The risky steady state and the interest rate lower bound. N°1913 / June 2016.
Date: 2016-06-09
Institutions: European Central Bank
Subjects: economic cycle, inflation, monetary policy, venture capital

Working Paper Series 

Timothy Hills, 
Taisuke Nakata and 
Sebastian Schmidt 

The risky steady state and the 
interest rate lower bound 

No 1913 / June 2016 

Note: This Working Paper should not be reported as representing the views of the European Central Bank (ECB). 
The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

 
 
 
Abstract

Even when the policy rate is not at the eﬀective lower bound (ELB), the possibility that

the policy rate will become constrained by the ELB in the future lowers today’s inﬂation by

creating tail risk in future inﬂation and thus reducing expected inﬂation. In an empirically

rich model calibrated to match key features of the U.S. economy, we ﬁnd that the tail risk

induced by the ELB causes inﬂation to undershoot the target rate of 2 percent by as much

as 45 basis points at the economy’s risky steady state. Our model suggests that achieving

the inﬂation target may be more diﬃcult now than before the Great Recession, if the recent

ELB experience has led households and ﬁrms to revise up their estimate of the ELB frequency.

JEL: E32, E52

Keywords: Deﬂationary Bias, Disinﬂation, Inﬂation Targeting, Risky Steady State, Tail Risk,

Zero Lower Bound.

ECB Working Paper 1913, June 20161Non-technical summary

Under conventional speciﬁcations of monetary policy, the possibility that future adverse

shocks will force the central bank to lower the policy rate to its eﬀective lower bound (ELB)

gives rise to a downward bias in inﬂation expectations by creating tail risk in future inﬂation.

The ELB thus weighs on current inﬂation outcomes through private sector expectations even if

the policy rate is currently not constrained by the lower bound. In this paper, we characterize

the drag that the mere possibility of future ELB events can have on current inﬂation in terms

of the risky steady state. The risky steady state is the point where the economy converges

to as current shocks have dissipated but agents account for the uncertainty associated with

future shocks. We ﬁrst use a stylized New Keynesian model to illustrate how, and why, the

risky steady state diﬀers from the deterministic steady state. We show that inﬂation and the

policy rate are lower, and output is higher, at the risky steady state than at the deterministic

steady state. This result obtains because the ELB makes the distribution of ﬁrms marginal

costs of production asymmetric; the decline in marginal costs caused by a large negative shock

is larger than the increase caused by a positive shock of the same magnitude. As a result, the

ELB constraint reduces expected marginal costs for forward-looking ﬁrms, leading them to

lower their prices even when the policy rate is not currently constrained. Reﬂecting the lower

inﬂation rate at the risky steady state, standard Taylor-type monetary policy rules imply that

the policy rate is below its deterministic steady state. Since under active monetary policy, the

central bank lowers the policy rate more than one-for-one with changes in the inﬂation rate,

in equilibrium, the expected real rate is lower at the risky steady than at the deterministic

steady state, and the output gap is positive as a result. We then turn to the main exercise

of our paper, which is to explore the quantitative importance of the wedge between the risky

and deterministic steady states in an empirically rich DSGE model calibrated to match key

features of the U.S. economy. We ﬁnd that the wedge between the deterministic and risky

steady states is non-trivial in our calibrated empirical model.

Inﬂation is about 25 basis

points lower than the target inﬂation of 2 percent at the risky steady state, with 18 basis

points attributable to the ELB constraint as opposed to other nonlinearities of the model.

The policy rate and the output gap are 50 basis points lower and 0.3 percentage points

higher, respectively, at the risky steady state than at the deterministic steady state. The

magnitude of the wedge depends importantly on the frequency of hitting the ELB, which

in turn depends importantly on the level of the long-run equilibrium policy rate.

If the

policy rate at the deterministic steady state is 40 basis points lower than our baseline of 3.75

percent, then the deﬂationary bias would increase to more than 50 basis points, with the ELB

risk contributing 45 basis points to the overall deﬂationary bias. Our result that the ELB

constraint has enduring eﬀects on the economy even after lift-oﬀ provides a cautionary tale

for policymakers aiming to overcome the problem of persistently low inﬂation. In particular,

according to our model, inﬂation at the risky steady state is tightly linked to how frequently

ECB Working Paper 1913, June 20162the policy rate will be constrained by the ELB in the future. Thus, our model suggests that

achieving the inﬂation target may be more diﬃcult now than before the Great Recession,

if the recent lower bound experience, together with the recent downward assessment of the

long-run growth rate of the economy and long-run equilibrium policy rate, have made the

private sector to increase its assessment of the likelihood of hitting the ELB in the future.

ECB Working Paper 1913, June 201631

Introduction

This paper characterizes the risky steady state in an empirically rich sticky-price model

with occasionally binding eﬀective lower bound (ELB) constraints on nominal interest rates.

The risky steady state is the “point where agents choose to stay at a given date if they expect

future risk and if the realization of shocks is 0 at this date” (Coeurdacier, Rey, and Winant

(2011)). The risky steady state is an important object in dynamic macroeconomic mod-

els: This is the point around which the economy ﬂuctuates, the point where the economy

eventually converges to when all headwinds and tailwinds dissipate.

We ﬁrst use a stylized New Keynesian model to illustrate how, and why, the risky steady

state diﬀers from the deterministic steady state. We show that inﬂation and the policy rate

are lower, and output is higher, at the risky steady state than at the deterministic steady

state. This result obtains because the lower bound constraint on interest rates makes the

distribution of ﬁrms’ marginal costs of production asymmetric; the decline in marginal costs

caused by a large negative shock is larger than the increase caused by a positive shock of

the same magnitude. As a result, the ELB constraint reduces expected marginal costs for

forward-looking ﬁrms, leading them to lower their prices even when the policy rate is not

currently constrained. Reﬂecting the lower inﬂation rate at the risky steady state, the policy

rate is lower at the risky steady state than at the deterministic steady state. In equilibrium,

the expected real rate is lower at the risky steady than at the deterministic steady state, and

the output gap is positive as a result. These qualitative results are consistent with those in

Adam and Billi (2007) and Nakov (2008) on how the ELB risk aﬀects the economy near the

ELB constraint under optimal discretionary policy.

We then turn to the main exercise of our paper, which is to explore the quantitative im-

portance of the wedge between the risky and deterministic steady states in an empirically rich

DSGE model calibrated to match key features of the U.S. economy. We ﬁnd that the wedge

between the deterministic and risky steady states is non-trivial in our calibrated empirical

model. Inﬂation is about 25 basis points lower than the target inﬂation of 2 percent at the

risky steady state, with 18 basis points attributable to the ELB constraint as opposed to other

nonlinearities of the model. The policy rate and the output gap are 50 basis points lower and

0.3 percentage points higher, respectively, at the risky steady state than at the deterministic

steady state. The magnitude of the wedge depends importantly on the frequency of hitting

the ELB, which in turn depends importantly on the level of the long-run equilibrium policy

rate. If the policy rate at the deterministic steady state is 40 basis points lower than our

baseline of 3.75 percent, then the deﬂationary bias would increase to more than 50 basis

points, with the ELB risk contributing 45 basis points to the overall deﬂationary bias.

The observation that inﬂation falls below the inﬂation target in the policy rule at the

risky steady state is diﬀerent from the well-known fact that the average inﬂation falls below

the target rate in the model with the ELB constraint. The decline in inﬂation arising from a

ECB Working Paper 1913, June 20164contractionary shock can be exacerbated when the policy rate is at the ELB, while the rise in

inﬂation arising from an expansionary shock is tempered by a corresponding increase in the

policy rate. As a result, the distribution of inﬂation is negatively skewed and the average inﬂa-

tion falls below the median. This fact is intuitive and has been well known in the profession

for a long time (Coenen, Orphanides, and Wieland (2004) and Reifschneider and Williams

(2000)). The risky steady state inﬂation is diﬀerent from the average inﬂation; it is the rate

of inﬂation that would prevail at the economy’s steady state when agents are aware of risks. It

is worth mentioning that the average inﬂation falls below the target even in perfect-foresight

models or backward-looking models where the inﬂation rate eventually converges to its tar-

get. On the other hand, for the risky steady state inﬂation to fall below the inﬂation target,

it is crucial that price-setters are forward-looking and take tail risk in future marginal costs

into account in their pricing decisions.

Our result that the ELB constraint has enduring eﬀects on the economy even after liftoﬀ

provides a cautionary tale for policymakers aiming to overcome the problem of persistently

low inﬂation.

In particular, according to our model, inﬂation at the risky steady state is

tightly linked to how frequently the policy rate will be constrained by the ELB in the future.

Thus, our model suggests that achieving the inﬂation target may be more diﬃcult now than

before the Great Recession, if the recent lower bound experience, together with the recent

downward assessment of the long-run growth rate of the economy and long-run equilibrium

policy rate, have made the private sector to increase its assessment of the likelihood of hitting

the ELB in the future.

The question of how the possibility of returning to the ELB aﬀects the economy has

remained largely unexplored. The majority of the literature adopts the assumption that the

economy will eventually return to an absorbing state where the policy rate is permanently

away from the ELB constraint, and analyzes the dynamics of the economy, and the eﬀects
of various policies, when the policy rate is at the ELB (Eggertsson and Woodford (2003),
Christiano, Eichenbaum, and Rebelo (2011)). While an increasing number of studies have

recently departed from the assumption of an absorbing state, the focus of these studies is

mostly on how diﬀerently the economy behaves at the ELB versus away from the ELB, instead
of how the ELB risk aﬀects the economy away from the ELB.1 With the federal funds rate
ﬁnally raised from the ELB constraint after staying there for seven years, and with the pace of

the policy tightening expected to be gradual, the question of how the possibility of returning
to the ELB aﬀects the economy is as relevant as ever.2

Our paper builds on the work by Adam and Billi (2007) and Nakov (2008) who ﬁrst

1For example, Gavin, Keen, Richter, and Throckmorton (2015) and Keen, Richter, and Throckmorton
(2015) ask how diﬀerently technology and anticipated monetary policy shocks aﬀect the economy when the
policy rate is constrained than when it is not, respectively. Schmidt (2013) and Nakata (2013a) ask how
diﬀerently the government should conduct ﬁscal policy when the policy rate is at the ELB than when it is not.
2According to a special question in the Survey of Primary Dealers on December 2015, a median respondent
attached 20 percent probability to the event that the federal funds rate returns to the ELB within two years
after liftoﬀ.

ECB Working Paper 1913, June 20165observed that the possibility of returning to the ELB has consequences for the economy even

when the policy rate is currently away from the ELB. Our work diﬀers from these papers in

two substantive ways. First, while they pointed out the anticipation eﬀect of returning to the
ELB on the economy when the policy rate is near the ELB and the economy is away from the
steady state, our work shows that the possibility of returning to the ELB has consequences

for the economy even when the policy rate is well above the ELB and the economy is at the

steady state. Second, while they studied the eﬀects of the ELB risk in a highly stylized model,

we quantify the magnitude of the eﬀects of the ELB risk in an empirically rich, calibrated

model.

Our work complements a body of work that explores the so-called deﬂationary steady

state in sticky price models. A seminal work of Benhabib, Schmitt-Grohe, and Uribe (2001)

shows the existence of a deﬂationary steady state with the policy rate stuck at the lower

bound and inﬂation below target in a standard sticky-price model with a Taylor rule. Some

have recently studied deﬂationary steady states with zero nominal interest rates in other

interesting models with a nominal friction and a Taylor rule (Benigno and Fornaro (2015);

Eggertsson and Mehrotra (2014); Schmitt-Grohe and Uribe (2013)). Bullard (2010) argues

that these deﬂationary steady states are relevant in understanding the Japanese economy over

the last two decades as well as what may happen in other advanced economies. The steady

state we focus on is similar to the deﬂationary steady state in that both entail below-target

inﬂation. However, unlike in the deﬂationary steady state, the nominal interest rate is above

the ELB in the risky steady state.

