NEW KEYNESIAN MODEL

The Basic New Keynesian Model

Josef Stráský[email protected]

12th May 2011

Josef Stráský The Basic New Keynesian Model

Today’s program:

Basic New Keynesian model

Calvo pricing

Equilibrium under an Exogenous Money Supply

Effect of Monetary Policy Shock

Effect of Technology Shock


New Keynesian Model

Basic assumptions:

imperfect competition in the goods market

differentiated goods (continuum of differentiated goods)

producer sets the price

constraints on the price adjustment (Calvo pricing)

effects of monetary policy shocks and technology shocks


Households I

Objective function:

E0

∞∑

t=0

βtU(Ct ,Nt) (1)

Nt - hours of workU(Ct ,Nt) - utility function, increasing in Ct , decreasing inNt and concave with respect to both variablesCt - consumption index

Ct =

(

∫ 1

0Ct(i)1− 1

ǫ di

)ǫ

ǫ−1

(2)

We assume existence of continuum of goods represented bythe interval [0,1].Ct(i) - quantity of goods i consumed by the household in periodt .


Households II

Households must decide how to allocate expenditure amongdifferent goods for any level of expendituresZt =

∫ 10 Pt(i)Ct(i)di .

L =

(

∫ 1

0Ct(i)1− 1

ǫ di

)ǫ

1−ǫ

− λ

(

∫ 1

0Pt (i)Ct (i)di − Zt

)

Let us compute:d

dCt (i)

(∫ 1

0Ct (i)

1− 1ǫ di)

ǫ

1−ǫ

=ǫ

1 − ǫ

(∫ 1

0Ct (i)

1− 1ǫ di)

11−ǫ

(

1 −

1

ǫ

)

Ct (i)− 1

ǫ

= C1ǫ

t Ct (i)− 1

ǫ

We used definition (2)

Ct =

(

∫ 1

0Ct (i)1− 1

ǫ di

)ǫ

ǫ−1

(3)

We have therefore F.O.C. for all i :

C1ǫ

t Ct(i)−1ǫ = λPt(i) (4)


Households III

Recall:

C1ǫ

t Ct (i)−1ǫ = λPt (i) (5)

Therefore:

Ct(i) = Ct(j)(

Pt (i)Pt (j)

)−ǫ

(6)

Compute:Ct (i)Pt (i)

ǫ

= Ct (j)Pt (j)ǫ

Ct (i)Pt (i) = Ct (j)Pt (j)ǫPt (i)

1−ǫ

Zt =

∫ 1

0Pt (i)Ct (i)di = Ct (j)Pt (j)

ǫ

∫ 1

0Pt (i)

1−ǫdi = Ct (j)Pt (j)ǫP1−ǫ

t

We introduced price index Pt :

Pt =

(

∫ 1

0Pt (i)1−ǫdi

)1

1−ǫ

P1−ǫt =

∫ 1

0Pt(i)1−ǫdi (7)


Recall:

Ct(i) = Ct(j)(

Pt (i)Pt (j)

)−ǫ

Zt = Ct(j)Pt (j)ǫP1−ǫt

It clearly follows that:

Ct(i) =(

Pt (i)Pt

)−ǫ Zt

Pt

Recall definition (2) and substitute:

Ct =

(∫ 1

0Ct (i)

e−1ǫ di

)ǫ

1−ǫ

=

(

∫ 1

0Pt (i)

1−ǫdiPǫ−1t Z

ǫ−1ǫ

t P1−ǫ

ǫ

t

) ǫ

ǫ−1

=

(

P1−ǫ

t Pǫ−1t Z

ǫ−1ǫ

t P1−ǫ

ǫ

t

) ǫ

ǫ−1

=

(

Zǫ−1ǫ

t P1−ǫ

ǫ

t

) ǫ

ǫ−1

= Zt P−1t

And therefore we have:

Zt =

∫ 1

0Pt (i)Ct (i)di = Pt Ct (8)


Households V

Recall:

Ct(i) =(

Pt (i)Pt

)−ǫ Zt

Pt

Zt = Pt Ct

We finally got the households demand schedule:

Ct(i) =(

Pt(i)Pt

)−ǫ

Ct


Households VI

Budget constraint:∫ 1

0Pt(i)Ct(i)di + QtBt ≤ Bt−1 + WtNt − Tt (9)

PtCt + QtBt ≤ Bt−1 + WtNt − Tt (10)

Ct(i) - consumption of good iPt(i) - price of good iCt - consumption indexPt - price indexWt - nominal wageTt - net taxation expressed in nominal termsBt - quantity of one-period bonds, purchased in period tQt - price of a bond

Each bond purchased in t matures in t + 1 and pays one unit ofmoney.


