Basic New Keynesian Model
Barbara Annicchiarico
Università degli Studi di Roma "Tor Vergata"
May 2012
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
business cycle driven by real shocks (mainly by productivity
shocks)
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wage (no nominal rigidities)
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wage (no nominal rigidities)
the competitive equilibrium is Pareto optimal
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
I
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wage (no nominal rigidities)
the competitive equilibrium is Pareto optimal
no role for stabilizing policies in general
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
I
I
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wage (no nominal rigidities)
the competitive equilibrium is Pareto optimal
no role for stabilizing policies in general
monetary policy has no role (money is a veil - classical
dichotomy)
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
I
I
I
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wage (no nominal rigidities)
the competitive equilibrium is Pareto optimal
no role for stabilizing policies in general
monetary policy has no role (money is a veil - classical
dichotomy)
rational expectations
Basic New Keynesian Model
Motivation
I
The basic RBC models
I
I
I
I
I
I
I
I
business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wage (no nominal rigidities)
the competitive equilibrium is Pareto optimal
no role for stabilizing policies in general
monetary policy has no role (money is a veil - classical
dichotomy)
rational expectations
no asymmetric information
Basic New Keynesian Model
Motivation
I
but....recall some business cycle facts:
I
The real wage is only slightly pro-cyclical (a-cyclical is good
characterization... deviations from labor mkt equilbrium?)
Basic New Keynesian Model
Motivation
I
but....recall some business cycle facts:
I
I
The real wage is only slightly pro-cyclical (a-cyclical is good
characterization... deviations from labor mkt equilbrium?)
Nominal interest rate procyclical
Basic New Keynesian Model
Motivation
I
but....recall some business cycle facts:
I
I
I
The real wage is only slightly pro-cyclical (a-cyclical is good
characterization... deviations from labor mkt equilbrium?)
Nominal interest rate procyclical
Real interest rates mildly countercyclical
Basic New Keynesian Model
Motivation
I
but....recall some business cycle facts:
I
I
I
I
The real wage is only slightly pro-cyclical (a-cyclical is good
characterization... deviations from labor mkt equilbrium?)
Nominal interest rate procyclical
Real interest rates mildly countercyclical
Positive correlations between in‡ation and GDP
Basic New Keynesian Model
Motivation
I
but....recall some business cycle facts:
I
I
I
I
I
The real wage is only slightly pro-cyclical (a-cyclical is good
characterization... deviations from labor mkt equilbrium?)
Nominal interest rate procyclical
Real interest rates mildly countercyclical
Positive correlations between in‡ation and GDP
Positive correlations between GDP and money stocks
Basic New Keynesian Model
Motivation
Basic New Keynesian Model
The possibile story behind the above …gure....
I Assume that the nominal interest rate increases (i.e. contractionary
monetary policy shock)
I For a given level of in‡ation, the real interest rate increases
I Households will …nd it optimal to postpone consumption
=)
aggregate demand (AD) decreases
I Since …rms cannot revise their price immediatly (prices adjust
slowly), they will have to meet AD decrease by reducing the supply
of goods (a sustained decline in GDP)
I Maximal decline of GDP roughly a year to a year and a half after
the contractionary policy shock.
I The GDP de‡ator is ‡at for about a year and a half after which it
declines.
I Is money neutral?
Basic New Keynesian Model
General Features
I Focus on the nominal side of the economy and on the stabilizing
role of monetary policy
I A new perspective on the nature of in‡ation dynamics
I Frictions and imperfections
I
I
I
Prices do not adjust instantaneously
Firms are not identical and have some market power
(monopolistic competition)
In…nitely lived identical price-taking households (only in the
basic version of the model)
I 3 agents: households, …rms, and the central bank (or more common
speci…cation: households, …nal good producers, intermediate goods
…rms and the central bank).
I Some references: Clarida et al. (1999); Galí (2003, 2008); Woodfod
(2003); Goodfriend and King (1997); Goodfriend (2007).
Basic New Keynesian Model
A short digression on monopolistic competition
I
Assumptions
I
I
I
I
Each …rm is able to di¤erentiate its product from those of its
rivals (it acts as a monopolist in its particular product);
Each …rm takes the prices charged by its rivals as given (no
strategic interaction);
Consumers love variety - goods are not perfect substitutes
(they will not rush to buy other …rms’products as a result of a
slight increase in the price charged by one producer) ! the
larger the market size the higher the demand faced by each
producer;
Firms have the same technology - symmetry ! we can focus
on a representative producer.
Basic New Keynesian Model
Households
A representative in…nitely-lived agent seeking to maximize her
conditional expected lifetime utility function:
∞
E0 ∑ βt U (Ct , Nt )
t =0
where U (Ct , Nt ) =
C t1 σ
1 σ
1 +φ
Nt
1 +φ
and Ct
R1
0
Ct (i )
1
1
ε
di
! ε ε1
(love for variety )
σ is the inverse of the intertemporal elasticity of substitution; 0 < β < 1
ε is the constant elasticity of substitution ε > 1 between di¤erent goods
i 2 [0, 1]
as ε ! ∞ higher and higher degree of substitution ! less market power
of producers
Basic New Keynesian Model
Households
The period budget constraint is
R1
Pt (i )Ct (i ) di + Rt 1 Bt = Bt
1
+ W t N t + T t + Dt
0
Bt represents purchases of riskless one-period bonds paying one unit of
the numéraire in the following period , while Bt 1 is the quantity of
bonds carried over from period t 1. Rt is the gross nominal return on
riskless bonds purchased in period t 1
Dt dividends from ownership of …rms
households are subject to a solvency constraint lim Et fBT g = 0
T !∞
Pt (i ) is the price of good i.
