a (very) short introduction to the canonical nk …...s1 s2 ts (bi) econ5300 october, 2018 16 / 58....

A (VERY) SHORT INTRODUCTION TO THECANONICAL NK MODEL

Tommy Sveen

BI Norwegian Business School

October, 2018

TS (BI) econ5300 October, 2018 1 / 58

Introduction

The canonical new Keynesian (NK) model

Framework that helps us understand both:

the transmission mechanism of monetary policy.the design of rules or guidelines for the conduct of monetary policy.

Core structure that corresponds to a closed-economy RBC model.

New-Keynesian features:

Monopolistic competition and nominal rigidities.Output is demand determined.


Empirical Evidence

Christiano, Eichenbaum and Evans (1999,2005)

Use short-run restrictions to identify a monetary policy rule and thecorresponding shock to policy:

it = f (Ωt ) + εt ,

where it is the federal funds rate, f is a linear function of theinformation set Ωt , and εt is the monetary policy shock.

Let the vector of variables be Yt = [Y1,t , it ,Y2,t ]′.

Variables in Y1,t (GDP, consumption, GDP deflator, investments, realwage, labor productivity) do not react to monetary policy shocks.Variables in Y2,t (real profits and growth in M2) do not belong to Ωt .


Empirical Evidence


The Basic New Keynesian ModelMotivation

The Failure of the Classical Model

Micro-evidence on price-setting behavior

Empirical evidence on effects of monetary policy shocks:

Persistent effects on real variablesSlow adjustment of aggregate price level


The Basic New Keynesian ModelKey Ingredients

Households:

Complete financial markets.

Perfectly competitive labor market.

Firms:

Monopolistic competition and sticky prices.

Cobb-Douglas production function with labour as the only input.

General equilibrium (market clearing):

Dynamic Stochastic General Equilibrium (DSGE) model


The Basic New Keynesian ModelHouseholds

Households

Maximize consumption and leisure given an intertemporal budgetconstraint.

Optimality conditions:

Intratemporal

Allocation between consumption and leisure.Allocation between different types of goods.

Intertemporal:

The consumption Euler equation.



Maximize discounted expected utility:

Et∞

∑k=0

βkU (Ct+k ,Nt+k ) ,

where

Ct ≡(∫ 1

0Ct (i)

ε−1ε di

) εε−1

Budget constraint:∫ 1

0Pt (i)Ct (i) di + Et Qt ,t+1Dt+1 ≤ Dt +WtNt − Tt ,

and a solvency constraint.



First we solve for the consumption bundle:

minCt (i )

∫ 1

0Pt (i)Ct (i) di ,

given the equation Ct ≡(∫ 1

0 Ct (i)ε−1

ε di) ε

ε−1.

Demand for good i :

Ct (i) =(Pt (i)Pt

)−ε

Ct

We can find an equation for the CPI, Pt :

Pt =(∫ 1

0Pt (i)

1−ε di) 1

1−ε



We can then rewrite the budget constraint as:

PtCt + Et Qt ,t+1Dt+1 ≤ Dt +WtNt − Tt

Let us use the following period utility function:

U (Ct ,Nt ) =C 1−γt

1− γ− N

1+νt

1+ ν,

where 1γ is the intertemporal elasticity of substitution and parameter ν

can be interpreted as the inverse of the Frisch labor supply elasticity.



The remaining first-order conditions associated with the household’sproblem are:

Cγt N

νt =

Wt

Pt≡ Ωt ,

Qt ,t+1 = β

(Ct+1Ct

)−γ ( PtPt+1

)We also have it ≡ 1/Et (Qt ,t+1)− 1, where it is the (risk-free)nominal interest rate.


The Basic New Keynesian ModelFirms

Firms

There is continuum of monopolistically competitive firms, indexed byi ∈ [0, 1].Each firm produces a differentiated good.

Identical production technology:

Yt (i) = AtNt (i)1−α ,

where Yt (i) and Nt (i) are firm i’s production and labor input, andlnAt ≡ at = ρaat−1 + εat .

