Lectures on Monetary Policy, In‡ation and the Business Cycle Monetary Policy Tradeo¤s: Discretion vs. Commitment
by Jordi Galí
The Monetary Policy Problem 1 X t min E0f [
yet2 +
y
t=0
subject to:
2 t ]g
(1)
= Etf t+1g + yet + ut where futg evolves exogenously according to t
ut =
u
ut
1
+ "t
In addition: yet =
1
(it
Etf
yt+1g rtn) + Etfe
t+1 g
Note: utility based criterion requires
y
=
(2)
Optimal Policy with Discretion Each period CB chooses (xt;
t) y
subject to
yet2 +
2 t
= yet + vt Etf t+1g + ut is taken as given. t
where vt
to minimize
Optimality condition:
yet =
Equilibrium
(3)
t y
= y q ut yet = q ut it = rtn + q [ (1 u) + t
where q
2+
1 y (1
u)
y u]
ut
(4) (5) (6)
Implementation: it = rtn + [(1 uniqueness condition:
y
u)
+ y
u]
t
> 1 (likely if utility-based:
> 1)
Alternatively, it = rtn + q [ uniqueness condition:
(1
u)
> 1:
+
y u]
ut +
(
t
yq
ut )
Optimal Policy with Commitment State-contingent policy fe yt ;
1 E0 2
1 t gt=0 1 X
that maximizes t
(
y
t=0
subject to the sequence of constraints: t
=
Et f
Lagrangean: L=
1
X 1 E0 2 t=0
t
[
y
First order conditions:
yet2 + y
t
for t = 0; 1; 2; :::and where
t+1 g
yet
+
1
2 t
+2
= 0.
t
(
=0
t 1
2 t)
yet + ut
+
t t
yet2 +
=0
t
yet
t+1 )]
Eliminating multipliers: ye0 =
for t = 1; 2; 3; :::..
yet = yet
(7)
0 y 1
t
(8)
y
Alternative representation:
for t = 0; 1; 2; :::where pbt
yet =
pt
p
y 1
.
pbt
(9)
Equilibrium pbt = a pbt for t = 0; 1; 2; :::where a
1
+ a Etfb pt+1g + a ut y
y (1+
)+
2
Stationary solution:
pbt =
for t = 0; 1; 2; :::where
pbt 1 + (1 p 2
1
1 4 a 2a
u)
ut
(10)
2 (0; 1):
! price level targeting ! yet =
for t = 1; 2; 3; :::as well as
yet
ye0 =
1
y (1
y (1
u)
u)
u0
ut
(11)
Optimal Monetary Policy: Discretion vs. Commitment
Appendix: Sources of Cost Push Shocks Variations in desired price markups. t Assumption: time varying desired markup: nt t 1 Log-linearized optimal price setting rule: 1 X pt = (1 ) ( )k Etf nt+k + mct+k + pt+k g = (1
)
k=1 1 X
(
k=1
where mc ft
mct + t
= = =
n t.
Thus,
Etf Etf Etf
+ t+1 g + t+1 g +
t+1 g
)k Etfmc f t+k + pt+k g mc ft mc c t + ( nt ) (yt y t) + ( nt
)
where y t equilibrium output under a constant price markup .
Exogenous Variations in Wage Markups mct = wt at = w;t + mrst at = w;t + ( + ') yt
(1 + ') at
Thus, mc c t = ( + ') (yt
y t) + (
w)
w;t
where y t : equilibrium output under a constant price and wage markup. Implied in‡ation equation: t
= Etf
t+1 g
+ (yt
y t) + (
w;t
w)
