Basic New Keynesian Model
Barbara Annicchiarico
Università degli Studi di Roma "Tor Vergata"
Presentation 5/A
Basic New Keynesian Model
Motivation
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The basic RBC models
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business cycle driven by real shocks (mainly by productivity
shocks)
perfect competition (no market power)
‡exible prices and wages (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
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but....recall some business cycle facts:
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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 (prices do not adjust immediatly), the
real interest rate increases
I Households will …nd it optimal to postpone consumption
=)
aggregate demand (AD) decreases
I Since …rms cannot revise their prices 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
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Prices do not adjust instantaneously
In the intermediate good sector …rms 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)
basic model usually without physical capital
I Some references: Clarida et al. (1999); Galí (2003, 2008); Woodfod
(2003); Goodfriend and King (1997); Goodfriend (2007).
Basic New Keynesian Model
Agents
I 4 agents:
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Households
Final-good producing …rms
Intermediate-good producing …rms
Central bank
Basic New Keynesian Model
Speci…c assumptions on the monopolistically competitive sector
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Assumptions
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Each intermediate-good producing …rm is able to di¤erentiate
its product from those of its rivals (it acts as a monopolist in
its particular product market);
Each intermediate-good producing …rm takes the prices
charged by its rivals as given (no strategic interaction);
Final good producers face a technology where intermediate
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 intermediate - good producer) ! the larger
the market size the higher the demand faced by each
intermediate-good producer;
Firms have the same technology
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
1 σ
1 +φ
Nt
where U (Ct , Nt ) = C1t σ
1 +φ
0 < β < 1 is the discount factor, φ > 0 is the inverse of the Frisch
elasticity
σ is the inverse of the intertemporal elasticity of substitution
Basic New Keynesian Model
Households
The period budget constraint is
Pt Ct + Rt 1 Bt = Bt
1
+ Wt Nt + Tt + Dt
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
Wt : nominal wage (given, since we have competitive in factor markets);
Tt : lump-sum transfers (for now Tt = 0)
households are subject to a solvency constraint
Basic New Keynesian Model
Households:Optimization problem
The intertemporal
2 Lagrangian
∞
6
L t = Et ∑ β j 6
4
j =0
FOCs
wrt Ct
wrt Nt
wrt Bt
+ 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
∂ Lt
= 0 =) Ct
∂Ct
σ
= λ t Pt
∂ Lt
φ
= 0 =) Nt = λt Wt
∂Nt
∂ Lt
= 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 - NEW
(
)
Ct +σ1 Pt
1
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
Production
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Production split into two stages
1. Intermediate goods: employ labour; nominal rigidities and
market power
2. Final goods: technology given by a a constant elasticity (CES)
bundler of intermediate goods; perfect competition
Basic New Keynesian Model
Final-good producing …rms
A continuum of perfectly competitive …rms producing …nal goods act
under perfect competition using a CES technology:
Yt
R1
0
Yt ( i ) 1
1
ε
di
! ε ε1
,
ε is the constant elasticity of substitution ε > 1 between di¤erent goods
i 2 [0, 1] ,
as ε ! ∞ higher degree of substitution ! less market power of
intermediate-good producers
Final goods can be used for private consumption
Basic New Keynesian Model - NEW
Final-good producing …rms
The representative …nal good producer chooses Yt (i ) so to max pro…ts
given the CES technology, the price Pt (i ) and the …nal good price Pt
At the optimum we have:
Yt ( i ) =
where Pt =
R1
0
R1
0
Pt ( i ) 1
ε
di
! 11 ε
Pt ( i )
Pt
ε
Yt
- ideal price index such that
Pt (i )Yt (i ) di = Pt Yt . The demand for the individual intermediate-
goods of type i depends negatively on the …rm-speci…c goods price
relative to the average price level and positively on the aggregate level of
spending Yt .
Basic New Keynesian Model - NEW
Final-good producing …rms (proof)
The problem of the …nal-good producing …rms is to choose Yt (i )
so as to
8 max pro…ts, i.e. ! ε
9
ε 1
<
=
1
1
R
R
1
Yt (i )1 ε di
max
Pt
Pt (i )Yt (i ) di
;
fY t (i )g :
0
0
FOC (for intermediate - good i )
! ε ε1 1
1
R
1
Yt (i )1 ε di
Pt ε ε 1
1
0
!
