New Keynesian Model

Walsh Chapter 8

1 General Assumptions

• Ignore variations in the capital stock

• There are differentiated goods with Calvo price stickiness

• Wages are not sticky

• Monetary policy is a choice of the nominal interest rate

2 Households

• Maximize expected utility which depends on a composite consumptiongood, real money, and leisure

Et

∞∑i=0

βi

C1−σt+i

1− σ

+γ

1− b

(Mt+i

Pt+i

)1−b− χ

N1+ηt+i

1 + η

— Composite consumption good is CES

Ct =

[∫ 1

0cθ−1θjt dj

] θθ−1

θ > 1

• Choose cjt to minimize the cost of buying Ct

mincjt

∫ 1

0pjtcjtdj

subject to [∫ 1

0cθ−1θjt dj

] θθ−1

≥ Ct

— set up a Lagrangian

L =∫ 1

0pjtcjtdj + Ψt

Ct −(∫ 1

0cθ−1θjt dj

) θθ−1

— First order condition with respect to cjt

pjt −Ψtθ

θ − 1

(∫ 1

0cθ−1θjt dj

) θθ−1−1

θ − 1

θc−1θjt = 0

pjt −Ψt

(∫ 1

0cθ−1θjt dj

) 1θ−1

c−1θjt = 0

pjt −ΨtC1θt c−1θjt = 0

cjt =

(pjt

Ψt

)−θCt

— Integrate over j to solve for Ct and eliminate the multiplier

Ct =

∫ 1

0

(pjtΨt

)−θCtdj

θ−1θ

θθ−1

= ΨθtCt

[∫ 1

0p1−θjt dj

] θθ−1

— Solve for Ψt

1 = Ψt

[∫ 1

0p1−θjt dj

] 1θ−1

Ψt =

[∫ 1

0p1−θjt dj

] 11−θ≡ Pt

Substituting into demand yields demand as a function of relative priceand of composite consumption

cjt =

(pjt

Pt

)−θCt

the price elasticity of demand is θ > 1

• Budget constraint for household

Ct +Mt

Pt+Bt

Pt=Wt

PtNt +

Mt−1

Pt+

(1 + it−1)Bt−1

Pt+ Πt

where Πt is firm profits

• Household optimization problem

— define

ωt =(mt−1 + (1 + it−1) bt−1)

1 + πt= Ct +mt + bt − wtNt − Πt

— substitute bt out by using

bt = ωt − Ct −mt + wtNt − Πt

— value function

V (ωt) = max

(C1−σt

1− σ

)+

γ

1− b

(Mt

Pt

)1−b− χN

1+ηt

1 + η

+βEtV (ωt+1)

ωt+1 =(mt + (1 + it) (ωt − Ct −mt + wtNt − Πt))

1 + πt+1

— first order conditions

C−σt − βEtVω (ωt+1)(1 + it)

1 + πt+1= 0

γm−bt − βEtVω (ωt+1)it

1 + πt+1= 0

−χNηt + βEtVω (ωt+1)wt

1 + it

1 + πt+1= 0

— using envelope condition

Vω (ωt) = βEtVω (ωt+1)1 + it

1 + πt+1= C−σt

— FO conditions become

C−σt = βEtC−σt+1

(1 + it)

1 + πt+1

γm−bt = βEtC−σt+1

it

1 + πt+1= C−σt

it

1 + it

χNηt = βEtC

−σt+1

wt (1 + it)

1 + πt+1= C−σt wt

• Firms maximize profits subject to constraints

— Constraints

∗ production function

cjt = ZtNjt EZt = 1

∗ demand curve

cjt =

(pjt

Pt

)−θCt

∗ firm can adjust price with probability 1− ω

— choose labor to minimize costs taking the real wage as given

minwtNjt + ϕt(cjt − ZtNjt

)

