new keynesian model walsh chapter 8 - university at albanybd892/walsh8.pdf3 linearized new keynesian...
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![Page 1: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/1.jpg)
New Keynesian Model
Walsh Chapter 8
![Page 2: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/2.jpg)
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
![Page 3: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/3.jpg)
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
![Page 4: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/4.jpg)
• 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
![Page 5: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/5.jpg)
— 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
![Page 6: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/6.jpg)
— 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
![Page 7: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/7.jpg)
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
![Page 8: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/8.jpg)
• 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
![Page 9: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/9.jpg)
— 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
![Page 10: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/10.jpg)
— 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
![Page 11: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/11.jpg)
• 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
)
![Page 12: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/12.jpg)
∗ 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
![Page 13: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/13.jpg)
∗ 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
![Page 14: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/14.jpg)
∗ 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
![Page 15: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/15.jpg)
∗ 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
![Page 16: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/16.jpg)
• 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
![Page 17: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/17.jpg)
— 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)
![Page 18: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/18.jpg)
• 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
![Page 19: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/19.jpg)
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
![Page 20: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/20.jpg)
— 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
![Page 21: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/21.jpg)
— 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
]
![Page 22: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/22.jpg)
— 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)
![Page 23: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/23.jpg)
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
![Page 24: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/24.jpg)
∗ 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
![Page 25: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/25.jpg)
· 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
![Page 26: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/26.jpg)
· 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
![Page 27: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/27.jpg)
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
]
![Page 28: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/28.jpg)
• 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
![Page 29: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/29.jpg)
• 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
![Page 30: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/30.jpg)
— 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
![Page 31: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/31.jpg)
— 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
![Page 32: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/32.jpg)
— 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
![Page 33: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/33.jpg)
— 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
![Page 34: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/34.jpg)
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
![Page 35: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/35.jpg)
∗ 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
![Page 36: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/36.jpg)
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
![Page 37: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/37.jpg)
• 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
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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
![Page 39: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/39.jpg)
— They appear in the roots
— They need to be large enough for both roots to be unstable (requiringthe jump)
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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)
![Page 41: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/41.jpg)
— 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
![Page 42: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/42.jpg)
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
![Page 43: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/43.jpg)
∗ Linearized around the zero inflation steady state
ct = Etct+1 −1
σ(ıt − Etπt+1) +
(σ − 1
σ
) (Etψt+1 − ψt
)
![Page 44: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/44.jpg)
— 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
![Page 45: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/45.jpg)
• 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?
![Page 46: New Keynesian Model Walsh Chapter 8 - University at Albanybd892/Walsh8.pdf3 Linearized New Keynesian Model New Keynesian Phillips Curve p t Pt = 1 Et P1 i=0! i i’ t+i Pt+i Pt C1](https://reader030.vdocuments.net/reader030/viewer/2022040410/5ecda6841676b15e5e3f54a0/html5/thumbnails/46.jpg)
∗ 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
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• 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.