stone-Geary-function

Exercise1.56(a)TheStone-GearyutilityfunctionisU(x)=Π(xj–aj)bjwhereΣbj=1.Wewanttheexpenditurefunctionsoweminimizeexpendituresubjecttotheutilitylevelbeingatleastu.TheLagrangianisL=Σpixi–λ[Π(xj–aj)bj–u].Thefirstorderconditionsarepi–λbi(xi–ai)-1Π(xj–aj)bj=0alli=1,2,…,n(FOC1)andΠ(xj–aj)bj=u.(FOC2)Bothxiandλarefunctionsofpandu.Thefirstorderconditionsgivepixi–piai=λbiualli=1,2,…,n.Equation(1)Soλu=(pi/bi)(xi–ai)foralli=1,2,…,n.Hencexj–aj=(pi/bi)(xi–ai)(bj/pj)alli,j=1,2,…,nasλuisindependentfromi.SubstitutethisinFOC2.Sou=Π[(pi/bi)(xi–ai)(bj/pj)]bjwheretheproductisoverj=1,2,…,n=Π[(pi/bi)(xi–ai)]bjΠ(bj/pj)]bj=(pi/bi)(xi–ai)Π(bj/pj)bjusingΣbj=1.Itfollowsthatpixi=piai+biuΠ(pj/bj)bjEquation(2)Summingequation(2)overiandusingΣbi=1,givese(p,u)=Σpixi=Σpiai+uΠ(pj/bj)bjastheexpenditurefunctionfortheStone-Gearyutilityfunction.Sincee(p,u)=y,itfollowsthatv(p,y)=(y-Σpiai)Π(bj/pj)bjistheindirectutilityfunctionfortheStone-Gearyutilityfunction.(b)ThefirstpartofthequestionconcernstheHicksiandemandsxi(p,u)butthispartisabouttheMarshalliandemandsx*i(p,y).Thereatleastthreewaystogoaboutansweringthispart.ThefirstwayistomaximizeU(x)subjecttothebudgetconstraint.Thesecondwayistousetheequalitybetweenxi(p,u)andx*i(p,y)wheny=e(p,u),sumoverEquation(1)togetΣpix*i–Σpiai=Σλbiu=λuandthensubstitutebackintoEquation(1)togettheresult.TheposhwayistouseRoy’sIdentity.Wehaveδv(p,y)/δy=Π(bj/pj)bjandδv(p,y)/δpi=[-(bi/pi)y–ai+(bi/pi)Σpiai]Π(bj/pj)bj.ByRoy’sIdentity,x*i(p,y)=-δv(p,y)/δpi/δv(p,y)/δyandtheresultpix*i(p,y)=piai+bi(y–Σpiai)follows.Exercise1.56MartinCaley