the new kenesian model - web.sgh.waw.plweb.sgh.waw.pl/~mbrzez/monetary_economics/nkm.pdf · 2018....

Motivation Model Simulations and properties Applications Extensions and summary

The new Kenesian model

Michaª Brzoza-Brzezina

Warsaw School of Economics

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Plan of the Presentation

1 Motivation

2 Model

3 Simulations and properties

4 Applications

5 Extensions and summary

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Introduction

The MIU model was not able to re�ect features of realeconomies: lagged and prolonged reaction to shocks

The MIU model showed full neutrality and superneutrality ofmoney

Moreover central banks have generally moved from controlingmonetary aggregates to controlling short-term interest rates.

We need better models that can reproduce the basic featuresof the economy and inttroduce interest rates as monetarypolicy instrument.

The New Keynesian Model goes in this direction

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Introduction (cont'd)

It is the standard workhorse model of today's macroeconomistsanalyzing monetary policy and business cycles.

Origin:- The NKM is a general equilibrium model (as MIU) and isbased on the principle of microbased optimization. This can betraced back to the Lucas Critique and Real Business Cycle(RBC) economics of the 1980's and to the MIU model.- The standard MIU/ RBC model has been modi�ed in twoways:1) the old postulate of the Keynesian school that nominalrigidities matter was incorporated2) the idea that central banks adjust interest rates (and notmoney) in reaction to deviations of in�ation and output fromtargets (Taylor 1993) was incorporated.

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Introduction (cont'd)

The NKM is the best (or at least most popular) we have, butwe should be aware of its weaknesses. Economist constantlywork on development of new models.

E.g. models with �nancial frictions, banking sector, labormarket etc.

The derivation follows (approximately) Gali (2008)

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1 Motivation

2 Model


4 Applications


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Di�erence to MIU

Three basic modi�cations with respect to the MIU model:

1 ignore endogenous capital adjustment as suggested byMcCallum and Nelson (1999): it does not matter much for theanalysis of business cycle �uctuations

2 incorporate di�erentiated goods produced by monopolisticallycompetitive �rms (Dixit and Stiglitz 1977) facing constraintsto price adjustments (Calvo 1983)

3 represent monetary policy as setting the nominal interest ratein reaction to deviations of in�ation and output from targets

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Framework

Households consume and provide labour services, save inone-period bonds

Final good producers operate under perfect competition andproduce consumption good from intermediate goods

Intermediate goods producers operate under monopolisticcompetition and produce di�erentiated intermediate goods

Central bank sets interest rates (Taylor rule)

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Household's problem - objective

maxU = E0∞

∑t=0

βt [c1−σt

1− σ− n

1+ϕt

1+ ϕ] (1)

where nt is the work e�ort and ct is the �nal consumption good,subject to

Ptct + Bt = Wtnt + Rt−1Bt−1 + Tt

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FOCs

Lagrangean:

L = E0∞

∑t=0

[βt [c1−σt

1− σ− n

1+ϕt

1+ ϕ] +

+λt

Pt(Wtnt + Rt−1Bt−1 + Tt − Ptct − Bt)]

First order conditions are:

ct : c−σt = λt (2)

Bt

Pt: −λt + Etβλt+1Rt

Pt

Pt+1

= 0 (3)

nt : −nϕt + λt

Wt

Pt= 0 (4)

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Household's equilibrium conditions

FOCs yield two equilibrium conditions.Intertemporal - choice between consumption today and tomorrow:

c−σt = Etβc−σ

t+1Rt

Pt

Pt+1

(5)

Intratemporal - choice between consumption and leisure (laborsupply):

nϕt

c−σt

=Wt

Pt≡ wt (6)

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Household's equilibrium conditions LL

FOCs yield two equilibrium conditions.Intertemporal - choice between consumption today and tomorrow:

ct = Et ct+1 −1

σ

(Rt − Etπt+1

)(7)

Intratemporal - choice between consumption and leisure (laborsupply):

ϕnt + σct = wt (8)

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Final good producers

The �nal good producers combine di�erentiated intermediate goodsinto one �nal consumption good. They act under perfectcompetition and solve:

maxPtyt −1∫

0

Pj ,tyj ,tdj (9)

subject to:

yt =

[∫1

0

yε−1

εj ,t dj

] εε−1

(10)

where ε can be thought of as elasticity of substitution between thegoods yj . The higher is ε the better substitutes are these goods.