Wt : nominal wage (given, since we have competitive in factor markets);
Tt : lump-sum transfers
Basic New Keynesian Model
Households: intratemporal optimization problem
Households choose Ct (i ) so to min the cost function given the
! ε ε1
R1
1
consumption index Ct
for any given level of
Ct (i )1 ε di
0
expenditures
R1
Pt (i )Ct (i ) di.
0
At the optimum:
Ct (i ) =
where Pt =
R1
0
R1
Pt (i )
1 ε
di
! 11 ε
Pt (i )
Pt
ε
Ct
- ideal price index such that
Pt (i )Ct (i ) di = Pt Ct . The demand for the individual goods of
0
type i depends negatively on the …rm-speci…c goods price relative
to the average price level and positively on the consumption index.
Basic New Keynesian Model
Households: intratemporal optimization problem (proof)1
The problem of the household is:
R1
min Pt (i )Ct (i ) di for a given level of expenditure, Pt Ct , where
0
! ε ε1
R1
1 1ε
Ct (i )
Ct =
di
(0
)
R1
min
Pt (i )Ct (i ) di Pt Ct
fC t (i )g
0
FOC (for good i )
Pt (i )
!
Pt ε
R1
0
1 For
ε
Ct (i )
1
ε 1
ε
R1
0
di
Ct (i )
! ε 11
1
1
ε
di
Ct (i )
! ε ε1
1
ε
=
1
1
1
ε
Ct (i )1
1
ε
1
=0
P t (i )
Pt
a di¤erent approach to solve the typical household problem, see Galí
(2008), Appendix 3.1
Basic New Keynesian Model
Households: intratemporal optimization problem!(proof)
ε
Now recalling Ct =
R1
Ct (i )1
1
ε
ε 1
di
0
1
ε
the FOC becomes Ct Ct (i )
1
ε
Ct (i ) =
=
P t (i )
Pt
which gives
ε
Pt (i )
Pt
Ct
Now …nd Pt . Use the budget constraint:
R1
Pt (i )Ct (i ) di = Pt Ct
0
Use the demand functions
ε
R1
P (i )
Pt (i ) Pt t
Ct di = Pt Ct simplify and solve for Pt so as to
0
obtain
Pt =
R1
0
Pt (i )
1 ε
di
! 11 ε
Basic New Keynesian Model
Households: intertemporal optimization problem
The intertemporal
2 Lagrangian
∞
6
L t = Et ∑ β j 6
4
j =0
+ λ t +j
1 +φ
C t1+jσ
1 σ
N t +j
1 +φ
+
Bt +j 1 + Wt +j Nt +j + Tt +j +
+Dt +j Pt +j Ct +j Rt +1j Bt +j
FOCs
wrt Ct
∂L
= 0 =) Ct
∂Ct
σ
= λt Pt
wrt Nt
∂L
φ
= 0 =) Nt = λt Wt
∂Nt
wrt Bt
∂L
= 0 =) λt Rt
∂Bt
1
= βEt λt +1
3
7
7
5
Basic New Keynesian Model
Households
Combine all FOCs
The stochastic Euler consumption equation
(
)
Ct +σ1 Pt
Rt = βEt
Ct σ Pt +1
labor supply
φ
Nt = Ct
σ Wt
Pt
where the wage is given, that is workers do not have any market
power (see instead Erceg et al. 2000).
1/φ is the Frisch elasticity: the elasticity of labor supply to the
wage rate, given a constant marginal utility of consumption (i.e.
given Ct σ )
Basic New Keynesian Model
Firms
Assume a continuum of …rms indexed by i 2 [0, 1] . Each …rm
produces a di¤erentiated good.
Identical technology, represented by the production function:
Yt (i ) = At Nt (i )1
α
α<1
All …rms face an identical isoelastic demand schedule and take the
aggregate price level and the aggregate demand as given
Yt (i ) =
Pt (i )
Pt
ε
Yt
Aside: Why do we need monopolistic competition with price
rigidities?
Monopolistic competition: No perfect substitutability between
di¤erent goods types ! di¤erent prices do not lead to a perfect
substitution. The larger Yt , the higher the demand for each variety.
Basic New Keynesian Model
Firms - under ‡exible prices
The …rm’s objective is to max pro…ts wrt Nt (i ) and Pt (i ), given
Wt , Yt (i ) = At Nt (i )1
The Lagrangian is
Lt = Pt (i ) Yt (i )
| {z }
P t (i )
Pt
ε
α
and Yt (i ) =
W t Nt ( i )
Yt
P t (i )
Pt
ε
Yt .
0
B
B
MCtN (i ) B
Yt (i )
@ | {z }
P t (i )
Pt
ε
At Nt (i )
Yt
1
C
C
A
1 αC
where MCtN (i ) is the Lagrange multiplier (to be interpreted as the
nominal marginal cost; MCtN (i ) gives how much nominal costs
change if the …rm has to produce an additional unit of its good).
Basic New Keynesian Model
Firms - under ‡exible prices
The FOC wrt Nt gives the labor demand:
Wt
(1
α)MCtN (i )Nt (i )
α
=0!
(1
Wt
α)At Nt (i )
α
= MCtN (i )
while the FOC wrt Pt (i ) is
Yt (i )
Yt (i )
εPt (i )
εYt (i )
P t (i )
Pt
ε 1
1
P t Yt
+MCtN (i )ε
+MCtN (i )εYt (i ) Pt1(i )
Pt (i ) =
ε 1
1
P t Yt
=0
= 0 which simplies to
ε
|ε {z 1}
P t (i )
Pt
MCtN (i )
desired markup
M
Remarks: decreasing returns imply …rm’s speci…c marginal costs.
! PWt (ti ) = ε ε 1 (1 α)At Nt (i ) α ! while in perfect competition
Wt
P t (i )
= (1 α)At Nt (i ) α . De…ne real marginal
costs as MCt (i ) = MCtN (i )/Pt (i ) ! MCt (i ) = M 1 = ε ε 1 .
we would have:
Basic New Keynesian Model
Modelling Price Rigidities
I
Modelling price rigidities?
I
I
I
I
Stochastic staggering à la Calvo (1983): each period a
constant fraction of randomly selected …rms can readjust their
prices.
Quadratic adjustment costs à la Rotemberg (1983): the larger
the price change, the more costly the adjustment.
Deterministic staggering à la Taylor (1980): prices are set on
the basis of x-period contracts in a staggered fashion.
In the basic NK model the Calvo scheme is used.
Basic New Keynesian Model
The Calvo pricing: the idea
In the ‡ex price case, …rms could change their prices each period as a
constant markup over marginal cost. Now:
I Each …rm may reset its price with probability
1
θ in any given
period
I Thus each period a fraction
1 θ of producers reset their prices,
while a fraction θ keep their price unchanged
I As a result the average duration of a price is given by
(1
θ)
1
and θ is an index of price stickiness (this hazard rate is constant
over time)
All …rms resetting prices will choose an identical price, say Pt .
Basic New Keynesian Model
The Calvo pricing: the logic behind the …rm’s problem
I Now let’s focus on the …rm which has the ability to adjust its price
at time t . In choosing the price Pt the …rm will maximize the
expected discounted value of future pro…ts, since it will be stuck
with this price Pt for more than just one period. In other words the
…rm will choose the price Pt so as to max the current market value
of the pro…ts generated while that price remains e¤ective.
I The …rm will discount future pro…ts by the gross nominal interest
rate between today, t , and any future date, t + k (for k = 0...∞).
From the households Euler equation this discount factor can be
written as:
Qt,t +k = β
k
Ct +k
Ct
σ
Pt
Pt +k
This is known as as the stochastic discount factor.
Basic New Keynesian Model
The Calvo pricing: the logic behind the …rm’s problem
I In addition, the …rm will discount future pro…ts taking into account
the probability θ that the price will remain e¤ective for a while. If θ
is small, then the …rm will be able to revise prices frequently and so
it will heavily discount future pro…ts. If θ is large instead, the
opposite will be true. Today pricing decisions will likely to a¤ect
future pro…ts.
Basic New Keynesian Model
The Calvo pricing: the …rm’s problem
A …rm reoptimizing in period t will choose the price Pt that max
the current market value of the pro…ts generated while that price
remains e¤ective
8
0
19
>
>
=
<
∞
B
C
k
max ∑ θ Et Qt,t +k @Pt Yt +k jt Ψt +k Yt +k jt A
>
P t k =0
{z
} >
| {z } |
;
:
revenues
costs
where Yt +k jt = output in period t + k for a …rm resetting its price
ε
at time t, Ψt +k ( ) = cost function, Yt +k jt = PPt +t k
Yt +k and
Qt,t +k is the nominal stochastic discount factor between period t
and t + k and is de…ned as Qt,t +k = βk
C t +k
Ct
σ
Pt
P t +k .
Basic New Keynesian Model
The Calvo pricing
At the optimum:
8
0
19
ε 1
>
>
P
P
t
<
Yt +k Pt +t k + C=
∞
B Yt +k jt ε Pt +k
k
∑ θ Et Qt,t +k @
A =0
ε 1
∂Ψt +k
>
>
k =0
:
;
+ ∂Y
ε PPt +t k
Yt +k Pt1+k
t +k jt
8
19
0
ε
Pt
<
=
Y
ε
Y
+
∞
t
+
k
t
+
k
j
t
P t +k
k
A
=0
∑ θ Et Qt,t +k @
ε
∂Ψt +k
:
;
k =0
ε PPt +t k
Yt +k P1
+ ∂Y
t +k jt
o t
n
∞
∂Ψ
k
=0
∑ θ Et Qt,t +k Yt +k jt 1 ε + ∂Y t +k ε P1
k =0
t +k jt
t
Basic New Keynesian Model
The Calvo pricing
the previous
8 condition can
0 be expressed as
19
>
>
>
>
>
B
>
>
C>
>
>
B
C
>
>
>
>
B
=
<
C
∞
ε
∂Ψ
B
t +k C
k
C =0
∑ θ Et Qt,t +k Yt +k jt B Pt
B |{z}
>
>
k =0
t +k jt C
>
>
|ε {z 1} |∂Y{z
Bopt. price
>
>
}C
>
>
>
>
@
A
desired markup
>
>
N
>
>
MC
.
