Rules
7
Monetary Policy Design in the NK
Frame-work
This section follows the derivations in Gali chapter 4. However, the focus of the document is on defining a simple monetary policy rule concerning model (1): namely, an interest rate rule. Due to the focus in the literature being predominately on Taylor-type interest rate rules, we treat those solely when simulating the model and replicating the table on welfare lose in Gali.
Before we formally determine the welfare criterion through which we assess monetary policy rules, we must first show what the efficient allocation is in the classical economy followed by reemphasising how the benchmark New-Keynesian model (1) differs from the allocation obtained in the classical economy. To show this, we first restate a number of elements of the model;
U(Ct, Nt) (7.1)
Where; Ct ≡ hR1
0 Ct(i) 1−1
di
i−1
, subject to the resource constraints Nt =
R1
0 Nt(i)diandYt(i) =AtNt(i)
1−α. Logically, it also must follow that{C
t, Nt}= {Ct(i), Nt(i)}for optimality to hold asCt(i)6=Ct(x) cannot be optimal for any givenCt. Therefore, we can write the planners problem thus;
max
{Ct,Nt} Et
∞
X t=0
βU(Ct, Nt) = max
{Ct,Nt} Et
∞
X t=0
βU
Z 1 0
Ct(i)1−1di
−1
,
Z 1 0
Nt(i)di
!
= max
{Ct,Nt} Et
∞
X t=0
βU
Z 1 0
Ct(i)di, Z 1
0
Nt(i)di
(7.2)
Due to the common goods market clearing condition in models absent capital and government,Yt=Ct, we can stateCt(i) =AtNt(i)1−α. Thus (7.2) can be written;
max
{Ct,Nt} Et
∞
X t=0
βU(Ct, Nt) = max
{Nt} Et
∞
X t=0
βU
AtNt(i)1−α, Z 1
0
Nt(i)di
(7.3)
With the first order condition with respect to Nt(i);
∂ ∂Nt(i)
U
AtNt(i)1−α, Z 1
0
Nt(i)di
=Uc(1−α)AtNt(i)−α+Un = 0 (7.4)
M PN =−
UN
UC
to time varying markups and also price (and by extension output) dispersion amongst firms. The latter being a second order problem as shown in model (1) equations 1.115 - 1.141.
Now we have defined the efficient allocation, we are able to derive the welfare criterion in order to assess the performance of a simple monetary policy rule (Taylor Rule). Following Woodford & Rotemberg (1999), the welfare criterion is based on a second order approximation to the utility losses incurred by the representative consumer due to the deviations away from the efficient allocation - i.e., due, in the above model, to the existence of monopolistic competition and Calvo pricing.
In what follows, we shall derive the welfare loss function in detail. First we state;
Zt−Z
Z 'zˆt+
1 2zˆ
2
t to a second order (7.6)
Ucn= 0 separable utility (7.7)
ˆ
ztx= 0 ∀x >2 (7.8)
Ut≡U(Ct, Nt) =
Ct1−σ 1−σ −
Nt1+ϕ
1 +ϕ period utility (7.9)
Where (7.8) follows as we are using a second order Taylor expansion around the efficient S.S., and so all higher monomial terms are equal to zero. A second order Taylor expansion ofUtaround S.S., (C, N) yields;
Ut−U 'Uc(Ct−C) +Un(Nt−N) + 1
2Ucc(Ct−C) 2+1
2Unn(Nt−N) 2
'Uc(Ct−C)
C
C +Un(Nt−N) N
N +
1
2Ucc(Ct−C) 2C
2
C2 + 1
2Unn(Nt−N) 2N
