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 − − − + = − =