Finally, this paper is related to recent papers that have emphasized the importance of

the eﬀect of risk on steady states in various nonlinear dynamic models. de Groot (2014) and

Gertler, Kiyotaki, and Queralto (2012) discuss how the degree of risk aﬀects the balance sheet

conditions of ﬁnancial intermediaries at the steady state. Coeurdacier, Rey, and Winant

(2011), Devereux and Sutherland (2011) and Tille and van Wincoop (2010) study optimal

portfolio choices at the risky steady state in open-economy models. Our work is similar

to theirs in analyzing the eﬀect of risk on the steady state. However, while the wedge

between the deterministic and risky steady states is driven by the nonlinearity of smooth

diﬀerentiable functions in their models, the wedge is driven by an inequality constraint in our

model. Reﬂecting the diﬀerence in the types of nonlinearity involved, the solution methods

employed are diﬀerent. While these authors solve their models by using local approximation

methods that take advantage of diﬀerentiability of policy functions, we use a global method
to solve the model.3

The rest of the paper is organized as follows. After a brief review of the concept of

the risky steady state in section 2, Section 3 analyzes the risky steady state in a stylized

New Keynesian economy. Section 4 quantiﬁes the wedge between the deterministic and risky

3See de Groot (2013) and Meyer-Gohde (2015) for recent progress in computing the risky steady state in

nonlinear diﬀerentiable economies.

ECB Working Paper 1913, June 20166steady states in an empirically rich DSGE model. After putting our analysis in the context

of the current policy debate in Section 5, section 6 concludes.

2 The Risky Steady State: Deﬁnition

The risky steady state is deﬁned generically as follows.
Let Γt and St denote vectors of endogenous and exogenous variables, respectively, in the
model under investigation. Let f (·, ·) denote a vector of policy functions mapping the values

of endogenous variables in the previous period and today’s realizations of exogenous variables
into the values of endogenous variables today.4 That is,

Γt = f (Γt−1, St)

(1)

The risky steady state of the economy, ΓRSS, is given by a vector satisfying the following
condition.

ΓRSS = f (ΓRSS, SSS)

(2)

where SSS denote the steady state of St.5 That is, the risky steady state is where the economy
will eventually converge as the exogenous variables settle at their steady state. In this risky

steady state, the agents are aware that shocks to the exogenous variables can occur, but the

current realizations of those shocks are zero. On the other hand, the deterministic steady
state of the economy, ΓDSS, is deﬁned as follows:

ΓDSS = fP F (ΓDSS, SSS)

(3)

where fP F (·, ·) denotes the vector of policy functions obtained under the perfect-foresight
assumption.

3 The Risky Steady State in a Stylized Model with the ELB

3.1 Model

We start by characterizing the risky steady state in a stylized New Keynesian model.

Since the model is standard, we only present its equilibrium conditions here. The details of

the model are described in the Appendix A.

t = βδtRtEtC−χc
C−χc

t+1 Π−1
t+1

(4)

4Note that the policy function does not need to depend on the entire set of the endogenous variables in
the prior period. It may not depend on any endogenous variables in the prior period at all, as in the stylized
model presented in the next section.

5There is no distinction between deterministic and risky steady states for St because St is exogenous.

ECB Working Paper 1913, June 20167Yt
Cχc

t (cid:20)

ϕ

(cid:18)

Πt
¯Π

− 1

(cid:19)

Πt
¯Π

Yt+1
Cχc
t+1

ϕ

Πt+1
¯Π

(cid:18)

− 1

(cid:19)

Πt+1
¯Π

wt = N χn

t Cχc

t

− (1 − θ) − θwt

= βδtEt

Yt = Ct +

(cid:21)
Πt
¯Π

ϕ
2

(cid:20)
Yt = Nt

2

− 1

Yt

(cid:21)

φπ

Rt = max

RELB,

¯Π
β

Πt
¯Π

Yt
¯Y

φy

#

(cid:19)

"

(cid:18)
(δt − 1) = ρδ(δt−1 − 1) + ǫδ
t

(cid:19)

(cid:18)

(5)

(6)

(7)

(8)

(9)

(10)

C, N , Y , w, Π, and R are consumption, labor supply, output, real wage, inﬂation, and

the policy rate, respectively. δ is an exogenous shock to the household’s discount rate, and

follows an AR(1) process with mean one, as shown in equation 10. Equation 4 is the consump-

tion Euler equation, Equation 5 is the intratemporal optimality condition of the household,

Equation 6 is the optimality condition of the intermediate good producing ﬁrms relating to-

day’s inﬂation to real marginal cost today and expected inﬂation tomorrow (forward-looking

Phillips Curve), Equation 7 is the aggregate resource constraint capturing the resource cost

of price adjustment, and Equation 8 is the aggregate production function. Equation 9 is the
interest-rate feedback rule where ¯Π and ¯Y are the central bank’s objectives for inﬂation and
output.

A recursive equilibrium of this stylized economy is given by a set of policy functions for

{C(·), N (·), Y (·), w(·), Π(·), R(·)} satisfying the equilibrium conditions described above.

The model is solved with a global solution method described in detail in the Appendix C.

Table 1 lists the parameter values used for this exercise.

Table 1: Parameter Values for the Stylized Model

Parameter Description
β
χc
χn
θ
ϕ
400( ¯Π − 1)
φπ
φy
RELB
ρ
σǫ

Discount rate
Inverse intertemporal elasticity of substitution for Ct
Inverse labor supply elasticity
Elasticity of substitution among intermediate goods
Price adjustment cost
(Annualized) target rate of inﬂation
Coeﬃcient on inﬂation in the Taylor rule
Coeﬃcient on the output gap in the Taylor rule
The eﬀective lower bound
AR(1) coeﬃcient for the discount factor shock
The standard deviation of shocks to the discount factor
*The implied prob. that the policy rate is at the lower bound

Parameter Value
1
1+0.004365
1.0
1
11
200
2.0
1.5
0
1
0.8
0.24
100
10%

ECB Working Paper 1913, June 20168%
d
e
z

i
l

a
u
n
n
A

3.5

3

2.5

2

1.5

1

0.5

0

1

1

0

−1

−2

−3

−4

−5

−6

1

%
d
e
z

i
l

a
u
n
n
A

Figure 1: Policy Functions from the Stylized Model

Nominal Interest Rate
4

Inflation

Output

%
d
e
z

i
l

a
u
n
n
A

2

1

0

−1

−2

−3

−4

1

1.002 1.004 1.006 1.008
δ

m
o
r
f

i

n
o
i
t
a
v
e
D
%

e
t
a
t
S
y
d
a
e
t
S

.
t
e
D
e
h
t

0.5

0

−0.5

−1

−1.5

−2

−2.5

1

1.002 1.004 1.006 1.008
δ

Real Wage

Exp. Real Rate

3

2.5

2

1.5

1

0.5

%
d
e
z

i
l

a
u
n
n
A

1.002 1.004 1.006 1.008
δ

0

1

1.002 1.004 1.006 1.008
δ

1.002 1.004 1.006 1.008
δ

With uncertainty
Without uncertainty

Deterministic St.St.
Deterministic St.St.
Deterministic St.St.
Deterministic St.St.
Risky St.St.
Risky St.St.
Risky St.St.
Risky St.St.

*The dashed black lines (“Without uncertainty” case) show policy functions obtained under the perfect-foresight as-
sumption (i.e., σǫ = 0).

3.2 Dynamics and the risky steady state

Before analyzing the risky steady state of the model, it is useful ﬁrst to look at the

dynamics of the model. Solid black lines in Figure 1 show the policy functions for the policy

rate, inﬂation, output, and the expected real interest rate. Dashed black lines show the

policy function of the model obtained under the assumption of perfect foresight. Under the

perfect-foresight case, the agents in the model attach zero probability to the event that the

policy rate will return to the ELB when the policy rate is currently away from the ELB.

Under both versions of the model, an increase in the discount rate makes households want to

save more for tomorrow and spend less today. Thus, as δ increases, output, inﬂation, and the

policy rate decline. When δ is large and the policy rate is at the ELB, an additional increase

in the discount rate leads to larger declines in inﬂation and output than when δ is small and

the policy rate is not at the ELB, as the adverse eﬀects of the increase in δ are not countered

ECB Working Paper 1913, June 20169 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
by a corresponding reduction in the policy rate.

When the policy rate is at the ELB, the presence of uncertainty reduces inﬂation and

output. This is captured by the fact that the solid lines are below the dashed lines for

inﬂation and output in the ﬁgure. The non-neutrality of uncertainty is driven by the ELB

constraint. If the economy is buﬀeted by a suﬃciently large expansionary shock, then the

policy rate will adjust to oﬀset some of the resulting increase in real wages. If the economy is

hit by a contractionary shock, regardless of the size of the shock, the policy rate will stay at

the ELB and the resulting decline in real wages will not be tempered. Due to this asymmetry,

an increase in uncertainty reduces the expected real wage, which in turn reduces inﬂation as

price-setters are forward-looking and thus inﬂation today depends on the expected real wage.

With the policy rate constrained at the ELB, a reduction in inﬂation leads to an increase in

the expected real rate, pushing down consumption and output today. These adverse eﬀects

of uncertainty at the ELB are studied in detail in Nakata (2013b).

When the policy rate is away from the ELB, the presence of uncertainty reduces inﬂation

and the policy rate, but increases output. If the economy is hit by a suﬃciently large con-

tractionary shock, the policy rate will hit the ELB and the resulting decline in real wages will

not be tempered. If the economy is hit by an expansionary shock, regardless of the size of the

shock, the policy rate will adjust to partially oﬀset the resulting increase in real wages. Thus,

the presence of uncertainty, by generating the possibility that the policy rate will return to

the ELB, reduces the expected real wage and thus today’s inﬂation. When the policy rate is

away from the ELB, its movement is governed by the Taylor rule. Since the Taylor principle

is satisﬁed (i.e., the coeﬃcient of inﬂation is larger than one), the reduction in inﬂation comes

with a larger reduction in the policy rate. As a result, the expected real rate is lower, and

thus consumption and output are higher, with uncertainty than without uncertainty.

Table 2: The Risky Steady State in the Stylized Model

Deterministic steady state
Risky steady state
(Wedge)
Risky steady state w/o the ELB
(Wedge)

3.75
3.32
(−0.43)
3.72
(−0.03)
*Output is expresed as a percentage deviation from the determistic steady state.

Inﬂation Output∗ Policy rate
0
0.03
(0.03)
−0.02
(−0.02)

2
1.71
(−0.29)
1.99
(−0.01)

While these eﬀects are stronger the closer the policy rate is to the ELB, they remain

nontrivial even at the economy’s risky steady state. In the stylized model of this section, in

which the policy functions do not depend on any of the model’s endogenous variables from

the previous period, the risky steady state is given by the vector of the policy functions

evaluated at δ = 1. That is, inﬂation, output, and the policy rate at the risky steady state

are given by the intersection of the policy functions for these variables and the left vertical

axes. As shown in Table 2, inﬂation and output are 29 basis points lower and 0.03 percentage

ECB Working Paper 1913, June 201610points higher at the risky steady state than at the deterministic steady state, respectively.

The risky steady state policy rate is 43 basis point lower than its deterministic counterpart.

