【高级微观经济学】15年习题一.pdf

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1、GS/ECON5010sectionB"'AnswerstoAssignment1October2015Q1.Arethepreferencesdescribedbelowstrictlymonotonic?Convex?Explainbrie y.Therearethreedierentgoods:potatoes,riceandnoodles.(Soaconsumptionbundle(x1;x2;x3)isabundlewithx1kilogramsofpotatoes,x2kilogramsofriceandx3kilogramsofnoodles.)Eachkilogr

2、amofpotatoeshas1000calories,eachkilogramofricehas800caloriesandeachkilogramofnoodlesalsohas800calories.Ifbundlexhasstrictlymorecaloriesthanbundley,thenthepersonprefersxstrictlytothebundley.Ifthebundlesxandyhavethesamenumberofcalories,thenthepersonprefersstrictlywhicheverbundlecontainsstrictlymo

3、rerice.Iftwobundleshavethesamenumberofcalories,andthesamequantityofrice,thenthepersonisindierentbetweenthem.A1.Tocheckstrictmonotonicity,weneedtocheckwhetherthefollowingtwopropertiesholdforthepreferencesdescribed:(i)iftherearetwobundlesxandzwithxz,thentheconsumermustrankxasatleastasgoodasz;(i

4、i)ifx>>z,thentheconsumermustpreferstrictlyxtoz.Bothpropertiesholdhere.Ifxz,then1000x1+800x2+800x31000z1+800z2+800z3,sothatbundlexhasatleastasmanycaloriesasbundlez.Aswell,ifxz,thenx2z2,sothatbundlexhasatleastasmuchriceasbundlez.Andifx>>z,then1000x1+800x2+800x3>1000z1+800z2+800z3,sothatbundle

5、xhasstrictlymorecaloriesthanbundlez,andmustbepreferredstrictlybytheconsumer.Tocheckconvexity,supposethatxisatleastasgoodasz.Preferencesareconvexif(andonlyif)anybundletx+(1t)zisrankedatleastasgoodasz,forsomescalartwith0t1.Nowifxisatleastasgoodasz,thenbundlexhasatleastasmanycaloriesasbundlez,a

6、ndatleastasmuchriceasinbundlez.Thenumberofcaloriesinanybundletx+(1t)zmustbesomewhereinbetweenthenumberofcaloriesinbundlexandthenumberofcaloriesinbundlez.[Thenumberofcaloriesinbundletx+(1t)zisaweightedaverageofthenumberofcaloriesinbundlesxandz,withweightsoftonxand(1t)onz.]Also,thequantityofri

7、ceinbundletx+(1t)zmustbesomewhereinbetweenthequantityofriceinbundlexandthequantityofriceinbundlez.Soifxisrankedatleastasgoodasbundlez,thenbundletx+(1t)zmusthaveatleastasmanycaloriesasbundlez,andatleastasmuchriceasisinbundlez,and