New Keynesian model
Marcin Kolasa
Warsaw School of Economics
Department of Quantitative Economics
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NK model
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Flexible vs. sticky prices
Central assumption in the (neo)classical economics:
Prices (of goods and factor services) are fully flexible
An increase in money supply immediately increases prices 1:1
Classical dichotomy: money is neutral and monetary policy has no real
effects
Consequences for models: we can abstract from money and nominal
variables
(New) Keynesian economics:
Prices are sticky, i.e. they adjust sluggishly to macroeconomic shocks
(including monetary shocks)
Classical dichotomy does not hold: monetary policy has real effects
Also, additional propagation channels for other shocks
Consequences for models: money and nominal variables important
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Sticky prices: empirical evidence
Price duration:
US: average time between price changes is 2-4 quarters (Blinder et al.,
1998; Bils and Klenow, 2004; Klenow and Kryvstov, 2005)
Euro area: average time between price changes is 4-5 quarters (Rumler
and Vilmunen, 2005; Altissimo et al., 2006)
The higher inflation, the more frequently price changes occur
Cross-industry heterogeneity
Prices of tradables less sticky than those of nontradables
Retail prices usually more sticky than producer prices
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Why are prices sticky?
Lucas (1972): imperfect information
Extensions: rational inattention (Sims, 2003; Mackowiak and
Wiederholt, 2009), sticky information (Mankiw and Reis, 2007)
Costs of changing prices (explicit or implicit):
Menu costs
Explicit contracts which are costly to renegotiate
Long-term relationships with customers
’Good’ causes of price stickiness: in a stable economic environment
agents trust in price stability
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NK model
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New Keynesian model - basic features
General equilibrium model
Two stages of production, at one of them firms are monopolistically
competitive - can set their prices
Firms are not allowed to reoptimize their prices each period - prices
are sticky
Hence, monetary policy has real effects and so needs to be described
within the model
In a nutshell - the RBC model with:
Sticky prices
Monetary authority operating via an interest rate feedback rule
Simplifications:
No capital accumulation - labour is the only factor
No trend productivity growth, only stationary stochastic shocks (hence,
no need to normalize trending real variables)
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Households I
Rent labour (the only production factor) to firms
Own firms and so get their profits Divt
Hold nominal bonds Bt , paying a nominal and risk-free (i.e.
determined in the previous period) interest rate Rt−1 (expressed in
gross terms)
Make optimal consumption-savings (by adjusting bond holdings) and
work-leasure decisions
Intertemporal budget constraint:
Wt Lt + Divt + Rt−1 Bt = Pt Ct + Bt+1
(1)
where Pt is the price level of consumption and Wt is the nominal
wage rate
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Households II
Households maximize their expected lifetime utility :
U0 = E0
∞
X
β t u(Ct , Lt )
(2)
t=0
with instantaneous utility function u(Ct , Lt ) given by:
u(Ct , Lt ) =
L1+ϕ
Ct1−θ
−κ t
1−θ
1+ϕ
The optimization is subject to the constraints:
Budget constraint (1)
Transversality condition:
E0 lim
t→∞
Marcin Kolasa (WSE)
t
Bt Y 1
Pt s=1 Rs
NK model
!
≥0
(3)
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Households’ optimization
Lagrange function:
LL = E0
∞
X
t=0
β
t
L1+ϕ
Ct1−θ
−κ t
+ λ̃t
1−θ
1+ϕ
Wt Lt + Divt − Pt Ct
+Rt−1 Bt − Bt+1
!
First order conditions:
∂LL
= 0 =⇒ Ct−θ = λ̃t Pt
∂Ct
∂LL
= 0 =⇒ κLϕ
t = λ̃t Wt
∂Lt
n
o
∂LL
= 0 =⇒ λ̃t = βEt λ̃t+1 Rt
∂Bt+1
Marcin Kolasa (WSE)
NK model
(4)
(5)
(6)
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Firms
Two stages of production:
Final-goods firms produce output by combining intermediate goods
Intermediate-goods firms produce using labour as the only input
Contrary to the RBC model, final-goods production non-trivial since
intermediate goods are not perfect substitutes. Therefore, the final
output is not a simple sum of intermediate goods production.
