Monetary EconomicsLecture 1
The New Keynesian model
Johan Söderberg
Spring 2012
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Output
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M2 Growth
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Inflation
Fed Funds Shock
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Fed Funds Rate
Code and data source: http://faculty.wcas.northwestern.edu/~lchrist/
research/ACEL/acelweb.htm 2 / 40
Why does money matter for the real economy?I (Current) conventional explanation: Prices (and wages) are
sticky
How sticky are prices?I Nakamura & Steinson (2008): median duration of a price
change is 11 months
Issues:I Huge heterogeneity between sectors (0.5-27 months)I How to treat sales? (4.5 months)
How sticky are wages?I Taylor (1999): wages are on average changed once per year
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Goal of the course:I Build a model that can help us understand the monetary
transmission mechanism...I ...and guide the design of monetary policy
Framework: DSGE model, i.e., RBC model grafted with stickyprices (wages)
I utility maximizing householdsI profit maximizing firms (monopolistic competition)I prices setting frictions imposed exogenously
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Households
The representative household derives utility from consumption ofdifferent goods, indexed i ∈ [0, 1], according to the consumptionindex:
Ct =
(∫ 1
0Ct (i)
ε−1ε di
) εε−1
,
which should be maximized for any given expenditure level∫ 1
0Pt (i) Ct (i) di ≡ Zt .
The problem is formalized by means of the Lagrangian
L =
(∫ 1
0Ct (i)
ε−1ε di
) εε−1− λ
(∫ 1
0Pt (i) Ct (i) di − Zt
).
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HouseholdsThe solution yields the set of demand equations
Ct (i) =
(Pt (i)Pt
)−εCt ,
where
Pt ≡ λ−1 =
[∫ 1
0Pt (i)1−ε di
] 11−ε
is an aggregate price index.
Note that total consumption expenditures can be written as aproduct of the price index and the consumption index∫ 1
0Pt (i) Ct (i) di = PtCt
∫ 1
0
(Pt (i)Pt
)1−εdi = PtCt
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Households
The representative household seeks to maximize
E0
∞∑t=0
βt 11− σC1−σ
t − 11 + ϕ
N1+ϕt
,
subject to the intertemporal budget constraint
PtCt + QtBt = Bt−1 + WtNt + Tt ,
for t = 0, 1, 2, ....
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Households
Optimality conditions:
Nϕt
C−σt=
WtPt,
Qt = βEt
C−σt+1C−σt
PtPt+1
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Firms
Good i is produced by a monopolist with production function
Yt (i) = Nt (i) At
Production function corresponds to α =0 in Gali
Price are set according to the mechanism in Calvo (1983):I A firm is only allowed to reset its price with probability 1− θ
in any given period.I In each period, only a subset of measure 1− θ of all firms
change their pricesI The average price remains fixed for (1− θ)−1 periods
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I Firms are owned by householdsI Future profits is discounted with the nominal stochastic
discount factor that values a dollar paid off at some futuredate in terms of a dollar today
I The nominal stochastic discount factor between dates t andt + 1 is the variable Qt,t+1 that satisfies the equation
1 = Et (Qt,t+1Rt,t+1)
where Rt,t+1 is the gross return in period t + 1I To calculate the stochastic discount factor between dates t
and t + k, we can use the recursion
Qt,t+k = Qt,t+1Qt+1,t+2Qt+2,t+3...Qt+k−1,t+k
I It follows from the Euler equation that the nominal stochasticdiscount factor between dates t and t + k is
Qt,t+k = βk(Ct+k
Ct
)−σ ( PtPt+k
)10 / 40
Firms
The firm’s problem is to maximize its discounted profit stream:
E0
∞∑t=0
Q0,t [Pt (i) Yt (i)−WtNt (i)]
subject to the sequence of demand constrains
Yt (i) = Ct (i) =
(Pt (i)Pt
)−εCt
and the technological constraint
Yt (i) = Nt (i) At
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Firms
I Consider a firm reoptimzing its price at time tI Let the firm’s optimal price be denoted P∗t (i)I Ignoring states in which reoptimization is allowed, the
objective is
(P∗t (i)− Wt
At
)(P∗t (i)Pt
)−εYt
+θEtQt,t+1
(P∗t (i)− Wt+1
At+1
)(P∗t (i)Pt+1
)−εYt+1
+θ2EtQt,t+2
(P∗t (i)− Wt+2
At+2
)(P∗t (i)Pt+2
)−εYt+2
+...
