微观经济学2010 micro完全替代互补satia

Consumer

TherearethreestepsinvolvedintheConsumerTodescribehowandwhypeoplepreferonegoodtoanotherBudget Givenpreferencesandlimitedes,whatamountandtypeofgoodswillbeWhatcombinationofgoodswillconsumersbuytoizetheirsatisfaction?

BudgetDescribebudgetDescribechangesinbudget ernmentprogramsandbudgetNon-linearbudgetcommodity1,x2unitsofcommodity2andsoonvector(x1,x2,…,xn).

PhysicalConsumptionset:X={(x1,…,xn)|x10,…,xn0AconsumptionsetisthecollectionofallphysicallypossibleconsumptionbundlestotheconsumerBudgetCommoditypricesarep1,p2,…,Q:Whenisabundle(x1,…,xn)p1x1+…+pnxnwheremistheconsumer’s

BudgetThebundlesthatareonlyjustaffordableformtheconsumer’sbudgetconstraint.Thisistheset{(x1,…,xn)|x10,…,xnp1x1+…+pnxnmBudgetSetandConstraintfor

BudgetSetandConstraintfor m

+

=

m

+

=m m BudgetSetandConstraintfor

BudgetSetandConstraintfor m

+

=

m

+

=Just

NotaffordableJustaffordablem m BudgetSetandConstraintfor

BudgetSetandConstraintfor m

+

=

m

+

=NotaffordableJustaffordable

theofallaffordablem m BudgetSetandConstraintfor

Budget p1x1+p2x2=m

Slopeis-xmx2

=-

+

soslopeis-

m Budget Opp.costofanextraunitofcommodity1isp1/p2foregoneofcommodity

Budget Opp.costofanextraunitofcommodity1isp1/p2foregoneofcommodity Opp.costofanunitofcommodity2

p2/p1unitsforegoneofcommodity1.BudgetSets&Constraints; andPriceChangesdependuponpricesand happensaspricesor echange?

constraintchange em2budgetset Newaffordableconsumption

dothebudgetsetandbudgetconstraintchangeas emdecreases?budgetset

Originalnewbudgetconstraintsareparallel(same

budgetset dothebudgetsetandbudgetconstraintchangeas emdecreases?

BudgetConstraints- budgetset

thatarenolongerOldandnewareparallel.

Increasesin emshifttheconstraintoutwardinaparallelmanner,therebyenlargingthebudgetsetandimprovingDecreasesin emshifttheconstraintinwardinaparallelmanner,therebyshrinkingthebudgetsetandreducingBudgetConstraints-

Budget

PriceNooriginalchoiceislostandnewchoicesareaddedwhen worseoff. edecreasemay(typicallywill)

WhathappensifjustonepriceSupposep1Howdothebudgetsetandbudgetconstraintchangeasp1decreasesfromp1’top1”?

Howdothebudgetsetandbudgetconstraintchangeasp1decreasesfromp1’top1”? Newaffordable budgetset

budgetset

Budget

Price

choiceislostandnewchoicesareadded,soreducinone consumerworseoffSimilarly,increasingonepricepivotstheconstraintinwards,reduceschoiceandmay(typicallywill)maketheconsumerworseoff.

atarateof5%increasesallpricesby5%,frompto(1+0.05)p1.05p.AnadvaloremsalestaxleviedatarateofincreasesallpricesbytpfromptoAuniformsalestaxisapplieduniformlytoallAuniformsalestaxleviedatratetchanges

Auniformsalestaxleviedatratetchangesp1x1+p2x2=

p1x1+p2x2=p1x1+p2x2=Sales

Sales p1x1+p2x2= p1x1+p2x2=

e m t m (1t)

(1t)

1

1 (1t) (1t) Sales Auniformad salestaxleviedatrateisequivalentto

Foodstampsarecouponsthatcanbelegallyexchangedonlyforfood. (1t)

taxleviedat

1t

Howdoesacommodity-specificgiftsuchasafoodstam (1t) of“othergoods”ispG=$1“Othergoods”isacompositeItsimplifies ysistoa2-good

F+G=100: ThebudgetconstraintisF+G

GF+G=100:

GF+G=100:

Budgetsetafter40foodstampsissued.