In our model, the ELB constraint is not the only source of nonlinearity. Our speciﬁcations

of the utility function and the price adjustment cost also make the model nonlinear, and thus

explain some of the wedge between the deterministic and risky steady states. To understand

the extent to which these other nonlinearities matter, Table 2 also reports the risky steady

state in the version of the model without the ELB constraint. Overall, the diﬀerences between

the deterministic and risky steady states would be small were it not for the ELB constraint.

Inﬂation and the policy rate at the risky steady state are only 1 and 3 basis points below

those at the deterministic steady state, respectively. Output at the risky steady state is about

2 basis points below that at the deterministic steady state. Thus, the majority of the overall

wedge between the deterministic and risky steady states is attributed to the nonlinearity

induced by the ELB constraint, as opposed to other nonlinearities of the model.

3.3 The risky steady state and the average

Figure 2: Unconditional Distribution of Inﬂation in the Stylized Model

Inflation Target: 2
RSS Inflation: 1.71
Average Inflation: 1.65

Skewness: −0.3380

ELB Binds

1.2

1.4

1.6

1.8

2

2.2

−6

−4

−2

0

2

4

6

8

*RSS stands for the Risky Steady State.

It is important to recognize that the risky steady state is diﬀerent from the average. Let’s

take inﬂation as an example. The risky steady state inﬂation is the point around which

inﬂation ﬂuctuates and coincides with the median of its unconditional distribution in the

model without any endogenous state variables like the one analyzed here. On the other hand,

the average inﬂation is the average of inﬂation in all states of the economy. Provided that the

probability of being at the ELB is suﬃciently large, the unconditional distribution of inﬂation

is negatively skewed and therefore the risky steady state inﬂation is higher than the average

ECB Working Paper 1913, June 201611 
 
inﬂation, as depicted in Figure 2. The observation that the ELB constraint pushes down the

average inﬂation below the median by making the distribution of inﬂation negatively skewed

is intuitive and has been well known for a long time (Coenen, Orphanides, and Wieland

(2004) and Reifschneider and Williams (2000)). This holds true even when price-setters form

expectations in a backward-looking manner. The result that the ELB risk lowers the median

of the distribution below the target is less intuitive and requires that price-setters are forward-

looking in forming their expectations.

The magnitude of the wedge between the deterministic and risky steady states depends

importantly on the probability of being at the ELB. The blue line in the left panel of Figure 3

illustrates this point for inﬂation. In this ﬁgure, we vary the standard deviation of the discount

rate shock to induce changes in the probability of being at the ELB. According to the blue

line, a higher probability of being at the ELB is associated with a larger deﬂationary bias at

the risky steady state. In this stylized model, the risky steady state increases from 29 basis

points to 38 basis points when the probability of being at the ELB increases from 10 percent

to 12 percent. Similarly, the average inﬂation is lower when the probability of being at the

ELB is higher, as shown by the red line. As discussed earlier, when the ELB probability

is suﬃciently high, the average inﬂation is below the risky steady state inﬂation. However,

when the ELB probability is suﬃciently low, the average inﬂation is above the risky steady

state inﬂation, which reﬂects the fact that other nonlinear features of the model make the

unconditional distribution of inﬂation slightly positively skewed.

Figure 3: Conditional and Unconditional Averages of Inﬂation in the Stylized Model

E[Π|R
 = 1]
t

2.1

2

1.9

1.8

1.7

1.6

)

%
d
e
z

i
l

a
u
n
n
A

(

n
o
i
t
a
l
f
n
I

E[Π]
E[Π|R
 > 1]
t
RSS Inflation

−0.4

−0.6

−0.8

−1

−1.2

−1.4

−1.6

)

%
d
e
z

i
l

a
u
n
n
A

(

n
o
i
t
a
l
f
n
I

1.5
0

2

6
4
 = 1] (in %)
Prob[R
t

8

10

12

−1.8
0

2

6
4
 = 1] (in %)
Prob[R
t

8

10

12

Figure 3 also plots the conditional averages of inﬂation away from the ELB (the black

ECB Working Paper 1913, June 201612 
 
 
 
 
 
 
 
line in the left panel) and the conditional average of inﬂation at the ELB (the dashed black

line in the right panel). Not surprisingly, the conditional average of inﬂation away from the

ELB is higher than the unconditional average of inﬂation, which in turn is higher than the

conditional average of inﬂation at the ELB. The conditional average of inﬂation at the ELB

monotonically declines with the probability of being at the ELB, as the risky steady state

inﬂation and the unconditional average inﬂation do.

In contrast, the conditional average of inﬂation away from the ELB is non-monotonic;

It increases when the ELB probability is low and declines when the probability is high. As

a result, the conditional average of inﬂation away from the ELB is above the target rate

of 2 percent when the ELB probability is suﬃciently low. This happens because, when the

ELB probability is suﬃciently low, the conditional distribution of inﬂation away from the

ELB excludes the lower tail of the unconditional distribution, which is centered around a

level that is only slightly below 2 percent. However, the conditional average of inﬂation away

from the ELB is below the target rate when the lower bound risk is suﬃciently high and the

unconditional distribution of inﬂation is centered around a point suﬃciently below 2 percent.

As described in the Appendix H, the conditional average of inﬂation away from the ELB is

always above the target rate of 2 percent in the perfect-foresight version of the model with the

ELB. Thus, the importance of the ELB risk manifests itself in the below-target conditional

average of inﬂation away from the ELB.

While the risky steady state inﬂation cannot be measured in the data, the conditional

average of inﬂation away from the ELB can be. Thus, the importance of the lower bound

risk in the data manifests itself in the extent to which the conditional average of inﬂation

away from the ELB falls below the target rate of inﬂation. We will later examine the average

inﬂation away from the lower bound in several advanced economies in Section 5.

3.4 The risk-adjusted Fisher relation

One way to understand the discrepancy between deterministic and risky steady states is
to examine the eﬀect of the ELB risk on the Fisher relation. Let RDSS and ΠDSS be the
deterministic steady state policy rate and inﬂation. In the deterministic environment, the

consumption Euler equation evaluated at the steady state becomes

RDSS =

ΠDSS
β

(11)

after dropping the expectation operator from the consumption Euler equation and eliminating

the deterministic steady-state consumption from both sides of the equation. This relation is

often referred to as the Fisher relation.

In the stochastic environment, the consumption Euler equation evaluated at the (risky)

ECB Working Paper 1913, June 201613Figure 4: The Risk-Adjusted Fisher Relation and the Taylor Rule

 R

t

1

Standard
Fisher Relation

DSS

Risk−Adjusted
Fisher Relation

RSS

Taylor Rule

 Π
t
†DSS stands for “deterministic steady state,” and RSS stands for “risky steady state.”

steady state can be written as

RRSS =

ΠRSS
β

·

ERSS[

1
CRSS
Ct+1

(cid:16)

(cid:17)

χc ΠRSS
Πt+1 ]

(12)

where RRSS, ΠRSS, and CRSS are the risky steady-state policy rate, inﬂation, and con-
sumption. ERSS[·] is the conditional expectation operator when the economy is at the risky
steady state today. In the stylized model with one shock and without any endogenous state
variables, ERSS[·] := Et[·|δt = 1]. We will refer to Equation 12 as the risk-adjusted Fisher
relation. Relative to the standard Fisher relation, there is an adjustment term that reﬂects
the discrepancy between today’s economic conditions and the expected economic conditions

next period. This term captures the eﬀect of the tail risk in future economic conditions

induced by the ELB constraint on household expectations. Notice that the adjustment term

is less than one,

1
CRSS
Ct+1

< 1,

χc ΠRSS
Πt+1 ]

ERSS[

(13)

(cid:17)
because of the fat tail on the lower end of the distributions of future inﬂation and consumption

(cid:16)

induced by the ELB constraint.

In equilibrium, the steady state is given by the intersection of the line representing the

Fisher relation and the line representing the Taylor rule. Since the risk-adjustment term is less

than one, the line representing the risk-adjusted Fisher relation crosses the line representing

ECB Working Paper 1913, June 201614the Taylor rule at a point below the line for the standard Fisher relation crosses it, as shown

in Figure 4. Thus, inﬂation and the policy rate are lower at the risky steady state than at
the deterministic steady state.6

4 The Risky Steady State in an Empirical Model with the

ELB

We now quantify the magnitude of the wedge between the deterministic and risky steady

states in an empirically rich model calibrated to match key features of the U.S. economy.

4.1 Model

Our empirical model adds four additional features on top of the stylized New Keynesian

model of the previous section: (i) a non-stationary productivity process, (ii) consumption

habits, (iii) sticky wages, and (iv) an interest-rate smoothing term in the interest-rate feed-

back rule. Since these features are standard, we relegate the detailed description of them to
the Appendix B and only show the equilibrium conditions of the model here. Let ˜Yt = Yt
At
˜Ct = Ct
be the stationary representations of output, consumption,
At
real wage, and marginal utility of consumption, respectively, where At is a (deterministic)
non-stationary productivity path. The stationary equilibrium is characterized by the follow-

, and ˜λt = λt
A−χc
t

, ˜wt = wt
At

,

ing system of equations:

˜λt =

β
aχc

δtRtEt˜λt+1

−1

Πp

t+1

˜λt = ( ˜Ct −

(cid:0)
˜Ct−1)−χc

(cid:1)

ζ
a

Nt ˜wt
˜λ−1
t

ϕw

(cid:20)

(cid:18)

Πw
t
¯Πw − 1
(cid:19)

t

Πw
t

¯Πw − (1 − θw) − θw N χn
˜λt ˜wt (cid:21)
˜wt
˜wt−1

t =

Πw

Πp
t

=

βϕw
aχc−1 δtEt

Nt+1 ˜wt+1
λ−1
t+1 (cid:18)

t+1

Πw
¯Πw − 1
(cid:19)

˜Yt
˜λ−1
t (cid:20)

ϕp

Πp
t
¯Πp − 1
(cid:19)

(cid:18)

Πp
¯Πp − (1 − θp) − θp ˜wt
t

=

βϕp
aχc−1 δtEt

(cid:21)

˜Yt = ˜Ct +

ϕp
2

(cid:20)

2

˜Yt +

Πp
t
¯Πp − 1
(cid:21)
˜Yt = Nt

ϕw
2

(cid:20)

Πw
t
¯Πw − 1
(cid:21)

t+1

Πp
¯Πp − 1
(cid:19)

Πp
t+1
¯Πp

˜Yt+1
˜λ−1
t+1 (cid:18)
2

˜wtNt

(14)

(15)

Πw
t+1
¯Πw
(16)

(17)

(18)

(19)

(20)

6The other intersection of the risk-adjusted Fisher relation and the Taylor-rule equation indicates that, in
a deﬂationary equilibrium, inﬂation is higher at the risky steady state than at the deterministic steady state.
We have conﬁrmed that this is indeed the case using a semi-loglinear model with a three-state discount rate
shock. We will leave an in-depth examination of the risky steady state in a deﬂationary equilibrium to future
research.

ECB Working Paper 1913, June 201615and

where

Rt = max [1, R∗
t ]

R∗
t
¯R

=

R∗
t−1
¯R

(cid:18)

ρR

(cid:19)

(cid:18)

Πp
t
¯Πp

(1−ρr)φπ

(cid:19)

(1−ρr)φy

˜Yt
¯Y !

and the following processes for the discount rate:

(δt − 1) = ρδ(δt−1 − 1) + ǫδ
t

(21)

(22)

(23)

ζ is the degree of consumption habits in the household’s utility function and a is the trend
growth rate of productivity. ϕp and ϕw are the price and wage adjustment costs. ρR is the
weight on the lagged shadow policy rate in the truncated interest-rate feedback rule. ¯Πp
and ¯Πw are price and wage inﬂation rates in the determistic steady state. In the truncated
interest-rate feedback rule, ¯R and ¯Y are the policy rate and output (normalized by At) at
the deterministic steady state and are functions of the structural parameters. ˜Yt/ ¯Y is the
deviation of the stationarized output from its deterministic steady state and will be referred
to as the output gap in this paper.7 In the Appendix G, we also study the risky steady state
in several extended models which add more frictions and shocks to the empirical model of

this section.

