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Chapter1.StaticanddynamicMarshalliandemand
functionanditsproperties
1.1.StaticMarshalliandemandfunction
Thereisaconsumergoodsmarketwhere:
.i1i2iłin-consumergoods(productsandservices),
1
IE.ntR+
l-aconsumer’sincome,
xi(xlix2iłixπ)ER+
π-abundle(basket)ofgoodsthattheconsumerwantstopurchase
(consumptionbundle),
XiR+
π-asetofallbundlesofgoodsavailableonthemarketalongwithametricspecified
onit(goodsspace),
pi(plip2łipπ)ER+
π-avectorofpricesofgoods,
bi(blib2iłibπ)ER+
π-avectorofsupplyofgoods,
D(piI)i{xER+
π|plxl+p2x2+ł+pπxπśI}-abudgetset(acompactandconvexset
ofallconsumptionbundleswhosevalueisnotgreaterthantheconsumer’sincome),
L(piI)i{xER+
π|plxl+p2x2ł+pπxπiI}CD(piI)–abudgetline(asetofall
consumptionbundlesthevalueofwhichisequaltotheconsumer’sincome),
Bi{xER+
π|x
lśblix2śb2iiłixπśbπ}-asupplyset(compactandconvexsetofall
bundlesofgoodsrealisticallyavailableonthemarket),
PiD(piI)nB-abudgetandsupplyset(acompactandconvexsetofallbundlesof
goodsmeetingboththebudgetandthesupplyconstraints),
2
uiR+
πąRl-anutilityfunctiondescribingarelationofconsumerpreferences,whichwe
assumebydefaulttobeincreasing,differentiableand(strictly)concave.
Theconsumer’sgoalistofindtheoptimalbundleofgoodsinthesetPCXiR+
πi.e.
suchabundleofgoodsx-EPthatVxEPiu(x-)?u(x),whichmeansthatx-isbetterthan
anyotherbundleofgoodsinthesetPCXiR+
π.
1Intheteachingmaterialswewillconsidern-dimensionalcommodityspacesinwhichthenumberofconsumer
goodsisfinite.However,forthesakeofsimplicity,wewilllimitourselvesto2-dimensionalgoodsspaces.
2Sincethedemandforanyproductreportedbyasingleconsumerisusuallymanytimeslowerthanitssupply,
inthetheoryofconsumerdemanditisassumedthattheonlylimitationintheconsumer’schoiceoftheoptimal
consumptionbundleisthebudgetconstraint,andthesupplyconstraintisnotbinding.Therefore,generally
PiD(piI)nBiD(piI).
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