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FOCs

Lagrange'an:

Lt = Ptyt −∫

1

0

Pj ,tyj ,tdj − λt

[∫

1

0

yε−1

εj ,t dj

] εε−1

− yt

(11)

FOC:

yj ,t : Pj ,t − λtε

ε− 1

[∫1

0

yε−1

εj ,t dj

] εε−1−1

ε− 1

εy

ε−1ε −1

j ,t

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Demand function

Solves to (details in the technical appendix):

yj ,t =

(Pj ,t

Pt

)−ε

yt

where:

Pt =

1∫0

P1−εj ,t dj

11−ε

is the aggregate price level.

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The intermediate goods �rms

The intermediate goods production process is where thenominal stickiness is introduced.

In particular the standard assumption in the NK model is thatprices are sticky.

Some �rms will not be able to adjust their prices.

This is the reason we assume products are di�erentiated.

Under perfect competition everybody has the same price(equal to marginal cost).

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General setting

There is an in�nite number of �rms of measure one

Each produces a unique good according to the followingtechnology: yt(ι) = atnt(ι).

When setting its price the �rm is subjct to a rigidity: eachperiod only a fraaction 1− θ of �rms are allowed to changetheir prices.

The �rm's problem will be solved in two stages

First choose optimal factor employment to minimize theproduction cost.

Next, choose optimal price to maximize pro�ts.

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Optimal factor employment

The (real) cost function is qt = wtnt(ι)

Lagrangean: Lt = wtnt(ι) + µt (yt(ι)− atnt(ι))

First order condition wrt. nt(ι):wt = µtat

Substitute into cost function: qt(ι) = µtatnt(ι) = µtyt(ι)

Real marginal cost is: mct(ι) ≡ δqt (ι)δyt (ι)

= µt =wtat

Note that the RHS is independent of ι. Hence, so is the LHS.The marginal cost is the same for every �rm.

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Log-linearization

Marginal cost: mct =wtat

Log-linear approximation: mc(1+ ˆmct) =wa (1+ wt − at)

Steady state: mc=wa

Divide to get: mct = wt − at

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Optimal price setting

This is somwhat more complicated. Assume that every period onlya fraction 1− θ of �rms are allowed to change their prices. The�rm maximizes expected lifetime pro�ts:

maxΠt = Et

∞

∑i=0

(βθ)iΛt,t+i

(P∗t (ι)

Pt+i−mct+i

)yt+i (ι)

subject to the demand functions of �nal good producers:

yt+i (ι) =

(P∗t (ι)

Pt+i

)−ε

yt+i

where P∗t is the price set by �rms that are allowed to reoptimize inperiod t. Note that �rms are owned by households. Hence, theirpro�ts are discounted with β and vauled according to thehousehold's marginal utility of consumption:

Λt,t+i ≡u′(ct+i )

u ′(ct)20 / 64


Price as mark-up

After some transformation (details in the technical appendix) wearrive at:

Et

∞

∑i=0

(βθ)iΛt,t+i

(P∗tPt+i

− ε

ε− 1mct+i

)y ∗t+i = 0

Note that under �exible prices (θ = 0) and monopolisticcompetition the price chosen in period t is set as a mark-up overnominal marginal cost:

P∗t = MPtmct

where M ≡ εε−1 is the gross markup.

Hence, under monopolistic competition the price is set as amark-up over marginal cost.

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Phillips curve

Further derivations bring us to the (log-linearized) new KeynesianPhillips curve

πt =(1− βθ)(1− θ)

θmct + βEt πt+1

In�ation depends on marginal cost and expected in�ation.The latter because �rms have to be forward looking. They do notknow when they will be able to reset prices.

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Monetary policy

To complete the model we have to decide upon monetary policyThe standard assumption is that it follows a Taylor ruleThis is motivated by empirical observations of central bank behavior

Rt = ρRt−1 + (1− ρ)(φππt + φy yt) + ε i ,t

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The NK model (large version)

We now have a complete model:

Euler equation ct = Et ct+1 − 1

σ

(Rt − Etπt+1

)Consumption-leisure choice: ϕnt + σct = wt

Marginal cost: mct = wt − at

Productivity: at = ρaat−1 + εa,t

Production function: yt = at + nt

Phillips curve: πt =(1−βθ)(1−θ)

θ mct + βEt πt+1

Taylor rule: Rt = ρRt−1 + (1− ρ)(φππt + φy yt) + ε i ,t

Market clearing: ct = yt

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Large version - comments

The large version is a good starting point for applied (say atcentral banks) DSGE models

Of course it still lacks many aspects of reality

Several additional elements are added to make them matchthe data better

On the other hand, for educational reasons the NK model isoften reduced to three equations

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The NK model (compact version)

Derivation in Gali (2008), ch. 3

The three-equation model:

