浙江大学高级宏观经济学详细公式推导(上)

1.BBK模型-无生产的交易经济1.1最优问题21012112212max..,0ttitititsttttttcssstcscszsyzsycscs1.2一阶条件建立拉格朗日函数：2112212100tttititttttttittsscsLsszsyzsycscs1111=0ttttttLscsscs①1222=0ttttttLscsscs②1.3均衡消费由①、②可得：11122=ttcscs，连列121122=ttttcscszsyzsy可得：1112221122=1+1=1+ttttttcszsyzsycszsyzsy，其中（111-2=）2.BBK模型2.1AD最优问题国家1：10111100111111111111111max..,,10tttttttsttttttttssttttcssstmsasPsbsmszsycsGasbsGabwawbcs2.2AD一阶条件对国家1建立拉格朗日函数：111111000tttttttttttttttssscsLsmszsymsasPsbs首先求两个偏导数：11ca、11cb：111111111111111111111111cwcaacwawbcwcbb111111111=0tttttttLscswcsasmsas①1111111111=0ttttttttLscswcsbsmsPsbs②1111111111,1,1,1,1,1=0,1tttttttLstcstwcstastmstast③2.3AD均衡价格由①、②可得：111111tADttbswPswas由①、③可得：111111111111,1,1,1,1,1ttttADtADtttttscscsasmsmststcstcstast2.4SM最优问题国家1：1011111111111111111111111101max..,1,1,,1,100ttttttsttttttttttsttttttcssstasPsbsQstBstzsyBscsGasbsGabwawbBstBBscs2.5SM一阶条件对国家1建立拉格朗日函数：10111111210+,1,1tttttttsttttttttttttsscsLssasPsbsQstBstzsyBs首先求两个偏导数：11ca、11cb：111111111111111111111111cwcaacwawbcwcbb111111111=0tttttttLscswcsassas①1111111111=0ttttttttLscswcsbssPsbs②1111,1,10,1ttttttLsQststBst③2.6SM均衡价格由①、②可得：111111tSMttbswPswas由①、③可得：1111111111111=,1,1,1,1,1ttttttttttscscsasQststcstcstast2.7综合讨论综合AD和SM市场均衡价格讨论：ADtSMtPsPs、1,1,1ADtADtttmsmstQst3.BBK模型-社会计划者问题3.1最优问题210121112221111111111111122222222max..,0,1,1ttitititsttttttititcssstasaszsybsbszsyasbscGabwawbcGabwbwa3.2一阶条件建立拉格朗日函数：2101112111200ttttitititsatttbtttttttsscsLszsyasaszsybsbs首先求四个偏导数：11ca、11cb、22ca和22cb：111111111111111111111111cwcaacwawbcwcbb112222112122222112222211cwcbbcwbwacwcaa111111111+=0tatLcwcaa①1112222221+=0tatLawcaa②1111111111+=0tbtLcwcbb③111222222+=0tbtLcwcbb④3.3稳定均衡由①、②可得：11122221112111111=cwcacwca⑤由③、④可得：1112222111211111=1cwcbcwcb⑥由⑤、⑥可得：1111112222222211111211111111111==1cwcacwcbcwcacwcb所以21221121=11awwbbaww连列限制条件1211aazy⑦、1222bbzy⑧，可以解出：1a、2a、1b、2b。2211=abba⑨，其中1221=11。由⑧、⑨可得：1212211tabszya，12222211=1aabzyaa。由⑦可得：2111azya将以上三式代入⑥可得：11111111211122211111211111211111122111111=1111zyaawzyawzyzyawzyaaawwawzyzyaa4.BBK模型-带有贸易费用4.1最优问题国家1：10111111111111111111111111101maxln..,1,1,,1,100ttttttsttttttabttatsttttttscsstqasqbsQstBstqzsyBscsGasbsGabwawbBstBBscs国家2：2022222222122221112222221202maxln..,1,1,,1,100ttttttsttttttabttatsttttttscsstqasqbsQstBstqsyBscsGasbsGabwawbBstBBscs4.2一阶条件对国家1建立拉格朗日函数：110111111111110ln,1,1tttttttstttttttatabtttssLscssqzsyBsqasqbsQstBst首先求两个偏导数：11ca、11cb：11111111111111111111cwcaacwawbcwcbb1111111110tttaLswcasqa11111111110tttbLswcbsqb对国家2同理有：11212222210tttaLswcasqa1121222220tttbLswcbsqb4.3稳定均衡由国家1的一阶条件可得：111111abqwaqwb由国家2的一阶条件可得：122221abwqaqwb又因为稳定均衡时不存在套利机会，所以12=aaqq、12=bbqq，所以12212=aabbqqqq所以，11122121=1wwaawbwb⇒-212121=abwbaw由于我们考虑的是对称国家，所以均衡时：12=ab、21=ab所以，-121=awaw由于均衡时市场出清，所以12+=1aa（=1itizsy）由此可以推出：-12-1==1+wwabww，21-1==1+abww4.4国家1进口GDP占比11-1121+baqbqaaww4.5推导真实汇率国家1：最优化问题：11111111111min..1,1apbstGabwawb对国家1建立拉格朗日函数：111111111111Lapbwawb11111111111110Lwawbwaa111111111111110Lpwawbwbb由此可得：11111=1wapwb⇒1111wabpw代入1111111wawb可得：111111wbwwpw所以1111111,1111+1ctwwPapbpwwpwpw国家2：最优化问题：22211122222min..1,1apbstGabwawb对国家1建立拉格朗日函数：111222222211Lapbwawb111121222221110Lwawbwaa11112122222210Lpwawbwbb由此可得：122211=wapwb⇒2221wabpw代入1112211wawb可得：112211wbwwpw所以，1122222,22+111ctwwPapbpwwpwpw所以，112222,11,1111+11111+1ctctwwpwwpwpwPRERPwwpwwpwpw4.6社会计划者问题210121122111111111111222222maxln..,0,1,1ttitititsttttttititscsstasaszsbsbszsasbscGabwawbcGabwbwa建立拉格朗日函数：2101112111200lnttttitititsatttbtttttttssLscszsyasaszsybsbs首先求四个偏导数：11ca、11cb、22ca和22cb：111111111111111111111111cwcaacwawbcwcbb112222112122222112222211cwcbbcwbwacwcaa11111111+=0tatLwcaac①112222211+=0tatLwcaac②111111111+=0tbtLwcbbc③11222221+=0tbtLwcbbc④由①、②可得：11221112111=wcawca⑤由③、④可得：1122111211=1wcbwcb⑥由⑤、⑥可得：1111222211111211111==1wcawcbwcawcb所以22211=1awbbaw连列限制条件121aaz⑧、122bbz⑨，可以解出：1a、2a、1b、2b。由⑦、⑧可得：1221211awbzaw，122222211=11awawbzawaw。由⑧可得：112aza将以上三式代入⑥可得：1+1112212222111121212121212211111=11awawwzwaawawwawzaawwzawzaw5.简单小型开放经济交易模型5.1最优问题01100maxln..10ttttttttttcstcbyrbbBcbb5.2一阶条件建立拉格朗日函数：100ln1ttttttttttLcyrbcb0ttttLcc①11110ttttLrb②5.3稳定均衡由①可以推出：+1+1=ttttcc由②可以推出：11=1tttr所以+11=1tttccr由约束条件可得：112111=111ttttttttttttcyrbbyrbcbyr⇒121111+==1111ttttttttttttcbycccyrbrrr⇒231212112+=1+11111ttttttttttttttbyycccyrbrrrrr…⇒-+11+1+111111TTTtttttTtttstscyrbbyrr又因为+1+111TTtbr⇒-1+1111TTtttttttstscyrbyr⇒111111Ttttttstscyrbyr5.4经常账户1tttttttCAbbrbyc111111TttttttttstsCArbyyrbyr年金价值111sttsstryyrr，t期财富折现值：111stttsstWrbyr⇒1tttWybrr11ttttCAyyWr6.带有生产的小型开放经济交易模型6.1最优问题10021111111211max1..1121lnln0,ttttttttttttttttttddttttthcEstrdckkkdykyAkhrreAANIID6.2一阶条件建立拉格朗日函数：12101111011112tttttttttttttttthcLEdAkhkrdckkk2111111102tttttttttttttLdAkhkrdckkk①0ttttthLcc②110ttttttttthLchAkhh③1110tttttttLErd④11111111211110tttttttttttttLkkEAkhkkk⑤6.3稳定均衡均衡时：1ttkkk，1ttyyy，1tthhh，1tt，1ttAAA，d、r外生给定。6.3.1利用②tLc、③tLh、⑤1tLk推出h、k由⑤可以推出：111-1-11hkA由②、③可以推出：11111-11hhAAk所以，11111-1-1-1khA6.3.2利用11lnlntttAA推出AlnlnAA⇒1A6.3.3利用①tL推出c2111111102tttttttttttdAkhkrdckkk⇒1cAkhrdk6.4Log线性化6.4.111ttttkhAh111ˆˆˆˆˆˆ111111ˆˆˆ1ttttttttttttthAkhAktttkhAhhAkheAekeheAkehA