R1
0
Yt ( i )
ε 1
ε
di
! ε 11
Yt ( i )
1
ε
1
ε
=
Yt ( i ) 1
P t (i )
Pt
1
ε
1
Pt ( i ) = 0
Basic New Keynesian Model
Final-good producing …rms (proof)
Now recalling Yt =
R1
Yt ( i ) 1
1
ε
0
1
the FOC becomes Yt ε Yt (i )
1
ε
Yt ( i ) =
=
di
! ε ε1
P t (i )
Pt
which gives
ε
Pt ( i )
Pt
Yt
Now …nd Pt . Use the budget constraint:
R1
Pt (i )Yt (i ) di = Pt Yt
0
Use the demand functions
ε
R1
P (i )
Pt (i ) Pt t
Yt di = Pt Yt simplify and solve for Pt so as to
0
obtain
Pt =
R1
0
Pt ( i )
1 ε
di
! 11 ε
Basic New Keynesian Model
Intermediate-good producing …rms
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? We need price setting.
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
Intermediate-good producing …rms - under ‡exible prices
The …rm’s objective is to max pro…ts wrt Nt (i ) and Pt (i ), given
W t , Yt ( i ) = At N t ( i ) 1
The Lagrangian is
L t = P t ( i ) Yt ( i )
| {z }
P t (i )
Pt
ε
α
and Yt (i ) =
Wt 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 )1
Yt
where MCtN (i ) is the Lagrange multiplier (to be interpreted as the
nominal marginal cost; MCtN (i ) measures how much nominal costs
change if the …rm has to produce an additional unit of its good).
NB: the superscript n in MCtN (i ) stands for Nominal.
1
C
C
A
αC
Basic New Keynesian Model
Intermediate-good producing …rms - under ‡exible prices
The FOC wrt Nt gives the labor demand:
Wt
(1
α)MCtN (i )Nt (i )
α
=0!
(1
Wt
α ) At N t ( 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 becomes
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
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Modelling price rigidities?
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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.
There is widespread agreement that Rotemberg and Calvo
pricing assumptions are somewhat equivalent models, because
they deliver equivalent dynamics. Yes, but only up to a
…rst-order approximation and in the absence of trend
in‡ation!!!!
Basic New Keynesian Model with Rotemberg Pricing
Quadratic adjustment costs à la Rotemberg
According to Rotemberg (1982) each monopolistic …rm faces a
quadratic cost of adjusting nominal prices, measured in terms of
the …nal-good:
ADJ_Pt (i ) =
γp
2
Pt ( i )
Pt 1 ( i )
2
1
Yt
where γp =degree of nominal price rigidities.
Firm i will not always choose to charge the optimal price (i.e.
Pt (i ) = ε ε 1 MCtN (i )) since it is assumed to face convex costs to
changing its price.
Basic New Keynesian Model with Rotemberg Pricing
Quadratic adjustment costs: the …rm’s pricing problem
The i -type …rm will set the price so as to max the present discounted
value of future"pro…ts
#
∞
Et ∑ Qt,t +k
k =0
P t + k ( i ) Yt + k ( i )
γp
2
Wt +k Nt +k (i )+
P t +k (i )
P t +k 1 (i )
P (i )
1
ε
2
Yt + k
t
Yt and the technology
given the demand Yt (i ) =
Pt
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...∞).
Basic New Keynesian Model with Rotemberg Pricing
Quadratic adjustment costs: the …rm’s pricing problem
The …rm discounts future nominal pro…ts by the gross nominal interest
rate between today and future dates
λ
From the households Euler equation we have Qt,t +k = βk λt +t k where
Qt,t +1 = β λλt +t 1 ,
Et β λλt +t 1 = Rt
1
and Ct σ = λt Pt
this is known as as the stochastic discount factor
frequently used in the asset pricing literature
Basic New Keynesian Model with Rotemberg Pricing
Quadratic adjustment costs: the …rm’s pricing problem
Lt =
∞
∑ Qt,t +k
k =0
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
Pt + k ( i )
Yt + k ( i )
| {z }
P (i )t +k
P t +k
ε
Y t +k
γp
P t +k (i )
20 P t +k 1 (i )
B
B
MCtN+k (i ) B
B
@
1
Yt + k ( i )
| {z }
P t +k (i )
P t +k
3
Wt +k Nt +k (i )+
ε
Y t +k
2
P t + k Yt + k
At +k Nt +k
1
C
C
C
A
(i )1 α C
7
7
7
7
7
7
7
7
7
7
7
7
7
5
Basic New Keynesian Model with Rotemberg Pricing
Quadratic adjustment costs: the …rm’s pricing problem
FOC wrt Pt (i )
Yt ( i )
εPt (i )
P t (i )
Pt
ε 1
1
P t Yt
ε 1
P t (i )
P (i )
1
γ p P t t 1 (i )
Pt
P t Yt
+γp Et Qt,t +1 PPt +t (1i()i ) 1 Yt +1 PPt +1i (i2) Pt +1
t( )
+εMCtN (i )
FOC wrt Nt (i ) (as under ‡ex prices!)
