Lecture Note
The Basic New Keynesian Model
-
University:
Boston University -
Course:
CAS EC 203 | Empirical Economics 1 Academic year:
2023
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Views:
211
Pages:
27
Author:
5CelestialChronicle
The Basic New Keynesian Model Motivation and Outline
Evidence on Money, Output, and Prices:
Short Run E¤ects of Monetary Policy Shocks
(i) persistent e¤ects on real variables
(ii) slow adjustment of aggregate price level
(iii) liquidity e¤ect
Micro Evidence on Price-setting Behavior: signi…cant price and wage
rigidities
Failure of Classical Monetary Models
A Baseline Model with Nominal Rigidities
monopolistic competition
sticky prices (staggered price setting)
competitive labor markets, closed economy, no capital accumulation Households
Representative household solves
max E0
1
X
t
U (Ct; Nt)
t=0
where
Ct
Z
1
Ct(i)
1
1
1
di
0
subject to
Z
1
Pt(i)Ct(i) di + QtBt
0
for t = 0; 1; 2; ::: plus solvency constraint.
Bt
1
+ W t Nt
Tt Optimality conditions
1. Optimal allocation of expenditures
Pt(i)
Pt
Ct(i) =
implying
Z
Ct
1
Pt(i)Ct(i) di = PtCt
0
where
Pt
Z
1
1
1
Pt(i)1 di
0
2. Other optimality conditions
Un;t Wt
=
Uc;t
Pt
Uc;t+1 Pt
Q t = Et
Uc;t Pt+1 Speci…cation of utility:
Ct1
U (Ct; Nt) =
1
Nt1+'
1+'
implied log-linear optimality conditions (aggregate variables)
pt = ct + 'nt
1
ct = Etfct+1g
(it Etf t+1g
wt
where it
log Qt is the nominal interest rate and
discount rate.
Ad-hoc money demand
mt
pt = yt
it
)
log
is the Firms
Continuum of …rms, indexed by i 2 [0; 1]
Each …rm produces a di¤erentiated good
Identical technology
Yt(i) = AtNt(i)1
Probability of resetting price in any given period: 1
across …rms (Calvo (1983)).
2 [0; 1] : index of price stickiness
Implied average price duration
1
1
, independent Aggregate Price Dynamics
Pt =
Dividing by Pt
1
(Pt 1)
1
+ (1
)(Pt )
:
1
t
= + (1
Pt
)
Pt 1
1
1
1
1
Log-linearization around zero in‡ation steady state
t
= (1
)(pt
pt 1 )
+ (1
)pt
or, equivalently
pt = pt
1
(1) Optimal Price Setting
max
Pt
subject to
1
X
k
t+k (Yt+kjt )
Et Qt;t+k Pt Yt+kjt
k=0
Yt+kjt = (Pt =Pt+k ) Ct+k
for k = 0; 1; 2; :::where
Qt;t+k
Optimality condition:
1
X
where
t+kjt
k
k
Ct+k
Ct
Et Qt;t+k Yt+kjt Pt
k=0
0
t+k (Yt+kjt )
and M
1
Pt
Pt+k
M
t+kjt
=0 Equivalently,
1
X
k
Et Qt;t+k Yt+kjt
k=0
where M Ct+kjt
t+kjt =Pt+k
and
Pt
Pt 1
MM Ct+kjt
t 1;t+k
Pt+k =Pt
=0
t 1;t+k
1
Perfect Foresight, Zero In‡ation Steady State:
Pt
=1 ;
Pt 1
t 1;t+k
=1 ;
Yt+kjt = Y
;
Qt;t+k =
k
;
MC =
1
M Log-linearization around zero in‡ation steady state:
1
X
pt pt 1 = (1
)
( )k Etfmc
c t+kjt + pt+k
k=0
where mc
c t+kjt
mct+kjt
pt 1 g
mc.
Equivalently,
pt =
+ (1
)
1
X
k=0
where
log
1.
(
)k Etfmct+kjt + pt+k g
Flexible prices ( = 0):
pt =
=)
mct =
+ mct + pt
(symmetric equilibrium) Particular Case:
= 0 (constant returns)
=) M Ct+kjt = M Ct+k
Rewriting the optimal price setting rule in recursive form:
pt =
)mc
c t + (1
Etfpt+1g + (1
Combining (1) and (2):
t
where
= Et f
(1
t+1 g
)(1
+ mc
ct
)
)pt
(2) Generalization to
De…ne
2 (0; 1) (decreasing returns)
mct
Using mct+kjt = (wt+k
(wt
pt )
(wt
pt )
pt+k )
mpnt
1
(at
1
1
1
yt )
(at+k
mct+kjt = mct+k +
= mct+k
log(1
yt+kjt)
(yt+kjt
1
(pt
1
log(1
)
),
yt+k )
pt+k )
(3)
Implied in‡ation dynamics
t
where
(1
= Et f
)(1
t+1 g
)
+ mc
ct
1
1
+
(4) Equilibrium
Goods markets clearing
Yt(i) = Ct(i)
for all i 2 [0; 1] and all t.
