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9.1. THE REDUX MODEL 275⇒where ĝt w ≡ G w t /C0 w . 5 Do the same for PPP (9.12) and the domesticand foreign price levels (9.10)-(9.11) to getŜ t = ˆP t − ˆP t ∗ , (9.44)ˆP t = nˆp t (z)+(1− n)(Ŝt +ˆp ∗ t (z∗ )), (9.45)ˆP t ∗ = n(ˆp t (z) − Ŝt)+(1− n)ˆp ∗ t (z∗ ). (9.46)Log-linearizing the world demand functions (9.20) and (9.21) givesŷ t (z) = θ[ ˆP t − ˆp t (z)] + Ĉw t +ĝ w t , (9.47)ŷ ∗ t (z ∗ ) = θ[ ˆP ∗t − ˆp ∗ t (z ∗ )] + Ĉw t +ĝ w t . (9.48)Log-linearizing the ‘labor supply rules’ (9.30) and (9.31) gives(1 + θ)ŷ t (z) = −θĈt + Ĉw t +ĝt w , (9.49)(1 + θ)ŷt ∗ (z∗ ) = −θĈ∗ t + Ĉw t +ĝt w . (9.50)Log-linearizing the consumption Euler equations (9.26)—(9.27) givesĈ t+1 = Ĉt +(1− β)ˆr t , (9.51)Ĉ ∗ t+1 = Ĉ∗ t +(1− β)ˆr t , (9.52)and Þnally, log-linearizing the money demand functions (9.28) and(9.29) gives"ˆM t − ˆP t = 1 Ĉ t − β ˆr t + ˆP t+1 − ˆP t²1 − βˆM t ∗ − ˆP t ∗ = 1 Ã"Ĉ t ∗ ²− β ˆr t + ˆP t+1 ∗ − ˆP t∗1 − βÃ!#!#, (9.53). (9.54)5 The expansion of the Þrst term about 0-steady state values is,∆n(p t (z)/P t )y t (z) = n(y 0 (z)/P 0 )(p t (z) − p 0 (z)) + n(p 0 (z)/P 0 )(y t (z) − y 0 (z)) −n[(p 0 (z)y 0 (z))/P0 2 ](P t − P 0 ). When you divide by C0 w ,notethatC0w = y 0 (z) andP 0 = p 0 (z) togetn[ˆp t (z) − ˆP t +ŷ t (z)]. Expansion of the other terms follows in ananalogous manner.