1 The model (equations and variables)
1.1 The model in brief
The model you will analyze is a simplied version of the New-Keynesian small open-
economy (SOE) model in Justiniano and Preston (2010), which in turns, is based on the
model in Monacelli (2005) and Gali and Monacelli (2005). Compared to the model in Jus-
tiniano and Preston, our simplied model assumes that the law of one price (LoP) holds
for all retail imported goods and there is no price indexation for these imported goods.
Aggregate utuations are also only driven by six exogenous shocks: risk premium, mone-
tary policy (money supply), preference ("consumer condence"), foreign in ation, foreign
output, and foreign interest rate.
The model can be derived from the ground-up (micro-foundations) based on optimizing
households, domestic rms and importers, etc., resulting in a set of non-linear equations.
We will instead work directly with the log-linearized equilibrium equations, listed
below.
1.2 The log-linearized equations
Consumption Euler-equation (the IS equation):
bct = h
1 + h
bct1 + 1
1 + h

Etbct+1 1

1 h
1 + h
hbit Etbt+1i+ g^t (1)
Goods-market clearing condition:
byt = (1 )bct + byt + (2 )bSt (2)
bqt = (1 )bSt (3)
Changes (growth rate) of the terms of trade:
bSt bSt1 = bF;t bH;t (4)
1
Domestic-price in ation (the "Phillips curve"):
(bH;t H ^H;t1) = Et (bH;t+1 H ^H;t) + (1 H)(1 H)
H
cmct (5)
The real marginal cost: cmct = 'byt + bSt + bct (6)
The wedge between CPI- and PPI-in ation:
bt = bH;t + bSt bSt1 (7)
The uncovered interest-parity (UIP) condition:
bit bit = Etbect+1 bat + Etbt+1 (8)
The net-foreign-assets position (the current account):
byt bct = bat 1bat1 +
(1 )bqt (9)
Imported-good in ation (based on the law of one price):
bF;t = bect + bt (10)
Monetary-policy (Taylor) rule:bit = ibit1 + bt + ybyt + ybyt + ebect "m;t (11)
Evolution of risk premium: bt = bt1 + ";t (12)
Evolution of foreign output:
byt = ybyt1 + "y;t (13)
Evolution of foreign in ation:
bt = bt1 + ";t (14)
Evolution of foreign interest rate:bit = ibit1 + "i;t (15)
Evolution of preference shock:
g^t = g g^t1 + "g;t (16)
2
Denition of variables and shocks NOTE: all hatted variables are in terms of log or
percentage deviation from the steady-state value, except for bit, bt, bH;t, bF;t, bt , and bit ,
which are in terms of level deviation from the steady state (e.g. bit it i).
bct consumption (per capita)bit nominal interest ratebt CPI in ationbyt outputbSt terms of trade (price of exports in terms of imports)bqt real exchange ratebH;t domestic-goods (PPI) in ationbF;t imported-goods in ationcmct real marginal costbat domestic-households' holding of foreign assetsbect domestic-currency depreciation rate (% change in the exchange rate)byt foreign outputbt foreign in ationbit foreign interest ratebt relative risk premiumbgt consumer preference
"m;t monetary-policy shock (i.i.d.)
"y;t foreign-output shock (i.i.d.)
";t foreign-in ation shock (i.i.d.)
"i;t foreign interest-rate shock (i.i.d.)
"g;t preference shock (i.i.d.)
3
D
e

n
it
io
n
o
f
p
a
ra
m
e
te
rs
a
n
d
th
e
ir
c
a
li
b
ra
ti
o
n
P
ar
am
et
er
s
D
e
n
it
io
n
V
a
lu
e

in
ve
rs
e
in
te
rt
em
p
o
ra
l
el
a
st
ic
it
y
o
f
su
b
st
it
u
ti
o
n
1

o
p
en
n
es
s
p
a
ra
m
et
er
0
.2
5

el
a
st
ic
it
y
o
f
su
b
st
it
u
ti
o
n
b
et
w
ee
n
d
o
m
es
ti
c
a
n
d
im
p
o
rt
ed
g
o
o
d
s
0
.8
0

su
b
je
ct
iv
e
d
is
co
u
n
t
fa
ct
o
r
0
.9
9
H
p
ro
b
a
b
il
it
y
o
f
p
ri
ce

x
it
y
fo
r
d
o
m
es
ti
c
g
o
o
d
s
0
.7
0

ri
sk
-p
re
m
iu
m
p
a
ra
m
et
er
0
.0
1
'
in
ve
rs
e
F
ri
sc
h
la
b
o
r-
su
p
p
ly
el
a
st
ic
it
y
1
.2
6
h
d
eg
re
e
o
f
h
a
b
it
fo
rm
a
ti
o
n
0
.2
5

