浙江大学：《高级微观经济学》课程PPT教学课件（英文版）Lecture 9 time and assets market

Lecture 9: time and assets market

Lecture 9: time and assets market

Contents Inter-temporal preferences Two periods Several periods Asset market CAPM APT Complete market Pure arbitrage

Contents • Inter-temporal preferences – Two periods – Several periods • Asset market – CAPM – APT – Complete market – Pure arbitrage

Inter-temporal preferences Utility function of inter-temporal U(…cn)=∑(c,) Every period consumption c, depend on how much he consumed and invested in period t-1

Inter-temporal preferences • Utility function of inter-temporal • Every period consumption ct depend on how much he consumed and invested in period t-1. 1 1 1 ( ) ( ) T t T t t U c c u c  − = = 

Inter-temporal preferences Two periods model In the case with out any uncertainty max(co,C=u(co)+Su(c) st.(v-c0)(1+r) First order condition 0=6(1+r) If co =ci means 8

Inter-temporal preferences • Two periods model : • In the case with out any uncertainty • First order condition: • If means 0 1 0 1 0 1 max ( , ) ( ) ( ) . . ( )(1 ) U c c u c u c s t w c r c = + − + = 0 1 ( ) (1 ) ( ) u c r u c     = +  1 1 r  = + 0 1 c c   =

Inter-temporal preferences Two periods model with uncertainty investment Endowment wealth w Period1: consume CI, invest the rest wealth in two assets, (1-x) percentage has a certain return of Ro and x pays a random return of R, Period2: C2=W2=(w-CuIRX+ro(1-x)]=(w-CR Utility function: U(C,C2=u(c)+SEu(c2)

Inter-temporal preferences • Two periods model with uncertainty investment. – Endowment wealth w. – Period1: consume c1 , invest the rest wealth in two assets, (1-x) percentage has a certain return of R0 and x pays a random return of – Period2: – Utility function: R1 2 2 1 1 0 1 c w w c R x R x w c R = = − + − = − ( )[ (1 )] ( ) 1 2 1 2 U c c u c Eu c ( , ) ( ) ( ) = +

Inter-temporal preferences Two periods model Indirect utility function of period 1 with w y(w)=maxu(c)+SEu(w-Cr First order condition u'(C=dEu(CR E'(a2(R3-R0)=0

Inter-temporal preferences • Two periods model: – Indirect utility function of period 1 with w. – First order condition: 1 1 1 , ( ) max ( ) ( ) c x V w u c Eu w c R = + −  1 2 2 1 0 ( ) ( ) ( )( ) 0 u c Eu c R Eu c R R  =   − =

Inter-temporal preferences several periods model Period t: consume c. invest the rest wealth in two assets,(1-x, percentage has a certain return of Ro and x, pays a random return of R Period+ 1 t+1 (w1-c,)R Utility function U(2…7)=∑oE(e) t=0

Inter-temporal preferences • several periods model – Period t: consume ct , invest the rest wealth in two assets, (1-xt ) percentage has a certain return of R0 and xt pays a random return of – Periodt+1: – Utility function: R1 1 1 ( ) t t t t c w w c R + + = = − 1 0 ( , ) ( ) T t T t t U c c Eu c  = = 

Inter-temporal preferences ° Several periods model Indirect utility function of period T-1 VI-(WT-1= max u(C-+dEu(w--C-dr CT-1,x7-1 First order condition U(CT-=SEu(CT)R E(a7n)(R1-R)=0

Inter-temporal preferences • Several periods model: – Indirect utility function of period T-1. – First order condition: 1 1 1 1 1 1 1 , ( ) max ( ) ( ) T T T T T T T c x V w u c Eu w c R  − − − − − − − = + − 1 1 0 ( ) ( ) ( )( ) 0 T T T u c Eu c R Eu c R R  − =   − =

Inter-temporal preferences ° Several periods model For period T-2, when we got(C-2,xr-2)then DR -So T-2(WT-2)=max u(c _2)+dEV-WT-2-CT2R The first order condition u'(C-2)+SEV(W-R=0 E(wn1)R1-R)=0

Inter-temporal preferences • Several periods model: – For period T-2, when we got then – So – The first order condition: 1 2 2 ( ) w w c R T T T − − − = − 2 2 ( , ) T T c x − − 2 2 2 2 2 1 2 2 , ( ) max ( ) ( ) T T T T T T T T c x V w u c EV w c R  − − − − − − − − = + − 2 1 1 1 0 ( ) ( ) 0 ( )( ) 0 T T T u c EV w R EV w R R  − − −   + =   − =

Asset market CAPM: Capital Asset Pricing Model Consumption of the next period depend on how to invest the wealth in different assets c=(-c∑xR=(W-c)xR+∑x2R R is the return of asset a and x, is the percentage of it. Asset 0 is the no risky

Asset market • CAPM: Capital Asset Pricing Model – Consumption of the next period depend on how to invest the wealth in different assets. – is the return of asset a and is the percentage of it. Asset 0 is the no risky. 0 0 0 1 ( ) ( )[ ] A A a a a a a a c w c x R w c x R x R = = = − = − +   R a a x