πt = κxt + βEt πt+1 + εc,t (12)

xt = Et xt+1 −1

σ

(Rt − Etπt+1

)+ εa,t (13)

Rt = ρRt−1 + (1− ρ)(φππt + φy xt) + εR,t (14)

where xt denotes the (welfare relevant) output gap and

κ ≡ (1−βθ)(1−θ)θ (σ + ϕ)

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1 Motivation

2 Model


4 Applications


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The NK model - demand shock

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The NK model - supply shock

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The NK model - monetary policy shock

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Important properties

Short-run non-neutrality condition is ful�lled

The (hybrid) NK model generates hump-shaped impulseresponse functions as evidenced in empirical studies

In reaction to demand shocks output and in�ation moove inthe same direction

In reaction to supply shocks output and in�ation moove inopposite directions

Taylor rule guarantees stability only if Taylor principle ful�lled(φπ > 1).

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Exercises (3eq NK model)

Use NKM.mod

Check the Taylor principle in practice (try φπ < 1)

How does a demand shock work? Check correlation of outputand in�ation when the standard deviation of the demand shockis increased.

Do the same for the supply shock.

How does the economy react to monetary policy when pricesare highly elastic?

What happens when they are very sticky?

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1 Motivation

2 Model


4 Applications


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The NK model - possible applications

The NK model (usualy somewhat extended) is used at centralbanks and academia

It o�ers several attractive applications

Examples:

Historical decompositionsForecastingCouterfactual simulationsOptimal policyRules vs. discretion

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Explaining the past (historical decompositions)

Every endogenous variable can be decomposed into e�ects ofpast shocks

To see this note, that the solution of a DSGE model is a VAR,and a VAR can be written in MA form

But, in contrast to most VARs all shocks in a DSGE have aneconomic interpretation

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Explaining the past - example

Role of �scal shocks during the �nancial crisis (Coenen,Straub, Trabandt, 2011; AER)

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Explaining the past - example cont'd

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Forecasting

A DSGE model (like a VAR) can be used to forecast

We have to make assumptions about future shocks

Unconditional forecast - all future shocks assumed to be zero

Conditional forecast - some shocks assumed for future periods

The crucial assumption is whether these are expected orunexpected

Expected shocks have an impact before they arrive

This often gives counterintuitive results (see e.g. Laseen &Svensson, 2011; IJCB)

Several central banks developped forecasting DSGE models:SIGMA (Fed), Ramses (Riksbank), Nemo (Norges Bank),NAWM (ECB), SOE.PL (NBP)

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Forecasting - example

GDP growth forecast (Kªos, 2016)

16:2 16:4 17:2 17:4 18:2 18:4 19:2 19:4-1

-0.5

0

0.5

1

1.5

Wkła

d zab

urzeń [

pkt. p

roc.]

stan ustalonyzagraniczne, globalne i marż w hzsektora publicznegorynku pracymonetarnekursu walutowegokonsumpcji prywatnejinwestycji prywatnychinne podaży

-1

-0.5

0

0.5

1

1.5

PKB

- linia

ciągła

[%]

PKB

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Counterfactual simulations

DSGE models are robust to the Lucas Critique

So, they can be used to simulate di�erent policies

In general, two possible types of counterfactual simulations

change some shocks in the pastchange policy in the past

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Counterfactual simulations - shocks

Role of �nancial shocks during the crisis (Brzoza-Brzezina &Makarski 2011, JIMF)

0 10 20 30 40 50 60 70−0.03

−0.02

−0.01

0

0.01

0.02

0.03

data & forecastcounterfactual scenariodifference

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Designing optimal / comparing alternative policies

DSGE models have a natural metric for optimality

This is houesehold welfare

E.g. see our discussion of optimal in�ation rate

More complicated experiments can be done numemericaly

But, optimal policy can have a very complicated design

Instead of looking for optimal policy we can compare howclose to optimal implementable policies are (e.g. Taylor rules)

We can look for robust policies

In particular we can test new policies (with no history to useeconometrics), e.g. macropru, ZLB

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Designing optimal / comparing alternative policies

Optimal monetary and macroprudential policy in the euro area(Bielecki, Brzoza-Brzezina, Kolasa, Makarski 2017)

