Does the New Keynesian Model Have A
Uniqueness Problem?
Lawrence J. Christiano

Martin Eichenbaum
Johannsen

Benjamin K.

Disclaimer: The views expressed here are those of the authors and do not necessarilyre‡ect the views of the Federal Reserve Board, the FOMC, or anyone else associated with
the Federal Reserve System.

Introduction
NK models have been enormously in‡uential in terms of theirpolicy implications.
Models’implications for …scal policy are particularly striking
when ZLB is binding.
Key results:– When ZLB binds, output fall is potentially very large.
– The multiplier is larger when the ZLB binds than when it
doesn’t.
– The more binding is the ZLB the larger is the drop in output
and the larger is the multiplier.

These results generated using linearized version of NK model,
e.g. EW, CER.

Non-uniqueness and policy

Non-linear NK models have multiple equilibria.
Policy prescriptions can vary a lot across equilibria (Mertens
and Ravn, Braun et. al. (2012), Cochrane (2015).
– At some ZLB equilibria, multiplier is small or even negative.
– At other ZLB equilibria, mutliplier very large.

So, in principle non-uniqueness of equilibria poses an enormous
challenge for policy analysis in NK models.

Is non-uniqueness a substantive problem?
Yes, if there’s no compelling way to select among di¤erent
equilibria that give di¤erent answers to critical policy questions.
Our argument starts from presumption that the assumption of
RE obviously wrong.
But it can be a useful modeling strategy for thinking about aworld where RE isn’t literally true.
“... the model described above ’assumes’that agents know
a great deal about the structure of the economy andperform some non-routine computations. It is in order toask, then: will an economy with agents armed with‘sensible’rules-of-thumb, revising these rules from time totime so as to claim observed rents, tend as time passes tobehave as described...” Lucas (1978)

Selecting among equilibria
Suppose agents make a ‘small’error in forming expectations
about variables relative to their values in a particular REE.
– Does economy converge to that REE ?
– If yes, the RE equilibrium is stable-under-learning.

In our view, learnability is a necessary condition for an REE tobe empirically interesting.
Non-learnable REE equilibria are best viewed as mathematicalcuriosities.

Non-linear Calvo model with binding ZLB

Apply learnability criterion to standard fully non-linear NK
model with Calvo pricing frictions.
Unlike linearized NK models, ZLB REE can’t be characterized
by a set of numbers.
– There’s an endogenous state variable (past price dispersion),so ZLB REE is a set of functions.
– Must think about how agents learn about these functions.

Key Results...
There’s multiple REE, including sunspot equilibria (Mertens
and Ravn).
– When we consider fundamental shocks that trigger ZLB
episodes, we …nd two minimum state variable ZLB equilibria.
– These equilibria converge to di¤erent in‡ation rates if the ZLB
episode lasts forever.

Impact of government consumption can be very di¤erent in thedi¤erent ZLB equilibria.
– For example, there are ZLB REE in which multiplier isnegative.

Key Results...

There exists a unique interior ZLB REE that’s
stable-under-learning.
– That REE that converges to a relatively low ZLB de‡ation rate.

Controversial predictions of linearized NK model about …scal
policy in the ZLB, are satis…ed at unique learnable ZLB REE.
We conclude Calvo model doesn’t have a substantive
uniqueness problem, at least for analysis of …scal policy.

What about the Rotemberg model?
Many authors used non-linear versions of the Rotemberg model
to proxy for Calvo model.
– Much easier to work with, no endogenous state variables in
ZLB.
– ZLB REE is a set of numbers (not functions).

Linearized versions of these models are the same.
But non-linear versions of the two models are potentially very
di¤erent.

The Rotemberg model...
Some properties of non-linear Rotemberg model are very
sensitive to how you formulate adjustment costs for prices.
Number of ZLB REE and their stability properties depend onwhether and exactly how you scale adjustment costs for growth.
But there always exists a unique ZLB REE that’s
stable-under-learning.
At that equilibrium, impact of …scal policy in the ZLB are sameas those implied by log-linear NK model.