2
N2
'Uc(ˆct+ 1 2ˆc
2
t)C+Un(ˆnt+ 1 2ˆn
2 t)N+
1
2Ucc(ˆct+ 1 2cˆ
2 t)2C2+
1
2Unn(ˆnt+ 1 2nˆ
2 t)2N2
'Uc(ˆct+ 1 2ˆc
2
t)C+Un(ˆnt+ 1 2ˆn
2 t)N+
1 2Ucc(ˆct)
2C2+1
2Unn(ˆnt) 2N2
Noticing 12Ucc(12ˆct2)2C2 = 12Ucc( 1 2cˆ
4
t)C2 ≈0 up to a second order. Collecting like terms we obtain;
'Uc(ˆct+ 1 2ˆc
2 t+
1 2
Ucc
Uc
(ˆct)2C)C+Un(ˆnt+ 1 2nˆ
2 t+
1 2
Unn
Un
(ˆnt)2N)N (7.10)
Using (7.9) we can show;
Uc=
∂U ∂C =C
−σ, U cc=
∂2U
∂C2 =−σC
−σ−1 therefore, −Ucc
Uc
C=σ
Un =
∂U ∂N =N
ϕ, U nn=
∂2U ∂N2 =ϕN
ϕ−1 therefore, Unn
Un
Substituting the above two results into (7.10), and utilising the goods market clearing condition, yields;
'Uc(ˆct+ 1 2cˆ
2 t−
1 2σ(ˆct)
2)C+Un(ˆn t+
1 2nˆ
2 t+
1 2ϕ(ˆnt)
2)N
'Uc
ˆ
ct+ 1−σ
2 cˆ 2 t
C+Un
ˆ
nt+ 1 +ϕ
2 nˆ 2 t
N
Ut−U 'Uc
ˆ
yt+ 1−σ
2 yˆ 2 t
C+Un
ˆ
nt+ 1 +ϕ
2 ˆn 2 t
N (7.11)
To proceed, it is now required that rewrite ˆnt in terms of output. This process has been derived in model (1) equations (1.109-114). Restating this in terms of
ntyields;
yt=at+ (1−α)nt+ (1−α) log "
Z 1 0
Pt(i)
Pt 1−−α
di
#
| {z }
≡dt
(1−α)nt=yt−at−dt
(1−α)ˆnt= ˆyt−at−dt (7.12)
Where equation (7.12) follows from subtracting the π = 0 S.S. In the basic model, we showed that up to a first orderdt'0. However, here dtis a second order term and therefore has implications for welfare in the current analysis. Revising what was shown in the basic model above, dt is a measure of price dispersion. We shall now derive in detailLemma 1 on page 87 in Gali.
Pt= Z 1
0
Pt(i)1−di
1−1
1 = "
Z 1 0
Pt(i)
Pt 1−
di
#
= Z 1
0
exp[(1−)(pt(i)−pt)di]
'1 + (1−)1 Z 1
0
e0(pt(i)−pt)1di
| {z }
≡Ei{pˆt(i)}
+1 2(1−)
2Z 1
0
e0(pt(i)−pt)2di
| {z }
≡Ei{pt(i)2} '1 + (1−)1Ei{pˆt(i)}+
1 2(1−)
2E
i{pˆt(i)2}
−(1−) 2
2
Ei{pˆt(i)2} '(1−)1Ei{pˆt(i)}
(−1)
2 Ei{pˆt(i) 2
} 'Ei{pˆt(i)} (7.13)
Where the second order approximation - (7.13) - has been obtained using the following relationships; ˆpt(i)≡pt(i)−pt, andR1
0 e
0(pt(i)−pt)1di≡E
Z 1 0
Pt(i)
Pt
−1−α di' Z 1 0 exp −
1−α
(pt(i)−pt)
'1− 1−α
Z 1 0
e0(pt(i)−pt)di+1 2
1−α
2Z 1 0
e0(pt(i)−pt)2di
'1−
1−αEi{pˆt(i)}+
1 2
1−α
2
Ei{pˆt(i)2} (7.14)
Combining (7.13) and (7.14) allows us to derive the following result;
'1−
1−αEi{pˆt(i)}+
1 2
1−α
2
Ei{pˆt(i)2}
'1− 1−α
(−1)
2 Ei{pˆt(i) 2
}+1 2
1−α
2
Ei{pˆt(i)2}
'1− "
(1−α)
(1−α)2 (−1)
2 −
1 2
1−α
2#
Ei{pˆt(i)2}
'1−
(1−α)(−1)− (1−α)
1 2
1−α
Ei{pˆt(i)2}
'1−
α−α−1 (1−α)
1 2
1−α
Ei{pˆt(i)2}
'1 +
1−α+α (1−α)
| {z }
≡1/Θ 1 2
1−α
Ei{pˆt(i)2}
'1 + 1 2
1−α
1
ΘEi{pˆt(i) 2}
Z 1 0
Pt(i)
Pt
−1−α
di'1 + 1 2
1−α
1
Θvari{pˆt(i)} (7.15)
Where Ei{pˆt(i)2} is the variance in the price level; i.e., price dispersion. We
found earlier that (1−α) log
R1 0
P
t(i)
Pt
1−−α di
≡ dt. Using this result we
obtain;
(1−α) ln "
Z 1 0
Pt(i)
Pt
−1−α di
#
'(1−α) ln(1) + (1−α) ln
1 2
1−α
1
Θvari{pˆt(i)}
dt≡
2Θvari{pˆt(i)} (7.16)
'Uc
ˆ
yt+ 1−σ
2 yˆ 2 t
C+Un
ˆ
nt+ 1 +ϕ
2 ˆn 2 t
N
'Uc
ˆ
yt+ 1−σ
2 yˆ 2 t
C+Un 1
(1−α)(ˆyt−at−dt) + 1 +ϕ
2
1
(1−α)(ˆyt−at−dt) 2!