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Firms
Differenciating w.r.t P∗t yields that the first-order condition for thisproblem is: [
(1− ε) + εWtAt
1P∗t (i)
](P∗t (i)Pt
)−εYt
+θEtQt,t+1
[(1− ε) + ε
Wt+1At+1
1P∗t (i)
](P∗t (i)Pt+1
)−εYt+1
+θ2EtQt,t+2
[(1− ε) + ε
Wt+2At+2
1P∗t (i)
](P∗t (i)Pt+2
)−εYt+2
+...
= 0
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Firms
which after rearranging can be written more compactly as
∞∑k=0
θkEt
Qt,t+k
(P∗t (i)Pt+k
)−εCt+k
(P∗t (i)−MWt+k
At+k
)= 0,
whereM = εε−1
Note that with flexible prices (θ = 0):
P∗t (i) =MWtAt
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Aggregation
I Note that whatever price a firm has had in the past does notinfluence its reset price
I All firms who get to reset chooses the same priceP∗t (i) = P∗t ∀i
I Let St denote the subset of firms not reoptimizing at time t
Pt =
[∫ 1
0Pt (i)1−ε di
] 11−ε
=
[∫St
Pt−1 (i)1−ε di +
∫SC
t
Pt (i)1−ε di] 1
1−ε
=
[θ
∫ 1
0Pt−1 (i)1−ε di + (1− θ)
∫ 1
0(P∗t )1−ε di
] 11−ε
=[θ (Pt−1)1−ε + (1− θ) (P∗t )1−ε
] 11−ε
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AggregationGoods market clearing: Yt (i) = Ct (i)
Letting
Yt ≡(∫ 1
0Yt (i)
ε−1ε di
) εε−1
,
it follows that: Yt = Ct
Labor market clearing:
Nt =
∫ 1
0Nt (i) di =
∫ 1
0(Yt (i) /At) di
=
(YtAt
)∫ 1
0
(Pt (i)Pt
)−εdi︸ ︷︷ ︸
Dt
=
(YtAt
)Dt
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Equilibrium
Yt =Ct
Qt =βEt
C−σt+1C−σt
PtPt+1
Nt =
(YtAt
)Dt
WtPt
=Nϕ
tC−σt
Pt =[θ (Pt−1)1−ε + (1− θ) (P∗t )1−ε
] 11−ε
∞∑k=0
θkEt
Qt,t+k
( P∗tPt+k
)−εCt+k
(P∗t −M
Wt+kAt+k
)= 0
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Log-linearization
We solve the model by calculating a log-linear approximationaround the model’s non-stochastic zero inflation steady state
A log-linear approximation to the function Yt around Y is given by
Yt = elogYt ≈ Y + Y [logYt − logY ]
Another example assuming Yt = F (Xt ,Zt) = F(elog Xt , elog Zt
)Yt ≈ F (X ,Z ) +Fx (X ,Z ) X [logXt − logX ]
+Fz (X ,Z ) Z [logZt − logZ ]
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Log-linearization
Let variables without time-subscripts denote steady state values
Notation:
xt = logXt
xt = logXt − logX = xt − x
Note that:
[logXt − logX ] = log(Xt
X
)≈ Xt − X
X
the percentage deviation of Xt from its steady state value X
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Log-linearization
The consumption Euler equation:
Qt =βEt
C−σt+1C−σt
PtPt+1
︸ ︷︷ ︸
Gt
Qt =βEtGt
Log-linearization yields:
Q + Q [logQt − logQ] = βEt G + G [logGt − logG ](Q − βG)︸ ︷︷ ︸
0
−Q (logQ − logG)︸ ︷︷ ︸log β
+Q (logQt − Et logGt) = 0
logQt = Et logGt + log β
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Log-linearization
logQt = log (1 + yieldt)−1 = − log (1 + yieldt) = −it ≈ −yieldt
logGt = log
C−σt+1C−σt
PtPt+1
= −σ log (Ct+1/Ct)− log (Pt+1/Pt)
Plugging this into
logQ = Et logGt + log β
yields
ct = Etct+1 −1σ
(it − Etπt+1 − ρ)
where πt+1 = log (Pt+1/Pt) and ρ = −logβ
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Log-linearizationFrom the definition of the price index:
1 =
∫ 1
0
(Pt (i)Pt
)1−ε
︸ ︷︷ ︸Rt
di
Rt ≈ R + R [logRt − logR] = 1 + (1− ε) log (Pt (i) /Pt)
Hence, up to a first order approximation
0 =
∫ 1