∗ first order condition

ϕt =wt

Zt=

wt

MPN= real marginal cost

— choose price(pjt)to maximize real discounted profits

Et

∞∑i=0

ωi∆i,t+i

[pjt

Pt+icjt+i − ϕt+icjt+i

]where discount factor is

∆i,t+i = βi(Ct+iCt

)−σ

∗ since cjt depends on price through demand, substitute demand curvefor cjt

Et

∞∑i=0

ωi∆i,t+i

( pjt

Pt+i

)1−θ− ϕt+i

(pjt

Pt+i

)−θCt+i∗ optimal pjt = p∗t since firms are identical in all ways except date atwhich last changed price: derivative with respect to pjt

Et

∞∑i=0

ωi∆i,t+i

(1− θ)

(pjt

Pt+i

)−θ+ ϕt+iθ

(pjt

Pt+i

)−θ−1 Ct+iPt+i

= 0

Et

∞∑i=0

ωi∆i,t+i

[(1− θ)

(p∗tPt+i

)+ ϕt+iθ

](1

p∗t

)(p∗tPt+i

)−θCt+i = 0

∗ first term

Et

∞∑i=0

ωi∆i,t+i (1− θ)p∗tPt+i

P θt+iCt+i

=p∗tPt

(1− θ)Et

∞∑i=0

ωi∆i,t+iPtPθ−1t+i Ct+i

∗ second term

Et

∞∑i=0

ωi∆i,t+iϕt+iθPθt+iCt+i = θEt

∞∑i=0

ωi∆i,t+iϕt+iPθt+iCt+i

∗ solve for relative price

p∗tPt

=(

θ

θ − 1

) Et∑∞i=0 ωi∆i,t+iϕt+iP

θt+iCt+i

Et∑∞i=0 ω

i∆i,t+iPtPθ−1t+i Ct+i

=(

θ

θ − 1

) Et∑∞i=0 ωi∆i,t+iϕt+i

(Pt+iPt

)θCt+i

Et∑∞i=0 ω

i∆i,t+i

(Pt+iPt

)θ−1Ct+i

where last equality multiplies numerator and denominator by P−θt

∗ use

∆i,t+iCt+i = βiC1−σt+i C

σt

p∗tPt

=(

θ

θ − 1

) Et∑∞i=0 ωiβiϕt+i

(Pt+iPt

)θC1−σt+i

Et∑∞i=0 ω

iβi(Pt+iPt

)θ−1C1−σt+i

• Flexible price equilibrium —every firm adjusts every period, so ω = 0; loseall but first term

p∗tPt

=(

θ

θ − 1

)ϕt = µϕt

— implying that price is a markup over marginal costs since θ > 1

— since price exceeds marginal cost, output is ineffi ciently low

— since all firms charge the same price

ϕ =1

µ

— firms choose labor such thatZt

µ= wt

— households choose labor such that

χNηt

C−σt= w

— in flexible price equilibrium

Ct = Yt = ZtNt =

(1

χµ

) 1σ+η

Z1+ησ+ηt

∗ Full employment output is potentially affected by shocks to

· productivity (Zt)

· tastes (χ)

· demand elasticity (markup) (µ)

• Aggregate price level

— recall [∫ 1

0p1−θjt dj

] 11−θ≡ Pt

[∫ 1

0p1−θjt dj

]= P 1−θ

t

— Aggregate price level

P 1−θt = (1− ω) (p∗t )1−θ + ωP 1−θ

t−1

∗ 1− ω firms adjust this period and charge the optimal price

∗ ω do not adjust, and since the adjusting firms are drawn randomly,the price level for non-adjusters is unchanged

