# 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 hcat_ 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 EtAlnCf+ 1 = hAlnCf. Then use

да да да

Yk'E, {lnNOM= K?KEt {AlnNOM}, ^KE, {AlnNOM)

i=1 & i=0 i=1

+A NOt = ^KlEt {AlnNOt+i] and add and subtract

l=0

да

(1 -y)(hK)^KlEt {AlnNOt+^ to obtain (8.11) in the main text:

l=0 where ft =(1 -y)——— ( ln ( _ Et _1A ln Cf'j. Solving under the 1 кН

assumption that AlnNOt=pAlnNOt - 1 + et gives the following current account reaction function: Solving under the assumption that lnNOt = p ln NOt— 1 + et, in which case AlnNOt = (p — 1) ln NOt — 1 + et and

• (1 -у)(Г1к))[к‘ (Et Et_!)AlnNOt+i =-(1 st and
• (1 -y)(1 - h^fjc'Et A ln NOt+i =-(1 -p)(1 - r№~ hK ln NOt-

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

dst

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