By-product: Linear Approximations

Use non-linear Calvo model to assess robustness of log-linear
approximations.
Log–linear approximations work reasonably well for analysis of
ZLB and …scal policy.
Evidence that quality of linear approximations is poor rests on
examples where output deviates by more than 15 percent fromits steady state.

The Neo-Fisherian View
Conclude talk with remarks about neo-Fisherian view ofmonetary policy.
– To acheive a high in‡ation rate, the monetary should target ahigh nominal interest rate.

Theoretical foundation for that view collapses in the face ofstability-under-learning criterion.
Non-linear ‡exible price (BSGU) and NK models (Calvo,
Rotemberg) have unique RE equilibrium that’s
stable-under-learning.
– First result eliminates BSGU based arguments for neo
Ficherian view
– Second result eliminates Cochrane arguments based on NK
model.

Standard NK modelHousehold maximizesh∞E0 ∑ dt log (Ct )t=0t

dt = ∏
j=0

χ 2i
h
2 t

1
1 + rj 1!

Discount factor rt can take on two values: ‘normal value’r andr` , where r` < 0.
– If rt = r` , it stays at that value with prob p.
– Once you switch back to r, you stay there forever.
Pt Ct + Bt

(1 + Rt 1 ) Bt 1 + Wt ht + Πt .

ModelFinal homogeneous good, Yt , produced by competitive andidentical …rms:
Yt =

Z 1
0

Yj,t

ε
ε 1

ε 1
ε

dj

, ε > 1.

Input j produced by …rm j using technology Yj,t = hj,t .
– competitive in factor markets– monopolist in product market.

ModelMonopolist j choosing P̃t to maximize∞

Et ∑ βk λt+k (1 + υ)P̃t

Pt+k st+k Yj,t+k

k =0– υ is a subsidy that removes steady state distortions owing tomonopoly power.

Monopolist j sets price, Pj,t , subject to demand curve for its
good and Calvo sticky price friction
Pj,t =

Pj,t 1 with probability θP̃twith probability 1θ.

ModelAggregate output
Yt = pt ht
pt is a measure of price dispersion2

pt = 4 ( 1θ)"θπ tε 111θAggregate resource constraint
Ct + Gt

#

3 1

ε
1 ε

+ θπ tε (pt 1 ) 1 5
Yt .

Monetary policy rule
Rt = max f1, 1 + r + α (π t– Max operator re‡ects ZLB and α > 1 + r.

1)g.

Solving the modelCan reduce equilibrium conditions to four non-linear equations.
There’s an endogenous state variable, pt 1 , and an exogenous
state variable, rt .
So a solution to the model is a set of functions which satisfy
these conditions.
Stage 1: solve for the equilibrium functions that obtain whenrt = r, i.e. after the economy has exited the ZLB.
Y(pt 1 ), π (pt 1 ), F(pt 1 ), p (pt 1 )Stage 2: solve for equilibrium functions that obtain when rt isequal to r`Y` (pt 1 ), π ` (pt 1 ), F` (pt 1 ) and p` (pt 1 )ZLB REE Steady State

Consider limit as t ! ∞ when the economy stays in the ZLB.
pt converges to a number, p̂, for any interior equilibrium.
System of equations collapses to a system of equations in four
unknowns,
π ` (p̂), Y` (p̂), p` (p̂), and F` (p̂).

Solving for steady state ZLB
Equations de…ning an interior steady-state ZLB equilibriumcollapse into one equation one unknown
f (π ` ) = 0.
In a slight abuse of notation we drop explicit dependence of π `on p̂.
A necessary condition for ZLB REE equilibrium to be unique– There’s a unique solution to this equation.

Parameterizing the model

Benchmark values:
ε = 7.0, β = 0.99, α = 2.0, p = 0.75,
r` =0.02/4, θ = 0.85, η g = 0.2.