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Final-goods firms I
Final-goods firms produce according to the CES production function
(Dixit-Stiglitz aggregator):

φ
 φ−1
Z1
Yt (i)
Yt = 
φ−1
φ
di 
(7)
0
where:
The continuum of intermediate-goods firms (indexed by i) is
normalized to 1
Yt (i) is output produced by intermediate-goods firm i
φ > 1 is elasticity of substitution between individual intermediate goods
Note: When φ → ∞, Yt is a simple sum of intermediate products
(like in the RBC model, where all producers are perfectly competitive)
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Final-goods firms II
Maximization problem of final-goods firms:


Z1


Pt Yt − Pt (i)Yt (i)di
max

Yt ,Yt (i) 
0
subject to constraint (7)
Final-goods firms are competitive, so they maximize their profits by
chosing the inputs Yt (i), taking all prices (Pt (i) and Pt ) as given
First order conditions:
Yt (i) =
Pt (i)
Pt
−φ
Yt
(8)
Equation (8) defines the demand for intermediate input i
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Intermediate-goods firms I
Linear production function in labour only:
Yt (i) = Xt Lt (i)
(9)
Productivity Xt is common to all firms and follows the first-order
autoregressive process:
ln Xt = ρ ln Xt−1 + εt
(10)
where: 0 ≤ ρ < 1 and ε ∼ iid(0, σ 2 )
Labour inputs rented from households, technology available for free
Prices are set according to the Calvo (1983) mechanism
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Price setting with flexible prices I
Maximization problem of intermediate-goods firm i:
max
Pt (i),Yt (i),Lt (i)
{Pt (i)Yt (i) − Wt Lt (i)}
subject to the demand function (8) and the production function (9)
Maximization problem rewritten using the demand and production
function constraints:
(
)
Wt
Pt (i) −φ
max
Pt (i) −
Yt
Xt
Pt
Pt (i)
Each intermediate-goods firms takes the economy-wide wage rate Wt
and output Yt as given
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Price setting with flexible prices II
First-order condition:
Pt (i) =
Note that
Wt
Xt
φ Wt
φ − 1 Xt
(11)
is marginal cost
So, imperfectly competitive intermediate-goods firms set their prices
as a (constant) mark-up over marginal costs, where the mark-up
φ
equals to φ−1
Note that since neither mark-ups nor marginal costs are firm-specific,
all intermediate-goods firms choose the same prices
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Price setting with sticky prices I
Calvo scheme:
Each period a constant proportion 1 − γ (0 < γ < 1) of randomly
selected intermediate-goods firms is allowed to reset their prices
The remaining intermediate-goods firms have to keep their prices
unchanged
Firms allowed to reset their price take into account that they may not
be allowed to do so in the future
The probability that in period t + s the price of intermediate-goods
firm i is still Pt (i) equals γ s
The expected time of a price remaining fixed equals (1 − γ)−1
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Price setting with sticky prices II
Maximization problem of intermediate-goods firm i:
( ∞
)
X
Wt+s
Pt (i) −φ
s s
max Et
λ̃t+s β γ Pt (i) −
Yt+s
Xt+s
Pt+s
Pt (i)
s=0
Note:
Profit maximization is dynamic: firms must take into account that they
may not have a chance to reset their prices in the future
Firms are owned by households, so they discount the utility value of
their future profits by the discount factor β
First-order condition:
∞
X
s s
Et
λ̃t+s β γ Pt (i) −
s=0
Marcin Kolasa (WSE)
φ Wt+s
φ − 1 Xt+s
NK model
Pt (i)
Pt+s
−φ
Yt+s = 0
(12)
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Price setting with sticky prices III
First-order condition (12) is the same for each firm allowed to reset
its price
Therefore, all firms allowed to reoptimize at time t choose the same
price, which we denote by P̃t
The aggregate price level Pt is then:
Pt
1
1
 1−φ
Z
=  Pt (i)1−φ di 
=
0
=
1−φ
(1 − γ)P̃t1−φ + γPt−1
1
1−φ
(13)
where the first equality follows from (8)
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Monetary policy
Prices are sticky, so monetary policy has real effects
Monetary authorities set the short-term (one period) nominal interest
rate according tho the Taylor-like feedback rule (see Taylor, 1993):
Rt = R + aπ (πt − π̄) + ay (ln Yt − ln Y )
(14)
where:
R = π̄β is steady state nominal (gross) interest rate
t
π = PPt−1
