Analysing shocks and policy in the NKM
Alex Karalis Isaac
University of Warwick
March 2022
Karalis (University of Warwick) EC924 March 2022 1 / 34
Preview of the lecture
1 IRFs and shocks in the BNKM
I Solving backward and forward looking difference equations
I Substituting the short rate through the NKM
I Gali’s matrix format
I Solution and responses to shocks (IRFs)
2 From IRFs to policy analysis
I Optimal Policy
I Determinacy and (optimal) policy in the NKM
I Welfare calculations
I An implementable Taylor Rule
F The implied policy-technology shock
I Welfare with implementable Taylor rules
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Solving difference equations
Before studying Gali’s NKM and policy it is useful to recall difference eqns
Consider the backward looking equation
yt = γyt−1 + σet (1)
where et ∼iid (0, 1) and |γ| < 1 Then
yt+s − Etyt+s =
s−1
∑
k=0
γkσet+s−k
⇒ ∂yt+s
∂et
= γsσ
γkσ are the dynamic multipliers of the shocks
Plotting γkσ as a function of k is referred to as the IRF
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Backward looking equation: lag operator solution
We can re-write (1)
(I − γL)yt = σet
where Lyt = yt−1. Provided |γ| < 1
yt = (I − γL)−1σet
= (I + γL+ γ2L2 + γ3L3 + . . . )σet
=
∞
∑
k=0
γkσet−k
To see this define
(I − γL)−1 = (I + ψ1L+ ψ2L2 + . . . )
and solve (I − γL)−1(I − γL) = I , by equating coefficients on Li to 0 for
i ≥ 1
Leading the last line forward s periods we again see ∂yt+s∂et = γ
sσ
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Solving forward looking difference equations
Macro models often have forward looking components
The scalar forward-looking difference equation
yt = φEtyt+1 + σut (2)
can be written in terms of F where Fyt = Etyt+1:
(1− φF )yt = σut
yt = (1− φF )−1σut
= σ
∞
∑
k=0
φkEtut+k
provided |φ| < 1, by a similar argument to the backward-looking equation.
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Solving forward looking difference equations
In forward looking equations it is typical to consider autoregressive shocks
ut = ρut−1 + et
where et is iid as before
Solution of the forward looking equation is completed by noting that
Etut+k = ρ
kut
So that
yt = σ
∞
∑
k=0
(φρ)kut
=
σ
1− φρut
a simple case of the result that the purely forward looking model implies no
endogeneous dynamics
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Policy and shocks in the NKM
Technology shocks drive movements in unobservable natural rates,
I Monetary policy based on observables implies policy errors.
While it is possible to add other shocks (demand, costs push) to the NKM
much policy analysis therefore assumes
I A technology shock (NKM followed from RBC literature)
I A monetary policy shock
Recall the NKM
y˜t = Et y˜t+1 − 1σ (rt − r
n
t )
pit = βEtpit+1 + κy˜t (3)
it = ρ+ φy y˜t + φpipit + vt
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Adding shocks to the NKM
There is a rule it = ρ+ φy y˜t + φpipit written for the nominal rate
I Is it a Taylor rule?
To which we have appended a shock
vt = ρvvt−1 + evt
which we later relate to policy errors arising from unobservability of y˜t
ynt = ϑya + ψ
n
yaat is the natural rate; y˜t = yt − ynt
Driven by at = ρaat−1 + ea,t
Welfare consequences turn on how nearly a monetery policy sets y˜t = 0
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Dynamic responses to monetary and technology shocks
Thus model responses to shocks ea,t , ev ,t drive results. Reprise:
To examine responses to shocks we can put (3) in a form similar to (2):
Substitute it into the IS eqn
y˜t = Et y˜t+1 − 1σ (φy y˜t + φpipit + vt − Etpit+1 − rˆ
n
t )
pit = βEtpit+1 + κy˜t[ σ+φy
σ
φpi
σ−κ 1
] [
y˜t
pit
]
=
[
1 1σ
0 β
]
Et
[
y˜t+1
pit+1
]
+
[
1
σ
0
]
(rˆnt − vt) (4)
where rˆnt = r
n
t − ρ are deviations in the natural rate due to ea,t
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Gali solution
Standard matrix algebra renders (4) in the format of section 3.4 Gali (2008):[
y˜t
pit
]
= ATEt
[
y˜t+1
pit+1
]
+ BT (rˆ
n
t − vt) (5)
Provided eigenvalues of AT are < 1 in absolute value then, by analogy to
(2), (5) solves forward: [
y˜t
pit
]
= [I2 − ρxAT ]−1 BT xt (6)
where ρx is the persistence of the composite shock xt = (rˆnt − vt)
IRFs follow by leading (6) forward and replacing future shocks with expected
values: xt+i = ρ
i
xxt
Note: in the book Gali derives (6) equation by equation, using undertermined coefficients.