Steady state output is normalized to 1 by setting χ = 1.25.
Sensitivity analysis in appendix.

ZLB Steady States

The function f (π ` ) has inverted U shape so there’s either two
interior steady-state ZLB equilibria or none.
Benchmark case: two steady-state ZLB REE equilibria– ‘high’and ‘low’in‡ation.

Number of minimum state variable ZLB REE equilibria
coincides with number of steady state ZLB REE equilibria.
– A numerical result, not a theorem.

ZLB REE Steady States
0.04

G` = GG` = 1. 05 × G

f(π` )

0.02

0.00

0.02

0.04
0.90

0.92

0.94

π`

0.96

0.98

1.00

Dynamic Response Functions

Unlike Rotemberg, ZLB REE isn’t a number because ofendogenous state variable, pt 1 .
Consider dynamic response of π t and Ct to rt shock when
economy converges to high and low π steady-state ZLB REE.
Refer to these paths as: high and low in‡ation ZLB REE.

Dynamic Response Functions

0.99

0.76

0.98

0.74

Inflation

0.96

Consumption

High-Inflation ZLB REE

0.97

Low-Inflation ZLB REE

0.95
0.94
0.93

High-Inflation ZLB REE

0.72
0.70
0.68
0.66

Low-Inflation ZLB REE

0.64
5

10

15

Period In ZLB

20

25

0.62

5

10

15

Period In ZLB

2025

Dynamic Response Functions
Along high π ZLB REE path,– Quarterly π and C initially drop by 1.5 and 6.35 percentage
points, respectively.
– After about 5 quarters π and C declines stabilize at 1.3 and6.3 percentage points.

Along low π ZLB REE path, quarterly π and C initially drop by
7.25 and 23.5 percentage points.
– After about 5 quarters π and C declines stabilize at
23.3 percentage points.

6.0 and

The MultiplierG` = 1.05 Gh , i.e. when economy is in ZLB, G rises by 1
percentage of steady state output.
Multiplier:∆ C ` ( pt 1 ) + G `G`.
(C` (pt 1 ) + G` )∆G`

If economy is in high π (low π) ZLB REE for low value of G,
it’s in high π (low π) ZLB REE for low value of G.
– A non-trivial assumption.

The Multiplier

High-Inflation ZLB REE

Low-Inflation ZLB REE

0.0

2.4
0.2

MultiplierMultiplier

2.2
2.0
1.8

0.4
0.6
0.8

1.6
5

10

15

Period in ZLB

20

25

1.0

5

10

15

Period in ZLB

20

25

ComparisonsMultiplier in high-π ZLB REE is large, exceeding two over thetime period displayed.
Multiplier is negative in low-π ZLB REE.
To understand this result, note that an increase in G` shiftsf (π ` ) upwards.
– So e¤ect of increase in G depends on which equilibrium wefocus on.

Dramatic illustration of basic result in Mertens and Raven(2011) where multiplier in one REE is a lot smaller than in theother REE.

ZLB REE Steady States
0.04

G` = GG` = 1. 05 × G

f(π` )

0.02

0.00

0.02

0.04
0.90

0.92

0.94

π`

0.96

0.98

1.00

Comparison to linearized model
Impact period of shock

Responses of linearized model similar tothose in high-π ZLB
REE.
Properties of low-π ZLB REE are very di¤erent.
Model
LinearNonlinear, high-in‡ation
Nonlinear, low-in‡ation

Output-2.18-2.84-17.87In‡ation-0.0066-0.0093-0.0734

Multiplier
1.63
2.24
-0.35

Comparison to linearized model
Basic qualitative results reported in CER using log-linear
approximation hold up when we focus on high-π ZLB REE.
Multiplier can be much bigger than 1 when ZLB binds.
When duration of ZLB increases or degree of ‡exibility of prices
increases,– Severity of output collapse and multiplier are larger.