Let us denote the marginal product of labor for firm i as

MPNt (i) ≡ (1− α)Yt (i) /Nt (i)



Price setting

We assume staggered price setting à la Calvo (1983): each firm facesa constant and exogenous probability, (1− θ), of getting toreoptimize its price in any given period.

Fraction (1− θ) of firms change their price in any given period.

On average firms change their price every 11−θ period.



1 θθ

θ 1 θ1 θθ

t+1

t+2 S11 S12 S21 S22

S1 S2



Firms choose prices, output and labor input to maximize:

max∞

∑k=0

Et Qt ,t+k [Pt+k (i)Yt+k (i)−Wt+kNt+k (i)] ,

s.t.

Yt+k (i) =

(Pt+k (i)Pt+k

)−ε

Yt+k ,

Yt+k (i) = At+kNt+k (i)1−α

Pt+k+1(i) =

P∗t+k+1(i) with prob. (1− θ)Pt+k (i) with prob. θ.



The Lagrangian for firm i

Lt (i) = Et∞

∑k=0

(θ)k Qt ,t+k [P∗t (i)Yt+k (i)−Wt+kNt+k (i)

−ζt+k (i)

(Yt+k (i)−

(P∗t (i)Pt+k

)−ε

Yt+k

)−ψt+k (i)

(Yt+k (i)− At+kNt+k (i)1−α

)]+... other Calvo-states.

The three first lines correspond to the nominal price P∗t (i).

ζt (i) is the shadow price on the demand constraint.

ψt (i) is the shadow price on the production constraint.



Optimality conditions for firm i

Wrt employment:

∂Lt∂Nt (i)

= −Wt + ψt (i)MPNt (i) = 0⇒ ψt (i) =Wt

MPNt (i)

Wrt output:

∂Lt∂Yt (i)

= P∗t (i)− ζt (i)− ψt (i) = 0⇒ ζt (i) = P∗t (i)− ψt (i)

Wrt the nominal price:

Lt (i) = Et∞

∑k=0

(θ)k Qt ,t+k [Yt+k (i)− εζt+k (i)Yt+k (i) /P∗t (i)]



Combining and simplifying we get:

Et∞

∑k=0

θkQt ,t+kYt+k (i)[P∗t (i)− µψt+k (i)

]= 0,

where µ = εε−1 .

Consider first the case with flexible prices (θ = 0):

P∗t (i) = µψt (i)

Prices are set as a mark-up over marginal costs.



Combining the first-order conditions for the period t price and outputwe get:

Et∞

∑k=0

θkQt ,t+kYt+k (i)[P∗t (i)− µψt+k (i)

]= 0,


With sticky nominal prices: the price is set as a mark-up over aweighted average of current and future expected marginal costs.

The future gets a lower weight, both due to discounting and theprobability of a new price.Periods with high demand get a high weight.


The Basic New Keynesian ModelMarket Clearing

Market Clearing

All markets clear:

Yt = Ct ,

Nt =∫ 1

0Nt (i) di =

∫ 1

0

(Yt (i)At

) 11−α

di

=

(YtAt

) 11−α∫ 1

0

(Pt (i)Pt

) −ε1−α

di

Note the following:

Firms set prices and production is demand determined (Keynesianassumption). For each type of goods Yt (i) = Ct (i).Demand for labor is given by the production function.


The Basic New Keynesian ModelLog-Linearized Model

Log-Linearized Model

Households

Log-linearizing labor supply and the consumer Euler equation gives:

ωt = νnt + γct ,

ct = Etct+1 −1γ(rt − ρ) ,

where we have used rt ≡ it − Etπt+1.

Firms and market clearing

Aggregate production is:

yt = at + (1− α) nt

Log-linearizing the goods market clearing condition gives:

ct = ytTS (BI) econ5300 October, 2018 23 / 58

The Basic New Keynesian ModelLog-Linearized Model

Consider the following set of equations:

yt = at + (1− α) ntyt = ct

ct = Etct+1 −1γ(rt − ρ)

ωt = νnt + γct

With flexible prices: Pt = µψt , or ωt = (yt − nt ), which gives us fiveequations in five unknown.