1
Pt
Pt
1 (i )
Yt
=0
Wt
(1 α )A t N t (i )
α
= MCtN (i )
Basic New Keynesian Model with Rotemberg Pricing
Quadratic adjustment costs: the …rm’s pricing problem
Drop the i -index (there’s symmetry since all …rms face the same
optimization problem)
Yt
N
t
εYt + ε MC
P t Yt
+γp Et Qt,t +1
γp
P t +1
Pt
Let Πt = Pt /Pt
1
Pt
Pt 1
1 Yt + 1
1
P t +1
Pt
Pt
P t 1 Yt +
2
=0
(gross in‡ation).
N
t
Yt εYt + ε MC
γ p ( Π t 1 ) Π t Yt +
P t Yt
γp Et Qt,t +1 (Πt +1 1) Yt +1 Π2t +1 = 0
N.B. γp = 0 ) Pt =
ε
N
ε 1 MCt
Basic New Keynesian Model with Rotemberg Pricing
Equilibrium in the goods’market
Aggregate demand is Yt
R1
Yt ( i ) 1
0
1
ε
di
! ε ε1
symmetry, mkt clearing requires
Yt = C t +
γp
( Πt
2
1 ) 2 Yt
which is the resource constraint of the economy
hence, given
Basic New Keynesian Model with Rotemberg Pricing
List of equations - I part
Euler consumption equation
σ
1
(1) Rt 1 = βEt CCt +t 1
Π t +1
labor supply
φ
t
(2) Nt = Ct σ W
Pt
production function
(3) Yt = At Nt1 α
labor demand
t
α)At Nt α (At exogenous productivity)
(4) W
P t = MCt (1
equilibrium in goods mkt (resource constraint)
γ
(5) Yt = Ct + 2p (Πt 1)2 Yt
Basic New Keynesian Model with Rotemberg Pricing
List of equations - II part
(6) in‡ation equation
N
t
Yt εYt + ε MC
γ p ( Π t 1 ) Π t Yt +
P t Yt
γp Et Qt,t +1 (Πt +1 1) Yt +1 Π2t +1 = 0
stochastic discount factor
σ
1
(7) Qt,t +1 = β CCt +t 1
Π +1
Basic New Keynesian Model with Rotemberg Pricing
Steady state - I part
Let Π = 1 (no trend in‡ation)
Euler consumption equation
(1) R 1 = β
labor supply
(2) N φ = C 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 with Rotemberg Pricing
Steady state - II part
in‡ation equation
(6) MC 1 = M = ε ε 1
stochastic discount factor
(7) Qt,t +1 = β = R 1
Basic New Keynesian Model with Rotemberg Pricing
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 xt
= Xe log (1 +
Xt 1
X
)'
Basic New Keynesian Model with Rotemberg Pricing
Loglinearization
(1) Euler equation
n
o
σ
Ce ct
1
1= Re rt βEt
Ce ct +1
Πe πt +1
using the steady - state condition
σ
1 = e rt Et (e ct ct +1 ) e π1t +1
take log
ct = E t f c t + 1 g
1
( rt
σ
Et fπ t +1 g)
Basic New Keynesian Model with Rotemberg Pricing
Loglinearization
(2) labor supply
φ
t
Nt = C t σ W
Pt
wt
(Ne nt )φ = (Ce ct ) σ We
Pe pt
using the steady - state condition
wt
(e nt )φ = (e ct ) σ ee pt
take log
wt
pt = wt = σct + ϕnt
Basic New Keynesian Model with Rotemberg Pricing
Loglinearization
production function
(3) Yt = At Nt1 α
using the same strategy as above
Ye yt = Ae at (Ne nt )1 α
using the steady state and taking logs
yt = at + (1
where at = ρa at
1
+ εa,t ,
εa,t
α ) nt
N 0, σ2a
i.i.d.