;
:
t +k jt
M
N.B. with θ = 0, ‡ex prices, we have:
N
t +k
Pt +k = ε ε 1 ∂Ψ
∂Y t +k = M MCt +k (see above)
Basic New Keynesian Model
The Calvo pricing
The optimal condition can be rewritten as
n
o
∞
k
N
M MCt +k jt
=0
∑ θ Et Qt,t +k Yt +k jt Pt
k =0
which can be re-written as
∞
k
∑ θ Et
k =0
(Pt )
ε
ε
Qt,t +k PPt +t k
Yt +k Pt
M MCtN+k jt
=0
n
o
∞
k
M MCtN+k jt
=0
∑ θ Et Qt,t +k Ptε+k Yt +k Pt
k =0
n
o
k
ε Y
N
Q
P
M
MC
θ
E
∑
t
t,t +k t +k t +k
t +k jt
∞
Pt =
k =0
∞
k
∑ θ Et Qt,t +k Ptε+k Yt +k
k =0
Basic New Keynesian Model
The Calvo pricing: the price index
R1
Recalling Pt
Pt (i )
1 ε
di
0
Pt =
"
R
Pt (i )1
! 11 ε
ε
:
di + (1
θ ) (Pt )1
S (t )
ε
# 11 ε
S (t ) set of …rms not reoptimizing their posted price. Since the
distribution of prices among …rms not-resetting their prices corresponds
to the distribution of prices prevailing in the previous period we have:
Pt
Π1t
ε
where Πt = Pt /Pt
=
h
θPt1
ε
1
= θ + (1
1
+ (1
θ)
(gross in‡ation).
θ ) (Pt )1
Pt
Pt 1
1 ε
ε
i 11 ε
Basic New Keynesian Model
Equilibrium in the goods’market
Mkt clearing in each good mkt requires
Yt (i ) = Ct (i )
Aggregate output is Yt
R1
Yt (i )1
0
Yt = Ct
1
ε
di
! ε ε1
hence we have
Basic New Keynesian Model
Equilibrium in the labor market
In equilibrium
labor supply = labor demand
Z 1
=
Nt
0
Nt (i )di
Take the production function Yt (i ) = At Nt (i )1
Yt (i ) =
P t (i )
Pt
Nt
ε
α
and recall that
Yt to have:
=
Z 1
0
=
Yt
At
1
1 α
Yt (i )
At
1
1 α
di
Z 1
|
0
Pt (i )
Pt
{z
ε
1 α
price dispersion
di
}
Price dispersion represents the resource costs due to relative price dispersion under Calvo pricing. Higher price
dispersion
) more labor inputs are required to produce a given level of output.
Basic New Keynesian Model
List of equations - I part
Euler consumption
n equation
o
C σ
1
(1) Rt = βEt Ct +σ1 Πt1+1
t
labor supply
φ
t
(2) Nt = Ct σ W
Pt
production function
R 1 P t (i )
(3) Yt = At Nt1 α 0
Pt
labor demand
t
(4) W
α)At Nt
P t = MCt (1
equilibrium in goods mkt
(5) Yt = Ct
α
ε
1 α
1 α
di
(At exogenous productivity)
Basic New Keynesian Model
List of equations - II part
price level
(6) Πt1 ε = θ + (1 θ )
price setting equation
∞
(7)Pt =
∑ θk Et
k =0
n
∞
Pt
Pt 1
1 ε
Q t,t +k P tε+k Y t +k M MC tN+k jt
∑ θ k Et fQ t,t +k Ptε+k Y t +k g
k =0
stochastic discount factor
σ
Pt
(8) Qt,t +k = βk CCt +t k
P t +k
o
Basic New Keynesian Model
Steady state - I part
Let Π = 1 (no trend in‡ation)
Euler consumption equation
(1) R 1 = β
labor supply
(2) N φ = Y σ W
P
production function
(3) Y = AN 1 α
labor demand
(4) W
α)AN α
P = MC (1
equilibrium in goods mkt
(5) Y = C
Basic New Keynesian Model
Steady state - II part
price level
(6) 1 = 1
price setting equation
(7) MC 1 = M = ε ε 1
stochastic discount factor
(8) Qt,t +k = βk
Basic New Keynesian Model
Loglinearization - background
Consider an economic variable Xt
Denote by xt the log deviation of Xt from its steady state X
xt = log (Xt )
NB: xt = log( XXt ) = log(1 + X tX 1 ) '
Now
Xt = e log (X t ) = XX e log (X t ) = Xe log (X t )
Xe log (1 +xt )
hence
Xt
Xt
log(X )
Xt 1
X
log (X )
' X (1 + xt )
or
' Xe log xt
= Xe log (1 +
Xt 1
X
)'
Basic New Keynesian Model
Loglinearization
(1) Euler equation
n
o
σ
Ce ct
1
1= Re rt βEt
Ce ct +1
Πe πt +1
using steady state condition
σ
1 = e rt Et (e ct ct +1 ) e π1t +1
take log
c t = Et f c t + 1 g
1
(rt
σ
Et fπ t +1 g)
Basic New Keynesian Model
Loglinearization
(2) labor supply
φ
t
Nt = Ct σ W
Pt
wt
(Ne nt )φ = (Ce ct ) σ We
Pe pt
using steady state condition
wt
(e nt )φ = (e ct ) σ ee pt
take log
wt
pt = wt = σct + ϕnt
Basic New Keynesian Model
Loglinearization
production function
(3) Yt = At Nt1
α
Dt where Dt =
R1
0
Pt
Pt
ε
1 α
1 α
di
using the same strategy as above:
yt = at + (1 α) nt + dt
it can be shown2 that dt is of second order (only if Π = 1)
hence
yt = at + (1
where at = ρa at
2 See
1
+ εa,t ,
Galí (2008), Appendix 3.3
εa,t
α) nt
N 0, σ2a
i.i.d.