is (gross) inflation and π̄ is the target (steady state) inflation
Y is steady-state output
aπ > 1, ay ≥ 0
Central bank can completely stabilize inflation by responding very
aggresively to deviations of inflation from the target (i.e. by choosing
a very large value for aπ )
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General equilibrium
Market clearing conditions:
Output produced by firms must be equal to households’ total spending
(consumption):
Yt = Ct
(15)
Labour supplied by households must be equal to labour demanded by
firms:
−φ
Z1 Z1
Z1
Pt (i)
Yt
Yt
Yt (i)
di =
di = ∆t
(16)
Lt = Lt (i)di =
Xt
Pt
Xt
Xt
0
where: ∆t =
0
0
R1 Pt (i) −φ
0
Pt
di ≥ 1 is a measure of price dispersion
(∆t = 1 ⇔ ∀i : Pt (i) = Pt )
Proceeding similarly to (13) one can show that:
!−φ
P̃t
Pt−1 −φ
∆t = (1 − γ)
+γ
∆t−1
Pt
Pt
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NK model
(17)
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Equilibrium dynamics
Equilibrium dynamics can be summarized by 9 equations (4), (5), (6),
(12), (13), (14), (15), (16) and (17), as well as the transversality
condition (3)
They describe the evolution of 9 endogenous variables: Ct , Lt , Wt ,
λ̃t , Yt , Lt , Pt , P̃t and Rt .
This system can actually be reduced to just 5 endogenous variables:
Yt , Pt , P̃t , ∆t and Rt
The only exogenous driving force in the model is stochastic
productivity Xt defined by (10)
In principle, the New Keynesian model usually includes also other
shocks
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Log-linear approximation of the model
Due to nonlinearities and presence of expectations, the model does
not have a closed-form solution
Standard technique: log-linear approximation of the model equations
around the (non-stochastic) steady-state
Non-stochastic steady-state: no productivity shocks and all variables
in the model constant
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New Keynesian Phillips curve
Log-linearized (12) and (13) imply the following New Keynesian
Phillips curve:
πt = βEt πt+1 +
(1 − γ)(1 − βγ)
mct
γ
(18)
where:
mct is log devation of real marginal cost (i.e. XWt Pt t ) from its
steady-state value
we assumed for simplicity that the target (and so steady-state) inflation
rate is zero
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New Keynesian IS curve
Log-linearized (4), (6) and (15) imply the following New Keynesian IS
curve:
1
yt = Et yt+1 − (Rt − R − Et πt+1 )
(19)
θ
where:
yt = ln Yt − ln Y is log devation of output from its steady-state value
we assumed for simplicity that the target (and so steady-state) inflation
rate is zero
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Calibration
Standard calibration:
θ = 2 (more reasonable than 1 if there is no capital)
β = 0.99
κ=1
ϕ=1
φ = 6 (implies a steady-state mark-up of 20%)
γ = 0.75
aπ = 1.5, ay = 0.5 (Taylor, 1993)
ρ = 0.95, σ = 0.01
κ and φ do not appear in the log-linearized version of the model
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Technology shock
GDP
Real marginal cost
0.8
0
0.7
-0.1
0.6
0.5
-0.2
0.4
-0.3
0.3
0.2
-0.4
0.1
-0.5
0
0
10
20
30
40
50
60
70
80
90
0
100
10
20
Inflation
30
40
50
60
70
80
90
100
80
90
100
Nominal interest rate
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
-0.6
-0.6
-0.7
-0.7
-0.8
-0.8
0
10
20
30
40
50
60
70
80
90
100
70
80
90
100
0
10
20
30
40
50
60
70
Productivity
1.2
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
Notes: grey line - flexible prices (γ → 0); black line - sticky prices (our baseline model); time unit - quarters; all variables
expressed as percentage (inflation and interest rate - percentage point) deviations from their steady-state values
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Monetary shock I
Monetary shock: an additive stochastic component RRt in the
interest rate rule, so that (14) becomes:
Rt = R + aπ (πt − π̄) + ay (ln Yt − ln Y ) + RRt
RRt follows a first-order autoregressive process:
RRt = ρR RRt−1 + εR,t
where 0 ≤ ρR < 1
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Monetary shock II
GDP
Real marginal cost
0
0
-0.1
-0.25
-0.5
-0.2
-0.75
-0.3
-1
-0.4
-1.25
-1.5
-0.5
0
10
20
30
40
50
60
70
80
90
0
100
10
20
Inflation
30
40
50
60
70
80
90
100
70
80
90
100
70
80
90
100
Nominal interest rate
0
0
-0.25
-0.25
-0.5
-0.5
-0.75
-0.75
-1
-1
-1.25
-1.5
-1.25
-1.75
-1.5
-2
-1.75
0
10