One interesting di¤erence:– For parameter values that imply linear multipliers explode, REE
ceases to exist in non-linear model.

Stability-under-learning
In Calvo model, …rms choose Pj,t , based in part on value of Pt .
But, Pt is a function of …rms’collective price decisions.
Firms can’t ‘know’Pt when they choose their own price, insense of actually observing it.
Standard assumption: …rms form a ‘belief’about Pt when they
make their decision.
In REE that belief is correct.

Stability-under-learning

If …rms don’t have rational expectations, it’s not natural toassume they see Pt when they choose their prices.
But if they don’t see Pt , they also don’t see Ct .
Must attribute to …rms views about equilibrium functions for
current and future aggregate π and C.

Stability-under-learninge,fx` (pt 1 , t 1) : …rm’s belief, formed using information up to
time t 1, about equilibrium function for x` .
To make time t decision, …rms must forecast values of future
variables as pt evolves.
e,f– FONC’s involve objects like x` (pt 1 , t

1) ande,fx ` ( pt + j , t1) for j 0.
– So …rms must have views about the entire function.

All these functions have time t

1 as argument– Re‡ects our assumption that …rms think they’re in stationary
environment.

Stability-under-learning
Firms’beliefs evolve according toe,fx` (pt , t) = ωx` (pt 1 , t

1) + (1e,fω )x` (pt 1 , t

1).

For ω > 0, this formulation embodies the heroic assumptionthat agents know time t 1 equilibrium function for x` .
– Paper reports sensitivity analysis to simpler rules.

In Rotemberg model, there’s no state variables in ZLB.
– Replace above rule with assumption that agents’expectations
evolve according to simple constant gain algorithm about thevalues of variables.

Stability-under-learning in the NK model
When households make their time t consumption decisions,…rms’actions have already determined π t .
So households can compute the time t equilibrium function for
π (again heroic!).
π e,h` (pt 1 , t) : households’belief, at time t, about equilibrium
function for π ` .
Given new information, households beliefs evolve according to
π e,h` (pt , t + 1) = ωπ ` (pt 1 , t) + (1ω )π e,h` (pt 1 , t).

A learning ZLB equilibrium
Assume agents know REE functions when economy isn’t in
ZLB.
A sequence of functions for all of endogenous variables thatsatisfy– Resource constraint,– Monetary policy rule,– Household and …rm optimality conditions for all t,e,fe,f– given initial set of beliefs π ` ( , 0), C` ( , 0), and π e,h` ( , 0)that evolve according to above rules.

Stability-under-learning
A ZLB REE is stable-under learning if a learning equilibriumwith initial beliefs close to, but not equal to, the REE functions,converges back to the ZLB REE equilibrium.
If an economy stays in ZLB forever, it will converge to a steadystate ZLB REE.
Learning equilibrium must also approach steady state ZLB REE
if initial ZLB REE is stable-under-learning.
Allows us to eliminate all of ZLB REE that lead to low-π
steady state ZLB REE.
– They’re not stable-under-learning.

Stability-under-learning
Consider a …rm that believes thate,f

– Steady state in‡ation rate is π ` .
– Economy is in steady state corresponding to that rate ofin‡ation.
e,f

The belief π ` isn’t an REE belief so the steady state
associated with it (including pt 1 ) isn’t a steady state ZLB
REE.
e,f

So f (π ` ) isn’t equal to zero.

Stability-under-learning
e,f

e,f

There’s an equivalence between a belief π ` and a value of p̃`e,fP̃= Ptt 1 that will be chosen by …rms who can update their price.
e,f

So we use function f (π ` ) to de…ne a new functionf̃ (p̃e` )that must be equal to zero at a steady state ZLB REE.
Write …rms’FONC for p̃t = P̃t /Pt 1 can be written, after
imposing all of equilibrium conditions, ase,fF̃ p̃t , p̃`= 0.