R1
1 1
Letting Yt
di
Y
(i)
0 t
1
,
Yt = Ct
for all t. Combined with the consumer’s Euler equation:
yt = Etfyt+1g
1
(it
Etf
t+1 g
)
(5) Labor market clearing
Nt =
Z
1
Nt(i) di
0
=
Z
1
0
=
Yt
At
Yt(i)
At
1 Z
1
1
1
di
1
0
Pt(i)
Pt
1
di
Taking logs,
(1
where dt
(1
) log
R1
0
)nt = yt
(Pt(i)=Pt)
1
at + dt
di (second order).
Up to a …rst order approximation:
yt = at + (1
) nt Marginal Cost and Output
mct = (wt pt) mpnt
= ( yt + 'nt) (yt
'+
yt
=
+
1
Under ‡exible prices
'+
mc =
+
ytn
1
=) ytn =
where
y
where yt
(
ytn
log(1 ))(1 )
+'+ (1 )
> 0 and
=) mc
ct =
yet is the output gap
nt) log(1
)
1+'
at log(1
1
1+'
at
1
y
log(1
)
(6)
)
(7)
+
ya
'+
+
1
ya at
1+'
+'+ (1
(yt
).
ytn)
(8) New Keynesian Phillips Curve
t
where
+ '+
.
1
= Etf
t+1 g
+ yet
(9)
Dynamic IS equation
1
yet = Etfe
yt+1g
(it
Etf
t+1 g
rtn)
(10)
where rtn is the natural rate of interest, given by
rtn
=
Missing block:
+
+
n
Etf yt+1
g
ya Et f at+1 g
description of monetary policy (determination of it). Equilibrium under a Simple Interest Rate Rule
it =
+
t
+
et
yy
+ vt
(11)
where vt is exogenous (possibly stochastic) with zero mean.
Equilibrium Dynamics:
yet
t
where
AT
and
1
+ y+
combining (9), (10), and (11)
= AT
Etfe
yt+1g
+ BT (b
rtn
Etf t+1g
1
+ ( +
y)
;
BT
vt )
(12)
1 Uniqueness () AT has both eigenvalues within the unit circle
Given
0 and
y
0, (Bullard and Mitra (2002)):
(
is necessary and su¢ cient.
1) + (1
)
y
>0 E¤ects of a Monetary Policy Shock
Set rbtn = 0 (no real shocks).
Let vt follow an AR(1) process
vt =
v vt
1
+ "vt
Calibration:
v
= 0:5,
= 1:5,
y
= 0:5=4,
= 0:99,
= ' = 1,
= 2=3,
= 4.
Dynamic e¤ects of an exogenous increase in the nominal rate (Figure 1):
Exercise: analytical solution E¤ects of a Technology Shock
Set vt = 0 (no monetary shocks).
Technology process:
at =
a at 1
+ "at:
Implied natural rate:
rbtn =
ya (1
Dynamic e¤ects of a technology shock (
Exercise: AR(1) process for
at
a
a )at
= 0:9) (Figure 2) Equilibrium under an Exogenous Money Growth Process
mt =
m
mt
1
+ "m
t
(13)
Money market clearing
where lt
mt
b
bit
lt = ybt
= yet + ybtn
(14)
(15)
bit
pt denotes (log) real money balances.
Substituting (14) into (10):
(1 + ) yet =
Furthermore, we have
Etfe
yt+1g + b
lt + Etf
b
lt
1
=b
lt +
t
t+1 g
mt
+ rbtn
ybtn
(16)
(17) Equilibrium dynamics
3
2
3
2 n 3
2
Etfe
yt+1g
yet
rbt
AM;0 4 t 5 = AM;1 4 Etf t+1g 5 + BM 4 ybtn 5
b
b
mt
lt 1
lt 1
where
2
3
2
3
1+
0 0
1
1 0 5 ; AM;1 4 0
0 5 ; BM
AM;0 4
0
1 1
0 0 1
(18)
2
3
1 0
40 0 0 5
0 0
1
1
Uniqueness () AM
AM;0
AM;1 has two eigenvalues inside and one
outside the unit circle. E¤ects of a Monetary Policy Shock
Set rbtn = ytn = 0 (no real shocks).