98 00 02 04 06 08 10 12 14 161

1.1

1.2

1.3GDP in periphery

98 00 02 04 06 08 10 12 14 161

1.05

1.1

1.15GDP in core

98 00 02 04 06 08 10 12 14 161

2

3

4Credit in periphery

98 00 02 04 06 08 10 12 14 161

1.2

1.4

1.6

1.8Credit in core

98 00 02 04 06 08 10 12 14 160.4

0.6

0.8

1LTV in periphery

98 00 02 04 06 08 10 12 14 160.65

0.7

0.75

0.8

0.85LTV in core

98 00 02 04 06 08 10 12 14 16-0.1

-0.05

0

0.05Net exports in periphery

98 00 02 04 06 08 10 12 14 161

1.005

1.01

1.015Interest rate

counterfactual historical

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Optimal rate of in�ation (long run)

In the MIU model optimal in�ation rate is −r ss .This is the rate that maximizes money balances subject to theZLB

In the NK model this motive is absent

The argument is di�erent: there are two distortions in themodel - monopolistic competition and price dispersion

The former cannot be eliminated by monetary policy (we do itwith taxes)

But the latter can be eliminated with monetary policy

Zero in�ation makes all prices optimal

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Price dispersion

Labor market clearing:

nt =∫

1

0

nt(i)di

Substitute from production function and demand equation

nt =∫

yt (i)

atdi =

ytat

∫ (Pt (i)

Pt

)−ε

di

yt =atnt∫ (Pt (i)Pt

)−εdi≡ atnt

∆H,t

where ∆t denotes price dispersion.

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Optimal ss rate of in�ation cont'd

Price dispersion is lowest when all prices are equal

This happens with zero in�ation

This is the optimal ss in�ation rate in the NK model

To see it we need a nonlinear model with a properly calculatedsteady state

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Welfare consequences of non-zero ss in�ation

Welfare loss is usually reported �in per cent of lifetimeconsumption�

Compare welfare of non-zero ss in�ation to welfare withπss = 0.

W alternative policy = W ss,baseline policy ((1+ x)css , nss)

W alternative policy =1

1− β

(((1+ x)css)1−σ

1− σ− (nss)1+ϕ

1+ ϕ

)

x =1

css

(1− σ)

[(1− β)W alternative policy +

(nss)1+ϕ

1+ ϕ

] 11−σ

− 1

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Exercise - optimal rate of in�ation

Find numerically the optimal steady state in�ation rate in theNK model

Use NKM_nonlinear.mod

This is the NK model in original (nonlinear) form, derivationscan be found in NKM_derivation.pdf

Add welfare to the model

Compare steady state welfare for various steady state in�ationrates

Show that welfare is maximized for zero steady state in�ation

Plot price dispersion as a function of steady state in�ation

Calculate the welfare loss (as % of lifetime consumption) fromannual 4% steady state in�ation

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Optimal monetary policy (short run)

If sticky prices are the only distortion then optimal monetarypolicy in the short run is to stabilize in�ation perfectly

But in the linearized model price dispersion disappears

To speak about optimal in�ation we need a higher orderapproximation

Woodford showed that second order approximation is su�cient

Attention: there may be other distortions, e.g. sticky wages

Then optimal policy becomes more complicated (e.g. it mayalso have to stalbilize wages).

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Exercise - optimal monetary policy

Show that stable in�ation is the optimal monetary policy inthe NK model

Use NKM_nonlinear.mod

Solve for optimal monetary policy (use command�Ramsey_policy�) with welfare being the planer's objective(see Dynare User Guide for details)

Compare the outcome to the policy that fully stabilizesin�ation (you can substitute the Taylor rule with πt = 1.

Note 1: given the way ramsey_policy is declared in Dynare,period utility and not welfare is the objective

Note 2: when you compare with stable in�ation policy youhave to look at the second order approximation to the model!

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Rules (commitment) vs. discretion debate

Old debate:

should monetary policy be bound by rules or should it be freeto do whatever it wants every period?Kydland & Prescot (1980) and Barro & Gordon (1983) showthat central bank pursuing an overly ambitious output goal willend up with in�ation biasagents know that the central bank prefers high output(positive gap) and adjust expectationsas a result in�ation is higher, but output at natural levelthus CB should credibly commit to keeping output at potential

Today:

we do not think of central banks as trying to keep permanentlypositive output gapsbut Clarida, Gali & Gertler (1999) show that even withoutsuch targets, commitment can be good

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Rules (commitment) vs. discretion debate

Back to optimal policyIn the simplest case of one distortion (price dispersion) zeroin�ation is optimalBut under more general assumptions optimal policy has tosolve trade o�sRotemberg & Woodford (1998) show that when realimperfections are present the second order approximation tosocial welfare is

W0 = E0

{∞

∑t=0

βt(π2t + λx2t

)}(15)

Trade-o� between between stabilizing in�ation and output gapThis is also consistent with behavior of central banks, who aimto stabilize both in�ation and output gapsIn this case the question arises whether policy should beconducted discretionary or under commitment 52 / 64


Optimal policy under discretion

Under optimal discretionary policy (ODP) the CB is not ableto in�uence expectations about future policy

Hence, optimizing boils down to solving static problems:

minπt ,xt ,it

1

2(π2

t + λx2t )

subject to (12) and (13).