4.2 Calibration

We calibrate our model to match key features of the output gap, inﬂation, and the policy

rate in the U.S. over the past two decades, which are shown in Figure 5. We focus on this

relatively recent past because long-run inﬂation expectations were low and stable and the

ELB was either a concern or a binding constraint to the Federal Reserve during this period.

As shown in Figure 6, the median of CPI inﬂation forecasts 5-10 years ahead in the Survey

of Professional Forecasters, a commonly used measure of long-run inﬂation expectations,
declined to 2.5 percent in late 1990s and has been relatively stable since then.8 Also, the
concern for the ELB surged in the U.S. in the second half of 1990s when the Bank of Japan

lowered the policy rate to the lower bound for the ﬁrst time in the Post WWII history among
major advanced economies.9

We set the time discount rate to 0.99875 so that the contribution of the discount rate to

7Note that our output gap is not the ﬂexible-price output gap that is the deviation of the (normalied)

output from its ﬂexible-price counterpart.

8The long-run inﬂation expectations measured by PCE inﬂation is available only from 2007. The average
diﬀerential between CPI and PCE inﬂation rates over the past two decades is about 50 basis points. Thus,
the stability of CPI inﬂation expectations at 2.5 percent can be interpreted as the stability of PCE inﬂation
expectations at 2 percent.

9Some of the earliest research on the ELB were initiated within the Federal Reserve System in this period.
for example, Clouse, Henderson, Orphanides, Small, and Tinsley (2003), Reifschneider and Williams

See,
(2000), and Wolman (1998).

ECB Working Paper 1913, June 201616 
Figure 5: Output Gap, Inﬂation, and Policy Rate†

Output Gap (%)

Inflation (Annualized %)

Policy Rate (Annualized %)

5

2.5

0

−2.5

−5

−7.5

3

2.5

2

1.5

1

0.5

8

7

6

5

4

3

2

1

−10

1995

2000

2010

2015

2005
Year

0
1995

2000

2010

2015

2005
Year

0
1995

2000

2005
Year

2010

2015

†The measure of the output gap is based on the FRB/US model and is available upon request. The inﬂation rate is
computed as the annualized quarterly percentage change (log diﬀerence) in the personal consumption expenditure core
price index (St. Louis Fed’s FRED). The quarerly average of the (annualized) federal funds rate is used as the measure
for the policy rate (St. Louis Fed’s FRED). Vertical lines mark the year when ELB started binding. Horizontal lines
represent target values for the respective variables.

Figure 6: Long-Run Inﬂation Expectations†

4

3.5

3

2.5

2

1.5
1990

1995

2000

2005

2010

2015

Year
†Calculations based on: Federal Reserve Board, Survey of Professional Forecasters, accessed November 2015,
https://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-forecasters/.

the deterministic steady state real rate is 50 basis points. We set the target inﬂation in the

interest-rate feedback rule to 2 percent as this is the FOMC’s oﬃcial target rate of inﬂation.

We set the trend growth rate of productivity to 1.25 percent so that the policy rate is 3.75

percent at the economy’s deterministic steady state.

In the household utility function, the degree of consumption habits, the inverse Frisch

labor elasticity, and the inverse intertemporal elasticity of substitution are set to 0.5, 0.5 and

1, respectively. These are all within the range of standard values found the literature.

ECB Working Paper 1913, June 201617Table 3: Parameter Values for the Empirical Model

Parameter
β
a
χc
ζ
χn
θp
θw
ϕp
ϕw
Interest-rate feedback rule
400( ¯Π − 1)
ρR
φπ
φy
400(RELB − 1)
Shocks
ρd
σǫ,δ

Description
Discount rate
Trend growth rate of productivity
Inverse intertemporal elasticity of substitution for Ct
Degree of consumption habits
Inverse labor supply elasticity
Elasticity of substitution among intermediate goods
Elasticity of substitution among intermediate labor inputs
Price adjustment cost
Wage adjustment cost

(Annualized) target rate of inﬂation
Interest-rate smoothing parameter in the Taylor rule
Coeﬃcient on inﬂation in the Taylor rule
Coeﬃcient on the output gap in the Taylor rule
(Annualized) eﬀective lower bound

AR(1) coeﬃcient for the discount factor shock
The standard deviation of shocks to the discount factor

Parameter Value
0.99875
1.25
400
1.0
0.5
0.5
11
4
1000
300

2.0
0.8
3
0.25
0.13

0.85
0.69
100

Following Erceg and Lind´e (2014), the parameters governing the steady-state markups for

intermediate goods and the intermediate labor inputs are set to 11 and 4 and the parameters

governing the price adjustment costs for prices and wages to 1000 and 300. In a hypothetical

log-linear environment, these values would correspond to 90 and 85 percent probabilities that

prices and wages cannot adjust each quarter in the Calvo version of the model, respectively.

High degrees of stickiness in prices and wages help the model to match the moderate decline

in inﬂation in the data while the federal funds rate was constrained at the ELB.

The coeﬃcients on inﬂation and the output gap in the interest-rate feedback rule are

set to 3 and 0.25. The coeﬃcient on the output gap, 0.25, is standard. The coeﬃcient on

inﬂation is a bit higher compared to the values commonly used in the literature. A higher

coeﬃcient serves two purposes. First, it reduces the volatility of inﬂation relative to the

volatility of the output gap. Second, a higher value makes the existence of the equilibrium
more likely.10 Erceg and Lind´e (2014) argue that an inﬂation coeﬃcient of this magnitude is
consistent with an IV-type regression estimate of this coeﬃcient based on a recent sample.

The interest-rate smoothing parameter for the policy rule is set to 0.8. This high degree of

interest-rate smoothing helps in increasing the expected duration of the lower bound episodes,

improving the model’s implication in this dimension. The ELB on the policy rate is set to

0.13 percent, the average of the annualized federal funds rate during the recent lower bound

episode (from 2009:Q1 to 2015:Q2).

The persistence of the discount rate shock is set to 0.85. This is a bit higher than the

10Richter and Throckmorton (2015) show that the model with occasionally binding ELB constraints may
not have minimum-state-variable solutions when this coeﬃcient is low even if the Taylor-principle is satisﬁed.

ECB Working Paper 1913, June 201618common value of 0.8 used in most existing studies of models with occasionally binding lower
bound constraints.11 A higher persistence of the shock helps in increasing the expected
duration of being at the ELB, as the higher interest-rate smoothing parameter in the policy

rule does. The standard deviation of the discount rate shock is chosen so that the standard

deviation of the policy rate in the model matches with that in the data.

Table 4 shows the key statistics for the output gap, inﬂation and the policy rate in the

model and in the data. The measure of the output gap is based on the estimate of potential

output based from the FRB/US model. As for the measure of inﬂation, we use core PCE

Price Index inﬂation.

Table 4: Key Moments

Moment

Variable

St.Dev.(·)

E(X|ELB)

ELB

Output gap
Inﬂation
Policy rate
Output gap
Inﬂation
Policy rate
Frequency

Model

3.0
0.31
2.34
−3.7
1.21
0.13
13.8%

Data†
(1995Q3–2015Q2)
2.9
0.52
2.34
−4.2
1.48
0.13
36%
26 quarters

Expected/Actual Duration 8.6 quarters

†The measure of the output gap is based on the FRB/US model. Inﬂation rate is computed as the annualized quarterly
percentage change (log diﬀerence) in the personal consumption expenditure core price index. The quarerly average of
the (annualized) federal funds rate is used as the measure for the policy rate.

The standard deviation of the output gap in the model is 3.0, which is in line with the

sample standard deviation from the data. The conditional mean of the output gap at the

ELB in the model is -3.7 percent, which is above, but similar to, the estimate from the

data. The standard deviation of inﬂation in the model is 0.31 percent, which is lower than

what’s observed in the data, while the ELB conditional mean of inﬂation in the model is

1.21 percent, which is somewhat lower than what’s observed in the data. It should be noted

that, given that inﬂation was remarkably stable during the recent lower-bound episode in the

U.S., there is a tension between matching the unconditional standard deviation of inﬂation

and the conditional average of inﬂation at the ELB. Since the conditional mean of inﬂation

is the average of the left-tail of the unconditional distribution of inﬂation, an increase in the

standard deviation of inﬂation would necessarily imply a decrease in the conditional mean

of inﬂation at the ELB. In our case, if we increase the standard deviation of inﬂation in our

model to get closer to what’s observed in the data, the conditional mean of inﬂation at the
ELB would decline further away from its empirical counterpart.12 Our calibration reﬂects

11See, for example, Adam and Billi (2007), Nakov (2008), and Fern´andez-Villaverde, Gordon, Guerr´on-Quintana, and Rubio-Ram

(2015).

12A similar tension exists for matching the standard deviation of the output gap and the conditional mean

ECB Working Paper 1913, June 201619a compromise of staying reasonably close to the data in these two dimension. It is worth

noting that our estimate of the deﬂationary bias would be higher if we ignored the observed

stability of inﬂation at the ELB and adjusted our calibration to match the standard deviation

of inﬂation, as a lower conditional average of inﬂation at the ELB induces ﬁrms to lower prices

by more when away from the ELB. Thus, our estimate of the deﬂationary bias can be seen

as a conservative estimate.

As previously mentioned, the standard deviation of the discount rate shock was chosen

so that the standard deviation of the policy rate in the model matches with that in the

data, which is 2.34 percent. The model-implied unconditional probability of being at the

ELB and the expected ELB duration are about 14 percent and 2 years, respectively. While

these numbers are substantially higher than those in other existing models with occasionally

binding ELB constraints, they are substantially lower than the empirical counterparts over

the past two decades in the U.S. In particular, the duration of the recent ELB experience is

seen by the model as surprisingly long. Consistent with this interpretation, the data on liftoﬀ

expectations shows that market participants have underpredicted how long the policy rate

will be kept at the ELB throughout the recent ELB episode, as described in the Appendix E.

4.3 Results

Table 5: The Risky Steady State in the Empirical Model

Inﬂation Output gap Policy rate

Deterministic steady state
Risky steady state
(Wedge)
Risky steady state w/o the ELB
(Wedge)
E[·|Rt > RELB]

2
1.74
(−0.26)
1.92
(−0.08)
1.78

0
0.30
(0.30)
0.05
(0.05)
0.85

3.75
3.26
(−0.49)
3.56
(−0.19)
3.85

Table 5 shows the risky and deterministic steady state values of inﬂation, the output

gap, and the policy rate from our empirical model. For this model, the risky steady state

is computed by simulating the model for a long period while setting the realization of the

exogenous disturbances to zero. All (stationarized) endogenous variables eventually converge

in that simulation, and that point of convergence is the risky state of the economy. By

construction, the deterministic steady state of inﬂation is given by the target rate of inﬂation

and the output gap is zero at the deterministic steady state. As explained earlier, parameter
values (β, χc and a) are chosen so that the deterministic steady state of the policy rate is
3.75 percent.

Consistent with our earlier analyses based on a stylized model, inﬂation and the policy rate

are lower, and the output gap is higher, at the risky steady state than at the deterministic

of the output gap at the ELB, but to a lesser extent.

ECB Working Paper 1913, June 201620steady state.