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Some features of the solution
The pure forward looking model displays clearly the property that Estrella
and Furher (2002) document:
I The two variables are not tied to past values, but are free to jump
down instantly in response to the shock
I Solution format (6) makes it clear the whole expected future impact of
the shock is accounted for in the initial response
Initially we examine effects of vt for at fixed
I Both the output gap and inflation jump down in response to the
contractionary shock
I The effect is muted for inflation compared to the gap, κ = 0.128, in
the std calibration
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Gali solution
Consider first the response to monetary shock vt
Specialise xt = vt , putting ea,t = 0
From (6) you can see, as Ch. 3
y˜t = ψvyvt
pit = ψypivt
Exercise: derive as in Gali and check against (6)
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Gali IRFs for monetary shock (interest rate)
0 2 4 6 8 10 12
-0.2
-0.1
0
output gap to vt
0 2 4 6 8 10 12
-0.05
0
to vt
0 2 4 6 8 10 12
-0.2
-0.1
0
Aunnual to vt
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Features of the response
Contractionary shock unambiguously raises real rate above its natural
counterpart
I The more persistent the policy shock the longer the deviations last
This induces an output gap which reduces employment and consumption
Inflation also falls on impact, exacerbating effect of any increase in nominal
rates on the real rate
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Response to a technology shock
Consider next the response to a technology shock. Putting vt = 0
xt = rˆ
n
t = r
n
t − r
= ρ+ σψnyaEt [∆at+1]− ρ
= σψnya(ρa − 1)at (7)
The technology shock boosts the natural rate of output
I But the effect of monopoloy and price dispersion keeps output response
smaller by about 10% in this calibration
I Employment also falls in this case; a substitution effect Gali suggests is
in keeping with recent evidence
The negative output gap induces a negative inflation rate and (see the
book) the central bank drops nominal and real rates to accomodate this
Although we set the monetary shock, vt = 0, policy rule above may not
close the output gap instantly
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Response to a technology shock
0 2 4 6 8 10 12
-0.2
-0.1
0
Output gap to at
0 2 4 6 8 10 12
-0.1
-0.05
t to at
0 2 4 6 8 10 12
-1
-0.5
0
Annualized t to at
Karalis (University of Warwick) EC924 March 2022 16 / 34
From IRFs to policy analysis
We want to consider how to design policy rules/actions
I What goes in the rule, what parameters to choose
Start from the ideal case and examine welfare losses
Which may arise from the presence of
I Monopoly
I Sticky prices
In the absence of policy
And in the presence of technology shocks which are key to RBC agenda
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Efficient Allocation
What is that? The best possible outcome:
Solve the planning problem
maxU(Ct ,Nt) s.t.
Ct = (
∫ 1
0
Ct(i)
e−1
e )
e
e−1
Ct(i) = AtNt(i)
1−α
Nt =
∫ 1
0
Nt(i)di
Why is this not a dynamic problem?
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Efficient Allocation
From the text book, or as an exercise, Efficiency requires us to :
I Equate the Marginal Rate of Substitution (between...) to the Marginal
Product of Labour
I Equalise consumption, production and employment across varieties
(because...)
