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AppendixDeriving the Linearized Budget ConstraintFirst, to derive Eq. (8.5), start with: then note that R_{t}, _{1} = 1, factor out CR and NO_{t} on the LHS and RHS, respectively:
Then divide the whole expression by NO_{t}
Now recall that 1/ (_{+1} ( + r^j and take logs and exponential in the infinite summations:
Applying the log to the interest rate factorial, it is easy to see that:
We can then add and subtract infinitely many lnCf s and lnNO_{t}’s:
which is Eq. (8.6) in the main text:
To linearize this expression, we use the standard formulaf(x) = f(a) +f (x) (x—a)
“1Г к!Г 1 " Now note that 1 + УК = 1 + кУК = 1 + = and _ м J L J L 1 К L1К that in the steady state, when all the hat variables take the value of 0, да да _{c} 1 _{+} ?K = 1 + ?K , then it is easy to obtain the final linearized form: _ i=1 _ _ i=1 _
Derivation of the Current Account Reaction Function with External Habits and a Constant World Real Interest RateWe start by linearizing the definition of aggregate consumption C = (1 y)CR + yCN^{r} to obtain
Following Obstfeld and Rogoff (1996), the current account can be expressed as:
which, noting that B_{t} = (1 y)B^{p} + B^{G}t and r = e^{ln(1}+^{r)} — 1, can be linearized as the following the current account to net output ratio:
CA where ca_{t} = . Now substitute Eq. (8.13) into Eq. (8.14), then into ' NO_{t 1} Eq. (8.15). Simplifying and using — = 1 and (e“ — 1) = r as in Kano (2008), yields: ^{e}
The constant world real interest rate assumption implies ln(1 + r_{t}) = 0 V i. We also add and subtract (1 — y)hca_{t}_ _{1} to obtain
Furthermore, loglinearizing the Euler equation gives
which can be expressed as
Substituting Eq. (8.17) into Eq. (8.16) gives
which corresponds to Eq. (8.8) in the main text. To further solve this expression, it is necessary to assume a stochastic process for Д1пЛ^О_{(}. Under the assumption of Д1пЛ^О,=рД1п^О,_ _{1} + e_{t} Equation (8.8) can be expressed as follows:
да where we replaced vc) ~y){^{1} ~)( ^{ E}t1))^{lnNO}t+i with i=0 ^{hK}(^{1}~y)(^{(} ~r) 2^{nd} (1 _^)1 _ h_{K}y_{K}^{i}E_{t} Д ln NO_{t}+i w^{ith}
Eq. (8.9) in the main text:
To derive Eq. (8.10) note that under the assumption that ln NO _{t}=p ln ln NO _{t} _ _{1} + e_{t}, Д ln NO _{t} = (p _ 1) ln NO'_{t} _ _{1} + e _{t} and follow the steps above. In this case

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