Basic New Keynesian Model
Loglinearization
labor demand
t
(4) W
P t = MCt (1
α)At Nt
α
becomes
wt = mct + at
equilibrium in goods mkt
(5) Yt = Ct becomes
yt = ct
αnt
Basic New Keynesian Model
Loglinearization
price equation
(6) Πt1
ε
= θ + (1
θ)
Pt
Pt 1
1 ε
Log-linearizing around Π = 1 in steady state
π t = (1
θ ) (pt
pt
Pt
Pt 1
=1:
1)
In‡ation results from the fact that …rms reoptimizing in any given
period choose a price, pt , that di¤ers from the economy’s average
price in the previous period, pt 1 . Under ‡ex prices:
π t = pt pt 1 .
Basic New Keynesian Model
Loglinearization
price setting equation (7) and discount factor (8)
log-linearizing around a zero in‡ation steady state given (6)
where Θ
π t = βEt fπ t +1 g + (1
|
1 α
1 α+αε .
Iterating forward:
1 θ
βθ )
Θmct
{z θ }
λ
∞
π t = λ ∑ βk Et fmct +k g
k =0
In words: in‡ation will be higher when …rms expect average markups
(marginal cost) to be lower (higher) than their steady state level Firms
having the opportunity to reset their prices will choose a price above the
economy average’s price level in order to readjust their markup closer to
the desired level. Nature of in‡ation: in‡ation here results from the
price-setting decisions of …rms setting their prices in response to current
and expected cost conditions.
Basic New Keynesian Model
Summing up
ct = Et fct +1 g σ1 (rt Et fπ t +1 g)
wt = σct + ϕnt
yt = at + (1 α) nt
wt = mct + at αnt
yt = ct
π t = βEt fπ t +1 g + λmct
get rid of ct and of wt
yt = Et fyt +1 g
1
(rt
σ
Et fπ t +1 g)
and solve for mct
mct =
σ+
ϕ+α
1 α
yt
1+ϕ
at
1 α
Basic New Keynesian Model
Summing up
ytn = natural level of output (output under ‡ex prices - with
markup>1 - second best output)! marginal costs are constant
mc = 0
1+ ϕ
ϕ+α
0 = σ + 1 α ytn 1 α at
ytn = ϑa at
where ϑa
Hence
1+ ϕ
1 α/
σ+
mct =
ϕ+α
1 α
σ+
ϕ+α
1 α
(yt
ytn )
Basic New Keynesian Model
The New Phillips Curve (NPC)
The traditional Phillips Curve is expressed in terms of the output
(or unemployment). Starting from π t = βEt fπ t +1 g + λmct ,
using the eq. just found, mct = σ +
ϕ+α
1 α
( yt
φ+α
π t = βEt fπ t +1 g + λ σ +
(yt
1 α
|
{z
}
κ
ytn ) , for mct :
ytn )
(NPC)
Basic New Keynesian Model
The New Phillips Curve (NPC)
Remarks
I
Forward looking (the traditional Phillips curve was backward
looking... see below)
I
Derived from microfoundations (the traditional Phillips curve
was a sort of empirical regularity)
I
The evolution of in‡ation depends on the level of economic
activity
I
In contrast with classical monetary models an increase in the
money supply has no direct e¤ect on prices. Its impact is
indirect, working through the induced changes in the level of
economic activity.
Basic New Keynesian Model
The New Phillips Curve (NPC)
The traditional Phillips Curve relates in‡ation to some cyclical
indicators as well as its own lagged values. A common speci…cation
takes the form:
π t = υπ t
1
+ ζ (yt
y ) + shockt
where past in‡ation matters for the determination of current
in‡ation and current in‡ation is positively correlated with past
output.
Basic New Keynesian Model
The New Phillips Curve (NPC)
The previous properties stand in contrast with those characterizing
the NPC. Iterating it forward yields:
∞
π t = κEt ∑ βi (yt +i
i =0
ytn+i )
Remarks
I
past in‡ation is not a relevant factor in determining current
in‡ation
I
in‡ation is positively correlated with future output (so
in‡ation leads output)
I
intuition: in a world with staggered price setting, in‡ation
results as a consequence of the price setting decisions by …rms
currently resetting their prices; price setting decisions depends
on current and expected marginal costs, not on past in‡ation.
Basic New Keynesian Model
The Model to be simulated
The IS
yt = Et fyt +1 g
The NPC
where ytn = ϑa at .
0
1@
rt
σ |{z}
?
1
Et f π t + 1 g A
π t = βEt fπ t +1 g + κ (yt
ytn )
Basic New Keynesian Model
The Model to be simulated
We have a model with two forward looking (jump) variables.
Blanchard-Kahn condition for the existence of a unique rational
expectations equilibrium requires a number of eigenvalues outside
the unit circle equal to the number of forward looking variables.
We need one more equation, namely, the conduct of monetary
policy.