Inﬂation falls 26 basis points below the target rate of inﬂation at the risky

steady state. This is large given the small standard deviation of inﬂation. The policy rate

at the risky steady state falls 49 basis points below its deterministic counterpart. While this

is a small number relative to its standard deviation, it is nevertheless signiﬁcant in light of

recent discussions among economists and policymakers regarding the long-run equilibrium
policy rate.13 Finally, the output wedge between the deterministic and risky steady states is
small, with the output gap standing at 0.30 percentage point at the risky steady state.

As explained in the previous section, the discrepancy between the deterministic and risky

steady states is not only driven by the lower bound constraint on policy rates, but is also

aﬀected by other nonlinear features of the model. To isolate the eﬀects of the lower bound

constraint, the fourth line of Table 5 shows the risky steady state of the model without the

lower bound constraint. Inﬂation, the output gap, and the policy rate are 1.92, 0.05, and 3.56

percent, respectively. Thus, most of the wedge between the deterministic and risky steady

states in the model with the ELB constraint is attributed to the nonlinearity associated with

the ELB constraint, as opposed to other nonlinear features of the model. For inﬂation, the

ELB risk accounts for 18 basis points of the overall deﬂationary bias.

4.4 Sensitivity Analyses

4.4.1 Long-Run Interest Rates

There are substantial uncertainties surrounding the level of the long-run real equilib-

rium interest rate. Many economists recently have argued that various structural factors—

including a lower trend growth rate of productivity, demographic trends, and global factors—
have contributed to a persistent downward trend in the long-run equilibrium interest rate.14
A lower long-run equilibrium interest rate means that the probability of hitting the ELB is

higher, which ceteris paribus increases the magnitude of the undershooting of the inﬂation

target at the risky steady state.

Figure 7 shows how sensitive the risky steady state of our empirical model is to alternative

assumptions about the deterministic steady state interest rate. In this exercise, we vary the

long-run deterministic steady-state policy rate by varying the trend growth rate. As shown

in the top-left panel, the probability of the policy rate being at the ELB increases as the

deterministic steady state policy rate declines. With the deterministic steady state policy

rate at 3.35 percent, the probability of being at the ELB is approximately 25 percent. A

higher probability of being at the ELB increases the wedge between the deterministic and

risky steady states. With the deterministic steady state policy rate at 3.35 percent, the risky

steady state inﬂation, output, and the policy rates are 1.47 percent, 0.64 percent, and 2.39
percent.15 Since the risky steady state does not depend on the deterministic steady state

13See, for example, Hamilton, Harris, Hatzius, and West (2015) and Rachel and Smith (2015).
14See, for example, the Council of Economic Advisers (2015) and IMF (2014).
15Note that an increase in the output gap does not necessarily mean an increase in the level of output

ECB Working Paper 1913, June 201621Figure 7: Long-Run Interest Rates and the Risky Steady State†

Probability of being at the ELB

RSS Inflation

26

22

18

14

10

6
3.35

0.7

0.6

0.5

0.4

0.3

0.2

0.1

2

1.9

1.8

1.7

1.6

1.5

With ELB Constraint
Without ELB Constraint

3.5

3.75

4

4.25

3.35

3.5

3.75

4

4.25

DSS Policy Rate

DSS Policy Rate

RSS Output Gap

RSS Policy Rate

4.5

4

3.5

3

2.5

0
3.35

3.5

3.75

4

4.25

DSS Policy Rate

2
3.35

3.5

3.75

4

4.25

DSS Policy Rate

†DSS stands for “deterministic steady state,” and RSS stands for “risky steady state.” Vertical lines mark the DSS
policy rate in the baseline calibration.

policy rate in the model without the ELB constraint, as shown in the dashed lines, a large

fraction of the overall wedge is explained by the ELB risk when the deterministic steady state

policy rate is lower. For inﬂation, the ELB risk accounts for 45 basis points of the overall
deﬂationary bias of 53 basis points.16

4.4.2 Policy Parameters

We have shown that, at the risky steady state, inﬂation falls below the target rate of 2

percent by a nontrivial amount in our empirical model. In our model where the prices and

wages are indexed to the target rate of inﬂation, such undershooting of the inﬂation target

is undesirable. A natural question to ask is what the central bank can do to mitigate the
deﬂationary bias.17

Figure 8 showd how the probability of being at the ELB and the risky steady-state

because output measures are stationarized by the trend growth rate.

16Hamilton, Harris, Hatzius, and West (2015) argue that any value between 0 and 2 percent is a plausible
value for the long-run real rate. Thus, the long-run nominal rate of 2.39 percent—or equivalently, the long-run
real rate of 0.39 percent—in this example is within their plausible range.

17In this paper, we examine this issue only in the context of a Taylor-type rule. Under price-level targeting
or nominal income targeting, which are known to mitigate the decline in inﬂation at the ELB, the deﬂationary
bias is likely to be smaller than under a conventional Taylor-type rule.

ECB Working Paper 1913, June 201622 
 
20

17

14

11

B
L
E

t

]

R
=
R
[
b
o
r
P

Figure 8: Alternative Policy Parameters†

17

2

24

2

2

1.9

1.8

1.7

1.6

n
o
i
t
a
l
f
n
I

S
S
R

1.5

2.5
2.5

3

2.5

2

1.5

n
o
i
t
a
l
f

n

I

S
S
R

1.9

1.8

1.7

1.6

3
3

3.5
3.5
Inﬂation Coeﬃcient (φπ)

4
4

16

15

14

13

12

4.5
4.5

1.8

1.6

1.4
0
0
Output Gap Coeﬃcient (φy)

0.1
0.1

0.2
0.2

0.3
0.3

20

2

15

1.9

10

1.8

5

1.7

16

8

0

16

12

8

4

0.75
0.75

0.8
0.8
Interest Rate Smoothing Parameter (ρr)

8
0.85
0.85

B
L
E

t

]

R
=
R
b
o
r
P

[

RSS Inflation (Left Axis)
ELB Frequency (Right Axis)

0
1
1.8 2 2.2 2.4 2.6 2.8 3
1.8 2 2.2 2.4 2.6 2.8 3
Inﬂation Target ( ¯Πp)

1.6
−1
−1
Eﬀective Lower Bound (RELB)

0
0 0.25
0 0.25

−0.5
−0.5

†In each panel, the vertical line marks the baseline parameter value. In the bottom-left panel, the thin black line shows
the 45 degree line.

inﬂation rate depends on parameters governing the interest-rate feedback rule. According

to the top-left panel, a higher coeﬃcient on inﬂation reduces the probability of being at the

ELB, raising the risky steady state closer to the inﬂation target. On the other hand, a higher

coeﬃcient on the output gap increases the probability of being at the ELB and reduces RSS

inﬂation. These results hold because a more aggresive response to inﬂation reduces both the

volatility of inﬂation and the policy rate in equilibrium, and are consistent with the analytical

results from a two-state shock model in Nakata and Schmidt (2014).

The top-right panel shows that a higher interest-rate smoothing parameter reduces the

probability of being at the ELB, thus reducing the deﬂationary bias at the risky steady state.

When the policy rate depends more on the lagged policy rate, the policy rate adjusts by less

in response to ﬂuctuations in the demand shock. Thus, higher inertia lowers the standard

deviation of the policy rate, leading to a lower ELB frequency.

Not surprisingly, a higher inﬂation target reduces the probability of being at the ELB and

reduces the diﬀerence in inﬂation between deterministic and risky steady states, as shown

in the bottom-left panel. With the inﬂation target at 1.8 percent, the deﬂationary bias at

ECB Working Paper 1913, June 201623 
 
 
 
the risky steady state is about 35 basis points. With the inﬂation target at 3 percent, the

deﬂationary bias at the risky steady state is about 12 basis points. This exercise demonstrates

the importance of taking into account the lower bound risk in the cost-beneﬁt analysis of
raising the inﬂation target.18 Finally, the last panel demonstrates that, if the eﬀective lower
bound is lower, the ELB binds less often and thus the deﬂationary bias is lower at the risky

steady state.

5 Discussion

Why should we care about what a theoretical model has to say about the ELB risk and

the resulting deﬂationary bias? In this section, we argue that we should care about it for

two reasons. First, the model with the ELB risk is consistent with the undershooting of the

inﬂation target observed in some economies even before the policy rate became constrained

by the ELB constraint. Second, the model with the ELB risk provides a cautionary tale for

policymakers aiming to achieve their inﬂation objectives in the current environment of low

long-run equilibrium real interest rates.

5.1 Low inﬂation before the Great Recession

Figure 9 shows the conditional averages of inﬂation over the past two decades when

policy rates were not constrained by the ELB in four advanced economies—economies that

have recently faced the ELB constraint for the ﬁrst time since WWII and target 2% inﬂation

symmetrically. According to the ﬁgure, the conditional averages of inﬂation away from the

ELB are below the target rate of 2 percent in all these economies. In the U.S and Canada,

inﬂation averaged about 20 and 40 basis points below 2 percent while the policy rate was

not constrained by the ELB. In the UK, the conditional average of inﬂation away from the

ELB is 60 basis points below 2 percent.

In Sweden, the average inﬂation rate is below 1

percent while the policy rate was above the ELB. In the Appendix F, we show that the

undershooting of the inﬂation target is robust to alternative starting dates of sample and the

diﬀerence between the conditional average and the target rate is statistically signiﬁcant for

all four economies.

While there are many explanations for this systematic undershooting of the inﬂation

targets before the ELB became a binding constraint—such as positive supply shocks from

emerging economies and persistent slack in the economies—it is interesting to note that this

undershooting is consistent with the prediction of the model with the ELB risk. As argued

earlier, one key prediction of the model with the ELB risk is that the conditional average

of inﬂation away from the ELB falls below the target rate of 2 percent, provided that the

probability of being at the ELB is suﬃciently high. In our empirical model, the deﬂationary

18The computation of the optimal inﬂation target is often conducted under the assumption of perfect-

foresight. See, for example, Williams (2009) and Coibion, Gorodnichenko, and Wieland (2012).

ECB Working Paper 1913, June 201624Figure 9: Conditional Averages of Inﬂation Away From the ELB†

2

1.5

1

0.5

United
States

Canada

United
Kingdom

Sweden

† The ﬁgure shows the average of the annualied quarterly inﬂation rate over the past two decades (since 1995Q3) for
each country during non-ELB quarters—when the country’s policy rate was not constrained by the ELB. For the U.S,
the measure of inﬂation is based on core PCE Price Index. For other economies, the measure of inﬂation is ased on core
CPI Index. ELB-binding quarters are from 2009Q1 to present in the U.S., from 2009:Q2 to 2010:Q1 in Canada, from
2009:Q2 to present in UK, from 2009:Q4 to 2010:Q2 and from 2014:Q4 to present in Sweden.
Data for inﬂation sourced from: for United States, Federal Reserve Bank of St. Louis, Personal Consumption Expendi-
tures: Chain-type Price Index Less Food and Energy, accessed December 2015, https://research.stlouisfed.org/fred2/;
for all other countries, OECD (2015), Inﬂation (Core CPI) (indicator), accessed November 2015, DOI: 10.1787/eee82e6e-
en.

bias is indeed large enough to push the conditional average of inﬂation away from the ELB
below the inﬂation target, as shown in the last row of Table 5. 19

Since the risk of hitting the ELB was widely seen as unlikely before the Great Recession,

we do not think that this ELB risk is a good explanation for the target undershooting in pre-

ELB era in reality. However, we think the ability of our model to generate the undershooting

in pre-ELB era is attractive. As explained in Section 3.3 and Appendix H, a version of our

model with the ELB constraint, but without the ELB risk, does not have this feature. More

research is certainly needed to better understand the sources of low inﬂation in advanced

economies and what policymakers can do to address this problem.