With sticky prices: need the Fisher equation (rt = it − Etπt+1), theinflation equation, and monetary policy.


The Basic New Keynesian ModelThe New-Keynesian Phillips Curve

The New-Keynesian Phillips Curve (NKPC)

Let steady-state inflation be zero.

Use the FOC and insert for the discount factor from the consumptionEuler-equation.

Let P∗t ≡P ∗tPtdenote the newly set relative price in period t (all price

setters are equal and will hence choose the same price).



The FOC for price-setting can be averaged over price-setters andrewritten as:

Et∞

∑k=0

(θβ)k(Ct+kCt

)−γ

Π−1t ,t+kY t+k |t[P∗t − µMC t+k |tΠt ,t+k

]= 0,

where:Y t+k |t is period t + k demand faced by firms that last changed theirprice in period t.

MC t+k |t ≡ψ t+k |tPt+k

is period t + k real marginal cost for those firms.

Πt ,t+k ≡ Pt+kPt

is gross inflation between t and t + k .

Next, we log-linearize the last equation around steady state (recallthat terms in bracket are zero in steady state):

Et∞

∑k=0

(βθ)k[p∗t −mc t+k |t − πt ,t+k

]= 0



The (price-setter’s) real marginal cost can be log-linearized as:

mc t+k |t = ωt+k −(y t+k |t − n t+k |t

)= ωt+k +

α

1− αy t+k |t −

11− α

at+k

= mct+k +α

1− α

(y t+k |t − yt+k

)= mct+k −

εα

1− α(p∗t − πt ,t+k ) ,

since mct+k = ωt+k +α1−αyt+k −

11−αat+k and

y t+k |t − yt+k = −ε (p∗t − πt ,t+k ).



Combining the latter two equations we get:

p∗t = (1− βθ)Et∞

∑k=0

(βθ)k (πt ,t+k +Θmct+k ) , (1)

where Θ ≡ 1−α1−α+εα ≤ 1.

With α > 0 (which implies that Θ < 1): Strategic complementaritiesin price setting.

If average MC increases, some firms will increase their price andthereby face lower demand.Those firms reduce production, but since there is DRTS those firmsface a lower than average MC. This reduces the incentive to increasethe price.



The NKPC is given by:

πt = βEtπt+1 + λ mct , (2)

where λ ≡ (1−θ)(1−βθ)θ Θ.

See below for details of how to get from eq. (1) to eq. (2)

Important results (Galí 2003):

“[T]he forward looking nature of inflation”.“[T]the important role played by variations in markups... as a source ofchanges in aggregate inflation”.


The Basic New Keynesian ModelThe New-Keynesian Phillips Curve and the Output Gap

The New-Keynesian Phillips Curve and the Output Gap

The “traditional”Phillips curve shows a relationship between “outputgap”and inflation. What about the NKPC?

Log-linearize the (average) real marginal cost MCt = ΩtMPNt

:

mct = ωt − (yt − nt ) = (1+ ν) nt + (γ− 1) yt

=

(γ+

α+ ν

1− α

)yt −

1+ ν

1− αat ,

where the second equality follows from labor supply and the thirdfrom aggregate production.

Increases with production.Decreases with productivity.



Flexible price equilibrium

With flexible prices: mct = 0.

Therefore, flex-price output is given by:

0 =

(γ+

α+ ν

1− α

)ynt −

1+ ν

1− αat

ynt =1+ ν

γ+ ν− α (γ− 1)at ,

which only depend on productivity.

Output gap

Define the output gap as:

xt ≡ yt − yntTS (BI) econ5300 October, 2018 31 / 58


The real marginal cost can be written as:

mct =(

γ+α+ ν

1− α

)xt

The NKPC can therefore be written as:

πt = βEtπt+1 + κxt ,

where κ ≡ λ(γ+ α+ν

1−α

).

Important result (Galí 2003):

The output gap plays a role, but “the notion of output gap found inthe recent literature bears little resemblance with the ad-hoc, largelyatheoretical output gap measures used in the traditional empiricalanalyses of inflation and monetary policy.”