N.B. if rt = r (peg) the Blanchard-Kahn condition is not satis…ed
! Self-ful…lling increases (or decreases) in in‡ation cannot be
ruled out (real indeterminacy).
Basic New Keynesian Model
The Monetary Policy
The central bank reacts to deviations from the natural level output
and from the in‡ation target (0 in‡ation) according to a simple
Taylor’s rule (Taylor 1993):
rt = φπ π t + φy (yt
ytn ) + ut
where ut is an exogenous stochastic component:
ut = ρu ut
1
+ ξ r ,t
Basic New Keynesian Model
The BK conditions (implementability)
Write the model as follows
h
yt = Et fyt +1 g σ1 φπ π t + φy (yt
ϑa at ) + ut
π t = βEt fπ t +1 g + κ (yt ϑa at )
or as
Et fyt +1 g = yt + σ1 φπ π t + φy (yt ϑa at ) + ut
Et fπ t +1 g = 1β π t 1β κ (yt ϑa at )
hence
Et fyt +1 g =
1 + φy +
Et f π t + 1 g
11
σ β κ yt
= 1β π t
+
1
σ
1
β κyt
φπ
1
β
+
1
β κϑa at
π t + ut
Et f π t + 1 g
1
σ Et
11
σ β κϑa at
i
f π t +1 g
φ y ϑ a at
Basic New Keynesian Model
The BK conditions (implementability)
The system can be witten in matrix notation as
Et fyt +1 g
=
Et f π t + 1 g
!
1
1
1 + φy + σ1 1β κ
φ
yt
π
σ
β
+
1
1
π
t
βκ
β
!
11
1
φy ϑ a
ut
σ β κϑa
+
1
at
0
κϑ
a
β
The relevant matrix for determinacy of the
! RE equilibrium is
1
1
11
1 + φy + σ β κ
σ φπ
β
A=
1
1
κ
β
β
Basic New Keynesian Model
The BK conditions (implementability)
Trace A: φy +
1
β
+
κ
σβ
+1 > 0
1
Determinant A: σβ
σ + κφπ + σφy > 0
See the Addendum to chapter 4 of the Woodford (2003) book.
The matrix A has two eigenvalues outside the unit circle if the
following condition holds:
κ (φπ
1) + (1
β ) φy > 0
Basic New Keynesian Model
The Monetary Policy: The Taylor Principle
In the basic setup if φy = 0, it must be φπ > 1 in order to have a
unique equilibrium (the monetary authority should respond to
deviations of in‡ation from its target level by adjusting the nominal
interest rate more than proportionally, so as to increase the real
rate) (i.e. the Taylor’s principle).
This condition rules out self-ful…lling in‡ationary (or de‡ationary)
dynamics (and in doing so, it ensures real determinacy - the
existence of a unique equilibrium).
Otherwise the condition is: κ (φπ 1) + (1 β)φy > 0.
We neglect the zero-lower bound problem (rt
0) ! and the
emergence of a liquidity trap at rt = 0 (see Benhabib,
Schmitt-Grohé and Uribe 2001).... problem arising also with
wealth e¤ects...
Galì (2008) ch. 4 p. 78
Basic New Keynesian Model
Where’s money?
I
We have a cashless economy
I
Money is not incorporated in the analysis: money as unit of
account
I
Assuming money in the utility function (with separability) we
will have an LM of the form: mt pt = vσ ct ηit . Here the
LM determines the quantity of money that the central bank
must supply in order to support the nominal interest rate
implied by the Taylor’s rule.
I
Cashless (or careless?) theory of monetary policy.
I
Even under the assumption of non-separability, the real e¤ects
of having real balances into the model are negligible. See
Walsh (2003), Woodford (2003).
I
However, when we have OLG real balances do play a role (see
Bénassy 2007, Piergallini 2006).
Basic New Keynesian Model
Calibration
α = 1/3
β = 0.99
σ=1
ϕ=1
ε=6
1 θ = 1/3
φπ = 1.5
φy = 0.5/4
ρZ = 0.95
ρu = 0.5
labor’s elasticity
discount factor
inverse of the IES!log utility
inverse of the Frisch elasticity
elasticity of substitution between goods
probability of resetting price! average duration 3 quarters
response to in‡ation
response to output gap
persistence of tech shock
persistence of the interest rate shock
E¤ects of a Technology Shock
Income
Inflation
1
Employment
-0.045
-0.03
-0.05
-0.035
0.9
-0.055
0.8
-0.04
%
-0.06
-0.045
-0.065
0.7
-0.05
-0.07
0.6
0.5
-0.055
-0.075
0
5
10
15
-0.08
0
quarters
5
10
15
-0.06
0
5
quarters
Consumption
Ouput gap
1
10
15
quarters
Nominal Interest Rate
-0.02
-0.07
-0.08
0.9
-0.025
-0.09
%
0.8
-0.03
-0.1
0.7
-0.11
-0.035
0.6
0.5
-0.12
0
5
10
quarters
15
-0.04
0
5
10
quarters
15
-0.13
0
5
10
quarters
15
E¤ects of a Positive Technology Shock
Propagation of the shock
Higher productivity ! lower (real) marginal costs ! lower
in‡ation ! the Central Bank partially accommodates the
improvement in technology by lowering the nominal interest rate
and the real rate; the policy is not su¢ cient to close the negative
output gap; income increases but less than its natural counterpart;
employment declines.