5.2 An implication for the future

It is quite likely that the perceived probability of hitting the ELB is higher now than

before the Great Recession. The Great Recession made clear that the economy can be hit by

19Note that this undershooting of the inﬂation target while the policy rate is above the ELB is not consistent
with the deﬂationary steady state of the sticky-price economy (Benhabib, Schmitt-Grohe, and Uribe (2001)
and Armenter (2014)). In the deﬂationary steady state, inﬂation is below the target, but the policy rate is at
the ELB.

ECB Working Paper 1913, June 201625shocks that are substantially larger than the macro shocks that hit the economy during the

Great Moderation. Moreover, several years of disappointing output growth in the aftermath

of the Great Recession have led many analysts to revise down their estimates of the trend

growth rate of productivity and long-run nominal interest rates. As shown in Section 4.4.1 a

lower long-run equilibrium interest rate would imply a higher ELB frequency.

As our earlier sensitivity analyses demonstrated, the size of the deﬂationary bias at the

risky steady state increases with the probability of being at the ELB. Thus, our model

provides a cautionary tale for policymakers: Achieving the target rate of inﬂation may have

become more diﬃcult now than before the Great Recession if the recent ELB experience has

led the private sector to revise its assessment of the likelihood of ELB events.

6 Conclusion

In this paper, we have examined the implications of the ELB risk—the possibility that

the policy rate will be constrained by the ELB in the future—for the economy when the

policy rate is currently not constrained. Using an empirically rich DSGE model calibrated to

capture key features of the U.S. economy over the past two decades, we have shown that the

ELB risk causes inﬂation to fall below the target rate of 2 percent by about 20 basis points

at the risky steady state. The deﬂationary bias induced by the ELB risk at the risky steady

state can be as much as 45 basis points under alternative plausible assumptions about the

long-run growth rate of the economy and monetary policy parameters. Our analysis suggests

that achieving the inﬂation target may be more diﬃcult now than before the Great Recession,

if the recent lower bound episode has led the private sector to increase their assessment of

the ELB risk.

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ECB Working Paper 1913, June 201629For Online Publication: Technical Appendix

A Details of the Stylized Model

This section describes a stylized DSGE model with a representative household, a ﬁnal good
producer, a continuum of intermediate goods producers with unit measure, and government
policies.

A.1 Household

The representative household chooses its consumption level, amount of labor, and bond
holdings so as to maximize the expected discounted sum of utility in future periods. As is
common in the literature, the household enjoys consumption and dislikes labor. Assuming
that period utility is separable, the household problem can be deﬁned by

max
Ct,Nt,Bt

E1

∞

t−1

βt−1

δs

t=1
X

h

s=0
Y

ih

C1−χc
t
1 − χc

−

N 1+χn
t
1 + χn

i

subject to the budget constraint

or equivalently

PtCt + R−1

t Bt ≤ WtNt + Bt−1 + PtΦt

Ct +

Bt
RtPt

≤ wtNt +

Bt−1
Pt

+ Φt

(A.1)

(A.2)

(A.3)

where Ct is consumption, Nt is the labor supply, Pt is the price of the consumption good, Wt
(wt) is the nominal (real) wage, Φt is the proﬁt share (dividends) of the household from the
intermediate goods producers, Bt is a one-period risk free bond that pays one unit of money
at period t+1, and R−1

is the price of the bond.

t

The discount rate at time t is given by βδt where δt is the discount factor shock altering
the weight of future utility at time t+1 relative to the period utility at time t. This shock
follows an AR(1) process:

δt − 1 = ρ(δt−1 − 1) + ǫt,

ǫt ∼ N (0, σǫ)

(A.4)

This increase in δt is a preference imposed by the household to increase the relative valuation
of future utility ﬂows, resulting in decreased consumption today (when considered in the
absence of changes in the nominal interest rate).

A.2 Firms

There is a ﬁnal good producer and a continuum of intermediate goods producers indexed
by i ∈ [0, 1]. The ﬁnal good producer purchases the intermediate goods Yi,t at the intermedi-
ate price Pi,t and aggregates them using CES technology to produce and sell the ﬁnal good

ECB Working Paper 1913, June 201630Yt to the household and government at price Pt. Its problem is then summarized as

max
Yi,t,i∈[0,1]

1

PtYt −

Pi,tYi,tdi

0

Z

subject to the CES production function

Yt =

0

h Z

1

θ−1
θ

i,t di

Y

θ
θ−1

.

i

(A.5)

(A.6)

Intermediate goods producers use labor to produce the imperfectly substitutable intermedi-
ate goods according to a linear production function (Yi,t = Ni,t) and then sell the product to
the ﬁnal good producer. Each ﬁrm maximizes its expected discounted sum of future proﬁts20
by setting the price of its own good. We can assume that each ﬁrm receives a production
subsidy τ so that the economy is fully eﬃcient in the steady state.21 In our baseline, however,
we set τ = 0. Price changes are subject to quadratic adjustment costs.

max
Pi,t

E1

such that

∞

t−1

βt−1

δs

λt

Pi,tYi,t − (1 − τ )WtNi,t − Pt

t=1
X

h

s=0
Y

i

h

ϕ
2

[

Pi,t
Pi,t−1 ¯Π

− 1]2Yt

i

Yi,t =

−θ

Pi,t
Pt (cid:21)

(cid:20)

Yt.22

(A.7)

(A.8)

t−1
λt is the Lagrange multiplier on the household’s budget constraint at time t and βt−1
s=0 δs
is the marginal value of an additional proﬁt to the household. The positive time zero price
i
is the same across ﬁrms (i.e. Pi,0 = P0 > 0).

h Q

λt

A.3 Government policies

It is assumed that the monetary authority determines nominal interest rates according to

a Taylor rule

Rt = max

1,

"

¯Π
β

Πt
¯Π

φπ

φy

Yt
¯Y

#

(A.9)

(cid:19)
Pt−1 and ¯Y is the steady state level of output. This equation will be modiﬁed in

where Πt = Pt
order to do an extensive sensitivity analysis of policy inertia and other rule speciﬁcations.

(cid:18)

(cid:18)

(cid:19)

20Each period, as it is written below, is in nominal terms. However, we want each period’s proﬁts in real
terms so the proﬁts in each period must be divided by that period’s price level Pt which we take care of further
along in the document.

21(θ − 1) = (1 − τ )θ which implies zero proﬁts in the zero inﬂation steady state. In a welfare analysis, this
would extract any inﬂation bias from the second-order approximated objective welfare function. τ therefore
represents the size of a steady state distortion (see Chapter 5 Appendix, Gal´ı (2008)).

22This expression is derived from the proﬁt maximizing input demand schedule when solving for the ﬁnal
good producer’s problem above. Plugging this expression back into the CES production function implies that
the ﬁnal good producer will set the price of the ﬁnal good Pt = h R

P 1−θ
i,t

1
1−θ

1
0

di

i

.

ECB Working Paper 1913, June 201631A.4 Market clearing conditions

The market clearing conditions for the ﬁnal good, labor and government bond are given

by

and

Yt = Ct +

1

ϕ
2

0

Z

Pi,t
Pi,t−1 ¯Π
1

h

Nt =

Ni,tdi

− 1

2

Ytdi

i

0

Z

Bt = 0.

(A.10)

(A.11)

(A.12)

A.5 Recursive equilibrium

Given P0 and a two-state Markov shock process establishing δt and γt, an equilibrium
consists of allocations {Ct, Nt, Ni,t, Yt, Yi,t, Gt}∞
t=1, and a policy in-
strument {Rt}∞
t=1 such that (i) given the determined prices and policies, allocations solve the
problem of the household, (ii) Pi,t solves the problem of ﬁrm i, (iii) Rt follows a speciﬁed
rule, and (iv) all markets clear.

t=1, prices {Wt, Pt, Pi,t}∞

Combining all of the results from (i)-(v), a symmetric equilibrium can be characterized

recursively by {Ct, Nt, Yt, wt, Πt, Rt}∞

t=1 satisfying the following equilibrium conditions:

Yt
Cχc

t (cid:20)

ϕ

(cid:18)

Πt
¯Π

− 1

(cid:19)

Πt
¯Π

− (1 − θ) − θ(1 − τ )wt

= βδtEt

Yt+1
Cχc
t+1

ϕ

Πt+1
¯Π

(cid:18)

− 1

(cid:19)

Πt+1
¯Π

t = βδtRtEtC−χc
C−χc

t+1 Π−1
t+1

wt = N χn

t Cχc

t

Yt = Ct +

(cid:20)
Yt = Nt

(cid:21)
Πt
¯Π

Πt
¯Π

ϕ
2

¯Π
β

2

− 1

Yt

(cid:21)

φπ

Yt
¯Y

φy

#

Rt = max

1,

"

(A.13)

(A.14)

(A.15)

(A.16)

(A.17)

(A.18)

(cid:19)

(cid:18)
Equation A.13 is the consumption Euler equation, Equation A.14 is the intratemporal op-
timality condition of the household, Equation A.15 is the optimal condition of the inter-
mediate good producing ﬁrms (forward-looking Phillips Curve) relating today’s inﬂation to
real marginal cost today and expected inﬂation tomorrow, Equation A.16 is the aggregate
resource constraint capturing the resource cost of price adjustment, and Equation A.17 is the
aggregate production function. Equation A.18 is the interest-rate feedback rule.

(cid:19)

(cid:18)

B Details of the Empirical Model

This section describes an extension of the stylized model with a representative household,
a ﬁnal good producer, a continuum of intermediate goods producers with unit measure, and
the government.

ECB Working Paper 1913, June 201632B.1 Household markets

B.1.1 Labor packer

The labor packer buys labor Nh,t from households at their monopolistic wage Wh,t and
resells the packaged labor Nt to intermediate goods producers at Wt. The problem can be
written as

1

subject to the following CES technology

max
Nh,t,h∈[0,1]

WtNt −

Wh,tNh,tdf

0

Z

1

θw −1
θw

h,t

N

dh

θw
θw −1

.

(cid:21)

Nt =

0

(cid:20)Z

The ﬁrst order condition implies a labor demand schedule

Nh,t =

−θw

Nt.23

Wh,t
Wt (cid:21)

(cid:20)

θw is the wage markup parameter.

B.1.2 Household

(B.1)

(B.2)

(B.3)

The representative household chooses its consumption level, amount of labor, and bond
holdings so as to maximize the expected discounted sum of utility in future periods. As is
common in the literature, the household enjoys consumption and dislikes labor. Assuming
that period utility is separable, the household problem can be deﬁned by

max
Ch,t,wh,t,Bh,t

E1

∞

t−1

βt−1

δs

# "

"

s=0
Y

t=1
X

subject to the budget constraint

(Ch,t − ζCa
1 − χc

t−1)1−χc

− A1−χc
t

N 1+χn
h,t
1 + χn #

(B.4)

PtCh,t+R−1

t Bh,t ≤ Wh,tNh,t−Wt

or equivalently

ϕw
2 "

Wh,t

aWh,t−1

¯Πw

1−ιw

ιw

Πw

t−1

(cid:0)

(cid:1)

(cid:0)

(cid:1)

2

− 1

#

Nt+Bh,t−1+PtΦt−PtTt

(B.5)

Ch,t+

Bh,t
RtPt

≤ wh,tNh,t−wt

ϕw
2 "

wh,t
awh,t−1

¯Πw

Πp
t
1−ιw

ιw

Πw

t−1

− 1

#

2

Nt+

Bh,t−1
Pt

+Φt−Tt (B.6)

(cid:0)
and subject to the labor demand schedule

(cid:1)

(cid:0)

(cid:1)

Nh,t =

−θw

Nt.

