyi?

E?

ThenecessaryandsucientFOCsforthisproblemare:

forvarietyi:

?

1?

Zn0y?

1?

idi?

1?

1?

1?

y?

1?

1i?

=?

piforvarietyj:

?

1?

Zn0y?

1?

idi?

1?

1?

1?

y?

1?

1j?

=?

pj1Thishandout:

CosimoBeverelli,HEI,Geneva.E-mail:

beverel1hei.unige.ch1TakingtheratiooftheFOCsweget:

?

yiyj?

1?

=pipj,yiyj=?

pipj?

（3）or,usingthelawsoflogarithms,ln?

yiyj?

=?

ln?

pipj?

（4）,ln（yi）?

ln（yj）=?

ln（pi）?

ln（pj）（5）Equation（3）representstherelativedemandforanytwovarietiesasafunc-tionofrelativepricesand?

.Wecannowbemoreexplicitontheparameter?

:

（i）j?

jisthe（constant）elasticityofsubstitutionbetweenvarieties:

seeequation（4）.（ii）j?

jisalsothe（constant）priceelasticityofdemand:

seeequation（5）.Nowwecanmanipulateequation（3）inordertogetanexpressionforyj=yj（pj;E;P）:

Multiplybothsidesbypi,toget:

piyiyj=p1?

ip?

jIntegratebetween0andn:

Rn0piyidiyj=Rn0p1?

idip?

jUsingthebudgetconstraint,wecanconvenientlyrewritethisexpressionasfollows:

Eyj=Rn0p1?

idip?

j,yj=Ep?

jRn0p1?

idi（6）Now,denethepriceindexasaCESaggregateofprices:

P?

Zn0p1?

idi?

11?

（7）Equation（6）thenbecomes:

yj=yj（pj;E;P）=Ep?

jP1?

（8）（demandforvarietyj）2?

IndirectUtilityStraightforwardmanipulationofequation（6）showsthatindirectutilitytakesthefollowingform:

V=EPLetsrewrite（6）as:

y?

1?

j=E?

1?

p?

（?

1）j?

Rn0p1?

idi?

1?

Integratebetween0andn:

Zn0y?

1?

jdi=E?

1?

Rn0p?

（?

1）jdi?

Rn0p1?

idi?

1?

?

Zn0y?

1?

jdi?

1=E?

Zn0p1?

jdi?

（1?

1?

）（?

1）,Y=E?

Zn0p1?

jdi?

11?

Y=V=EP（9）-ProductionsideTherearenproducersofvarieties.Forreasonsthatwillbeclearinaminute,eachofthemisamonopolistintheproductionofasinglevariety.Firmjhasthefollowingcostfunction:

TCj=（cyj+F）w（10）Thiscostfunctionhasthefollowingcharacteristics:

-laboristheonlyfactorofproduction.Therewardtolaborisequaltothewageratew-totalcostscomprisevariablecosts（functionofthequantityproduced）andxedcosts-thecostfunctionissubadditive.RecallfromyourMicroeconomicsIclassthat-inthecaseofsingleoutputtechnology-thecostfunctionissubadditiveiaveragecosts（AC）declineatalllevelsofoutput.Thisiseasilychecked:

ACj?

TCjyj=cw+Fyj3ACjyj=?

Fwy2j0.Inpercapitaterms,YL=?

L1?

1?

（17）As（17）shows,anincreaseinLiswelfare-increasing（denitionofscaleeect）7