Note that expectation terms are taken as given, since the CBis assumed not to in�uence them.

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Optimal policy under discretion

FOCs:

πt : πt + µ1,t = 0

xt : λxt − κµ1,t + µ2,t = 0

it : µ2,t = 0

This yields

πt = −λ

κxt

This is called targeting rule (in contrast to instrument rules)After an in�ationary shock the CB allows the output gap tobecome negative 54 / 64


Optimal policy under commitment

Under (credible) commitment the CB is able to in�uenceexpectations about future policy

Hence, it minimizes

minπt ,xt ,it

1

2E0

∞

∑t=0

βt(π2t + λx2t )

subject to current and future period (12) and (13)

Lagranean:

E0∞

∑t=0

βt{1

2(π2t + λx2t ) + µ1,t (πt − κxt − βπt+1 − εc,t ) + µ2,t

(xt − xt+1 +

1

σ(Rt − πt+1)− εa,t

)}

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Optimal policy under commitment (today's FOCs)

FOCs for t = 0:

π0 : π0 + µ1,0 = 0

x0 : λx0 − κµ1,0 + µ2,0 = 0

i0 : µ2,0 = 0

This yields

π0 = −λ

κx0 (16)

Same as under discretion56 / 64


Optimal policy under commitment (future FOCs)

FOCs for t ≥ 1:

πt : βπt + βµ1,t − βµ1,t−1 +β

σµ2,t−1 = 0

xt : βλxt − βκµ1,t + βµ2,t − µ2,t−1 = 0

it : µ2,t = 0

This yields

πt = −λ

κ(xt − xt−1) (17)

Di�erent than in period t = 0Takes past developments into account.

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Commitment and time inconsistency

So optimal commitment policy (OCP) means doing smthtoday and promising to do smth di�erent from tomorrow on

But tomorrow will be today tomorrow!

OCP is time inconsitent

Possible solutions?

1 appoint very credible central bankers

2 or act in �timeless perspective� - pretend that OCP has beenapplied long ago and use (17) even in t = 0

Note that the time inconsistency problem is similar to the oneof forward guidance under a ZLB

58 / 64


Commitment vs discretion

What is better: OCP or ODP?

Walsh shows that neither invokes an in�ation bias

But ODP generates a stabilization bias (the economy is morevolatile and welfare lower)

We'll show it numerically

The superiority of commitment calls for a credible, long-termarrangement for the central bank

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Commitment vs discretion

1 2 3 4 5 6 7 8 9 10−6

−4

−2

0x 10

−3 x

1 2 3 4 5 6 7 8 9 10−5

0

5

10x 10

−3 pi

1 2 3 4 5 6 7 8 9 10−8

−6

−4

−2

0x 10

−3 x

1 2 3 4 5 6 7 8 9 100

0.002

0.004

0.006

0.008

0.01pi

60 / 64


Exercise - commitment vs. discretion

Show numerically that ODP generates a less stable economyand lower welfare than OCP

Use 3-eq NK model NKM.mod

Switch o� output and in�ation persistence and shockautocorrelation

Add welfare approximation to the model

Substitute Taylor rule with targeting rules for ODP or OCP(timeless)

Compare impulse responses to supply shocks

Compare volatilities and mean welfare

61 / 64



1 Motivation

2 Model


4 Applications


62 / 64


The NK model - possible extensions

The NK model misses several aspects of reality

Some standard extensions introduced to applied NK models:

habits in the utility function add a backward looking term tothe IS equationindexation in the price-setting process adds a backward lookingterm in the Phillips curveinvestment adjustment costs soften the response of investmentto shocks

More complicated extensions:

�nancial frictions (Bernanke, Gertler, Gilchrist 1999, Kiyotaki& Moore (1997), Iacoviello 2005)unemployment (Gali 2010)

63 / 64


Summary

The new Keynesian model is a state-of-the-art DSGE model

In addition to microfoundations it features monopolisticcompetition and sticky prices

Monetary policy a�ects the real economyZero is the optimal in�ation rateCommitment policy is better than discretion

Widely used at central banks and in academic literature

64 / 64

the new kenesian model - web.sgh.waw.plweb.sgh.waw.pl/~mbrzez/monetary_economics/nkm.pdf · 2018....

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