Dynamic IS equation

Combine the (linearized) Euler equation with ct = yt . Using thedefinition of the output gap, we can write:

xt = Etxt+1 −1γ[it − Etπt+1 − ρ] + uxt ,

where uxt = Etynt+1 − ynt = − (1− ρa)

1+νγ+ν−α(γ−1)at (remember that

at = ρaat−1 + εat ).

Woodford (2003) writes the equation as follows:

xt = Etxt+1 −1γ[(it − Etπt+1)− rnt ] ,

where rnt is the so-called Wicksellian real rate.



Monetary policy

Let monetary policy be given by:

it = ρ+ φππt + φy xt + uit ,

where uit = ρiuit−1 + εit and εit is white noise.

This rule is often denoted "Taylor-type rule", after John Taylor (1993).

Next, we want to analyze the dynamic consequences of shocks to theeconomy for our model:

xt = Etxt+1 −1γ[it − Etπt+1 − ρ] + uxt ,

πt = βEtπt+1 + κxt ,

it = ρ+ φππt + φy xt + uit


The Basic New Keynesian ModelSolving Linear RE Models —MUC

Method of Undetermined Coeffi cients

Our model can be written in the form:

A1

[πtxt

]= A2Et

[πt+1xt+1

]+ C1

[uxtuit

]and since A1 is non-singular, we have[

πtxt

]= AEt

[πt+1xt+1

]+ C

[uxtuit

]where A = A−11 A2 and C = A

−11 C1.

Restriction on A-matrix: all eigenvalues inside the unit circle. Thisimplies that

κ (φπ − 1) + φx (1− β) > 0



Consider a simple univariate example: zt = aEtzt+1, where a is a(positive) parameter.

One solution to this equation is that z = 0 for all t. When are theremore than one non-explosive solution?

Assume we have found one such solution zt = b. We then have

Etzt+1 =ba, Etzt+2 =

ba2, etc

If a > 1, then limk→∞ Etzt+k = 0, and the solution is non-explosive.If a = 1, then Etzt+k = b for all k .If a < 1, then limk→∞ Etzt+k = ∞ and the solution is explosive. Inthis case we have only one solution!



Our basic NK model is given by zt = AEtzt+1 + Cut .

We guess on the reduced-form: zt = ΨutTherefore

Etzt+1 = ΨEtut+1 = ΨRut ,

and

zt = AEtzt+1 + Cut = AΨRut + Cut = (AΨR + C ) ut

Undetermined-coeffi cient reasoning implies

Ψ = AΨR + C

which can be solved using vec (Ψ) =[I −

(RT ⊗ A

)]−1vec (C ) .


The Basic New Keynesian ModelThe Dynamic Consequences of Shocks

The Dynamic Consequences of Shocks

The dynamic consequences of:

Monetary policy shocksProductivity shocks


The Basic New Keynesian Model


Monetary Policy in the Basic NK ModelEffi cient Allocation

Effi cient AllocationSocial planner’s problem

Social planner maximizes (representative) household’s welfare:

maxU (Ct ,Nt )

subject to:

Ct =

(∫ 1

0Ct (i)

ε−1ε di

) εε−1

Ct (i) ≤ AtNt (i)1−α ∀i ∈ [0, 1]

Nt =∫ 1

0Nt (i) di


Monetary Policy in the Basic NK ModelEffi cient Allocation

Optimality Conditions

The social planner problem implies the following optimality conditions:

Ct (i) = Ct ∀i ∈ [0, 1]Nt (i) = Nt ∀i ∈ [0, 1]

Consume the same amount of every good, which implies that the useof labor is equal across firms.

In addition we have:

−UN (Ct ,Nt )UC (Ct ,Nt )

= (1− α)YtNt

MRS = MRTLHS: Marginal cost (in units of consumption goods) of increasing theuse of labor in production.RHS: Marginal increase in production of increasing the use of labor inproduction.


