第一讲偏好与效用

11•–•–•–•2•X={x,y,z,….}–X={(x1,x2)∈R+2:x10;x20}•–XxyXxyxyxy3•–R(x,y)“xy”•••–R(x,y)R(y,x)–x,y,z∈X,R(x,y)R(y,z)R(x,z)4•–•xy•x•–••x–……5•–X•(Completeness)x,yxyyx,yyy•(Transitivity)x,y,z,xyyz,xz62•≻–x≻yxyyx–;(y)={x∈X:x;y}•–xyxyyx–(y)={x∈X:x~y}——•(y)={x∈X:xy}=(y)∪;(y)7•(Lexicographicpreference)–x=(x1,x2)–x1x2,•(50)(4100)121212111122,,xxorxxandxx=≥•(5,0)(4,100)–(5,0)(4,100)–(4,100)(5,0)–(5,0);(4,100)8•(Lexicographicpreference)L–(x0)={(1,1)}–≻(x0)={x|x11x1=1x21}–≺(x0)={x|x11x1=1x21}9x1x2x0=(1,1)u(·):R+nÆR∀x0,x1∈R+n,x0x1⇔u(x0)≥u(x1)X∈R+nRu(·)∈+x0x1u010x0x1•u(x)∀x0,x1∈R+n,x0x1⇔u(x0)≥u(x1)–v(x)=u(x)+8()()–v(x)=100*u(x)+8–v(x)=u(x)3–v(x)=u(x)211•v(x)=f(u(x))f:RÆRu•1.2–u(x)()v(x)u(x),v(x)•–x0x1–iffu(x0)≥u(x1)[u]–ifff(u(x0))≥f(u(x1))[f]–iffv(x0)≥v(x1)123•–•–X–X13∀A⊆Xa,∃a∈Axa∀x∈A•–AA–An+1x∈A–A-{x}nA-{x}y–xyyA–yxxA14X•X1X•X-X1X2X-X1•….()()•X-(X1∪X2…∪Xn-1);XnX-(X1∪X2…∪Xn-1)•X=X1∪X2…,∪Xk•Xk|X|,Xn,n=1,2,…,k•u(x)=k,ifx∈Xk–ab,a∉X1∪X2…∪Xu(b)-1,u(a)u(b)15•u(x)–{xn}Xu(x1)=0.–{x1,…,xn-1}u(x1)…,u(xn-1),xkxl⇔u(xk)u(xl)*Xxkxl⇔u(xk)u(xl)–∃kn,xn∼xku(xn)=u(xk)–{u(xk)|xn;xk}∪{-1}{u(xk)|xk;xn}∪{1}u(xn)•∀kn,xkxn⇔u(xk)u(xn)•u(x)–∀x,y∈X,{xn}xk=x,xl=y,–n=Max{k,l}xkxn⇔u(xk)u(xn)•L(Lexicographicpreference)–x=(x1,x2)–x1x2,•[0,1][0,1]L–u(x)L–a∈[0,1],(a,1);L(a,0),,u(a,1)u(a,0)121212111122,,xxorxxandxx=≥[,],(,)L(,),,(,)(,)–r(a)u(a,1)q(a)u(a,0)–q(a)[0,1]–q(a)•ba(b,0);L(a,1)u(b,0)u(a,1)•q(b)u(b,0)u(a,1)q(a)q(b)≠q(a)–17•x2x1•1D1–x1≻x2Bδ(x1)Bδ(x2)x∈B(x1),y∈B(x2)x≻y18Bδ(x1)x1x2xyBδ(x2)x1x211(1,1)x2x′x1x1≻x2≻x14•2(D2)–(J-R)x∈X(x)X–(MWG),1n,xnyn21{(x,y)}nnn∞=linlin2xylimxxnn→∞=limyynn→∞=19{xn}{yn}xyx(x){yn}yD1D2•D1ÆD2–D1{xn,yn}xnynlimnÆ∞xn=x,limnÆ∞yn=y–xyy≻x.–D1B(y)B(x)z∈B(y),w∈B(x)z≻w.–N{xn,yn}nN,xn∈B(x)yn∈B(y)–yn≻xn–20D1D2•D2ÆD1–D2x≻yB(x,r)B(y,r)–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–21u(x)––x1≻x2,u(x1)u(x2)–ε=[u(x1)-u(x2)]/2–∃δ011•d(x,x1)δxu(x)u(x1)-ε,•d(x,x2)δxu(x)u(x2)+ε,–B(x1)B(x2)x∈B(x1),y∈B(x2)x≻y–D122•–xn=(1+1/n,1),yn=(1,2)•Æxnyn–x=limnÆ׿xn=(1,1),y=limnÆ׿yn=(1,2)•Æyx–23x1x211{xn}xyn=(1,2)†Debru245•Rubinstein,2009,LecturenotesinMicroeconomics,–Lecture1-2•Mas-Colelletal.1995,MicroeconomicTheory,–Ch1,Ch3C•JehleandP.Reny,AdvancedMicroeconomicTheory–Ch125