co
n
su
m
p
ti
o
n
el
a
st
ic
it
y
o
f
p
re
fe
re
n
ce
sh
o
ck
0
.1
2
H
d
eg
re
e
o
f
p
ri
ce
in
d
ex
a
ti
o
n
0
.2
0

i
T
ay
lo
r-
ru
le
in
te
re
st
sm
o
o
th
n
es
s
0
.7
5

T
ay
lo
r-
ru
le
re
sp
o
n
se
to
in

a
ti
o
n
2
.0
1

y
T
ay
lo
r-
ru
le
re
sp
o
n
se
to
o
u
tp
u
t
0
.0
8

y
T
ay
lo
r-
ru
le
re
sp
o
n
se
to
o
u
tp
u
t
g
ro
w
th
0
.6
7

e
T
ay
lo
r-
ru
le
re
sp
o
n
se
to
ex
ch
a
n
g
e-
ra
te

u
ct
u
a
ti
o
n
s
0

A
R
(1
)
co
e
ci
en
t
o
f
ri
sk
-p
re
m
iu
m
sh
o
ck
0
.9
5

y

A
R
(1
)
co
e
ci
en
t
o
f
fo
re
ig
n
-o
u
tp
u
t
sh
o
ck
0
.5
5

A
R
(1
)
co
e
ci
en
t
o
f
fo
re
ig
n
-i
n

a
ti
o
n
sh
o
ck
0
.3
5

i
A
R
(1
)
co
e
ci
en
t
o
f
fo
re
ig
n
in
te
re
st
-r
a
te
sh
o
ck
0
.6
5

g
A
R
(1
)
co
e
ci
en
t
o
f
p
re
fe
re
n
ce
sh
o
ck
0
.8
0

m
,

,

y
,

,

i
,

g
st
a
n
d
a
rd
d
ev
ia
ti
o
n
s
o
f
sh
o
ck
s
1
4
The Questions
1. Solve the model described above using Dynare. Obtain the impulse response for 10
periods to a one-time 1% shock to
a. money supply or the domestic interest-rate shock ("m;t);
b. preference ("g;t);
c. foreign output ("y;t).
Analyze (i.e. explain the dynamics) and plot the eect of each of these shocks to
domestic output (byt), consumption (bct), interest rate (bit), in ation (bt), domestic-currency
nominal depreciation (bect), and the "shocked" variable (e.g. if it's a foreign output shock,
plot byt ). Relate your analysis to what you have learned in the rst half of the course (the
"qualitative model"). For the money supply or the domestic interest-rate shock, do you
observe an overshooting of the nominal exchange rate?
[Extra points: (i) plot the six variables above in a 3x2 plot (with 3 rows and 2
columns); (ii) plot the evolution of the nominal exchange rate and the current account in
a separate gure and explain the dynamics.]
2. In the model above, we assume a fully- exible ( oating) exchange rate system.
Suppose that the central bank also directly intervenes in the foreign exchange market, i.e.
it's operating under a managed oating exchange rate. This policy can be analyzed within
our model by assuming that
e = 0:65 > 0
Redo #(1.c) above, i.e. plot and analyze the impulse response to a 1% foreign-output shock,
under this alternative policy. Particularly, analyze the eect of this policy on the size of
the uctuations of CPI-in ation, output, and the nominal exchange rate, in comparison to
the case with e = 0.
3. Now assume that the central bank is operating under a xed exchange-rate system.
Specically, the monetary policy rule in equation (11) is replaced with the following policy
rule: bect = 0
This policy rule eectively (and credibly) xes the nominal exchange rate at a specied
level. Redo #(1.c) above, i.e. plot and analyze the impulse response to a 1% foreign-
output shock, under this policy. Particularly, analyze the eect of this policy on the size
of the uctuations of CPI-in ation, output, the nominal interest rate, and the nominal
5
exchange rate, in comparison to the oating exchange-rate system (both when e = 0 and
e = 0:65).
[Notes: (i) Dynare does not plot the impulse response of a variable if that variable is
always constant (zero deviation from the steady state), (ii) since the foreign-debt holding,bat, enters the UIP condition in equation (8), you will generally not nd bit = bit , unless
= 0)]. [Extra points: plot the variables under the three policies in one gure, e.g. the
plot for output should include three dierent impulse responses.]
6
References
 Gali, J. and T. Monacelli. 2005. "Monetary policy and exchange rate volatility in a
small open economy". Review of Economic Studies 72: 707-734.
 Justiniano, A. and B. Preston. 2010. "Monetary policy and uncertainty in an empirical
small open-economy model". Journal of Applied Econometrics 25: 93-128.
 Monacelli, T. 2005. "Monetary policy in a low pass-through environment". Journal of
Money, Credit, and Banking 37(6): 1019-1045.
7 