International macroe.. - Free

294CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSFrom the money demand functions it follows that the exchange rate isS 0 = M 0. (9.140)M0∗Log-linearizing around the 0-steady state. The log-expansion of (9.114)and (9.115) around 0-steady state values gives 11ˆM t − ˆP t = 1 β² Ĉt +²(1 − β) ˆδ t , (9.141)ˆM t ∗ − ˆP t ∗ = 1 ² Ĉ∗ t + β²(1 − β) [ˆδ t + Ŝt+1 − Ŝt]. (9.142)Log-linearizing the consolidated budget constraints (9.136) and (9.137)with B 0 = G 0 = G ∗ 0 =0gives 12(191-192)⇒Ĉ t = n[ˆp t (z)+ˆx t (z)− ˆP t ]+(1−n)[ˆq ∗ t (z)+Ŝt +ˆv t (z)− ˆP t ]−βˆb t , (9.143)Ĉt ∗ =(1−n)[ˆp ∗ t (z ∗ )+ˆx ∗ t (z ∗ )− ˆP t ∗ ]+n[ˆq t (z ∗ )−Ŝt+ˆv t ∗ (z ∗ )− ˆP t ∗ ]+βnt .1 − nˆb(9.144)Log-linearizing (9.128)—(9.133) givesŷ t (z) = nˆx t (z)+(1− n)ˆv t (z), (9.145)ŷt ∗ (z∗ ) = (1− n)ˆx ∗ t (z∗ )+nˆv t ∗ (z∗ ), (9.146)ˆx t (z) = θ[ ˆP t − ˆp t (z)] + Ĉt, (9.147)11 Taking log-differences of the money demand function (9.114) givesˆM t − ˆP³ ´t = 1 ² [Ĉt−(ln(1−δ t )−ln(1−δ 0 ))]. But ∆(ln(1−δ t )) ' −δ0 δ t−δ 01−δ 0 δ 0= −β ˆδ 1−β t ,which together gives (9.141).12 Write (9.136) as C t = p t(z)x t (z)∆C t = C t − C 0 = ∆the Þrst term is ∆hpt(z)x t(z)P t+ S tqt ∗ (z)v t(z)P t−h i h δtBtP tStqt ∗ (z)vt(z)P t− ∆δt B tP t. It follows thati. The expansion ofiP t+ ∆h ipt(z)x t(z)P t= x 0 (z)[ˆx t +ˆp t − ˆP t ] because P 0 = p 0 (z). The expansionof the second term follows analogously. h i To expand the third term, notingthat P 0 = 1, δ 0 = β, and B 0 =0gives∆ δtB tP t= βB t . After dividing through byC0w = y 0(z), and noting that x 0 (z)/y 0 (z) =n, andv 0 (z)/y 0 (z) =(1 − n), we obtain(9.143).