Households VII

i) Intratemporal substitution Optimal household’s planconditions

Uc,tdCt = −Un,tdNt (11)

PtdCt = WtdNt (12)

Intratemporal substitution condition:

−Un,t

Uc,t=

Wt

Pt(13)

i) Intertemporal substitution

Uc,tdCt = βEt(Uc,t+1)dCt+1 (14)

PtdCt = QtPt+1dCt+1 (15)

Intertemporal substitution condition:

Qt = βEt

(

Uc,t+1

Uc,t

Pt

Pt+1

)

(16)


Households IV

Let us assume simple separable utility function:

U(Ct ,Nt) =C1−σ

t

1 − σ−

N1+ϕt

1 + ϕ(17)

Uc,t(Ct) = C−σt (18)

Un,t(Nt) = −Nϕt (19)

We plug the utility function into optimality conditions:

Wt

Pt= Cσ

t Nϕt (20)

Qt = βEt

((

Ct

Ct+1

)σ Pt

Pt+1

)

(21)

The first condition can be exactly log-linearised:

wt − pt = σct + ϕnt (22)


Log-linearization of second condition I

Recall:

Qt = βEt

((

Ct

Ct+1

)σ Pt

Pt+1

)

(23)

Let us define:

it = − log Qt - nominal interest rate (logarithm of grossyield of bond)

ρ = − log β - household’s discount rate

πt+1 = log Pt+1Pt

- inflation rate

∆ct+1 = ct+1 − ct = log Ct+1Ct

- growth

The optimality condition can be now equivalently rewritten as:

1 = Et (exp (it − σ∆ct+1 − πt+1 − ρ)) (24)


Log-linearization of second condition II

Recall:

1 = Et (exp (it − σ∆ct+1 − πt+1 − ρ)) (25)

For perfect foresight steady state, we assume constant inflationπ and constant growth γ. We therefore have following steadystate condition:

i = ρ+ π + σγ (26)

Now we can log-linearize the condition:

exp(it − σ∆ct+1 − πt+1 − ρ) ∼

∼ 1 + (it − i)− σ(∆ct+1 − γ)− (πt+1 − π)

∼ 1 + it − σ∆ct+1 − πt+1 − ρ

We can now derive log-linearized Euler equation:

1 = Et (1 + it −∆ct+1 − πt+1 − ρ)

ct = Et(ct+1)−1σ(it − Et(πt+1)− ρ) (27)


Firms I

Production function:

Yt(i) = AtNt(i)1−α (28)

All firms face identical stochastic demand schedule:

Ct(i) =(

Pt(i)Pt

)−ǫ

Ct

Firms take aggregate price level Pt and aggregate consumptionindex Ct as given.


Calvo pricing

Introduction of price stickiness as proposed by Calvo (1983).

Each firm can reset its price with probability (1 − θ) inany given period independently of the time elapsed sincethe last adjustment.Consequently, in each period (1 − θ) fraction of firms resettheir prices, whereas θ firms cannot change the price inthis period.θ is natural index of price stickinessAverage duration of a price is (1 − θ)−1

All firms that are allowed to reset the price in the periodface the same optimality problem. Therefore they allchoose new optimal price P∗

t

Let us employ following strategy:1 Let us first investigate Aggregate price dynamics as if we

know new optimal prices P∗t

2 Afterwards we investigate Optimal price settings by firmsJosef Stráský The Basic New Keynesian Model

Aggregate price dynamics I

Price index in any given period can be written as:

Pt =

(

θ

∫ 1

0Pt−1(i)1−ǫdi + (1 − θ)(P∗

t )1−ǫ

)1

1−ǫ

Pt =(

θP1−ǫt−1 + (1 − θ)(P∗

t )1−ǫ)

11−ǫ

P1−ǫt = θP1−ǫ

t−1 + (1 − θ)(P∗t )

1−ǫ

P1−ǫt

P1−ǫt−1

= θ + (1 − θ)(Pt∗)

1−ǫ

P1−ǫt−1

Π1−ǫt = θ + (1 − θ)

(

P∗t

Pt−1

)1−ǫ

We obviously used Π1−ǫt =

(

PtPt−1

)1−ǫ


Aggregate price dynamics II

Recall:

Π1−ǫt = θ + (1 − θ)

(

P∗t

Pt−1

)1−ǫ

Now let us log-linearize this condition around steady state withzero inflation: Pt = Pt−1 = P∗

t = P and Πt = 1.

log(

e(1−ǫ) logΠt

)

= log(

θ + (1 − θ)e(1−ǫ)(log P∗t −log Pt )

)

(1 − ǫ)πt = (1 − θ)(1 − ǫ)(p∗t − p)− (1 − θ)(1 − ǫ)(pt − p)

πt = (1 − θ)(p∗t − pt)