3 Linearized New Keynesian Model

• New Keynesian Phillips Curve

p∗tPt

=(

θ

θ − 1

) Et∑∞i=0 ωiβiϕt+i

(Pt+iPt

)θC1−σt+i

Et∑∞i=0 ω

iβi(Pt+iPt

)θ−1C1−σt+i

P 1−θt = (1− ω) (p∗t )1−θ + ωP 1−θ

t−1

— Let the relative price the firm chooses when he adjusts be

Qt =p∗tPt

— Q = 1 in steady state and when all firms can adjust every period

— dividing second equation by P 1−θt

1 = (1− ω)Q1−θt + ω

(Pt−1

Pt

)1−θ

— expressed in percent deviations about steady state with Pt−1Pt

= 1

1 = (1− ω) (1 + (1− θ) qt) + ω (1− (1− θ) πt)

qt =ω

1− ωπt

— rewrite first equation as

QtEt

∞∑i=0

ωiβi(Pt+iPt

)θ−1

C1−σt+i = µEt

∞∑i=0

ωiβiϕt+i

(Pt+iPt

)θC1−σt+i

— Approximate the left hand side as

C1−σ

1− ωβ+

(C1−σ

1− ωβ

)qt

+C1−σ∞∑i=0

ωiβi[(1− σ)EtCt+1 + (θ − 1) (Etpt+i − pt)

]

— Approximate the right hand side as

µ

(C1−σ

1− ωβ

)ϕ

+µϕC1−σ∞∑i=0

ωiβi[(1− σ)EtCt+1 + θ (Etpt+i − pt) + Etϕt+i

]

— Equating and noting that µϕ = 1(1

1− ωβ

)qt +

∞∑i=0

ωiβi [(−1) (Etpt+i − pt)] =∞∑i=0

ωiβi[Etϕt+i

](

1

1− ωβ

)qt =

∞∑i=0

ωiβi[Et(ϕt+i + pt+i

)]−(

1

1− ωβ

)pt

p∗t = qt + pt = (1− ωβ)∞∑i=0

ωiβi[Et(ϕt+i + pt+i

)]the optimal nominal price equals the expected discounted value of fu-ture nominal marginal costs

— can write equation as

ωβEt (qt+1 + pt+1) = qt + pt − (1− ωβ) (ϕt + pt)

Solving for qt

qt = (1− ωβ) ϕt + ωβ [Et (qt+1 + pt+1)− pt]

qt = (1− ωβ) ϕt + ωβEt (qt+1 + πt+1)

— using qt = ω1−ωπt to eliminate qt

ω

1− ωπt = (1− ωβ) ϕt + ωβEt

(ω

1− ωπt+1 + πt+1

)

πt =(1− ωβ) (1− ω) ϕt

ω+ βEtπt+1

πt = κϕt + βEtπt+1

— Differences between the New Keynesian and old Keynesian PhillipsCurves

∗ no backward-looking terms, expected future inflation matters, notlagged inflation

∗ marginal cost instead of output gap — under some restrictions thesame

· from household’s labor supply decision, real wage must equal mar-ginal rate of substitution between leisure and consumption

wt − pt = ηnt + σyt

· using

ct = yt = nt + zt

· marginal costs equals real wage divided by marginal product oflabor (Zt)

ϕt = wt − pt − zt = wt − pt − (yt − nt)= ηnt + σyt − zt = η (yt − zt) + σyt − zt

= (η + σ)

[yt −

1 + η

(η + σ)zt

]where

yft =

1 + η

(η + σ)zt

implying that

ϕt = (η + σ)[yt − yft

]= γ

[yt − yft

]· New Keynesian Phillips Curve becomes

πt = κxt + βEtπt+1

· more complicated when do not have constant returns to scale, butprinciple is same

• IS curve

yt = Etyt+1 −1

σ(ıt − Etπt+1)

— ∗ expressed in terms of the output gap xt = yt − yft

xt = Etxt+1 −1

σ(ıt − Etπt+1) + ut

where ut = Etyft+1 − y

ft

• Taylor Rule for nominal interest rate

ıt = δππt + δxxt + vt

4 Uniqueness of the equilibrium

• Substitute for the interest rate to get a two-equation system in two un-knowns

Etπt+1 =1

β[πt − κxt]

Etxt+1 = xt +1

σ[δππt + δxxt + vt]− ut −

1

σβ[πt − κxt]

