金融经济学课后答案-给学生-2012

22.1U(c)V(c)c2Rn+V(x)=f(U(x))f(¢).U(c)º8c1;c22RN+U(c1)¸U(c2),c1ºc2f(¢)V(c1)=f(U(c1))¸f(U(c2))=V(c2),U(c1)¸U(c2)V(c1)¸V(c2),c1ºc2V(c)º2.2*1ababc=[c0;c1a;c1b]U(c)=logc0+12(logc1a+logc1b)U(c)=11¡°c1¡°0+12³11¡°c1¡°1a+11¡°c1¡°1b´U(c)=¡e¡ac0¡12¡e¡ac0+e¡ac0¢.(a)c¸c0c0¸c00;c1a¸c01a;c1b¸c01blog(¢)log(c0)¸log(c00);log(c1a)¸log(c01a);log(c1b)¸log(c01b)U(c)¸U(c0)cºc022(b)fc(n)g11R3limn!1c(n)=c80;9±jc0i¡cij·±jlog(c0i)¡log(ci)j3;i=0;1a;1b±9Nn¸Nkc(n)¡ck=q(c0¡c(n)0)2+(c1a¡c(n)1a)2+(c1b¡c(n)1b)2±jU(c(n))¡U(c)jlimc(n)!cU(c(n))=U(c)(c)U(C)U(C0)log(c0)+12log(c1a)+12log(c1b)log(c00)+12log(c01a)+12log(c01b)8®2(0;1)U(®C0+(1¡®)C)=log(®c00+(1¡®)c0)+12log(®c01a+(1¡®)c1a)+12log(®c01b+(1¡®)c1b)¸®(log(c00)+12log(c01a)+12log(c01b))+(1¡®)(log(c0)+12log(c1a)+12log(c1b))=®U(C0)+(1¡®)U(C)U(C0)2.3U(c)=c¡12ac2c2R+aU(c)a/cU(c).a=1U(12)=38U(1)=12U(3)=¡1:5U(c)a=0U(c)=ca0c2[0;1a]U(c)2.41abcabc12X1=[1;1;1]X2=[1;2;3]S1S2(a)(b)X(c)µ11µ22c°3(d)K1122(e)(f)(g)3X3=[0;0;1].(a)R3(b)X=2411121335=[X1;X2](c)µ1X1+µ2X2=[µ1+µ2;µ1+2µ2;µ1+3µ2]µ1S1+µ2S2(d)K12K2[3K;5K;7K]KS1+2KS2(e)M=fY2R2:Y=µ1X1+µ2X2;µ1;µ22Rg(f)M+=fY2R2:Y=µ1X1+µ2X2;µ1;µ22R+g(g)X=2411012013135=[X1;X2;X3]M=fY2R3:Y=µ1X1+µ2X2+µ3X3;µ1;µ2;µ32Rg2.52.412µ11µ22.fC2R3+:C=®1X1+®2X2;®1S1+®2S2·µ1S1+µ2S2g2.62.43S33012312.fC2R3+:C=®1X1+®2X2+®3X3;®1S1+®2S2+®3S3·µ1S1+µ2S2+µ3S3g2.7*11010011[100;1]c°42U(c0;c1)=logc0+½logc1:½0111+rFrF(a)Ua(b)²w²c00sUb/UbwrF½²rF½(c)Ub¸Ua(d)gUb(w¡g)=Ua:gg½grF.