Monetary Policy in the Basic NK ModelIneffi ciencies in the NK Model — Flexible Prices

Flexible Prices

Let us assume that the fiscal authorities pay an employment subsidyτw to firms per unit of labor. In that case their first-order conditionbecomes:

Pt = µ(1− τw )Wt

(1− α)Yt/Nt,


Use the labor supply equation to get rid of the real wage:


=1− α

µ (1− τw )

YtNt.

We therefore get the effi cient allocation if τw = 1− 1µ .


Monetary Policy in the Basic NK ModelIneffi ciencies in the NK Model — Sticky Prices

Sticky Prices

In the sticky-price model there are two sources of ineffi ciencies:

1. Fluctuations in the mark-up over marginal costs. Let us define themark-up as:

µt =Pt

(1−τw )Wt(1−α)Yt/Nt

= µ(1− α)Yt/Nt−UN (Ct ,Nt )UC (Ct ,Nt )

,

where we have assumed an optimal employment subsidy.Rewriting gives:


=µ

µt(1− α)Yt/Nt

2. Due to staggered price setting (not all firms change their price in agiven period), we will have Pt (i) 6= Pt (j) for any pair of goods (i , j)whose prices are not adjusted in the same period.


Monetary Policy in the Basic NK ModelA Measure of Welfare (Loss)

Loss function

We can do a second order approximation to household’s welfare in thecase of an employment subsidy:

Wt =12Et

∞

∑k=0

βk[(

γ+α+ ν

1− α

)x2t+k +

ε

λπ2t+k

],

where λ ≡ (1−θ)(1−βθ)θ

1−α1−α+αε .

The loss function is increasing in the variance of the output gap andinflation. The former is due to the variability of mark-ups; the second isdue to cost of price dispersion.


Monetary Policy in the Basic NK ModelThe Divine Coincidence

The Divine Coincidence

The loss function is:

Wt =12Et

∞

∑k=0

βk[(

γ+α+ ν

1− α

)x2t+k +

ε

λπ2t+k

],

The model economy can be written as:

xt = Etxt+1 −1γ(it − Etπt+1 − rnt ) ,

πt = βEtπt+1 + κ xt

The divine coincidence:

Set interest rates such that xt = 0 for all t.This implies that πt = 0.

The nominal interest rate must be such that rt = it − Etπt+1 = rnt .TS (BI) econ5300 October, 2018 47 / 58

Monetary Policy in the Basic NK ModelImplementation — Indeterminacy

Indeterminacy

What is wrong with the following rule:

it = rnt

The rule is consistent with optimal policy:

Inflation and output gap are zero. The nominal and real interest rateequals the natural real rate.

The rule is also consistent with many other outcomes (multipleequilibria).

Remember the restriction on the A−matrix:κ (φπ − 1) + φx (1− β) > 0.



The Taylor principle: "Adjust the nominal interest rate more thanone-for-one with changes in inflation"

What happens if the central bank does not follow the Taylor principle?

The Taylor principle is important in the design of monetary policyrules. Avoids that the central bank becomes a source of unnecessaryfluctuations in economic activity.

Clarida, Galí, and Gertler (2000) and Lubik and Schorfeide (2004):change from passive to active monetary policy in the early 1980’s canexplain the observed stabilization of macroeconomic outcomes in theUS.



Consider the following rule:

it = rnt + φππt + φxxt

Let us consider the following parameter values:

α ν γ ε β θ λ κ

1/3 1 1 6 0.99 2/3 0.0425 0.1275

We let the parameters in the policy rule be φπ = 0.8 and φx = 0.

We plot impulse responses to a sunspot shock as in Clarida, Galí andGertler (2000).



0 2 4 6 8 10 121.5

1

0.5

0

inflation

Impulse Responses to a Sunspot Shock

0 2 4 6 8 10 121.5

1

0.5

0

output gap

0 2 4 6 8 10 121

0.5

0

nominal rate

0 2 4 6 8 10 120

0.2

0.4

real rate

See Galí (1997), “Solving Linear Dynamic Models with Sunspot Equilibria: A note”TS (BI) econ5300 October, 2018 51 / 58


The following simple rule therefore implements optimal policy:

it = rnt + φππt + φxxt

if κ (φπ − 1) + φx (1− β) > 0.