0log (Pt (i) /Pt) di
or
logPt =
∫ 1
0logPt (i) di
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Log-linearization
Dt =
∫ 1
0
(Pt (i)Pt
)−εdi ≈ 1− ε
∫ 1
0log (Pt (i) /Pt) di = 1
Using this, log-linearization of
Nt =
(YtAt
)Dt
yields
yt = nt + at
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Log-linearization
Dividing the price level with Pt−1 yields( PtPt−1
)1−ε= θ + (1− θ)
( P∗tPt−1
)1−ε
Log-linearizing we get
1 + (1− ε) log (Pt/Pt−1) = θ + (1− θ) [1 + (1− ε) log (P∗t /Pt−1)]
or
πt = (1− θ) (p∗t − pt−1)
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Log-linearization
The first order condition for price-setting can be written as∞∑
k=0(θβ)k Et ∆t+k [(P∗t /Pt+k)−MMCt+k ] = 0,
where MCt = (Wt/Pt) A−1t .
Note that MC =M−1. Hence
∆t+k (P∗t /Pt+k) ≈∆ + ∆ [log (P∗t /Pt+k)− log 1] + [log∆t+k − log∆]∆t+kMMCt+k ≈∆ + ∆ [logMCt+k − logMC ] + [log∆t+k − log∆]
and it follows that
∆t+k [(P∗t /Pt+k)−MMCt+k ] ≈ ∆ [(p∗t − pt+k)− mct+k ]
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Log-linearization
Using the above, the log-linearized version of the price-settingcondition reads∞∑
k=0(θβ)k Et [(p∗t − pt+k)− mct+k ] = 0
Rearranging
p∗t = µ+ (1− θβ)∞∑
k=0(θβ)k Et mct+k + pt+k
where µ = logM≈M− 1
A resetting firm will choose a price that correspons to the desiredmarkup over a weighted average of current and expected futurenominal marginal costs.
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Log-linearizationIt is convenient to rewrite the price setting condition morecompactly as a difference equation
p∗t = (1− θβ)∞∑
k=0(θβ)k Et mct+k + pt+k
= (1− θβ) (mct − pt) + θβ (1− θβ)∞∑
k=0(θβ)k Et mct+k+1 + pt+k+1
The price setting condition for t + 1 is
p∗t+1 = (1− θβ)∞∑
k=0(θβ)k Et+1 mct+k+1 + pt+k+1
Taking time t expectations, using the law of iterated expectations,and substituting into the period t condition, we get
p∗t = (1− θβ) (mct − pt) + θβEtp∗t+1
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Log-linearized equilibrium conditions
yt = ct
ct = Etct+1 −1σ
(it − Etπt+1 − ρ)
yt = nt + at
wt − pt = ϕnt + σct
πt = (1− θ) (p∗t − pt−1)
p∗t = (1− θβ) (mct + pt) + θβEtp∗t+1
mct = wt − pt − at
πt = pt − pt−1
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Log-linearized equilibrium conditions
The system of equations can be reduced to
yt = Etyt+1 −1σ
(it − Etπt+1 − ρ)
πt = λmct + βEtπt+1
mct = (ϕ+ σ) yt − (1 + ϕ) at
where λ = (1−θ)(1−θβ)θ
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Inflation dynamics
Iterating forward, the Phillips curve can be written as
πt = λ∞∑
k=0βkEtmct+k
I Inflation is purely forward-lookingI The average markup in the economy is µt = −mctI If firms expect average markups to be below their steady state
level µ, those firms who adjust choose a price above theeconomy’s average price
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The natural level of outputThe natural level of output yn
t is the equilibrium level of outputunder flexible prices
In this case the price setting condition is given by
p∗t − pt = µ+ mct
Since p∗t = pt in a flexible price equilibrium, we get
mc = −µ = (ϕ+ σ) ynt − (1 + ϕ) at
In the natural equilibrium, marginal cost is constant at its steadystate value.
