Exchange Rate Economics: Theories and Evidence

256 The new open economy macroeconomics part 1
If it is assumed that all producers in a country are symmetric,that is,they set the
same price and output in equilibrium,then (10.3a) and (10.3b) may be simplified to:
P =
P ∗ =
[
np t (h) 1−θ + (1 − n)[S t p ∗ t (f )]1−θ ] 1/(1−θ)
,(10.3a ′ )
[
n[p t (h)/S t ] 1−θ + (1 − n)p ∗ t (f )]1−θ ] 1/(1−θ)
. (10.3b ′ )
10.1.2 Log-linearising around the steady state
In solving the model we follow Obstfeld and Rogoff and log-linearise the earlier
equations around the steady state. The log-linear approximations of (10.3a ′ ) and
(10.3b ′ ) around the initial steady state are:
and
ˆP t = nˆp t (h) + (1 − n)[Ŝ t + ˆp ∗ t (f )],(10.3a′′ )
ˆP ∗
t = n[ˆp t (h) − Ŝ t ]+(1 − n)[ˆp ∗ t (f )],(10.3b′′ )
where the hats denote a percentage change around the steady-state equilibrium:
that is, ˆX t = dX t / ¯X 0 ,where ¯X 0 is the initial steady-state value.
The log-linear first-order conditions for consumption (from equation (10.11)) in
the home and foreign country are:
Ĉ t+1 = Ĉ t + (1 − β)ˆr t ,(10.14)
Ĉ ∗
t+1 = Ĉ ∗
t + (1 − β)ˆr t ,(10.15)
and the corresponding log-linear money demand equations are:
(
ˆM t − ˆP t = 1 ɛ Ĉt − β ˆr t + ˆP
)
t+1 − ˆP t
,(10.16)
1 − β
(
ˆM t
∗ − ˆP t
∗ = 1 ɛ Ĉ t
∗ − β ˆr t + ˆP t+1 ∗ − )
ˆP t
∗
. (10.17)
1 − β
Comparing these equations to money market relationships used in other chapters,
we note that a key feature of these equations is that the scale variable is consumption
rather than income. Log-linear versions of equation (10.9) and its foreign
counterpart are:
ŷ t = θ[ ˆP t − ˆp t (h)]+C w
t ,(10.18)
ŷ ∗ t = θ[ ˆP ∗
t
− ˆp ∗ t
(f )]+C
w
t . (10.19)