πt and xt are both zero for it = rnt . The two last terms are thereforeboth zero.

The central bank only use the two terms as a "threat" of a strongresponse to an eventual deviation of the output gap and inflationfrom target.


Monetary Policy in the Basic NK ModelIndeterminacy — Interest Rate Smoothing

Interest Rate Smoothing

Often we see rules specified as follows:

it = φi it−1 + (1− φi ) [rnt + φππt + φxxt ]

The condition is still:

κ (φπ − 1) + φx (1− β) > 0.

The real interest rate must eventually increase!


Monetary Policy in the Basic NK ModelIndeterminacy —Forward-Looking Rule

Forward-Looking Rule

A forward-looking rule would be:

it = rnt + φπEtπt+1 + φxEtxt+1.

The conditions are then:

κ (φπ − 1) + φx (1− β) > 0,

κ (φπ − 1) + φx (1+ β) < 2γ (1+ β)

The reaction cannot be too large. Why?

Let us consider the impulse responses for φπ = 40 and φx = 0.


Monetary Policy in the Basic NK ModelIndeterminacy —Forward-Looking Rule

0 2 4 6 8 10 120.04

0.02

0

0.02

inflation

Impulse Responses to a Sunspot Shock

0 2 4 6 8 10 120.5

0

0.5

output gap

0 2 4 6 8 10 120.5

0

0.5

nominal rate

0 2 4 6 8 10 120.5

0

0.5

real rate

See Galí (1997), “Solving Linear Dynamic Models with Sunspot Equilibria: A note”TS (BI) econ5300 October, 2018 55 / 58

Derivation of the NKPC

We start with

p∗t= (1− βθ)Et∞

∑k=0

(βθ)k (πt ,t+k+Θmc t+k )

Forward the last equation one period:

Et p∗t+1= (1− βθ)Et∞

∑k=0

(βθ)k (πt+1,t+k+1+Θmc t+k+1)

Subtract βθE t p∗t+1 from the next to last equation:

p∗t−βθE t p∗t+1= (1− βθ)Et

∞

∑k=0

(βθ)k (πt ,t+k+Θmc t+k )

−βθ (1− βθ)Et∞

∑k=0

(βθ)k (πt+1,t+k+1 +Θmct+k+1)



Some algebra for the real marginal cost:

∞

∑k=0

(βθ)k mc t+k = mc t+∞

∑k=1

(βθ)k mc t+k= mc t+βθ∞

∑k=0

(βθ)k mc t+k+1,

mc t =∞

∑k=0

(βθ)k mc t+k−βθ∞

∑k=0

(θβ)k mc t+k+1

Some algebra for the rate of inflation πt ,t+k = πt+1 + πt+2 + ..+ πt+k :

∞

∑k=1

(θβ)k πt ,t+k =∞

∑k=1

(θβ)k (πt+1 + πt+1,t+k )

=βθ

1− βθπt+1 + βθ

∞

∑k=1

(θβ)k πt+1,t+k+1,

βθ

1− βθπt+1 =

∞

∑k=1

(βθ)k πt ,t+k − βθ∞

∑k=1

(βθ)k πt+1,t+k+1.



We had

p∗t − βθEt p∗t+1 = (1− βθ)Et∞

∑k=0

(βθ)k (πt ,t+k +Θmct+k )

−βθ (1− βθ)Et∞

∑k=0

(βθ)k (πt+1,t+k+1 +Θmct+k+1)

Combining the above we get:

p∗t = βθEt p∗t+1 + βθEtπt+1 + (1− βθ)Θmct .

We get a relationship between p∗t and πt from log-linearizing the CPI:

p∗t =θ

1− θπt .

The NKPC is therefore given by:

πt = βEtπt+1 + λ mct ,

where λ ≡ (1−θ)(1−βθ)θ Θ.


a (very) short introduction to the canonical nk …...s1 s2 ts (bi) econ5300 october, 2018 16 / 58....

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