# Derivation of the Current Account Reaction Function with Internal Habits and a Constant World Real Interest Rate

Start with the following expression for the current account

In order to solve out for the growth rate of consumption in the current account equation, we add and subtract *hca _{t}_*

_{1}on both sides of the equation. This yields

Now note that with internal habits in consumption, Eq. (8.4) can be log-linearized as E_{t}AlnCf+ _{1} = hAlnCf. Then use

да да да

Yk'E, {ln*NO*_{M}= *K?KE**_{t}* {Aln

*NO*

_{M}},

*^KE,*{Aln

*NO*

_{M})* i=1 & i*=0 i=1

*+A
NO _{t} = ^K^{l}E_{t}* {Aln

*NO*

_{t}

**+***i]*and add and subtract

l=0

да

(1 *-y)(hK)^K ^{l}E_{t}* {Aln

*NO*to obtain (8.11) in the main text:

_{t}+^l=0

where *f _{t}* =(1

*-y)———*( ln (

*_ E*ln

_{t}__{1}A*Cf'j.*Solving under the 1

*кН*

assumption that AlnNO_{t}=pAlnNO_{t - 1} + *e _{t}* gives the following current account reaction function:

Solving under the assumption that lnNO_{t} = *p* ln *NO _{t—}* 1 + e

_{t}, in which case AlnNO

_{t}= (p — 1) ln

*NO*1 +

_{t —}*e*and

_{t}- (
^{1 -}*у)(***Г***1к))[к‘*(*E*_{t}^{—}**E**)A_{t}_!^{lnNO}*t+i*=^{-}(^{1}*s*_{t}^{and} - (1 -y)(1 -
**h^fjc'E***t*A ln**NO***t+i*=-(1 -p)^{(1 - r}№~^{hK}ln**NO***t-*

i=0 ^{1 - pK}

yields the following current account reaction function:

It is easy to show that for log-level net output shocks, ^{dCa}‘* >* 0, and

*ds,*

*d ca*

that for log-difference net output shocks, ^{L} < 0, meaning that this

*ds _{t}*

model provides identical identification restrictions to the model derived under the assumption of external habits.