Inter-temporal preferences Two periods model Indirect utility function of period 1 with w y(w)=maxu(C)+SEu(w-CR First order condition (c1)=6El(C2)R El(2)(R-R)=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 Periodt+ 1: C=W=(W-CR Utility function (1…n)=∑oEl(c) 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 V_(Wr-1)=max u(c-1+SEu(w DR T-12x7-1 First order condition U(CTD=SEu(CR E(xn)(R1-R0)=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, x-2)then T-2T-2 DR So VI-2(Wr-2)=max u(C-2)+SEVT-WT-2-CT-R CT-2 T-2 The first order condition u(C -2)+SEV(W-R=0 E(w1)(R1-R0)=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 − − − + = − =