Lectures on Monetary Policy, In‡ation and the Business cycle Monetary Policy Design in the Basic New Keynesian Model
by Jordi Galí
The E¢ cient Allocation
where Ct
hR
1 1 0 Ct (i)
1
max U (Ct; Nt) i 1 di subject to:
Ct(i) = At Nt(i)1 ; all i 2 [0; 1] Z 1 Nt = Nt(i) di 0
Optimality conditions:
Ct(i) = Ct, all i 2 [0; 1]
Nt(i) = Nt, all i 2 [0; 1] Un;t = M P Nt Uc;t where M P Nt
(1
) AtNt .
Sources of Suboptimality of Equilibrium 1. Distortions unrelated to nominal rigidities: Monopolistic competition: Pt = M =)
Wt M P Nt ,
where M
" " 1
>1
Un;t Wt M P Nt = = < M P Nt Uc;t Pt M
t Solution: employment subsidy : Under ‡exible prices, Pt = M (1M P )W Nt .
=) Optimal subsidy: M(1
Un;t Wt M P Nt = = Uc;t Pt M(1 ) ) = 1 or, equivalently,
= 1" .
Transactions friction (economy with valued money): assumed to be negligible
2. Distortions associated with the presence of nominal rigidities: Markup variations resulting from sticky prices: Mt = Pt M Wt =M P Nt (assuming optimal subsidy)
(1
Pt )(Wt =M P Nt )
Un;t Wt M =) = = M P Nt 6= M P Nt Uc;t Pt Mt Optimality requires that the average markup be stabilized at its frictionless level. Relative price distortions resulting from staggered price setting: Ct(i) 6= Ct(j) if Pt(i) 6= Pt(j). Optimal policy requires that prices and quantities (and hence marginal costs) are equalized across goods. Accordingly, markups should be identical across …rms/goods at all times.
=
Optimal Monetary Policy in the Basic NK Model Assumptions: optimal employment subsidy =) ‡exible price equilibrium allocation is e¢ cient no inherited relative price distortions, i.e. P 1(i) = P i 2 [0; 1]
1
for all
=) the e¢ cient allocation can be attained by a policy that stabilizes marginal costs at a level consistent with …rms’desired markup, given existing prices: no …rm has an incentive to adjust its price, i.e. Pt = Pt 1 and, hence, Pt = Pt 1 for t = 0; 1; 2; :::As a result the aggregate price level is fully stabilized and no relative price distortions emerge. equilibrium output and employment match their counterparts in the (undistorted) ‡exible price equilibrium allocation.
Equilibrium under the Optimal Policy yet = 0
=0 it = rtn t
for all t. Implementation: Some Candidate Interest Rate Rules Non-Policy Block: yet =
1
(it t
Etf =
t+1 g
Et f
t+1 g
rtn) + Etfe yt+1g +
yet
An Exogenous Interest Rate Rule it = rtn Equilibrium dynamics: yet
t
where
Etfe yt+1g Etf t+1g
= AO
AO
1
1
+ Shortcoming: the solution yet = t = 0 for all t is not unique: one eigenvalue of AO is strictly greater than one. ! indeterminacy. (real and nominal).
An Interest Rate Rule with Feedback from Target Variables it = rtn + Equilibrium dynamics: yet
t
where AT
= AT
1 + y+
t
+
y
yet
Etfe yt+1g Etf t+1g 1 + ( +
y)
Existence and Uniqueness condition: (Bullard and Mitra (2002)): (
1) + (1
)
y
>0
Taylor-principle interpretation (Woodford (2000)): di = =
d + +
de y (1 )
y y
d
Figure 4.1 2.1 2
1.8
1.6 Determinacy 1.4
φπ
1.2
1
0.8
0.6 Indeterminacy 0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
φy
1.4
1.6
1.8
2
A Forward-Looking Interest Rate Rule it = rtn + Equilibrium dynamics: yet
t
where
Etf
t+1 g
= AF
+
y
Etfe yt+1g
Etfe yt+1g Etf t+1g
1 1 1 ( 1) y AF 1 1 (1 ( 1) y) Existence and Uniqueness conditions (Bullard and Mitra (2002):
( (
1) + (1 ) 1) + (1 + )
y y
> 0 < 2 (1 + )
Figure 4.2
35
30
25
20
φπ
Indeterminacy
15
Determinacy 10
5
0
0
0.5
1
1.5
φy
2
Shortcomings of Optimal Rules they assume observability of the natural rate of interest (in real time). this requires, in turn, knowledge of: (i) the true model (ii) true parameter values (iii) realized shocks Alternative: “simple rules”, i.e. rules that meet the following criteria: the policy instrument depends on observable variables only, do not require knowledge of the true parameter values ideally, they approximate optimal rule across di¤erent models
Simple Monetary Policy Rules Welfare-based evaluation: 1 X Utn t Ut W E0 UcC t=0 =)
1
X 1 = E0 2 t=0
t
expected average welfare loss per period: 1 [ var(e yt) + var( t)] L= 2
See Appendix for Derivation.
yet2 +
2 t
A Taylor Rule it =
+
t
+
+
y
y
Equivalently: it = where vt
y
ybtn
+
t
Equilibrium dynamics: yet
t
where
Exercise:
1 + y+
at
yet + vt
Etfe yt+1g + BT (b rtn Etf t+1g 1 + ( +
AT and
= AT
ybt
: Note that rbtn
;
y)
y
ybtn =
y
ybtn)
1
BT n ya
[ (1
AR(1) + modi…ed Taylor rule it = +
a)
+
y]
t+
y
at yt
Money Growth Peg mt = 0 money market clearing condition b bit lt = yet + ybtn where lt process
De…ne lt+
mt
lt
pt and
t.