Very much as in the RBC (think UG - tangency of PPF and IC)
There are two sources of distortions
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Distortions
Distortions from monopoly power
I A constant distortion
I Tackeld by a subsidy not monetary policy
Distortions from price sticky-ness
I Aggregate distortion
I Price dispersion
I Target of monetary policy to eliminate these suboptimalities
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Distortions
Monopoloy
Inefficiently low production, employment, consumption
Set price as mark up over Marginal Cost
Pt =M Wt
MPNt
So that
−Uct
Unt
=
Wt
Pt
=
MPNt
M
First equality is delivered by markets; second violates efficiency
Subsise employment at rate τ to restore desired equaliy
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Distortions
Pricy stickiness
Aggregate inefficiency: output will differ from the flexible price equilibrium
due to variations marginal costs across firms
I A time-varying wedge between the MRS and the MPN
Mt = Pt
(1− τ)(Wt/MPNt) =
PtM
Wt/MPNt
so that MRS=MPN can only be restored by stabilising mark-up at the
frictionless level M
Price Dispersion:
I Products enter preferences symmetrically, but price stickyness induces
relative prices 6= 1 so that demand is asymetric.
I Although technology is symmetric, production is not, and some firms
are too large (high MC) or small (low MC).
From symmetric equilibrium MP to remove incentive to change prices
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From IRFs to policy analysis
To compare the performance of monetary rules a Welfare criterion is useful
Rotemburg and Woodford (1999) show that an appropriate criterion is
W =
1
2
E0
∞
∑
t=0
[(
σ+
ϕ+ α
1− α
)
y˜2t +
e
λ
pi2t
]
following from a second order approximation to lifetime utility U
Output gap and inflation deviations reduce welfare with
I Output gap deviations worse for higher risk aversion and Frisch labour
elasticity, which amplify wedge between MRS and MPN
I Counterintuitively, inflation deviations are more harmful for increases in
substitutability (which reduces monopoly power) due to an increased
utility loss from price dispersion in this case
I Inflation losses are also increasing in average contract duration, i.e. in
θ, as this increases the price dispersion resulting from any given
inflation deviation
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Optimal policy
In an ideal world welfare losses would be 0
What is the policy that attains such an outcome? Consider
it = r
n
t
This will always instantly close the output gap
With y˜t = 0 marginal costs do not vary, prices are fixed
Price stickiness is non binding
While simple there are two problems
I Determinacy
I Implementability
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Optimal policy: indeterminacy
Bullard and Mitra (2002) show that stability of AT under the rule in (3)
requires
κ(φpi − 1) + (1− β)φy > 0 (8)
⇒ φpi > 1 if φy = 0
Our optimal policy is consistent with sunspot equilibria!
Such equililbria are very bad for welfare
To stabilise optimal policy introduce feedback which ‘leans against the wind’
it = r
n
t + φy y˜t + φpipit
The threat of policy action deters coordination on alternative equilibria
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A simple rule
Unfortunately this is not an implementable rule as rnt is unobservable
I For obvious reasons, estimating the natural rate is an active field
Attention therefore shifts to implementable rules, based on observables
The classic Taylor rule is one example
it = ρ+ φy yˆt + φpipit (9)
where yˆt = yt − y , assuming y is known, or estmated from time-series data
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Simple rules and monetary shocks
In the case that at variation drives natural rate variation
The observable rule induces a monetary policy shock
φy (yt − y) = φy (yt − ynt + ynt − y)
= φy y˜t + φy yˆ
n
t
= φy y˜t + vt
yˆnt is the deviation in the natural rate induced by the tech shock and missed
by the observable rule
The standard deviation of the policy shock is now proportional to the weight
on the output gap in the Taylor rule
Karalis (University of Warwick) EC924 March 2022 27 / 34
Simple rules and monetary shocks
We have a model where there are tehcnology shocks, at = ρaat−1 + ea,t
and monetary shocks vt
We can analyse these using the dynamic responses captured in (5) as the at
shock enters through the natural rate deviation rˆnt
Both the deviation of the natural rate in the IS eqn,rˆnt , and the monetary
shock, vt , are driven by the same underlying process
Use the natural real rate and output equations
rnt = ρ+ σϕ
n
yaEt∆at+1
ynt = ϕ
n
yaat + ϑ
n
y
to define yˆnt and rˆ
n
t , then combine xt = rˆ
n
t + vt to see that
xt = ϕ
n
ya(σ(ρa − 1)− φy )at (10)
so the shock is really a re-scaled technology shock, c.f. (7)
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Exploring policy rules
The parameters φpi and φy are central bank choices
What values should the central bank choose?