Intuition: such a response of employment is due to price rigidities
preventing an immediate adjustment of the price level to the new
level of costs.
The economy’s response to a positive technology shock is in
contrast with the results of the standard RBC model (comovement
between productivity, output and employment in response to
technology shocks).
However, empirical evidence points to a negative relationship
between employment and productivity
E¤ects of a Positive Technology Shock
Weaker response (baseline Taylor rule in red)
φπ = 1.1; φy = 0
Income
Inflation
1
Employment
0
-0.05
0.8
-0.1
0.7
-0.15
0.6
-0.2
-0.1
%
0.9
0
-0.2
0.5
0
5
10
15
-0.3
-0.25
0
5
quarters
10
15
-0.4
0
5
quarters
Consumption
Output Gap
1
15
Nominal Interest Rate
0
-0.05
-0.05
0.9
10
quarters
-0.1
-0.1
0.8
-0.15
%
-0.15
-0.2
0.7
-0.2
-0.25
0.6
0.5
-0.25
-0.3
0
5
10
quarters
15
-0.35
0
5
10
quarters
15
-0.3
0
5
10
quarters
15
E¤ects of a Monetary Policy Shock
Income
Inflation
0
0
-0.2
-0.05
-0.1
-0.6
-0.15
-0.8
-0.2
-1
-0.25
-1.2
-0.3
-0.5
%
-0.4
Employment
0
-1
-1.4
0
5
10
15
-0.35
-1.5
0
5
quarters
10
15
-2
0
5
quarters
Consumption
10
15
quarters
Ouput gap
Nominal Interest Rate
0
0
0.5
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
0.3
-0.8
-0.8
0.2
-1
-1
-1.2
-1.2
%
0.4
-1.4
0
5
10
quarters
15
-1.4
0.1
0
5
10
quarters
15
0
0
5
10
quarters
15
E¤ects of a Monetary Policy Shock
Propagation of the shock
The presence of price rigidities is a source of nontrivial real e¤ects
of monetary policy shocks.
Firms cannot adjust the price of their good when they receive new
information about costs or demand conditions.
The shock corresponds to an increase of 1% in ξ r ,t ! the shock
generates an increase in the real rate, a decrease in in‡ation,
output and employment; the natural level of output is not a¤ected
by the monetary policy shock.
The increase in the nominal rate is less than 1% because of the
decline in in‡ation and in the output gap.
New Keynesian Model with Indexation of the NPC
Indexation of the New Phillips Curve (NPC)
The NPC is theoretically appealing, however it cannot account for
many features of the data that motivated the traditional Phillips
curve speci…cation.
In particular, cross-correlations between in‡ation and detrended
output observed in the data suggest that output leads in‡ation
(not the opposite). In this sense, empirical evidence tends to be
more consistent with a traditional, backward-looking Phillips curve
than with the new version. This evidence seems reinforced by
many of the estimates of hybrid Phillips curves of the form
π t = γb π t
1
+ γf Et fπ t +1 g + κ (yt
ytn )
How can we "microfound" this hybrid NPC? In the periods
between price reoptimizations …rms mechanically adjust their prices
according to the previous period in‡ation! This speci…cation
generates a certain degree of in‡ation persistence (observed in the
data).
E¤ects of a Technology Shock (baseline model in red)
Income
Inflation
Employment
1
-0.03
0
0.9
-0.04
0.8
-0.05
-0.06
0.7
-0.06
-0.08
0.6
-0.07
-0.02
%
-0.04
-0.1
0.5
0
5
10
15
-0.08
-0.12
0
5
quarters
10
15
-0.14
0
5
quarters
Consumption
Output Gap
1
0
0.9
-0.02
0.8
-0.04
0.7
-0.06
10
15
quarters
Nominal Interest Rate
-0.07
-0.08
%
-0.09
-0.1
-0.11
0.6
-0.08
0.5
-0.1
0
5
10
quarters
15
-0.12
0
5
10
quarters
15
-0.13
0
5
10
quarters
15
E¤ects of a Monetary Policy Shock
Income
Inflation
0.2
0
Employment
0.5
0
-0.05
-0.2
-0.1
-0.4
-0.15
-0.5
-0.6
-0.2
-1
-0.8
-0.25
-1
-0.3
%
0
-1.2
0
5
10
15
-0.35
-1.5
0
5
quarters
10
15
-2
0
5
quarters
Consumption
Output Gap
0.2
10
15
quarters
Nominal Interest Rate
0.2
0.6
0
0.5
-0.2
0.4
-0.4
-0.4
0.3
-0.6
-0.6
0.2
-0.8
-0.8
0.1
-1
-1
0
-1.2
-1.2
-0.1
%
0
-0.2
0
5
10
quarters
15
0
5
10
quarters
15
0
5
10
quarters
15
An NK model with public consumption
Equilibrium equations
the IS
yt = Et fyt +1 g σ1 (rt Et fπ t +1 g) + 1
the NPC
π t = βEt fπ t +1 g + κ (yt ytn )
where
ytn = ϑa at + ϑg gt
rt = φπ π t + φy (yt ytn ) + ut
ut = ρu ut 1 + ξ i ,t
gt = ρg gt 1 + ξ g ,t
at = ρa at 1 + ξ a,t
ρg gt
E¤ects of a Government Spending Shock
Inflation
Income
0.4
%
0.35
Employment
0.055
0.55
0.05
0.5
0.045
0.45
0.04
0.4
0.035
0.35
0.3
0.25
0.2
0
5
10
15
0.03
0
5
quarters
10
15
0
5
quarters
Consumption
10
15
quarters
Ouput gap
Nominal Interest Rate
-0.3
0.028
0.09
-0.35
0.026
-0.4
0.024
-0.45
0.022
0.07
-0.5
0.02
0.06
-0.55
0.018
-0.6
0.016
%
0.08
-0.65
0
5
10
quarters
15
0.014
0
0.05
5
10
quarters
15
0.04
0
5
10
quarters
15
E¤ects of a Government Spending Shock
Propagation of the shock
An increase in government spending leads to an increase in AD !