• In matrix form[Etπt+1Etxt+1

]=

1β

−κβ

βδπ−1σβ 1 + βδx+κ

βσ

[ πtxt

]+

[0

vtσ − ut

]

• For uniqueness of equilibrium, since there are two forward-looking variables,the roots of the characteristic equation must both be greater than one

— equivalently, the system must be unstable

— letting roots be θ, characteristic equation[1

β− θ

] [1 +

βδx + κ

βσ− θ

]+κ

β

[βδπ − 1

σβ

]= 0

• Stability using phase diagram

— Set expected changes to zero (strictly correct if shocks permanent)

Etπt+1 − πt =1− ββ

πt −κ

βxt = 0

Etxt+1 − xt =βδπ − 1

σβπt +

βδx + κ

βσxt +

vt

σ− ut = 0

along Etπt+1 − πt = 0 πt =κ

1− βxt

— slope of ∆π = 0 is positive

along Etxt+1 − xt = 0 πt =βδx + κ

1− βδπxt +

β

1− βδπ(vt − σut)

— slope of ∆x = 0 depends on sign of 1− βδπ

— Case 1: 1− βδπ < 0

∗ slope of ∆x = 0 is negative, implying that slope of ∆π = 0 > slopeof ∆x = 0

κ

1− β>βδx + κ

1− βδπ

κ (1− βδπ) < (βδx + κ) (1− β)

−κβ (δπ − 1)− βδx (1− β) < 0

κ (δπ − 1) + δx (1− β) > 0 (1)

∗ model is globally unstable, equivalently both roots are outside theunit circle

∗ unique equilibrium at intersection and otherwise explosion

— Case 2: 1− βδπ > 0 and slope of ∆π = 0 > slope of ∆x = 0

∗ sign of (1) is reversed

∗ model is saddlepath stable

∗ can begin anywhere along the saddlepath and model approacheslong-run equilibrium

— Case 3: 1− βδπ > 0 and slope of ∆π = 0 < slope of ∆x = 0

∗ equation (1) holds (begin with sign reversed and do not flip on secondline)

∗ model is globally unstable

∗ unique equilibrium at intersection

— Equation (1) is necessary and suffi cient for global instability and uniqueequilibrium

∗ uniqueness of equilibrium requires a strong enough response of theinterest rate to deviations of inflation and output from their steady-state values

• Positive interest rate shock vt increases

— Assume 1− βδπ < 0

— ∆x = 0 shifts down and inflation and output gap fall

— as the shock slowly disappears, ∆x = 0 shifts back up and output andinflation return to long-run values

— if shock were permanent, output and inflation would never return tosteady-state values

— note output and inflation move together —no policy tradeoffs betweenkeeping inflation on target and output on target if no shocks to supply

4.1 Cochrane’s Critique

4.1.1 Determinacy —How can we rule out all unstable equilibria?

• Cochrane’s answer is that the Fed promises to "blow up the economy" ifprice and output do not jump to allow non-explosive equilibrium

— Can the Fed blow up the economy? that is, make price and/or outputexplode?

∗ Output cannot reach ±∞

∗ Price can reach 0 or ∞

∗ Interest rate cannot go negative

∗ Cochrane views output trajectory as model flaw — should price takeoff on an unstable trajectory with explosive output, the Calvo fairywould visit more often, as he does in Argentina

∗ Yes, the Fed can blow up the economy

— Would the Fed blow up the economy?