Money growth process
mt =
where
m
2 [0; 1)
Figure 3 (based on
m
m
mt
1
+ "m
t
= 0:5)
E¤ects of a Technology Shock
Set
mt = 0 (no monetary shocks).
Technology process:
at =
Figure 4 (based on
Empirical Evidence
a
= 0:9).
a at 1
+ "at Technical Appendix
Optimal Allocation of Consumption Expenditures
R1
Maximization of Ct for any given expenditure level 0 Pt (i) Ct (i) di
L=
Z
1
1
1
Ct (i)
1
di
0
Zt can be formalized by means of the Lagrangean
Z
1
Pt (i) Ct (i)di
Zt
0
The associated …rst order conditions are:
Ct (i)
1
1
Ct =
Pt (i)
for all i 2 [0; 1]. Thus, for any two goods (i; j) we have:
Ct (i) = Ct (j)
Pt (i)
Pt (j)
which can be plugged into the expression for consumption expenditures to yield
Ct (i) =
Pt (i)
Pt
Zt
Pt
for all i 2 [0; 1]. The latter condition can then be substituted into the de…nition of Ct , yielding
Z 1
Pt (i) Ct (i) di = Pt Ct
0
Combining the two previous equations we obtain the demand schedule:
Ct (i) =
Log-Linearized Euler Equation
Pt (i)
Pt
Ct We can rewrite the Euler equation as
1 = Et fexp(it
In a perfect foresight steady state with constant in‡ation
ct+1
and constant growth
i=
+
(19)
)g
t+1
we must have:
+
with the steady state real rate being given by
r
i
=
A …rst order Taylor expansion of exp(it
exp(it
ct+1
ct+1
t+1
+
) around that steady state yields:
) ' 1 + (it
= 1 + it
t+1
i)
( ct+1
ct+1
)
(
t+1
)
t+1
which can be used in (19) to obtain, after some rearrangement of terms, the log-linearized Euler equation
1
ct = Et fct+1 g
(it
Et f
t+1 g
)
Aggregate Price Level Dynamics
Let S(t) [0; 1] denote the set of …rms which do not re-optimize their posted price in period t. The aggregate price level
evolves according to
Pt =
Z
1
1
Pt 1 (i)
di + (1
1
)(Pt )
S(t)
=
(Pt 1 )1
+ (1
)(Pt )1
1
1
1 where the second equality follows from the fact that the distribution of prices among …rms not adjusting in period t
corresponds to the distribution of e¤ective prices in period t 1, with total mass reduced to .
Equivalently, dividing both sides by Pt 1 :
1
t
= + (1
Pt
Pt 1
)
1
(20)
where t PPt t 1 . Notice that in a steady state with zero in‡ation Pt = Pt 1 :
Log-linearization around a zero in‡ation ( = 1) steady state implies:
t
= (1
)(pt
(21)
pt 1 )
Price Dispersion
From the de…nition of the price index:
1 =
Z
1
0
=
Z
Pt (i)
Pt
1 "
di
1
expf(1
0
' 1 + (1
)
)(pt (i)
Z
pt )gdi
1
(pt (i)
pt )di +
)2
(1
2
0
Z
1
(pt (i)
0
thus implying the second order approximation
pt ' Ei fpt (i)g +
(1
)
2
Z
0
1
(pt (i)
pt )2 di
pt )2 di where Ei fpt (i)g
In addition,
R1
pt (i) di is the cross-sectional mean of (log) prices.
0
Z
0
1
Pt (i)
Pt
1
di =
Z
1
exp
1
Z 1
0
' 1
1
0
1 (1
' 1+
2 1
1
= 1+
2 1
1
' 1+
2 1
where
1
1
+
)
(pt (i)
2
1
pt )di +
2
(pt (i)
Z
pt ) di
1
1
1
pt )2 di +
2
(pt (i)
0
Z
1 1
(pt (i)
Z
1
(pt (i)
0
2
1
Z
pt )2 di
1
(pt (i)
0
pt )2 di
0
1
vari fpt (i)g > 1
, and where the last equality follows from the observation that, up to second order,
Z 1
Z 1
2
(pt (i) pt ) di '
(pt (i) Ei fpt (i)g)2 di
0
0
vari fpt (i)g
Finally, using the de…nition of dt we obtain
dt
(1
) log
Z
0
1
Pt (i)
Pt
1
di '
1
2
vari fpt (i)g
pt )2 di
The Basic New Keynesian Model
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