消费者问题偏好关系消费者问题：偏好关系

1•A•A•B•C2•–•X–n+Xφ∉⊆R–X–X+φ∉⊆X–0X••nn+{x:0for1,2,...,}iXxin==∈≥=RR3•–X–••–∼(x0)={x:x∼x0}(0){0}•(x0)={x:xx0}4•x2≻(x0)∼(x0)x4*1kx0()x1x3x2x5x*=x1+kex2≺(x0)1*1+k[(11)k]•x1∼x*=x1+ke[e=(1,..,1),k]•x2≻x3,x3≫x2455t1(1)45•x4≻x5,x5=tx1+(1-t)x4•4(LocalNonsatiation)()–∀x0∈X∀ε0∃x∈Bε(x0)∩X,x;x0–x2x14x2≻(x0)∼(x0)x*=x1+kexx0x=x1+ke≺(x0)6x2•4–(0)≻(x0)∼(x0)x0≺(x0)7()≻(x0)x3x2≻(x0)∼(x0)xx00(x)≺(x)≺≺(x0)8•5(StrictlyMonotonicity)(yy)–∀x1,x2∈Xx1x2x1x2x1x2x1;x2,,–Æ9•x2x2x2x1≻(x0)•x0≻(x0)x3≺(x0)x∼(x0)10x1•xt=tx1+(1-t)x2,t∈[0,1]x22x1tx2≻(x0)•x1xtx0•≺(x0)x∼(x0)11x1•6–D1∀x1,x2∈Xx1x2,xt=tx1+(1-t)x2x2t∈[0,1]•Æx2x2x1xtx2≻(x0)•x0t0(1)3(0)≺(x0)x3xt=tx0+(1-t)x3t∈[0,1]12x1∼(x0)6•D1∀x1,x2∈Xx1x2,xt=tx1+(1-t)x2x2t∈[01]t∈[0,1]•D2(x0)D1ÆD2•D1ÆD2–D1x1,x2∈(x0)102012–x1x0,x2x0,x1x2–D1xt=tx1+(1-t)x2x2tt1+(1t)20t∈(0)–xt=tx1+(1-t)x2x0.xt∈(x0)•D2ÆD1D21(0)–D2x1∈(x0)–x1x0D21(1)00–D2tx1+(1-t)x0x013•–xt=tx1+(1-t)x0∼x0t∈(0,1)x2≻(0)x1xt≻(x0)xtx0≺(x0)∼(x0)x1()14•7–x1≠x0x1x0,xt≻x0t∈(0,1)Æ•Æx22x1≻(x0)x0xt≺(x0)∼(x0)•15x1•212dxMRSd≡−–⇒1201uudx=–⇒x2x2x2x1≻(x0)x2≻(x0)()x10()x3x0≺(x0)∼(x0)x0≺(x0)∼(x0)16x1x1••–1–2Æ–3–4ÆÆ4–5–6ÆÆÆ6–7ÆÆ17†Debru†DebruXu(·)()†-Step1:-Step2:18:•u(x):RnÆR–t*·e∈EA∩EBt*·e∼x0–u(x)≡t*–Æt*–x2E={x|x=t·e,t≥0}Æt*2t*·eEA={x|x=t·ex0,t≥0}e=(1,1)tex0t*·e∈A∩B•0xEB={x|x0x=t·e,t≥0}019x1•–∼(x0){x|x∼x0}={x|u(x)=u(x0)}–(x0){x|xx0}={x|u(x)≥u(x0)}(x){x|xx}{x|u(x)≥u(x)}———u(x)0(x)uux20012(x)(x)0uudxdx∂∂+=(x)uu=02112(x)/dxuxMRS∂∂≡−=1212xx∂∂x0(0)•120012(x)/MRSuudxux=∂∂20x1∼(x0)•–f(x)f(x)()•u(x)–u(x)()•Æ(x0){x|xx0}={x|u(x)u(x0)}•Æu(x)Æu(x)–u(x)21•-(C-D)()–u(x1,x2)=x1αx2β–MRS=αxα-1xβ/βxαxβ-1=αx/βxMRS12αx1x2β/βx1x2βαx2/βx1–ln[u(x1,x2)]=αlnx1+βlnx2x2tx1x1tx2k=αx2/βx1x2∼(x0)22x1()•–u(x1,x2)=αx1+βx2–MRS12=α/βMRS12=α/β•–u(x1,x2)=Min{αx1,βx