(a)c0=100;c1=1;Ua=log(100)+½log(1)=log(100)(b)fC2R2+:c0=100¡S;c1=1+S(1+rF);S2Rgw=100+11+rFmaxSlog(100¡S)+½log(1+S(1+rF))S=100½(1+rF)¡1(1+½)(1+rF)c0=100(1+rF)+1(1+½)(1+rF)=11+½w;c1=½(100(1+rF)+1)(1+½)(1+rF)=½(1+rF)1+½wc°5Ub=log(w1+½)+½log(½(1+rF)1+½w)c0c1rF;½rF1½101(c)S=0UaSUbUb¸Ua(d)Ub(w¡g)=log(w¡g1+½)+½log(½(1+rF)1+½(w¡g))g=w¡½¡¡½1+½(1+rF)¡½1+½10011+½(1+½)g½rFg½rFgc°62c°3Arrow-Debreu3.1*1abab(a)Arrow-DebreuÁ(b)021Arrow-Debreu(c)(d)U(c0;c1a;c1b)=¡e¡c0¡12¡e¡c1a+e¡c1b¢:(e)Arrow-DebreuÁ(f)/.(a)Arrow-DebreuXa=[1;0];Xb=[0;1](b)2Xa+Xb(c)w=2Áa+Ábfc2R3+:c0+Áac1a+Ábc1b=wg(d)max[c0;c1a;c1b]¡e¡c0¡12(e¡c1a+e¡c1b)83ARROW-DEBREUs.t.c0+Áac1a+Ábc1b=w=2Áa+Ábc¤0=2Áa+Áb+Áalog(2Áa)+Áblog(2Áb)1+Áa+Ábc¤1a=2Áa+Áb+Áblog(2Áb)¡log(2Áa)¡Áblog(2Áa)1+Áa+Ábc¤1b=2Áa+Áb+Áalog(2Áa)¡log(2Áb)¡Áalog(2Áb)1+Áa+Ábc¤Áa;ÁbÁac¤1aÁac¤0;c¤1bc¤0;c¤1b(e)Áa=Áb=0:1c¤0=¡0:018203.211010011[100;1]U(c0;c1)=logc0+½logc1:0111+rFrF(a)rF(b)1(c)(d)½.(a)maxµlog(100¡µ)+½log(1+µ(1+rF))µ¤=100½(1+rF)¡1(1+½)(1+rF)c°9c¤0=100(1+rF)+1(1+½)(1+rF);c¤1=½(100(1+rF)+1)1+½(b)µ¤=0(c)rF=1100½¡1(d)rF013.3*3.221[1;100](a)2/(b)12/(c)(d)1(e)Pareto.(a)2maxµlog(1¡µ)+½log(100+µ(1+rF))µ¤=½(1+rF)¡100(1+½)(1+rF)c¤0=(1+rF)+100(1+½)(1+rF);c¤1=½((1+rF)+100)1+½(b)/c°103ARROW-DEBREU(c)100½(1+rF)¡1(1+½)(1+rF)+½(1+rF)¡100(1+½)(1+rF)=0(d)rF=1½¡1(e)1(½log(c¤1))0(log(c¤0))0=½c¤0c¤1=½11+rF2½11+rFPareto3.4*1abcArrow-Debreu12e1:100000e2:020010050U(c)=¡1c0¡13µ1c1a+1c1b+1c1c¶:(a)/(b)(c)(d)ck;1!=e1!k=1;2e1!!=a;b;c1/.(a)1maxc0;c1¡1c0¡13(1c1a+1c1b+1c1c)s.t.c0+Áac1a+Ábc1b+Ácc1c=1001c¤0=1001+pÁa=3+pÁb=3+pÁc=3c°11c¤1a=100p3Áa(1+pÁa=3+pÁb=3+pÁc=3)c¤1b=100p3Áb(1+pÁa=3+pÁb=3+pÁc=3)c¤1c=100p3Ác(1+pÁa=3+pÁb=3+pÁc=3)2c¤0=200Áa+100Áb+50Ác1+pÁa=3+pÁb=3+pÁc=3c¤1a=200Áa+100Áb+50Ácp3Áa(1+pÁa=3+pÁb=3+pÁc=3)c¤1b=200Áa+100Áb+50Ácp3Áb(1+pÁa=3+pÁb=3+pÁc=3)c¤1c=200Áa+100Áb+50Ácp3Ác(1+pÁa=3+pÁb=3+pÁc=3)(b)100p3Áa(1+pÁa=3+pÁb=3+pÁc=3)+200Áa+100Áb+50Ácp3Áa(1+pÁa=3+pÁb=3+pÁc=3)=200100p3Áb(1+pÁa=3+pÁb=3+pÁc=3)+200Áa+100Áb+50Ácp3Áb(1+pÁa=3+pÁb=3+pÁc=3)=100100p3Ác(1+pÁa=3+pÁb=3+pÁc=3)+200Áa+100Áb+50Ácp3Ác(1+pÁa=3+pÁb=3+pÁc=3)=50Áa=112Áb=13Ác=43(c)ÁaÁbÁc12c¤1ac¤1bc¤1c(d)12c¤1ac¤1b=sÁbÁa=2=200100c¤1ac¤1c=sÁcÁa=4=20050c°44.1145XX=2664110001210013210143213775:(a)(b)(c)(d)Arrow-Debreu(e)(f).5X1=[1;1;1;1];X2=[1;2;3;4];X3=[0;1;2;3];X4=[0;0;1;2];X5=[0;0;0;1](a)X1;X2;X3;X44(b)Á=[Á1;Á2;Á3;Á4]Á1+Á2+Á3+Á4=1Á1+2Á2+3Á3+4Á4=2:5Á2+2Á3+3Á4=1:5Á3+2Á4=0:75Á4=0:25Á=[14;14;14;14]144(c)1213[1;1;1;1]1(d)2345X2¡2X3+X4=[1;0;0;0]X3¡2X4+X5=[0;1;0;0]X4¡2X5=[0;0;1;0]X5=[0;0;0;1](e)5µ1;µ2;µ3;µ4;µ5µ1+µ2;µ1+2µ2+µ3;µ1+3µ2+2µ3+µ4;µ1+4µ2+3µ3+2µ4+µ5(f)R44.21r12r2r1r2.2124.3iriaribi=1;2(a)ria¸rib(b).(a)riaribi1i001ribria02ria¸rib(b)r1ar2b;r1br2a4.44.15123/211/2(a)(b)123.(a)1425001[0;0;1;0]0(b)21113120121013c°154.51f!1;!2;!3gabcXa=2411035;Xb=2401135;Xc=241=211=235:0Sc=Sb=Sc=1Á=[Á1;Á2;Á3](a)[2;¡1;2](b)(c)(d)[1=2;1=2;1=2](e)(f)a1=2.(a)2£1+(¡1)£1=1;(¡1)£1+2£1=1;2£12+(¡1)£1+2£12=1[2;¡12;2](b)(c)(d)12£1+12£1=1;12£1+12£1=1;12£12+12£1+12£12=1[12;12;12](e)c°1644.6Forward101F0111Stt=0;1rF11²1F0²S011111S1(a)S1S1rF(b)111yy.(a)F01+rFS0F01+rF=S0F0=(1+rF)S0(b)F0(1+rF)S0F0+y=(1+rF)S04.7/S0S0rArB1111²1F01²S0xrB11CoveredInterestRateParity.0F01+rA0S01+rBF01+rA=S01+rBF0=1+rA1+rBS04.84.1125121/2(a)(b)c°17(c)5.(a)Á=[Á1;Á2;Á3;Á4]Á1+Á2+Á3+Á4=0:8Á1+2Á2+3Á3+4Á4=2Á2+2Á3+3Á4=1:2Á4=0:2Á=[0:2;0:2;0:2;0:2]40.6rF=1Á1+Á2+Á3+Á4¡1=0:25(b)Q=[0:25;0:25;0:25;0:25](c)1EQ[X1]1+rF=11+0:25=0:8Notes12350.821.20.2c°66.1U(c)=u0(c0)+X!2­¼!u1(c1!).cc0!0x=[c0;cx]cºc0U(c)=u0(c0)+X!2­¡f!0g¼!u1(c1!)+¼!0u1(cx)¸u0(c0)+X!2­¡f!0g¼!u1(c01!)+¼!0u1(cx)=U(c0),X!2­¡f!0g¼!u1(c1!)¸X!2­¡f!0g¼!u1(c01!)y=[c0;cy]U(c)=u0(c0)+X!2­¡f!0g¼!u