Stability-under-learning
De…ne the best-response functione,fp̃t = g(p̃` ).
This function has the property that,
e,f

e,f

F̃ g(p̃` ), p̃`In a steady state RE ZLB equilibriume,fp̃t = p̃`= 0.

Stability-under-learning
Following …gure plots typical …rm’s best response function, i.e.
e,fp̃t as a function of p̃` .
Steady state ZLB REE equilibria correspond to the two points
where best response function intersects 45 degree line.
e,fGiven any belief, p̃` , between RE steady state beliefs, best
e,f

e,f

response g(p̃` ) is greater than p̃` .
Follows that realized π will exceed beliefs about π.
So learning equilibrium will move towards high-π ZLB REE
steady state.

Best Response Function
1.00g(p̃ e,` f )

0.95
0.90
0.85
0.80
0.75
0.75

0.80

0.85

p̃ e,` f

0.90

0.95

1.00

Best Response Functione,f

Now consider any belief, p̃` , that exceeds high-π REE ZLB
steady state.
e,f

e,f

– Best response function g(p̃` ) is less than p̃` .
– So realized π will be lower than beliefs about π.
– Learning equilibrium will move towards high-π ZLB REE
steady state.
e,fFinally, consider any belief, p̃` , that’s less than low-π ZLB
REE ZLB steady state.
e,f

e,f

– Here best response function g(p̃` ) is less than p̃` .
– So realized π will be lower than beliefs about in‡ation.
– Learning equilibrium will move away from the low-π ZLB REE
steady state.

Stability-under-learning
Previous discussion focused on limiting point of ZLB REE.
To be stable-under-learning, functions de…ning a learningequilibrium must converge point-wise to functions de…ning a
ZLB REE for every possible for pt , including steady state valueof p` .
Just showed that any ZLB REE that converges to low-π steadystate ZLB REE doesn’t satisfy this condition.
So those equilibria aren’t stable-under-learning.

Stability-under-learning
Previous discussion doesn’t establish that a ZLB REE thatconverges to high-π steady state REE ZLB isstable-under-learning.
Our solution algorithm parameterizes ZLB REE functions witha …nite number of parameters, zt .
Learning algorithm de…nes a mapping from current values of
those parameters to next period’s values:
zt+1 = s (zt ) .
De…ne

"

#
dsi (z)S(z̃) =jz̃ ,dzj

for all i, j < N where N is number of parameters.

Stability-under-learning
Evaluate S for the parameters of the high-π REE.
Max eigenvalue is less than one in absolute value.
– So, locally, functions in neighborhood of high-π ZLB REE will
converge to those REE functions in a learning equilibrium.

Repeat analysis for parameters of the low-π ZLB REE.
– Maximum eigenvalue is greater than one in absolute value.
– So, locally, functions in neighborhood of low-π ZLB REE,
equilibrium will diverge from those REE functions in a learningequilibrium.

Learning equilibria
Dynamic Paths

At t = 1 economy is in high- π ZLB REE steady state wherep 1 = p` .
At time 0,e,fx` (p 1 ,1) = x` (p 1 ) + x̄` , x̄` > 0.

Agents think that if ZLB ends, economy will be in REE thatconverges to high-π steady state.
– Also assume parameter ω = 1 (tomorrow will be like today).

x̄` = (-.02,-.01, 0, 0.01, 0.02).

Learning equilibria

Red line: REE value in high in‡ation ZLB REE
1.000

0.965
0.960
0.955

Inflation

Inflation

0.995
0.990
0.985

0.950
0.945
0.940
0.935

0.980

0.930
0.975

0

10

20

30

Period In ZLB

40

50

0.925

0

10

20

30

Period In ZLB

4050

Learning equilibria

Regardless of value of x̄` , π converges to high π ZLB REE.
Establishes that learning equilibrium converges to equilibrium
function de…ning an REE when evaluated at the p` .
Second panel: begin from the low-π ZLB REE.
In‡ation diverges from that equilibrium in the learningequilibrium.