Optimal price settings I

Any firm reoptimizing in period t chooses such price P∗t that

maximizes the current market value of the profits generateduntil the price remains effective:

maxP∗

t

∞∑

k=0

θkEt[

Qt,t+k(

P∗t Yt+k |t −Ψt+k (Yt+k |t)

)]

(29)

Yt+k |t - product in period t + k of a firm that lastreoptimized in tPsi(·) - cost function

Qt,t+k = βk(

Ct+kCt

)σ PtPt+k

- stochastic discount factor for

nominal payoffs

Reoptimizing firm is subject to the sequence of demandconstraints for all k :

Yt+k |t =

(

P∗t

Pt+k

)−ǫ

Ct+k


Optimal price settings II

Recall the optimality problem, substitute the constraint,differentiate by P∗

t and compute:

maxP∗

t

∞∑

k=0

θk Et[

Qt,t+k(

P∗t Yt+k |t −Ψt+k (Yt+k |t )

)]

maxP∗

t

∞∑

k=0

θk Et

[

Qt,t+k

(

P∗t

(

P∗t

Pt+k

)−ǫ

Ct+k −Ψt+k

(

(

P∗t

Pt+k

)−ǫ

Ct+k

))]

∞∑

k=0

θk Et

[

Qt,t+k

(

(1 − ǫ)Yt+k |t + ǫ ψt+k |t(

Yt+k |t)

P∗(−1−ǫ)t

Ct+k

P−ǫt+k

)]

= 0

∞∑

k=0

θk Et

[

Qt,t+k

(

Yt+k |t +ǫ

1 − ǫψt+k |t(Yt+k |t ) P∗(−1−ǫ)

tCt+k

P−ǫt+k

)]

= 0

∞∑

k=0

θk Et

[

Qt,t+k

(

Yt+k |t P∗t +

ǫ

1 − ǫψt+k |t(Yt+k |t ) Yt+k |t

)]

= 0

∞∑

k=0

θk Et[

Qt,t+k Yt+k |t(

P∗t +M ψt+k |t(Yt+k |t )

)]

= 0


Optimal price settings III

Recall:∞∑

k=0

θk Et[

Qt,t+k Yt+k |t(

P∗t +M ψt+k |t (Yt+k |t )

)]

= 0

ψt+k |t = dΨt+k (Yt+k |t )/dYt+k |t - nominal marginal costs

M = ǫǫ−∞

Note that for limit case of no price rigidities (θ = 0):

P∗t = Mψt|t (30)

M can then be interpreted as desired (frictionless) markup (if firmsmay change price in any period the would always choose suchmarkup).We can further rewrite optimality condition:

∞∑

k=0

θk Et

[

Qt,t+k Yt+k |t

(

P∗t

Pt−1+M MCt+k |t (Yt+k |t )

Pt+k

Pt−1

)]

= 0

where MCn+k |t =ψt+k|t

Pt+kare real marginal costs.


Optimal price settings IV

Recall:∞∑

k=0

θk Et

[

Qt,t+k Yt+k |t

(

P∗t

Pt−1+M MCt+k |t (Yt+k |t )

Pt+k

Pt−1

)]

= 0

Let us log-linearize this condition around zero inflation:P∗

tPt−1

=Pt+kPt−1

= 1. This implies that all firms will produce thesame quantity of output: Yt+k |t = Y and MCt+k |t = MC. Inconstant price environment, there is no effect of price rigiditiesand therefore MC = 1

M . Finally in steady states Qt+k |t = βk

holds.∞∑

k=0

(βθ)k [p∗t − pt−1 + Et (mc t+k |t + pt+k − pt−1)

]

= 0

p∗t − pt−1 = (1 − βθ)

∞∑

k=0

(βθ)k Et (mct+k |t + pt+k − pt−1)

p∗t = µ+ (1 − βθ)

∞∑

k=0

(βθ)k Et (mct+k |t + pt+k )


Goods Market Equilibrium

Equilibrium condition:

Yt(i) = Ct(i) (31)

We define aggregate output similarly to consumption index:

Yt =

(

∫ 1

0Yt(i)

e−1ǫ di

)ǫ

1−ǫ

(32)

Therefore:

Yt = Ct (33)

From intertemporal optimality condition and equilibriumcondition:

yt = Et(yt+1)−1σ(it − Et(πt+1)− ρ) (34)


Labour Market Equilibrium

Clearing of labour market requires:

Nt =

∫ 1

0Nt(i)di

Nt =

∫ 1

0

(

Yt(i)At

)1

1−α

di

Nt =

(

Yt

At

)1

1−α

∫ 1

0

(

Pt(i)Pt

)− ǫ

1−α

di

Second row follows directly from production function and thethird one employs consumption schedule with Yt → Ct .In log terms (exactly):