We can compare performance of alternative choices underW
I W microfounds the old literature of LQR p policy assessment
Social welfare losses,W are increasing in y˜2t , pi
2
t
To gain intuition about the performance of different rules we can look at
their IRFs in response to (10)
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Candidate Taylor Rules
Gali (2008) considers four candidate Taylor rules on yˆt , pit :
I (φpi, φy ) = (1.5, 0.125) ‘Taylor (1993)’
I (φpi, φy ) = (1.5, 0) ‘No output gap’
I (φpi, φy ) = (5, 0) ‘Hawk?’
I (φpi, φy ) = (1.5, 1) ‘Dove?’
Which regime entails the most aggressive action against inflation?
Which induces the smallest ouptut gap?
Which will be the best regime?
I The one which prodcues the smallest deviations in the weighted
average of y˜t , pit
Karalis (University of Warwick) EC924 March 2022 30 / 34
Candidate Taylor Rules
Gali (2008) considers four candidate Taylor rules on yˆt , pit :
I (φpi, φy ) = (1.5, 0.125) ‘Taylor (1993)’
I (φpi, φy ) = (1.5, 0) ‘No output gap’
I (φpi, φy ) = (5, 0) ‘Hawk?’
I (φpi, φy ) = (1.5, 1) ‘Dove?’
Which regime entails the most aggressive action against inflation?
Which induces the smallest ouptut gap?
Which will be the best regime?
I The one which prodcues the smallest deviations in the weighted
average of y˜t , pit
Karalis (University of Warwick) EC924 March 2022 30 / 34
IRFs to unit at shock, alternative Taylor rules
0 5 10 15 20 25 30 35 40
-0.6
-0.4
-0.2
0
output gap to obs policy
Base
No y
=5
y=1
0 5 10 15 20 25 30 35 40
-0.8
-0.6
-0.4
-0.2
0
to obs policy
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Welfare losses, alternative Taylor rules
Gali runs a simulation exercise and compares losses underW :
Taylor Rules
φpi 1.5 1.5 5 1.5
φy 0.125 0 0 1
σ(y˜) 0.55 0.28 0.04 1.40
σ(pi) 2.60 1.33 0.21 6.55
Loss 0.30 0.08 0.002 1.92
The results confirm the picture from the IRF analysis
A strong response to inflation closes the inflation and output gaps
I UnderW there are no trade offs between stablisations
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Discussion of these results
To stabilise inflation is to stabilise output
I The Phillips Curve is a function of the output gap
Output stabilisation is not improved by targeting the output gap
High φy does not mean CB cares more about employment stabilisation
I At least, not in the BNKM
I In the BNKM underW there is no such thing as Hawks and Doves
I There is no reason here to be ‘smooth’ with the interest rate
I Things may be different in real life!
But σ(v) = φy so raising φy increases variance of policy-error component
This reduces the performance of the rule
Output gap is unobserved, but pit is an excellent signal
⇒ target the signal, don’t put weight on your noise!
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Optimal policy
A policy which could ensure y˜t = pit = 0 ∀ t would suffer no losss
At the natural rate of output, the economy attains the efficient allocation
I Recall the RBC roots and the consumer optimisation problem
At least if a subsidy is in place to offset ss. costs of monopoly
I Gali suggests a subsidy to equate the real wages and MPNt
In the presence of sticky prices, price stability pi = 0 is the only way to avoid
costs of price dispersion and attain the flexible price equilibrium
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