current output increases by more than natural output !
employment increases.
Firms will revise their prices upward (as real marginal costs are
higher)! the Central Bank will react to in‡ation by increasing the
nominal interest rate more than proportionally ! as a result the
real interest rate will increase (lean against the wind policy ).
What about consumption? A positive government spending shock
leads to a decline in consumption!
Discussion
I
The basic New Keynesian model generates some implausible
results, such as:
I
I
I
No in‡ation persistence
No output persistence
No comovement between consumption and public spending
Discussion
I
Lack of in‡ation persistence: in this forward looking model
current in‡ation is a jump variable, but actual in‡ation
displays highly serially correlated behavior. Possible solutions:
I
I
I
I
I
Indexation (see e.g. Christiano et al., 2005)
Deviations from the assumption of full-information in price
setting
Rule of thumb …rms (Galí and Gertler, 1999) - each …rm is able
to adjust its price in any given period with a …xed probability
1 θ. A fraction 1 ω of the …rms set prices optimally. The
remaining fraction of …rms, ω, instead use a simple rule of
thumb that is based on the past aggregate price behavior.
Wage rigidities
Variable capital utilization (marginal costs less sensitive to
output variations). See Christiano et al. (2001)
Discussion
I
Lack of output persistence: the response of output to shock is
hump shaped. Possible solutions:
I
I
I
I
Adjustment costs on labor and investments
Price and wage staggering
Habit formation
Diminishing returns
Discussion
I
No comovement between private consumption and public
consumption. Possible solutions:
I
I
Rule of thumb consumers (Galí et al. 2007).
Wealth e¤ects and accommodating monetary policy.
Discussion
In the basic New Keynesian model there is no trade-o¤ between
output and in‡ation stabilization (in the absence of cost push
shock).
In addition, the "divine coincidence": stabilizing the in‡ation rate
stabilizes the welfare relevant output gap too (see Galí and
Blanchard 2007), where the welfare relevant output gap=distance
between the actual level of output and the e¢ cient level of output
(…rst best level).
Under real wage rigidities the divine coincidence does not hold.
The gap between the …rst and the second best output is not
constant.
Discussion
Empirically the basic model does a poor job (too much forward
looking...).
Maybe we need a more structural model.....
References
I
Bénassy, J.P. (2007), Money, Interest, and Policy, The MIT Press.
I
Benhabib, J. & Schmitt-Grohe, S., Uribe, M., (2001), The Perils of Taylor Rules, Journal of Economic
Theory, 96(1-2).
I
Blanchard, O., Galí J., (2007), Real Wage Rigidities and the New Keynesian Model, Journal of Money,
Credit, and Banking, 39(s1),35-65
I
.
Calvo, G., (1983), Staggered Prices in a Utility-Maximizing Framework, Journal of Monetary Economics,
12(3).
I
Christiano, L., Eichenbaum, M., Evans C. (1998) Monetary Policy Shocks: What Have We Learned and to
What End ?,in J.B. Taylor and M. Woodford eds., Handbook of Macroeconomics, volume, 1A, 65-148.
I
Christiano, L., Eichenbaum, M., Evans C. (2005), Nominal Rigidities and the Dynamic E¤ects of a Shock
to Monetary Policy, Journal of Political Economy, 113(1).
I
Clarida, R., Gali, J., Gertler, M., (1999), The Science of Monetary Policy: A New Keynesian Perspective,
Journal of Economic Literature, 37(4).
I
Erceg, C. J. & Henderson, D. W. & Levin, A. T., (2000). Optimal monetary policy with staggered wage
and price contracts, Journal of Monetary Economics, 46(2), . 281-313.
I
Galí, J. (2003), New Perspectives on Monetary Policy, In‡ation, and the Business Cycle, in: Dewatriport,
M. et al. (Eds.), Advances in Economics and Econometrics: Theory and Applications; eighth World
Congress. 2003; 151-197.
References
I
Goodfriend, M., King, R., (1997), The New Neoclassical Synthesis and the Role of Monetary Policy, NBER
Macroeconomics Annual.
I
Piergallini, A., (2006), Real Balance E¤ects and Monetary Policy, Economic Inquiry, 44(3).
I
Rotemberg, J., (1983), Aggregate Consequences of Fixed Costs of Price Adjustment, American Economic
Review, 73(3).
I
Taylor, J., (1980), Aggregate Dynamics and Staggered Contracts, Journal of Political Economy, 88(1).
I
Walsh, C.E. (2003), Monetary Theory and Policy, The MIT Press.
I
Woodford, M. (2003), Interest & Prices, Princeton University Press, chapter 4.