Basic New Keynesian Model with Rotemberg Pricing
Loglinearization
labor demand
t
(4) W
P t = MCt (1
α)At Nt
wt
α
becomes
pt = mct + at
αnt
equilibrium in goods mkt (since trend in‡ation is set to zero)
γ
(5) Yt = Ct + 2p (Πt 1)2 Yt becomes
y t = ct
Basic New Keynesian Model with Rotemberg Pricing
Loglinearization
in‡ation equation (6) given (7)
1 ε + εMCt γp (Πt 1) Πt +
σ
γp Et β CCt +t 1
(Πt +1 1) YYt +t 1 Πt +1 = 0
log-linearizing around Π = 1 by using the approx xt ' dXt /X
totally di¤erentiate around the steady state
εdMCt γp d Πt + γp Et βd Πt +1 = 0
using MC = ε ε 1
t
ε dMC
γp d Πt + γp Et βd Πt +1 = 0
MC MC
dMC t
1) γp d Πt + γp Et βd Πt +1 = 0
MC ( ε
we have
ε 1
π t = βEt fπ t +1 g +
mct
γp
Basic New Keynesian Model with Rotemberg Pricing
Loglinearization
Iterating forward: π t = βEt fπ t +1 g +
πt =
1
ε
γp
ε 1
γp mct
∞
k
∑ β Et fmct +k g
k =0
In words: in‡ation will be higher when …rms expect average marginal cost
(markups) to be higher (lower) than their steady-state level. Firms 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 with Rotemberg Pricing
Summing up
(1)
(2)
(3)
(4)
(5)
(6)
ct = Et fct +1 g σ1 (rt Et fπ t +1 g)
wt pt = σct + ϕnt
wt pt = mct + at αnt
yt = at + (1 α) nt with at = ρa at
yt = ct
π t = βEt fπ t +1 g + εγ 1 mct
p
1
+ εa,t
Basic New Keynesian Model with Rotemberg Pricing
Summing up
Use (5) into (1) to get rid of ct
1
(rt Et fπ t +1 g)
σ
use (2) and (3) to get rid of wt pt
σyt + ϕnt = mct + at αn
use (4) yt = at + (1 α) nt ! nt = y1t aαt
use nt = y1t aαt in σyt + ϕnt = mct + at αnt
and solve for mct
yt = Et fyt +1 g
mct =
σ+
ϕ+α
1 α
yt
1+ϕ
at
1 α
(IS)
(mc)
Basic New Keynesian Model with Rotemberg Pricing
Summing up
De…ne ytn as the natural level of output (output under ‡ex prices with markup>1 - second best output)! marginal costs are
constant mc = 0
from (mc)
ϕ+α
1+ ϕ
0 = σ + 1 α ytn 1 α at
ytn = ϑa at
1+ ϕ
ϕ+α
α
where ϑa
1 α/ σ + 1
Hence at = ytn /ϑa and so
mct =
σ+
ϕ+α
1 α
(yt
ytn )
Basic New Keynesian Model with Rotemberg Pricing
The New Phillips Curve (NPC)
Starting from π t = βEt fπ t +1 g +
found, mct = σ +
ϕ+α
1 α
π t = βEt fπ t +1 g +
1
|
γp
using the eq. just
ytn ) , for mct :
(yt
ε
ε 1
γp mct ,
φ+α
σ+
(yt
1 α
{z
}
ytn )
(NPC)
κR
Remark: the traditional Phillips Curve was expressed in terms of
the output (or unemployment).
Basic New Keynesian Model with Rotemberg Pricing
The New Phillips Curve (NPC)
Remarks
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Forward looking (the traditional Phillips curve was backward
looking... see below)
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Derived from microfoundations (the traditional Phillips curve
was a sort of empirical regularity)
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The evolution of in‡ation depends on the level of economic
activity
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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 with Rotemberg Pricing
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 with Rotemberg Pricing
The New Phillips Curve (NPC)
The previous properties stand in contrast with those characterizing
the NPC. Iterating it forward yields:
∞
π t = κ R Et ∑ βi (yt +i
i =0
ytn+i )
Remarks
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past in‡ation is not a relevant factor in determining current
in‡ation
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in‡ation is positively correlated with future output (so
in‡ation leads output)
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intuition: in a world with staggered price setting, in‡ation
results as a consequence of the price setting decisions by …rms
resetting their prices; price setting decisions depends on
current and expected marginal costs, not on past in‡ation.
Basic New Keynesian Model with Rotemberg Pricing
The Model to be simulated
The IS
yt = Et fyt +1 g
The NPC
0
1@
rt
σ |{z}
?
Et f π t + 1 g A
π t = βEt fπ t +1 g + κ R (yt
where ytn = ϑa at .
1
ytn )
References
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Bénassy, J.P. (2007), Money, Interest, and Policy, The MIT Press.
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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,
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