20
30
40
50
60
70
80
90
100
0
10
20
Monetary shock
30
40
50
60
Real interest rate
1.2
0.1
1
0.08
0.8
0.06
0.6
0.04
0.4
0.02
0.2
0
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
Notes: ρR = 0.90; time unit - quarters; all variables expressed as percentage (inflation and interest rate - percentage point)
deviations from their steady-state values
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Government spending shock I
Government spending Gt fully financed by lump sum taxes Vt levied
on households, so that Gt = Vt holds every period
Modifications to the model:
Households’ budget constraint (1) becomes:
Wt Lt + Divt + Rt−1 Bt = Pt Ct + Bt+1 + Vt
Goods market clearing condition (15) becomes:
Yt = Ct + Gt
It is easy to verify that first-order conditions of households’
maximixation problem (4)-(6) remain unchanged
We assume that government spending is stochastic and follows a
first-order autoregressive process:
ln Gt = (1 − ρG )G + ρG ln Gt−1 + εG ,t
where:
0 ≤ ρG < 1
G is the steady state level of government spending
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Government spending shock II
GDP
Real marginal cost
0.16
0
0.14
-0.01
0.12
-0.02
0.1
-0.03
0.08
-0.04
0.06
-0.05
0.04
-0.06
0.02
-0.07
0
0
10
20
30
40
50
60
70
80
90
0
100
10
20
Inflation
30
40
50
60
70
80
90
100
70
80
90
100
Nominal interest rate
0
0
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
-0.1
-0.1
-0.12
-0.14
-0.12
0
10
20
30
40
50
60
70
80
90
100
0
10
20
Government spending
30
40
50
60
Consumption
1.2
0
1
-0.02
0.8
-0.04
0.6
-0.06
0.4
-0.08
0.2
0
-0.1
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
Notes: ρG = 0.95; government spending share in output is 20%; time unit - quarters; all variables expressed as percentage
(inflation and interest rate - percentage point) deviations from their steady-state values
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Shock decomposition in a standard medium-sized
closed-economy model (Smets and Wouters, 2007)
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Shock decomposition in a standard medium-sized
closed-economy model (Smets and Wouters, 2007)
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Shock decomposition in a standard medium-sized
open-economy model (Christoffel et al., 2008)
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Shock decomposition in a model with financial frictions
(Christiano et al., 2010)
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The role of expectations
Equation (18) implies that current inflation is affected by inflation
expectations
Modern monetary policy: management of expectations
Woodford: For not only do expectations about policy matter, (...)
but very little else matters
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Optimal monetary policy
Equation (16) implies that price dispersion (i.e. ∆t > 1) is costly
Price dispersion can be eliminated if the central bank chooses to
stabilize inflation at zero (i.e. sets the inflation target to zero and
responds very aggresively to any deviations from the target)
Hence, a policy strictly stabilizing inflation can replicate the flexible
price equilibrium
However, monetary policy may face a trade-off between stabilizing
inflation and keeping output at a desired (not constant, in general)
level
This trade-off vanishes if:
steady state output is efficient (i.e. distortions related to monopolistic
competition are eliminated, e.g. by proper subsidies to firms)
there are no cost-push shocks (i.e. shocks to the Phillips curve)
In this case perfect price stability is optimal
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New Keynesian model - summary
Very simple dynamic stochastic general equilibrium model (DSGE)
with monopolistic competition and sticky prices
Monopolistic power of firms =⇒ decentralized allocations are not
Pareto optimal (production not at an efficient level)
Price stickiness restores the role of monetary policy:
Monetary policy has real effects (affects output, consumption, real
wages)
The case for price stabilization: price stability eliminates price distortion
Pursuing strict price stabilization is optimal if steady state distortions
(due to monopolistic competition) are eliminated (e.g. by production
subsidies)
The workhorse model in central banks
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