N
'Uc
ˆ
yt+ 1−σ
2 yˆ 2 t
C+ Un (1−α)
ˆ
yt−at−dt+ 1 +ϕ
2(1−α)(ˆyt−at−dt) 2
N
Substituting equation (7.16) into the above equation yields;
'Uc
ˆ
yt+ 1−σ
2 yˆ 2 t
C+ Un (1−α)
ˆ
yt−at−
2Θvari{pˆt(i)}+ 1 +ϕ
2(1−α)
ˆ
yt−at−
2Θvari{pˆt(i)} 2
N
(7.17)
notice that the final term is higher than second order term. Therefore, the above equation can be written thus;
'UcC
ˆ
yt+ 1−σ
2 yˆ 2 t
+ UnN (1−α)
ˆ
yt−at−
2Θvari{pˆt(i)}+ 1 +ϕ
2(1−α)(ˆyt−at) 2
+t.i.p
Ut−U
UcC
'yˆt+ 1−σ
2 yˆ 2 t+
UnN
UcC(1−α)
ˆ
yt−at−
2Θvari{pˆt(i)}+ 1 +ϕ
2(1−α)(ˆyt−at) 2
+t.i.p
(7.18)
using the following relationships−Un
Uc =M Pn,M Pn= (1−α)
Y N
andY =C, we can write express the above equation thus;
' 1−σ 2 yˆ
2 t+
2Θvari{pˆt(i)} −
1 +ϕ
2(1−α)(ˆyt−at) 2
+t.i.p
' −1 2
Θvari{pˆt(i)} −(1−σ)ˆy 2 t +
1 +ϕ
(1−α)(ˆyt−at) 2
+t.i.p
' −1 2
Θvari{pˆt(i)} −(1−σ)ˆy 2 t +
1 +ϕ
(1−α) yˆ 2
t−2ˆytat+ ˆa2t
+t.i.p (7.19)
' −1 2
Θvari{pˆt(i)} −(1−σ)ˆy 2 t +
1 +ϕ
(1−α) yˆ 2
t−2ˆytat
+t.i.p
collecting ˆy2 t yields;
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
ˆ
y2t−2
1 +ϕ 1−α
(ˆytat)
+t.i.p
(7.20)
referring back to model (1) it can be shown that;
ynt =
(1 +γ) (1−α)σ+ (γ+α)
| {z }
Φn ya
at−
(µ−ln(1−α)) (1−α) (1−α)σ+ (γ+α)
| {z }
Therefore we can state;
ˆ
ynt ≡ynt −ynt =
(1 +γ)
(1−α)σ+ (γ+α)at (1−α)σ+ (γ+α)
(1 +γ) yˆ n
t =at (7.21)
substituting the above equation into equation (7.20) and simplifying yields;
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
ˆ
y2t −2
1 +ϕ
1−α
(1−α)σ+ (γ+α) (1 +γ) yˆ
n tyˆt
+t.i.p
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
ˆ
y2t−2ˆytnyˆt
+t.i.p
(7.22)
as noted by Gali, ˆyt−yˆtn = ˜yt. We are able to manipulate equation (7.22) to create the term present in Gali - ˜y2t - thus;
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
ˆ
yt2−2ˆytnyˆt+ ˆytn2−yˆ n2 t
+t.i.p
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
(ˆyt−yˆnt)(ˆyt−yˆtn)−yˆ n2 t
+t.i.p
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
(˜yt)(˜yt)−yˆnt2
+t.i.p
' −1 2
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
˜
yt2
+t.i.p
(7.23)
Note: in equations (7.17), (7.18) and (7.19) the elements 2Θ vari{pˆt(i)}2, at and ˆa2
t become included in the succeeding equations final element labelledt.i.p. This is due to in the case of the (7.17) the component being of an order>2, and in the case of the latter two equations, technology being unaffected by policy. Also, note that in equation (7.23) that natural rat is affected by technology and so this is subsumed intot.i.p. Rewriting as in Gali page 89
W=E0
∞
X t=0
βt
U t−U
UcC
=−1 2E0
∞
X t=0
βt
Θvari{pˆt(i)}+
σ+α+ϕ 1−α
˜
y2t
(7.24)
the final representation of the welfare loss function is required to be in the terms of the output gap and inflation. We must necessarily express the price dispersion component as a function of π. To do this we can make use of the following Lemma;
∞
X t=0
βtvari{pˆt(i)}=
θ
(1−βθ)(1−θ)
∞
X t=0
Therefore, equation (7.24) becomes;
W=−
1 2E0
∞
