Monetary EconomicsLecture 1

The New Keynesian model

Johan Söderberg

Spring 2012

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0 5 10 15−0.2

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Output

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0.2

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0.6

M2 Growth

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0.1

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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

0 2 4 6 8 10 12−0.4

−0.3

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0.8Nominal Rate

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0.2

0.4

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0.8Real Rate

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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

0 2 4 6 8 10 12−0.5

−0.4

−0.3

−0.2

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0.1

0.2Money growth

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The effects of a technology shock

at = 0.9at−1 + εat

0 2 4 6 8 10 12−0.2

−0.1

0Output gap

0 2 4 6 8 10 12−1

−0.5

0Inflation

0 2 4 6 8 10 120

0.5

1Output

0 2 4 6 8 10 12−0.2

−0.1

0Employment

0 2 4 6 8 10 12−1

−0.5

0Nominal Rate

0 2 4 6 8 10 12

−0.4

−0.2

0Real Rate

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