International macroe.. - Free

9.1. THE REDUX MODEL 283changes in consumption is∆Ut c = Ĉt +β Ĉ. (9.98)1 − βTo determine the effect on utility of leisure, in the 0-steady stateU y t−1 = −(ρ/2)[y0 2 +(β/(1 − β))y0]. 2 Directly after the shock,Ut y = −(ρ/2)[yt 2 +(β/(1 − β))y 2 ]. Using the Þrst-order approximation,yt 2 = y0 2 +2y 0 (y t − y 0 ), it follows that, ∆Uty = −(ρ/2)[(yt 2 − y0)+ 2 ⇐(178)(β/(1 − β))(y 2 − y0)]. 2 Dividing through by y 0 yields∆U y t= −ρ"y 2 0ŷt +Now use the fact that C 0 = y 0 = C0wà !(θ − 1)∆Ut c + ∆U ty = Ĉt − ŷ t +θAnalogously, in the foreign country∆U c∗t+ ∆U y∗t = Ĉ∗ t − Ã(θ − 1)θ!β(1 − β) y2 0ŷ#= ³ ´1/2,θ−1ρθ togetβ(1 − β)ŷ ∗ t +"Ĉ −"βĈ ∗ −(1 − β). (9.99)#(θ − 1)ŷ . (9.100)θ#(θ − 1)ŷ ∗ .θ(9.101)To evaluate (9.101), Þrst note that ŷ t = θ(1−n)Ŝt+Ĉw t which followsfrom (9.75). From (9.89) and (9.90) it follows that Ĉt = bŜt +Ĉ∗ t whereb =[r(θ 2 − 1)/(r(1 + θ) +2θ)]. Eliminate foreign consumption usingĈt ∗ =(Ĉw t − nĈt)/(1 − n) togetĈ t = (1 − n)r(θ2 − 1)Ŝ t +r(1 + θ)+2θĈw t . (9.102)Now plug (9.102) and (9.93) into (9.77) to get the long-run effect onconsumptionĈ = r(1 − n)(θ2 − 1)(9.103)[(r(1 + θ)+2θ)]Ŝt.Substitute Ĉ into (9.65) to get the long-run effect on home outputŷ =−rθ(1 − n)(θ − 1)Ŝ t . (9.104)r(1 + θ)+2θ