Wh,t
Wt (cid:21)

(cid:20)

(B.7)

23This implies that the labor packer will set the wage of the packaged labor to Wt = hR

1
0

W 1−θw
h,t

dh

i

1
1−θw

.

ECB Working Paper 1913, June 201633–or equivalently

−θw

(cid:20)

Nt.

(B.8)

Nh,t =

wh,t
wt (cid:21)
where Ch,t is the household’s consumption, Nh,t is the labor supplied by the household, Pt is
the price of the consumption good, Wh,t (wh,t) is the nominal (real) wage set by the house-
hold, Wt (wt) is the market nominal (real) wage, Φt is the proﬁt share (dividends) of the
household from the intermediate goods producers, Bh,t is a one-period risk free bond that
pays one unit of money at period t+1, Tt are lump-sum taxes or transfers, and R−1
is the
price of the bond. Ca
t−1 represents the aggregate consumption level from the previous period
that the household takes as given. The parameter 0 ≤ ζ < 1 measures how important these
external habits are to the household. Because we are including wage indexation, measured by
the parameter ιw, we assume the household takes as given the previous period wage inﬂation,
Πw

aWt−1 = wtPt

awt−1Pt−1 = wt

t−1, where Πw

awt−1 Πp
t .

t = Wt

t

The discount rate at time t is given by βδt where δt is the discount factor shock altering
the weight of future utility at time t+1 relative to the period utility at time t. δt is assumed
to follow an AR(1) process

(δt − 1) = ρδ(δt−1 − 1) + ǫδ
t

∀t ≥ 2

(B.9)

and δ1 is given. The innovation ǫδ
t is normally distributed with mean zero and standard devi-
ation σδ. It may therefore be interpreted that an increase in δt is a preference imposed by the
household to increase the relative valuation of the future utility ﬂows, resulting in decreased
consumption today (when considered in the absence of changes in the nominal interest rate).

At is a non-stationary total factor productivity shock that also augments labor in the utility
function in order to accommodate the necessary stationarization of the model later on. See
the next section for more details on this process.

B.2 Producers

B.2.1 Final good producer

The ﬁnal good producer purchases the intermediate goods Yf,t at the intermediate price
Pf,t and aggregates them using CES technology to produce and sell the ﬁnal good Yt to the
household and government at price Pt. Its problem is then summarized as

max
Yf,t,f ∈[0,1]

1

PtYt −

Pf,tYf,tdi

0

Z

subject to the CES production function

1

θp−1
θp

f,t

Y

di

θp
θp−1

.

(cid:21)

Yt =

0

(cid:20)Z

θp is the price markup parameter.

(B.10)

(B.11)

ECB Working Paper 1913, June 201634B.2.2 Intermediate goods producers

There is a continuum of intermediate goods producers indexed by f ∈ [0, 1]. Intermediate
goods producers use labor to produce the imperfectly substitutable intermediate goods ac-
cording to a linear production function (Yf,t = AtNf,t) and then sell the product to the ﬁnal
good producer. Each ﬁrm maximizes its expected discounted sum of future proﬁts24 by set-
ting the price of its own good. Any price changes are subject to quadratic adjustment costs.
ϕp will represent an obstruction of price adjustment, the ﬁrm indexes for prices—measured
by ιp—and takes as given previous period inﬂation Πp
t−1, and ¯Πp represents the monetary
authority’s inﬂation target.

max
Pf,t

E1

∞

t−1

βt−1

λt

δs

#

"

"

s=0
Y

t=1
X

such that

Pf,tYf,t − WtNf,t − Pt

ϕp
2

[

1−ιp

¯Πp

Pf,t
Πp

t−1

ιp Pf,t−1

(cid:0)

(cid:1)

(cid:0)

(cid:1)

Yf,t =

−θp

Yt.25

Pf,t
Pt (cid:21)

(cid:20)

− 1]2Yt

#
(B.12)

(B.13)

t−1
λt is the Lagrange multiplier on the household’s budget constraint at time t and βt−1
s=0 δs
is the marginal value of an additional proﬁt to the household. The positive time zero price
i
is the same across ﬁrms (i.e. Pi,0 = P0 > 0).

hQ

λt

At represents total factor productivity which follows a random walk with drift:

ln(At) = ln(a) + ln(At−1) + at.

(B.14)

a is the unconditional rate of growth of productivity. at is a productivity shock following an
AR(1) process:

ln(at) = ρaln(at−1) + ǫA
t .

(B.15)

where ǫA
is normally distributed with mean zero and standard deviation σA. This growth
t
factor will imply that some of the variables will acquire a unit root, meaning the model will
have to be stationarized. Monetary policy will also have to accommodate this growth factor
as well.

B.3 Government policies

It is assumed that the monetary authority determines nominal interest rates according to

a truncated notional inertial Taylor rule augmented by a speed limit component.

Rt = max [1, R∗
t ]

(B.16)

24NOTE: Each period, as it is written below, is in nominal terms. However, we want each period’s proﬁts
in real terms so the proﬁts in each period must be divided by that period’s price level Pt which we take care
of further along in the document.

25This expression is derived from the proﬁt maximizing input demand schedule when solving for the ﬁnal
good producer’s problem above. Plugging this expression back into the CES production function implies that
the ﬁnal good producer will set the price of the ﬁnal good Pt = hR

P 1−θp
f,t

1
1−θp

1
0

di

i

.

ECB Working Paper 1913, June 201635where

R∗
t
¯R

=

R∗
t−1
¯R

Πp
t
¯Πp

ρR

(1−ρr)φπ

Yt
At ¯Y

(1−ρr)φy

exp(ǫR
t )

(B.17)

(cid:19)
Pt−1 is the inﬂation rate between periods t − 1 and t, ¯R =

(cid:18)

(cid:18)

(cid:19)

(cid:18)

(cid:19)

¯Πpaχc
β

(see the section
t represents white noise monetary policy shocks with

t = Pt

where Πp
on stationarization to see why), and ǫR
mean zero and standard deviation σR.

B.4 Market clearing conditions

The market clearing conditions for the ﬁnal good, labor and government bond are given by

Yt = Ct +

0

Z
1

1

ϕp
2 "

1−ιp

¯Πp

Pf,t
Πp

t−1

(cid:0)
ϕw
2 "

(cid:1)
wh,t
awh,t−1

(cid:0)
¯Πw

2

− 1

#

Ytdf + ...

2

ιw

− 1

#

Ntdh

ιp Pf,t−1
Πp
(cid:1)
t
1−ιw

Πw

t−1

... +

wt

0

Z

and

B.5 An equilibrium

Nt =

1

(cid:0)

(cid:1)
Nf,tdi

(cid:0)

(cid:1)

0

Z

Ca

t = Ct =

1

0

Z

Ch,tdh

1

Bt =

Bh,tdh = 0.

0

Z

(B.18)

(B.19)

(B.20)

(B.21)

Given P0 and stochastic processes for δt, an equilibrium consists of allocations {Ct, Nt,

Nf,t, Yt, Yf,t, Gt}∞

t=1, prices {Wt, Pt, Pf,t}∞

t=1, and a policy instrument {Rt}∞

t=1 such that

(i) allocations solve the problem of the household given prices and policies

∂Ch,t : (Ch,t − ζCa

t−1)−χc − λt = 0

(B.22)

∂wh,t : θwA1−χc

t

wh,t
wt (cid:19)

N 1+χn
t

wt (cid:18)
+(1 − θw)λt

−θw(1+χn)−1

wh,t
wt (cid:19)

(cid:18)
Πp
t
1−ιw

−θw

Nt

ιw

Πw

t−1

−λtwtϕw

wh,t
awh,t−1

¯Πw

Πp
t
1−ιw

ιw

Πw

t−1

Nt

− 1

!

awh,t−1

¯Πw

+βδtEtλt+1wt+1ϕw

wh,t+1
awh,t

(cid:0)
¯Πw

Πp
(cid:0)
(cid:1)
t+1
1−ιw (Πw
t )ιw

(cid:1)
− 1

!

Nt+1

wh,t+1
aw2
h,t

(cid:0)
¯Πw

Πp
(cid:0)
(cid:1)
t+1
1−ιw (Πw
t )ιw

(cid:1)
= 0

(cid:0)

(cid:1)

∂Bh,t : −

λt
RtPt

+ βδtEt

λt+1
Pt+1

= 0

(cid:0)

(cid:1)

(B.23)

(B.24)

ECB Working Paper 1913, June 201636 
 
(ii) Pf,t solves the problem of ﬁrm i

By making the appropriate substitution (the intermediate goods producer’s constraints in
place of Yf,t and subsequently in for Nf,t) and by dividing each period’s proﬁts by that pe-
riod’s price level Pt so as to put proﬁts in real terms (and thus make proﬁts across periods
comparable) we get the following:

∂Pf,t : λt

Yt
Pt

−θp wt

At (cid:18)

Pt
Pf,t (cid:19)

¯Πp
h
1+θp
(cid:0)

1−ιp

Pt
Πp

t−1

(cid:0)
= βδtEt

(cid:1)

i

ιp Pf,t−1
λt+1Yt+1
(cid:1)
Pt+1

ϕp

1−ιp

¯Πp

Pf,t
Πp

t−1

ιp Pf,t−1

− 1

!

− (1 − θp)

(cid:0)

Pt+1ϕp

(cid:1)

Pf,t+1
(cid:0)
(cid:1)
1−ιp (Πp

t )ιp Pf,t

¯Πp

− 1

!

¯Πp

"

(cid:0)

(cid:1)

(cid:0)

(cid:1)

−θp

Pf,t
Pt (cid:19)

t )ιp P 2
(B.25)

f,t #

(cid:18)
Pf,t+1
1−ιp (Πp

(iii) Pf,t = Pj,t ∀i 6= j

Yt
λ−1
t "

ϕp

Πp
t

1−ιp

¯Πp

ιp

Πp

t−1

− 1

!

Πp
t

1−ιp

¯Πp

(cid:0)

(cid:1)

(cid:0)
(cid:1)
... = βδtEt

(cid:1)

Yt+1
(cid:0)
ϕp
λ−1
t+1

¯Πp

ιp

Πp
t−1
Πp
(cid:0)
(cid:1)
t+1
1−ιp (Πp

− (1 − θp) − θp wt
At #
Πp
t+1
1−ιp (Πp

− 1

¯Πp

!

t )ιp

t )ιp

= ...

(B.26)

(iv) Rt follows a speciﬁed rule

(cid:0)

(cid:1)

(cid:0)

(cid:1)

and

(v) all markets clear.

Combining all of the results derived from the conditions and exercises in (i)-(v), a sym-
t=1 satisfying

metric equilibrium can be characterized recursively by {Ct, Nt, Yt, wt, Πp
the following equilibrium conditions:

t , Rt}∞

λt = βδtRtEtλt+1

−1

Πp

t+1

(cid:1)
λt = (Ct − ζCt−1)−χc

(cid:0)

(B.27)

(B.28)

Nt
λ−1
t "

ϕw

Πw
t

1−ιw

¯Πw

ιw

Πw

t−1

− 1

!

Πw
t

1−ιw

¯Πw

ιw

Πw

t−1

t N χn
− (1 − θw) − θw A1−χc
λtwt

t

= ...

#

(cid:0)

(cid:1)

(cid:0)

(cid:1)

... = βδtEt

(cid:1)
ϕw

Nt+1
(cid:0)
λ−1
t+1

(cid:0)
¯Πw

Πw
(cid:1)
t+1
1−ιw (Πw

t )ιw

− 1

!