Budgetsetafter40foodstampsissued.Thefamily’sbudgetsetisenlarged. (F-40)+G=100forF40G=100forF<40

(F-40)+G=100forF40G=100forF<40Whatiffoodstampscanbetradedona

F+G=100: Budgetconstraintafter40foodstampsissued.Budgetconstraintwithblackmarkettrading. F+G=100+0.5(40-F)for(F-40)+G=100forF

Budgetconstraintafter40foodstampsissued.Blackmarkettradingmakesthebudgetsetlargeragain. 、 username:

ShapesofBudgetQ:WhatmakesabudgetconstraintaA:Astraightlinehasaconstantslopeandp1x1+…+pnxn=msoifpricesareconstantsthenaconstraintisastraightline.ShapesofBudgetButwhatifpricesarenotE.g.bulkbuyingdiscounts,orpricepenaltiesforbuying“toomuch”.Thenconstraintswillbe

ShapesofBudgetConstraints-tyDiscountsSupposep2isconstantat$1butthatp1=$2for0x120andp1=$1fortyDiscountsSupposep2isconstantat$1butthatp1=$2for0x120andp1=$1forx1>20.Thentheconstraint’sslope2,for0x1

ShapesofBudgetConstraintswithatyDiscount m= Slope=-2/1=-(p1=2,Slope=-1/1=-p1/p21,forx1>

(p1=1,

ShapesofBudgetConstraintswithatyDiscount m= Slope=-2/1=-(p1=2,Slope=-1/1=-1(p1=1,p2=1)

BudgetConstraintswitha Theconstraint2x1+x2=mfor0x120

ShapesofBudgetConstraintswith ShapesofBudgetConstraintswith

ty

m=

ty

Budget

Budget Budget

Indifferencecurves(无差异曲线Marginalrateofsubstitution(边际替代率

RationalityinBehavioralAdecisionmakeralwayschoosesitsmostpreferredalternativefromitssetofSotomodelchoicewemustmodeldecision-makers’preferences. odifferentconsumptionstrictpreference(严格偏好)xismorepreferredthanyIndifference(无差异):x actlyaspreferredasweakpreference弱偏好)xisasatleastaspreferredasisy

AssumptionsaboutPreferencebundlesxandyitisalwayspossibletomakethecomparisonbetweenxandyReflexivity(反身性):AnybundlexisalwaysatleastaspreferredasitselfAssumptionsaboutPreference 传递性):xisatleastaspreferredasy,andyisatleastaspreferredasz,then

Indifference’Takeareferencebundlex.Thesetofallbundlesequallypreferredtox’istheindifferencecurvecontainingx’;thesetofallbundlesyx’.’Indifference Indifference

x’x”

y WP theset bundlespreferredto y

AllbundlesinI1arestrictlypreferredtoallinI2.zAllbundlesinI2 strictlypreferred allinWeaklyPreferredSet(弱偏好集 StrictlyPreferredSet(严格偏好集 WP theset bundles preferredto

IndifferenceCurvesCannot

SlopesofIndifferenceI I1xyz

Whenmoreofacommodityisalwayspreferred,thecommodityisagood.Ifeverycommodityisagoodthenindifferencecurvesareneativelsloed.SlopesofIndifferenceanegativelyslopedindifferencecurve.

