International macroe.. - Free

298CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSTo solve for the exchange rate take Ŝ from (9.171) and plug into(9.174) to get"1+#β ³ˆMt −²(1 − β)ˆM´t∗ 1 =³Ĉt −²Ĉ∗ t´+Using (9.177) to eliminate Ĉ − Ĉ∗ ,yougetβ² 2 (1 − β)³Ĉ− Ĉ ∗´+ β²(1 − β)Ŝt.Ŝ t =β + ²(1 − β)β(² − 1)h²( ˆMt − ˆM ∗ t ) − (Ĉt − Ĉ∗ t )i . (9.178)This is not the solution because Ĉt − Ĉ∗ t is endogenous. To get thesolution, you have from the consolidated budget constraints (9.143)and (9.144)Ĉ t = nˆx t (z)+(1− n)[Ŝt +ˆv t (z)] − βˆb t , (9.179)Ĉ ∗ t = (1− n)ˆx ∗ t (z ∗ )+n[ˆv ∗ t (z ∗ ) − Ŝt]+β n1 − nˆb t , (9.180)(201-202)⇒and you have from (9.147)—(9.150)ˆx t (z) =Ĉt; ˆx ∗ t (z∗ )=Ĉ∗ t ; ˆv t(z) =Ĉ∗ t ; ˆv∗ t (z∗ )=Ĉt. (9.181)Subtract (9.180) from (9.179) and using the relations in (9.181), youhaveβŜ t =(Ĉt − Ĉ∗ t )+2(1 − n) 2ˆb t . (9.182)Substitute the steady state change in relative consumption (9.170) into(9.177) to get2θ(1 − n)ˆb = −β(1 + θ) [Ĉt − Ĉ∗ t − Ŝt], (9.183)and plug (9.183) into (9.182) to getĈ t − Ĉ∗ t − Ŝt =2θ(1 + θ) [Ĉt − Ĉ∗ t − Ŝt].It follows that Ĉt −Ĉ∗ t −Ŝt =0. Looking back at (9.183), it must be thecase that ˆb = 0 so there are no current account effects from monetaryshocks. By (9.164) and (9.165), you see that Ĉ = Ĉ∗ =0,andby