Notes on Galì 2002 NBER Eleni Iliopulos PSE, University of Paris 1
1
The model
Here I present the main equations of the model (the numbering corresponds to the NBER version of Galì’s paper) Technology shock (p. 6): at =
a
at
gt =
g gt 1
1
+ "at
Demand shock (p.6): + "gt
New Keynesian PC (19, p. 9): t
= Et
t+1
+ kxt
Euler equation (20, p.9): xt =
1
(rt
Et
rrt ) + Et xt+1
t+1
Money demand (5, p.5) mt
pt = yt
rt
Equilibrium real output (11, p 7): yt =
+
a at
+
g gt
Equilibrium real (natural) rate (14, p 7): rrt =
+
a a
at + 1
1
g
1
g
gt
; g = +' ; = +' (where = where = log ; a = 1+' +' subsidy, =mark-up and le’s assume = : the equilibrium allocation under ‡exi price coincides with the e¢ cient allocation ! = 0):Moreover, =intertemporal elasticity of substitution in consumption and ' =intertemporal elasticity of substitution in labor supply. Finally, xt = yt yt =output gap. To close the model we now need to specify a monetary policy.
2
Monetary policy shocks (section 4.1, p 14)
The money growth rate is supposed exogenous: Money supply process (25, p. 14): mt =
mt
m
1
+ "m t
The model is thus composed by the following equations: (19, p 9), (20, p 9), (5, p. 5), (25, p 14). On the top of that we need an equation to link the above rate of growth of nominal money, to real money balances,(5, p. 5), i.e. we de…ne: mt where yt
t
pt )
(mt
pt 1 ) +
1
t
pt 1 : Finally, substitute (5, p 5) into (20, p.9) to get:
= pt
yt =
(mt
1 1
(yt
(mt
pt ))
Et
rrt + Et fyt+1
t+1
yt+1 g
For the moment we just focus on money supply shocks. In the Soderlind codes we can write (see galimodel_simple_rules.m): 3 2 3 2 3 32 32 2 0 0 1 0 0 0 mt+1 mt 1 m0 7 6 7 6 6 0 1 0 0 7 6 mt pt 7 6 1 1 0 1 7 6 mt 1 pt 1 7 6 0 7 76 7=6 6 1 +4 7 "m 1 5 4 0 1 1= 5 4 yt+1 5 4 0 0 1 + 5 0 54 yt 0 t 0 0 0 0 0 0 k 1 t+1 t 1st line: money supply; 2nd line: money growth; 3rd line: Euler eq.; 4rth line: NKPC.
3
Taylor rule
We use the following Taylor rule (34, p 24): rt =
+
t
+
x xt
Substituting it into (20, p 9), we get: xt = xt 1 +
x
=
1 1
( +
t
( +
t
+
x xt
Et 2
t+1
Et
t+1
rrt ) + Et xt+1
rrt ) + Et xt+1
Expand rrt to explicit the shocks: xt 1 + Et xt+1 +
1
Et
x
=
t+1
1
+
Et
t x
= xt 1 +
+
t+1
a a
1 t
at
a a
at 1
1
g
1
g
gt
1
g
g
gt + Et xt+1 (1)
The model is thus made of eq (19 p 9), the two dynamic equations for the shocks at p 6 and 1 2 32 3 100 0 at+1 6 0 1 0 0 7 6 gt+1 7 6 76 7 4 0 0 1 1= 5 4 xt+1 5 000 t+1 32 2 3 2 3 0 0 0 at 10 a 7 6 6 0 7 6 7 a 0 1 g g 7 6 t 7 + 6 0 1 7 "tg =6 x 1 5 4 x t 5 4 0 0 5 "t 4 1 1 a a g g 1+ 00 0 0 k 1 t But the Soderlind codes do all this for you!! You just need to expand the Euler equation to write explicitly all shocks: xt = Et xt+1 +
1
Et
t+1
1
= xt +
rt 1
Et (rt
+
t+1
)
at
a a
at +
a a
1
g
1 1
1
g
g
gt
The model can take the form (see Lecture 2): Xt+1 = (A indeed: 2 100 0 60 1 0 0 6 4 0 0 1 1= 000
where
32
3 2 at+1 7 6 gt+1 7 6 76 7 6 5 4 xt+1 5 = 4 t+1
BF ) Xt + "t
0
a
0
g
1
a a
2
0
3 0 6 0 7 7 +6 4 1= 5 [rt 0 ut =
F xt 3
g
0
1 2
g
3 10 6 0 1 7 "at 7 ]+6 4 0 0 5 "gt 00
0 0 1 k
3 32 0 at 6 7 17 7 6 gt 7 0 5 4 xt 5 1 t
g
gt
+ Et xt+1
and [rt
00
]=
2
3 at 6 gt 7 6 7 4 xt 5
x
t
see …le galìmodel_taylor_rule.m Note that: 2
3 100 0 60 1 0 0 7 7 A=6 4 0 0 1 1= 5 000 2 6 =6 4
1
2 6 6 4
a
a a
0
6 B F =6 4
1
k
1
g
2
3 2 0 0 6 0 7 6 0 6 7 6 4 1= 5 = 4 1= 0 0 2 10 60 1 =6 x 40 0 00 0
0
g
0
1
3 0 0 0 17 7 +1 1 5 k
a
a a
g
0
0
3 100 0 60 1 0 0 7 7 B =6 4 0 0 1 1= 5 000 2 3 0 6 0 7 7 B F =6 4 1= 5 0 0 0
1
g
0
g
0
2
g
1
g
1
2
A
0
0
a a
0
a
1
g
3 7 7 5
1 g
x
4
3 0 17 7 0 5 1
1
0 0 x 1= 0
0 0 + k +1 k
0
0 0 1 k
3 0 7 0 7 1= 5 0 0 1 1
1 1
3 7 7 5