t
t
is a money demand shock following the
t
=
t 1
=) bit = 1 (e yt + ybtn b lt+ + lt+ 1 = b
t
+ "t
b lt+) t
Equilibrium dynamics: 3 2 3 2 n 3 2 Etfe yt+1g rbt yet AM;0 4 t 5 = AM;1 4 Etf t+1g 5 + BM 4 ybtn 5 lt+ 1 lt+ t where AM;0
2 4
1+ 0
3
0 0 1 05 1 1
;
AM;1
2
3
1 4 0 05 0 0 1
Simulations and Evaluation of Simple Rules
;
BM
2
3
1 0 40 0 05 0 0 1
Table 4.1: Evaluation of Simple Monetary Policy Rules Taylor Rule Constant Money Growth 1:5 1:5 5 1:5 0:125 0 0 1 y ( ; ) (0; 0) (0:0063; 0:6) (e y)
0:55
0:28
0:04
1:40
1:02
1:62
( )
2:60
1:33
0:21
6:55
1:25
2:77
welf are loss
0:30
0:08
0:002
1:92
0:08
0:38
Technical Appendix: Derivation of Second-Order Approximation of Welfare around the Undistorted Flexible Price Equilibrium Allocation We derive a second order approximation of utility around the e¢ cient equilibrium allocation. Under our assumptions the latter corresponds to the ‡exible price equilibrium allocation. All along we assume that utility is separable in consumption and hours (i.e., Ucn = 0 ). In order to lighten the notation we de…ne Ut U (Ct ; Nt ), Utn U (Ctn ; Ntn ), and U U (C; N ). A second order Taylor of expansion of Ut yields: Ut
n Utn = Uc;t Ctn
Ctn
Ct
Ctn
1 n + Ucc;t (Ctn )2 2
Letting x et
log
Xt Xtn
n + Un;t Ntn
Ctn
Ct
2
Ctn
Nt
Ntn Ntn
1 n + Unn;t (Ntn )2 2
Nt
Ntn
2
Ntn
denote log-deviations from ‡exible price equilibrium values, we can write: Ut
n Ctn Utn = Uc;t
where we use the approximation
e ct + Xt
1 2 Xtn
Xtn
e c2t
n Ntn + Un;t
'x et +
1 2 x e 2 t
n et +
1+' 2 n et 2
The next step consists in rewriting n et in terms of the output gap. Using the fact that Nt = we have (1
)n et = yet + dt
1
Yt At
1
R1 0
Pt (i) Pt
1
di ,
where dt
(1
) log
R1 0
Pt (i) Pt
1
di : The following lemma shows that dt is proportional to the cross-sectional
variance of relative prices and, hence, of second order. Lemma 1: up to a second order approximation, dt = present appendix.
1
vari fpt (i)g, where
2
1
+
. See proof at the end of the
Accordingly we have: Ut
n Utn = Uc;t Ctn
yet +
1 2
yet2
+
n Ntn Un;t 1
yet +
1+' ye2 + 2(1 ) t 2
vari fpt (i)g
where we have made use of the market clearing condition e ct = yet for all t.
Finally, we recall that when the optimal subsidy is in place, the ‡exible price allocation is e¢ cient, thus implying n n Un;t Ntn = Uc;t Ctn (1
)
Hence, up to second order, we have Ut
Utn =
1 n n U C 2 c;t t
+ ' + (1 1
)
yet2 +
(1 + ( 1
n Next we derive a …rst order approximation to Uc;t Ctn about the steady state:
n Ctn = Uc C + (Ucc C + Uc ) Uc;t
= Uc C + Uc C (1 It follows that, up to second order,
)
Ctn
C C
b cnt
1))
vari fpt (i)g
Utn =
Ut
+ ' + (1 1
1 Uc C 2
)
yet2 +
(1 + ( 1
1))
vari fpt (i)g
Accordingly, we can write a second order approximation to the consumer’s welfare losses resulting from deviations from the e¢ cient allocation, expressed as a fraction of steady state consumption (or output), as: W
E0
1 X t=0
t
Ut Utn Uc C
=
X 1 E0 2 t=0 1
t
+ ' + (1 1
)
yet2 +
(1 + ( 1
Lemma 2: up to second order and additive term independent of policy„ 1 X t=0
t
vari fpt (i)g =
(1
)(1
)
1 X t=0
Proof: Woodford (2003, chapter 6) Combining the previous lemma with the expression above we get X 1 W= E0 2 t=0 1
Hence the average period welfare loss will be given by: L=
Proof of Lemma 1 Let pbt (i) pt (i) pt . Notice that,
t
yet2 +
2 t
1 [ var(e yt ) + var( t )] 2
t
2 t
1))
vari fpt (i)g
1
Pt (i) Pt
) pbt (i)g
= expf(1
) pbt (i) +
= 1 + (1 Furthermore, from the de…nition of Pt , we have 1 = expression implies
R1 0
(
Ei fb pt (i)g = In addition, a second order approximation to Pt (i) Pt
=1
1
0
1
1
Pt (i) Pt
1
#
di
2
1
pbt (i)2
di. Hence, a second order approximation to this
Ei fb pt (i)2 g
yields:
1
Combining the two previous results, it follows that "Z
1) 2
1
Pt (i) Pt
Pt (i) Pt
)2
(1
pbt (i) +
= 1+
1 2
= 1+
1 2
1 2
2
pbt (i)2
1
1 1 1 1
Ei fb pt (i)2 g vari fpt (i)g
and where the second equality holds up to second order, given that (Ei fb pt (i)g)2 is of higher order. R1 1 Thus, we have ut = (1 ) log 0 PPt (i) di ' 2 vari fpt (i)g, up to a second order approximation. QED. t
where
1
+