∗ Dynamic inconsistency —optimal to promise to blow up the economyto prevent it from embarking on an unstable path

∗ Not likely to actually carry out promise

∗ More likely policy is that economy can get on unstable path and theFed has plans to move it to a more favorable equilibrium

∗ Latter does not rule out the path

4.1.2 New Keynesian Model does not have Old Keynesian dynamics

• Monetary authority wants to offset positive demand shock

• Old Keynesian dynamics

—M ↓ i ↑ demand ↓ with sticky prices Y ↓

— over time P ↓ due to low demand, implying MP back up

— model is stable, and economy returns to equilibrium over time

• New Keynesian dynamics

— i ↑ Y and π both jump down

— once shock goes away, everything returns to equilibrium

— no dynamics other than dynamics of shock

— model is unstable, so economy always in equilibrium

— if output and inflation do not jump correct amount, off on explosivepath

4.1.3 Identification

• Taylor Rule

it = δππt + δxxt + vt

• When vt increases, both πt and xt must jump to rule out explosions

— Therefore error and rhs variables are correlated

— There are no good instruments as need it to respond to the variablescaused by the error

• δπ and δx do not show up in the dynamics of the model

— They appear in the roots

— They need to be large enough for both roots to be unstable (requiringthe jump)

5 Monetary Transmission Mechanism

• Write IS curve as a function of real interest rate gap

xt = Etxt+1 −1

σ(ıt − Etπt+1) + ut

xt = Etxt+1 −1

σ(rt − rt) where rt = σut

— rt is the Wicksellian real interest rate

— solve forward

xt = −1

σ

∞∑i=0

Et (rt+i − rt+i)

— output depends negatively on expected future real interest rate devia-tions

— if interest rates are always expected to equal the Wicksellian real rate,then output gaps are zero

• Where is money in the model?

mt − pt =(

1

biss

)(σyt − ıt)

— money adjusts to get the desired interest rate

6 Economic Disturbances

• Demand

— Taste

∗ Amend utility function to have a taste shock

Et

∞∑i=0

βi

ψ1−σt+i C

1−σt+i

1− σ

+γ

1− b

(Mt+i

Pt+i

)1−b− χ

N1+ηt+i

1 + η

∗ Euler equation becomes

ψ1−σt C−σt = βEtψ

1−σt+1 C

−σt+1

(1 + it)

1 + πt+1

∗ Linearized around the zero inflation steady state

ct = Etct+1 −1

σ(ıt − Etπt+1) +

(σ − 1

σ

) (Etψt+1 − ψt

)

— Government spending

∗ Resource constraint changes

yt =(C

Y

)ssct +

(G

Y

)ssgt

∗ IS curve becomes

xt = Etxt+1 −1

σ(ıt − Etπt+1) + ξt

ξt =(σ − 1

σ

)(C

Y

)ss (Etψt+1 − ψt

)−(G

Y

)ssEt (gt+1 − gt)

+(Ety

ft+1 − y

ft

)1

σ=

1

σ

(C

Y

)ss· expected changes matter

• Supply

— If no shocks to supply, then stabilization of inflation stabilizes outputgap

— Solving New Keyneisan Phillips Curve forward yields

πt = κ∞∑t=0

βiEtxt+i

— Implies that if expected future output gaps are zero, then inflation isat its target of zero

— Supply shocks change this, raising inflation for each level of output andproducing policy tradeoffs

— What might supply shocks be?

∗ Equation determining inflation before linearization, contained tastes,marginal costs, and productivity

∗ Shocks to these variables can add shocks to the Phillips curve

· However, these shocks constitute shocks to the output gap evalu-ated at changing values of full-employment output

· Remember the output gap is evaluated relative to a fixed steadystate, so the equilibrium output gap responds to supply shocks

· Monetary authority should be stabilizing output around the fluctu-ating output gap, yielding no policy tradeoffs

• Sticky wages and prices

— If only prices are sticky, optimal policy adjusts to keep prices from everhaving to adjust (zero inflation)

— If wages are sticky, and there are real shocks, then a policy which keepsprices from ever having to adjust could increase the output gap

∗ Negative productivity shock reduces the marginal product of labor

∗ Rigid nominal wages and a monetary policy to keep prices from ad-justing would imply MPL(N0) < W

P . Firms would reduce employ-ment and output.

∗ Now, there is a tradeoff between stabilizing inflation and the outputgap.