2}MRS12=-∞MRS1212MRS12x2=αx1/βMRS=023MRS12=0•(HomotheticPreferences)(f)–x1x2∀α≥0,αx1αx2–L–u(x1x2)=x1αx2βαβ0x21u(x1,x2)x1x2,α,β0–kαx2αx1x1x2αxx1x∼(x0)24x1•––•1.1u(x)=t*(x),x∼t*(x)e()•αx∼αt*(x)e•u(x)u(αx)=αt*(x)=αu(x)25•(quasilinearpreference)(qp)–1x1~x2∀α≥0,x1+αe1~x2+αe1e1=(1,0,…,0)–•u(x1,x2)=v(x2)+x1140080010001200x1x3=(x11+α,x21)4(2+2)αα200400600x2x4=(x12+α,x22)α123452611u(x)=x1+v(x2,…,xn)•(x2,…,xn)v(x2,…,xn)(0,x2,…,xn)∼(v(x2,…,xn),0,…,0)•(,2,n)((2,n),,,)–x(x1x2x)∼(x1+v(x2x)00)(x1,x2,…,xn)(x1+v(x2,…,xn),0,…,0)–1x1+v(x2x)x1+v(x2,…,xn)27•:CES1/1212(,)[]uxxxxρρραβ=+1.1ifρ=1212(,)uxxxxαβ=+2.0ifρ→1212(,)αβαβαβ++=uxxxxfρ1212(,)3.ifρ→−∞1212(,)min{,}uxxxx=fρ1212(,)min{,}uxxxx281D1D2•D1ÆD2–D1{xn,yn}xnynlimÆxn=xlimÆyn=ylimnÆ∞xx,limnÆ∞yy–xyy≻x.–D1B(y)B(x)z∈B(y),w∈B(x)z≻w.–N{xn,yn}nN,xn∈B(x)yn∈B(y)x∈B(x)y∈B(y)–yn≻xn–291D1D2•D2ÆD1–D2x≻yB(x,r)B(y,r)B(xr)B(yr)z∈B(y)w∈B(x)–B(x,r)B(y,r)z∈B(y),w∈B(x)zw.–xn∈B(x,1/n),yn∈B(y,1/n),ynxn–limnÆ∞xn=x,limnÆ∞yn=y–D2yx–3022.3•Step1:u(x)(111)–e=(1,1,…,1)•t·e∈X,∀t∈R+–{0x}Atte≡≥{0x}Btte≡≥–AB–•A•BAB–AB[,)At=∞[0,]Bt=3122.3•Step1:u(x0)–[0,][,)ABtt+=∞=∪∪R[,][,)+∪∪Æ≥ttÆABφ≠∩*tAB⇒∃∈∩*t–t*ext⇒∼t.•u(x0)≡t*,t*·e∼x0P.13222.3•Step2:u(x)–∀x1,x2∈R+n,(P.1)u(x1)u(x2)–x1x2xx–u(x)(1)(2)u(x1)·eu(x2)·e–u(x1)≥u(x2)–u(x)u(x)333CES1/1212(,)[]uxxxxρρραβ=+1212ln[]ln(,)xxuxxρραβρ+=ρ121200ln[]limln(,)limxxuxxρρρραβρ→→+=ρρρ1122120[lnln]limlnlnxxxxxxxxρρρρραβαβαβαβαβ→+==++++012xxραβαβαβ→+++lim()uxxxxαβ⇒=12120lim(,)uxxxxρ→⇒=343CES1/1212(,)[]uxxxxρρραβ=+ln[]xxρραβ+1212ln[]limln(,)limxxuxxρρρραβρ→−∞→−∞+=llρρβ121,1xx112212[lnln]limxxxxxxρρρρραβαβ→−∞+=+1212112[ln(/)ln]limlnif(/)xxxxxxxxxρρραβαβ→−∞⎧+=≥⎪+⎪211212221(/)[(/)lnln]limlnifxxxxxxxxxρραβαβ→+⎪=⎨+⎪=⎪22112(/)xxρραβ→−∞⎪+⎩1212lim(,)min{,}uxxxx⇒=351212(,){,}ρ→−∞