Learning equilibria

Till now we’ve assumed that agents belief that once ZLB isover, economy will go to REE that converges to the high-π
steady state.
Redo analysis assuming agents think that post-ZLB, economy
will go to REE that converges to low-π steady state.
All of qualitative results hold, e.g.REE that converge to low π
steady state are not stable-under-learning.

Fiscal policy in learning equilibria

Initially assume that agents think that when ZLB episode isover, economy goes to REE that converges to high-π steadystate.
Economy begins in high-π steady state.
At time 0, p 1 = 1, r falls to r` .
Firms obey learning laws discussed above.

0.770
0.765
0.760
0.755
0.750
0.745
1.000
0.998
0.996
0.994
0.992
0.990
0.988
0.986
0.984

Multiplier

Inflation Consumption

Fiscal policy in learning equilibria
Learning Equilibrium
RE Equilibrium
0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50

2.0
1.5
1.0

Period In ZLB

Fiscal policy in learning equilibria
C and and π converge to high π ZLB REE from above.
Reason that they initially take on higher values is that initialexpectations about higher future π and C spur demand now.
As expectations adjust downward with realized π and C, they
push C and π down further.
Multiplier starts out low because ZLB isn’t binding in …rst few
periods.
Once ZLB starts to bind, the multiplier quickly rises above 1.

Alternative experiment

After rt shock, …rms and household have beliefs near the low-π
ZLB REE.
C and π converge to high π ZLB REE from below.
Reason that they initially take on lower values is that
expectations about low future in‡ation and consumption
depress demand in the present.

Alternative experimentMultiplier starts out around 1 and then rises after that.
Multiplier rises because …scal expansion helps quickly move
expectations toward those associated with the high-π ZLB
REE.
– Without change in G , expectations remain close to
low-in‡ation ZLB REE for some time.

After many periods, the multiplier eventually approaches thehigh-in‡ation steady state ZLB REEvalue.

Mertens and Ravn (2015)

Report multiplier is small when they analyze a learningequilibrium near the low in‡ation steady state RE ZLB
equilibrium.
This result is very di¤erent than ours - we just argued that therelevant multiplier is large.
Why?

Mertens and Ravn (2015)
When we calculate the multiplier we initially consider an
economy in which agents initial expectations about in‡ationdi¤er by eπ from the low π steady state ZLB REE.
We then consider a separate economy with shocks that set r tor` and a shock to G.
Expectations start in the same place for the two economies.
We then use di¤erence in output between the two economies tocalculate the multiplier.

Mertens and Ravn (2015)
In their experiment when G increases, the rate of in‡ation in
the steady state of the ZLB REE falls by e0π .
When they raise G in the learning equilibrium, they also
decrease agents’expectations about in‡ation by e0π .
– In and of itself this fall in in‡ation reduces output in the ZLB.

Next …gure displays multiplier if we adopt their assumption.
We obtain a negative multiplier that persists for roughly 10
years.
Change in expectations is quantitatively much more important
than the increase in G.

Mertens and Ravn (2015)
0.5
0.4

Multiplier

0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4

0

10

20

30

Periods After Shock40

50

Rotemberg model
Scaling term of price-adjustment costs can have large e¤ect onproperties of the equilibria that we …nd that aren’t
stable-under-learning.
Φ t+k

Pj,t+kPj,t+k 1

1!2

But there’s always a unique stable under learning equilibrium in
our examples.
Adj. CostφΦt = 2φΦt = 2 (Ct + Gt )φΦt = 2 Yt

Stable Equilibrium
1.56
1.70
1.65

Unstable Equilibrium
0.98
0.36
1.07

Concluding Remarks

Non-uniqueness of equilibria in NK models does not pose asubstantive challenge to key conclusions about the e¢ cacy of…scal policy in ZLB episodes.
A close derivative of our analysis is that the ‘neo-Fisherian’views of monetary policy based on NK models of ‡exible pricemodel like BSGU are not empirically relevant.