SlopesofIndifferenceIflessofacommodityisalwayspreferredGoodSlopesofIndifferenceOnegoodandone positivelysloped

Perfectsubstitutes(完全替代Perfectcomplements(完全互补Satiation餍足BadExtremeCasesofIndifference

ExtremeCasesofIndifferenceCurves:PerfectSlopesareconstantat-BundlesinI2allhaveatotalof15units

o9

Eachof(5,5),and(9,5)contains5pairssoeachisequally l-BehavedConvexity:l-BehavedConvexity:Mixturesofbundlesare(atleastweakly)preferredtothebundlesthemselves.E.g.,the50-50mixtureofthebundlesxandyiszisatleastaspreferredasxor Well-BehavedPreferences--

Well-BehavedPreferences-- x

z=+ isstrictly

z=(tx1+(1-t)y1,tx2+(1- 2

1

tobothxandy.

ispreferredtoxandyfr <1.y Well-BehavedPreferences--

Preferencesarestrictly whenallmixtures

weaklyconvexif are preferredtobundlesxandy.y

leastonemixturezisequallypreferredx toa Non-Convex MoreNon-ConvexThemixture islesspreferredhnx

Themixture islesspreferredhnx MarginalRateofMarginalRateofdx2=’therateatwhichtheconsumerisonlyjustwillingtoexchangecommodity2forasmallamountofcommodity1.MarginalwillingnesstoMRSatx’lim-

2

=-dx2/dx1atMarginalRateof MRS&Ind.CurveClothing

AMRS=-B1

C

MRS= MRS esmoreasx1increases 64

D1-1-

MRS=EG

MRS=

MRSisnotalwaysincreasing

x1

U(x1,x2)2 2

MRS=

dUdU(x1,x2)

dx1

dx2= MRS=

MRSdx2

/_/

u

UtilityFunctionand’Aconsumerspreferencescanberepresentedbyautilityfunction’The“perfectsubstitutes”preferencecanbereresentedbUx The“perfectcomplements”preferences:U(x,y)=min{x,y}Istheutilitfunctionreresentationuniue

UtilityFunctionandTheutilityrepresentationisnotApositivemonotonictransformationofautilityfunctionrepresentsthesamereferenceastheoriinalutilitfunctionAslongastheutilityfunctionsgivethesameorderingoftheconsumptionbundles,Theutilityisordinal,ratherthanConsumerConsumerComputingordinaryV(x1,x2)

f[u(x1,x2

f'(u)MRSv/x1

f'(u)u/f'(u)u/

u/TheprincipalbehavioralpostulateisthattoTheavailablechoicesconstitutethechoiceset.Howisthemostpreferredbundleinthe

RationalConstrainedMorepreferredRationalConstrained preferred

RationalConstrainedThemostpreferredaffordablebundleisDEMAND（orDEMAND）atthegiven

Ordinarydemandswillbedenotedbyx1*(p1,p2,m)andx2*(p1,p2,m).* *RationalConstrained RationalConstrainedWhen

*>0and

*>0the

(x1*,x2*)is bundleisIfbuying(x1*,x2*)costs$mthenthebudgeti

(x1*,x2*)exhauststhe* *RationalConstrained (x1*,x2*)is(x1*,x2*)exhauststhebudget;p1x1*+p2x2*=m.

RationalConstrained (x1*,x2*)isinteriorTheslopeofthecurveat(x1*,x2*)equalstheslopeofthebudget ** **RationalConstrained(x1*,x2*)satisfiestwo(a)thebudget p1x1*+p2x2*=btanenc:thesloeofthebud

MeaningoftheTangency onsumptionbundle(x1,Theconsumeriswillingtogiveup2unitofx2constraint,-p/p,andtheslopeof

exchangeforanadditionalunitof◦ Themarketallowshertogiveuponly1unitofindifferencecurvecontaining(x1*,x2*)

toobtainanadditionalequalat

(x1,x2)isnotoptimalShecanbebetteroffincreasingherconsumptionofSolvefor2simultaneousBudgetTheconditionsmaybeobtainedby theLagrangianmultipliermethod, constrainedoptimizationin locate(x1*,x2*)forgivenp1,p2andm?

SupposethattheconsumerhasCobb-U(x,x)