X t=0
βt
λπ 2 t +
σ+α+ϕ 1−α
˜
yt2
(7.26)
Whereλ≡Θ(1−βθθ)(1−θ). Thus yielding the final expression for theWelfare Loss function. Theaverage welfare loss per period is given by the linear combination of the variances of ˜ytandπt:
L=−
1 2
λvar(πt) +
σ+α+ϕ 1−α
var(˜yt)
(7.27)
We can now proceed by augmenting the .mod file for model (1) to allow for welfare analysis. Using the code labelled ’model (7)’ one is able to replicate Table 4.1 in Gali.
Above is the.modfile for model (1) is augmented threefold: firstly through the inclusion of the composite variablesloss Y tildeandloss inflwithin the welfare loss function; secondly through the replacement of the monetary policy shockV withφyπnyaat- implyingvarexois reduced toea and removing
ρv as a parameter - as in Gali; and thirdly through an Optimal Simple Rules block at the end of the .modfile.
Notice that the file is not strictly identical to that laid out in the above section. It has also been written with the inclusion of the variable N, and equation Y = A+ (1−α)N - the production function. Refer to the previous section for details. Notice also that in the parameter block the variables phi in and phi y now have paired sequences of values as found in the specification in Table 4.1 in Gali.
Table 1: Evaluation of Simple Monetary Policy Rules - Taylor Rule
Taylor-type Interest Rate Rule
φπ 1.5 1.5 5 1.5
φy 0.125 0 0 1
σ(˜y) 0.5546 0.2847 0.0466 1.3975
σ(π) 2.6092 1.3396 0.2192 6.5752
welf are loss 0.3076 0.0811 0.0022 1.9531
moments delivered by the model are quarterly and so must be multiplied by to obtain the figures in the table above.
Table 2 below shows so-called ’divine coincidence’: under an a optimal simple rule one is able to simultaneously achieve zero variation in both inflation and the output gap. However, this result is merely a theoretical curiosity; policy tradeoff’s exist in the real world and are an important consideration of policy.
Table 2: Divine Coincidence
φy = 0.125 and φπ = 1.5 φy = 0 and φπ= 1.5
Variable Mean STD. dev Variance Variable Mean STD. dev Variance
INFL 0 0.6523 0.4255 INFL 0 0.3349 0.1122
Y tilde 0 0.5546 0.3076 Y tilde 0 0.2847 0.811
N 0 0.8315 0.6913 N 0 0.4269 0.1822
φy= 0 and φπ= 5 φy = 1.5 and φπ = 1
Variable Mean STD. dev Variance Variable Mean STD. dev Variance
INFL 0 0.0548 0.0030 INFL 0 1.6438 2.7022
Y tilde 0 0.0466 0.0022 Y tilde 0 1.3975 1.9531
N 0 0.0699 0.0049 N 0 2.0952 4.3900
Optimal Weights (φy=−0.1 and φπ= 1.57911)
Variable Mean STD. dev Variance
INFL 0 0 0
Y tilde 0 0 0
N 0 0 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2
0.4 0.6 0.8 1 1.2
variance of inflation
variance of output gap
Varying φ
π (φy fixed at 0.125)
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0 0.5 1 1.5
Value of φ π
Welfare loss
0.5 1 1.5 2 2.5
0.5 1 1.5
variance of inflation
variance of output gap
Varying φ
y (φπ fixed at 1.5)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2
Value of φy