Rearranging the equation above, we get
ynt =
(1 + ϕ)
(ϕ+ σ)at −
µ
(ϕ+ σ)
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The natural level of output
We can now write mct as a function of the output gap yt ≡ yt − ynt
mct =mct −mc= [(ϕ+ σ) yt − (1 + ϕ) at ]− [(ϕ+ σ) yn
t − (1 + ϕ) at ]
= (ϕ+ σ) (yt − ynt )
= (ϕ+ σ) yt
Substituting this into the Phillips curve:
πt = κyt + βEtπt+1
where κ = λ (ϕ+ σ)
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The natural level of output
We can also rewrite the consumption Euler equation in terms ofthe output gap
yt = Etyt+1 −1σ
(it − Etπt+1 − ρ)
yt − ynt = Et
(yt+1 − yn
t+1)
+ Et(yn
t+1 − ynt)− 1σ
(it − Etπt+1 − ρ)
yt = Et yt+1 −1σ
(it − Etπt+1 − ρ− Etσ
(yn
t+1 − ynt))
yt = Et yt+1 −1σ
(it − Etπt+1 − rnt ))
where rnt ≡ ρ+ σEt
(yn
t+1 − ynt)
= ρ+ σ 1+ϕϕ+σEt (at+1 − at)
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Equilibrium under an interest rate rule
yt = Et yt+1 −1σ
(it − Etπt+1 − rnt ) (DIS)
πt = κyt + βEtπt+1 (NKPC)it = ρ+ φππt + φy yt + νt (Taylor rule)
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Equilibrium under an interest rate rule
Substituting the interest rate rule into DIS, we can write thesystem on the matrix representation:[
yt
πt
]= AT
[Et yt+1
Etπt+1
]+ BT (rt
n − νt)
where rtn = rn
t − ρ, and
AT = Ω
[σ 1− βφπσκ κ+ β (σ + φy )
],BT = Ω
[1κ
]
with Ω = 1σ+φy+κφπ
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Equilibrium under an interest rate rule
Stability requires that AT has both eigenvalues inside the unitcircle. A sufficient condition is that
κ (φπ − 1) + (1− β)φy > 0
Note that if φy = 0, this condition states that
φπ > 1
The Taylor principle! The central bank must raise the nominalinterest rate more than the increase in inflation.
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Calibration
Calibration:I β = 0.99I σ = 1I ϕ = 1I ε = 6I θ = 2/3I φπ = 1.5I φy = 0.5/4
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The effects of a monetary policy shock
νt = 0.5νt−1 + ενt
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The effects of a monetary policy shockI What about money?I Assume that money demand is given by
mt − pt = yt − ηitI Money demand only determines the quantity of money the
central bank needs to supply in order to support theequilibrium interest rate implied by the policy rule
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The effects of a technology shock
at = 0.9at−1 + εat
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0Inflation
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1Output
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0Employment
0 2 4 6 8 10 12−1
−0.5
0Nominal Rate
0 2 4 6 8 10 12
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−0.2
0Real Rate
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