数理经济学1new

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1、1BasicMathematicalConcepts1.1Concave,quasi-concaveandhomogenousfunctionsGiventwosetsXandY,ifeachelementofXcanbeassociatedwithanelementofY,whichwedenotebyf(x);thenwesayfisafunctionfromXintoY,denotedbyf:X!Y:ThesetsXandYaresaidtobethedomainandrangeoff;respectively.AsetXisconvexifforanyx;x02X;tx+(1

2、t)x02Xforallt2[0;1].(Note:thereisnoconcavesuchathingasconcaveset.)Afunctionf:X!R,whereXisaconvexset,isif,convexforanyx;y2X;and10;f(x+(1)y)f(x)+(1)f(y)Iffistwicecontinuouslydi¤erentiable,thenitisconcaveifandonlyiftheHessianoff01f11(x)


f1n(x)B@.......CA..fn1(x)


fnn(x)isnegative(p

3、ositive)semi-de…niteforallx2X(seebelow).Afunctionf:Rn+!Risquasiconcaveiftheuppercontourssetsofthefunctionfx:f(x)>agareconvexsetsforallvalueofa.Afunctionf:Rn+!Risquasiconconvexiffisquasi-concave.quasiconcaveAnalternativede…nitionforquasiconcavefunctionisthatfisifandonlyifquasiconvexforanytwod

4、istinctpointsxandx0inthedomainoff,andfor0<<1,0
0
f(x)f(x)fx=)fx+(1)x0f(x)Iffistwicecontinuouslydi¤erentiable,thenitisquasi-concaveifandonlyifthenaturallyorderedprincipalminoroftheborderedHessianoff010f1(x)


fn(x)BBf1(x)f11(x)


f1n(x)CCB........C@....Afn(x)fn1(x)


fnn(x)hassigns0f1

5、(x)


fj(x)0f1(x)f2(x)f(x)f(x)


f(x)1111j0oddf1(x)f11(x)f12(x)0;:::;........0ifjisevenforalljn:f2(x)f21(x)f22(x)....fj(x)fj1(x)


fjj(x)1Afunctionf:Rn+!Rishomogeneousofdegreekiff(tx)=tkf(x)forallt>0:Ineconomics,manyusefulfunctionsarehomogenousofdegree0or1,e.g.thedemandfunctionofaconsum

6、erishomogeneousofdegreezeroinpricesandincome.Euler’sTheorem:Supposefisahomogeneousfunctionofdegreek.Letx=tx.Di¤erenti-atingf(tx)=tkf(x)togetXn@f(tx)@f(tx)@xik1==ktf(x)@t@xi@ti=1Theequationholdsforallt.Inparticular,theequationistruefort=1;Xn@f(x)xi=kf(x):@xii=1ConsideranapplicationoftheEulerThe

7、orem.Letf(K;L)beaproductionfunctionwhereKandLarecapitalandlaborrespectively.Forconstantreturnstoscale,k=1andassumethepriceoftheoutputisequaltoone.Inacompetitivemarket,thefactorpriceisequaltothevalue@f@fofmarginalproduct.TheEulerth