(1 − α)nt = yt − at + dt (35)

where dt = (1 − α) log∫ 1

0

(

Pt (i)Pt

)− ǫ

1−α di is price dispersion. It

might be found out that dt ∼ var(pt(i)), which is equal to zero inthe first order approximation around zero inflation steady state(see Gali - Chapter 3, Appendix).Josef Stráský The Basic New Keynesian Model

Marginal Costs

Economy’s average real marginal costs:

mct = (wt − pt)− mpnt

mct = (wt − pt)− (at − αnt )− log(1 − α)

mct = (wt − pt)−1

1 − α(at − αyt)− log(1 − α)

(We used twice production function); Similarly:

mct+k |t = (wt+k − pt+k )− mpnt+k |t

mct+k |t = (wt+k − pt+k )−1

1 − α(at+k − αyt+k |t)− log(1 − α)

Put together:

mct+k |t = mct+k +α

1 − α(yt+k |t − yt+k )

mct+k |t = mct+k −αǫ

1 − α(p∗

t − pt+k )

Second row follows from demand schedule: Yt+k |t =(

P∗t

Pt+k

)−ǫ

Ct+k


Inflation I

Recall:

mct+k |t = mct+k −αǫ

1 − α(p∗

t − pt+k )

p∗t − pt−1 = (1 − βθ)

∞∑

k=0

(βθ)kEt(mct+k |t + pt+k − pt−1)

(37)

Substitute for mct+k |t and compute:

p∗t − pt−1 = (1 − βθ)

∞∑

k=0

(βθ)kEt(Θmct+k + pt+k − pt−1)

p∗t − pt−1 = (1 − βθ)

(

Θ

∞∑

k=0

(βθ)k Et(mct+k ) +

∞∑

k=0

(βθ)kEt(πt+k )

)

where Θ = 1−α1−α+αǫ

≤ 1. The above condition can be found as asolution of following difference equation:

p∗t − pt−1 = (1 − βθ)Θmct+k + βθEt(π

∗t+1 − pt)πt


Inflation II

Recall:

p∗t − pt−1 = (1 − βθ)Θmct+k + βθEt(π

∗t+1 − pt)πt

πt = (1 − θ)(p∗t − pt)

It might be now derived that:

πt = βEt(πt+1) + λmct (38)

where λ = (1−θ)(1−βθ)θ

ΘInflation results from purposeful price-setting decisions.(Contrary to classical model where inflation arises frommonetary policy rule.)Solving forward:

πt = λ

∞∑

k=0

βk Et(mct+k ) (39)


Natural output

Recall and compute:

mct = (wt − pt )− mpnt

mct = (σyt + φnt )− (yt − nt)− log(1 − α)

mct = (σ +σ + α

1 − α)yt −

1 + φ

1 − αat − log(1 − α)

mc = (σ +σ + α

1 − α)yn

t −1 + φ

1 − αat − log(1 − α)

The second row follows from household’s intratemporal optimalitycondition and the third one follows from yt = (1 − α)nt + at . Lastequation assumes flexible prices for that mct = mc and we denoteequilibrium level of output under flexible prices as natural level ofoutput yn

t . We may write:

ynt = ψn

yaat + ϑny

Where ψnya = 1+φ

σ(1−α)+φ+α and ϑny = −

(1−α)(µ−log(1−α)σ(1−α)+φα . Notice that

when µ = 0 (perfect competition) the natural level corresponds to theclassical equilibrium level of output. The firm’s market power lowersthe output uniformly without changing sensitivity to technology.


New Keynesian Phillips Curve

By subtracting mct − mc = mct we may write:

mct = (σ +σ + α

1 − α)(yt − yn

t )

mct = (σ +σ + α

1 − α)yt

We introduced output gap yt .Recall:

πt = βEt(πt+1) + λmct (40)

We finally get New Keynesian Phillips Curve

πt = βEt(πt+1) + κyt (41)

where κ = λ(

σ + φ+α1−α

)


Dynamic IS equation

Dynamic Investment-Savings equation follows from intertemporaloptimality condition:

yt = Et (yt+1)−1σ(it − Et (πt+1)− rn

t ) (42)

where rnt is natural rate of interest, given by:

rnt = ρ+ σEt (∆yt+1)

rnt = ρ+ σ ψn

ya Et (∆at+1)

By solving forward we may write:

yt = −1σ

∞∑

k=0

(rt+k − rnt+k ) (43)

where rt = it − Et (πt+1).Inflation is determined by output gap through New Keynesian PhilipsCurve and the output gap is given by path of evolution of real interestrate. Real interest rate might be evaluated only by description ofmonetary policy. Monetary policy is then non-neutral in contrast to theclassical model.


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NEW KEYNESIAN MODEL