¯Πw

Πw
t+1
1−ιw (Πw

t )ιw

Πw

t =

(cid:1)
(cid:0)
wt
Πp
t
awt−1
Πp
t

(cid:0)

(cid:1)

Yt
λ−1
t "

ϕp

Πp
t

1−ιp

¯Πp

ιp

Πp

t−1

− 1

!

1−ιp

¯Πp

(cid:0)

(cid:1)

(cid:1)
(cid:0)
... = βδtEt

(cid:1)

Yt+1
(cid:0)
ϕp
λ−1
t+1

¯Πp

ιp

Πp
t−1
Πp
(cid:1)
(cid:0)
t+1
1−ιp (Πp

− (1 − θp) − θp wt
At #
Πp
t+1
1−ιp (Πp

− 1

¯Πp

!

t )ιp

t )ιp

= ...

(cid:0)

(cid:1)

(cid:0)

(cid:1)

wt+1
wt
(B.29)
(B.30)

(B.31)

ECB Working Paper 1913, June 201637 
 
 
 
 
 
 
 
Yt = Ct +

ϕp
2 "

Πp
t

1−ιp

¯Πp

ιp

Πp

t−1

(cid:0)

(cid:1)

(cid:0)

(cid:1)

2

Yt +

− 1

#

ϕw
2 "

Πw
t

1−ιw

¯Πw

ιw

Πw

t−1

2

− 1

#

wtNt

(B.32)

Yt = AtNt
(cid:0)
Rt = max [1, R∗
t ]

(cid:1)

(cid:0)

(cid:1)

where

R∗
t
¯R

=

R∗
t−1
¯R

Πp
t
¯Πp

ρR

(1−ρr)φπ

Yt
At ¯Y

(cid:18)

(1−ρr)φy

exp(ǫR
t )

(cid:19)

(cid:19)
and given the following processes (∀t ≥ 2):

(cid:18)

(cid:19)

(cid:18)

and

(δt − 1) = ρδ(δt−1 − 1) + ǫδ
t

ln(At) = ln(a) + ln(At−1) + at.
ln(at) = ρaln(at−1) + ǫA
t .

B.6 A stationary equilibrium

, ˜Ct = Ct
At

Let ˜Yt = Yt
At

, and ˜λt = λt
A−χc
t
consumption, real wage, and marginal utility of consumption respectively. The stationary
symmetric equilibrium can now be characterized by the following system of equations.

be the stationary representations of output,

, ˜wt = wt
At

˜λt =

β
aχc

δtRtEt˜λt+1

Πp

t+1

−1 exp(−χcǫA

t+1)

(cid:1)
˜λt = ( ˜Ct − ˜ζ ˜Ct−1exp(−ǫA
t ))−χc,

(cid:0)

˜ζ =

ζ
a

(B.39)

(B.40)

Nt ˜wt
˜λ−1
t

ϕw

"

Πw
t

1−ιw

¯Πw

ιw

Πw

t−1

− 1

... =

βϕw
Nt+1 ˜wt+1
(cid:1)
(cid:0)
(cid:0)
aχc−1 δtEt
λ−1
t+1  

(cid:1)

¯Πw

1−ιw

¯Πw

!
Πw
(cid:1)
(cid:0)
t+1
1−ιw (Πw
t )ιw

Πw
t

− (1 − θw) − θw N χn
t
˜λt ˜wt #

= ...

ιw

Πw

t−1

(cid:0)
− 1

!

(cid:1)
¯Πw

Πw
t+1
1−ιw (Πw

t )ιw

exp

(1 − χc) ǫA

t+1

(cid:0)

(cid:0)

(cid:1)
Πw
t =

˜wt
˜wt−1

Πp

t exp

(cid:1)

(cid:0)
ǫA
t

˜Yt
˜λ−1
t "

ϕp

Πp
t

1−ιp

¯Πp

ιp

Πp

t−1

(cid:0)

Πp
t

1−ιp

¯Πp

ιp

Πp

t−1

− 1

!

(cid:1)
− (1 − θp) − θp ˜wt

= ...

#

... =

βϕp
(cid:0)
(cid:1)
aχc−1 δtEt

˜Yt+1
(cid:0)
˜λ−1
t+1  

(cid:1)
¯Πp

Πp
(cid:0)
(cid:1)
t+1
1−ιp (Πp
t )ιp

− 1

(cid:0)

!

(cid:1)

Πp
t+1
1−ιp (Πp

t )ιp

¯Πp

exp

(1 − χc) ǫA

t+1

(cid:0)

(cid:1)

2

˜Yt = ˜Ct +

ϕp
2 "

(cid:0)
Πp
t

(cid:1)

1−ιp

¯Πp

ιp

Πp

t−1

(cid:0)

(cid:1)

(cid:0)

(cid:1)

2

˜Yt +

(cid:0)
ϕw
2 "

− 1

#

(cid:1)

Πw
t

1−ιw

¯Πw

ιw

Πw

t−1

˜wtNt

(B.44)

− 1

#

˜Yt = Nt

(cid:0)

(cid:1)

(cid:0)

(cid:1)

(B.45)

(B.33)

(B.34)

(B.35)

(B.36)

(B.37)

(B.38)

(cid:1)
(B.41)

(B.42)

(B.43)

ECB Working Paper 1913, June 201638 
 
and

where

Rt = max [1, R∗
t ]

R∗
t
¯R

=

R∗
t−1
¯R

Πp
t
¯Πp

ρR

(1−ρr)φπ

(cid:19)
and given the following processes (∀t ≥ 2):

(cid:19)

(cid:18)

(cid:18)

(1−ρr)φy

˜Yt
¯Y !

exp

ǫR
t

(cid:0)

(cid:1)

and

(δt − 1) = ρδ(δt−1 − 1) + ǫδ
t

(B.46)

(B.47)

(B.48)

B.7 Stationary deterministic steady-state values

For each variable, Xt, we denote its corresponding stationary deterministic steady-state
value as ¯X. The following is a list of analytical expressions for the stationary steady states
for each of the variables of the model.

¯Πp = ¯Πp,

(this parameter is set exogenously by the monetary authority)

¯Πw = ¯Πp
aχc ¯Πp
β

¯R =

¯w =

θp − 1
θp

1
χc+χn

¯C =




¯λ =

χc 

θw

¯w (θw − 1)
1 − ˜ζ
(cid:17)
¯C

(cid:16)
1 − ˜ζ
(cid:17)

h(cid:16)
i
¯N = ¯Y = ¯C


−χc

C Solution Method

We describe our solution method using the stylized model analyzed in the main text. The

extension of the method to the empirical model is straightforward.

The problem is to ﬁnd a set of policy functions, {C(·), N (·), Y (·), w(·), Π(·), R(·)}, that

ECB Working Paper 1913, June 201639 
solves the following system of functional equations.

C(δt)−χc = βδtR(δt)EtC(δt+1)−χcΠ(δt+1)−1
w(δt) = N (δt)χnC(δt)χc
N (δt)
C(δt)χc

− 1

ϕ

− (1 − θ) − θw(δt)

Π(δt)
¯Π
N (δt+1)
C(δt+1)χc

(cid:16)

Π(δt)
¯Π
Π(δt+1)
¯Π

(cid:17)
ϕ

h

... = βδtEt

Π(δt+1)
¯Π

− 1

(cid:17)
Y (δt)

Y (δt) = C(δt) +

Y (δt) = N (δt)

R(δt) = max

1,

"

ϕ
2

¯Π
β

h

(cid:16)
Π(δt)
¯Π

Π(δt)
¯Π

(cid:20)

2

− 1

i

φπ

(cid:21)

(cid:20)

Y (δt)
¯Y

φy

#

(cid:21)

...

i

(C.1)

(C.2)

(C.3)

(C.4)

(C.5)

(C.6)

Substituting out w(·) and N (·) using equations (C.2) and (C.5), this system can be reduced
to a system of four functional equations for C(·), Y (·), Π(·), and R(·).

C(δt)−χc = βδtR(δt)EtC(δt+1)−χcΠ(δt+1)−1
Y (δt)
C(δt)χc

− 1

ϕ

Π(δt)
¯Π
Y (δt+1)
C(δt+1)χc

(cid:16)

Π(δt)
¯Π
Π(δt+1)
¯Π

(cid:17)
ϕ

h

... = βδtEt

− (1 − θ) − θY (δt)χnC(δt)χc

Y (δt) = C(δt) +

R(δt) = max

1,

"

ϕ
2
¯Π
β

(cid:16)
Π(δt)
¯Π
Π(δt)
¯Π

h

(cid:20)

2

− 1

φπ

i

(cid:21)

(cid:20)

Π(δt+1)
¯Π

− 1

(cid:17)
Y (δt)

Y (δt)
¯Y

φy

#

(cid:21)

...

i

(C.7)

(C.8)

(C.9)

(C.10)

Following the idea of Christiano and Fisher (2000) and Gust, L´opez-Salido, and Smith
(2012), I decompose these policy functions into two parts using an indicator function: One
in which the policy rate is allowed to be less than zero, and the other in which the policy
rate is assumed to be zero. That is, for any variable Z,

Z(·) = {R(·)≥1}Zunc(·) + (1 − {R(·)≥1})ZELB(·).

(C.11)

The problem then becomes ﬁnding a set of a pair of policy functions, {[Cunc(·), CELB(·)],
[Yunc(·), YELB(·)], [Πunc(·), ΠELB(·)], [Runc(·), RELB(·)]} that solves the system of functional
equations above. This method can achieve a given level of accuracy with a considerable less
number of grid points relative to the standard approach.26

The time-iteration method starts by specifying a guess for the values policy functions
take on a ﬁnite number of grid points. The values of the policy function that are not on
any of the grid points are interpolate or extrapolated linearly. Let X(·) be a vector of policy
functions that solves the functional equations above and let X (0) be the initial guess of such
policy functions.27 At the s-th iteration and at each point of the state space, we solve the

26A systematic analysis of the beneﬁts of using the Christiano-Fisher approach is available upon request.
27For all models and all variables, we use ﬂat functions at the deterministic steady-state values as the initial

guess.

ECB Working Paper 1913, June 201640system of nonlinear equations given by equations (C.7)-(C.10) to ﬁnd today’s consumption,
output, inﬂation, and the policy rate, given that X (s−1)(·) is in place for the next period.
In solving the system of nonlinear equations, we use Gaussian quadrature to evaluate the
expectation terms in the consumption Euler equation and the Phillips curve, and the value
of future variables not on the grid points are evaluated with linear interpolation. The system
is solved numerically by using a nonlinear equation solver, dneqnf, provided by the IMSL
Fortran Numerical Library.
If the updated policy functions are suﬃciently close to the
guessed policy functions, then the algorithm ends. Otherwise, using the updated policy
functions just obtained as the guess for the next period’s policy functions, we iterate on this
process until the diﬀerence between the guessed and updated policy functions is suﬃciently
∞ < 1e-11 is used as the convergence criteria). The solution
small (
method can be extended to models with multiple exogenous shocks and endogenous state
variables in a straightforward way.

vec(X s(δ) − X s−1(δ))

(cid:13)
(cid:13)

(cid:13)
(cid:13)

For the stylized model, we used equally spaced 201 grid points on the interval between
[1 − 4.5σδ, 1 + 4.5σδ] and 9 grid points for the Gaussian quadrature. For the empirical model,
we used 27 grid points for the discount rate shock on the interval between [1−4.5σδ, 1+4.5σδ],
15 grid points for the lagged consumption on the interval between -12 and 8 percent from
the steady state (normalized) consumption, 15 grid points for the lagged real wage on the
interval between -2.5 and 2 percent from the steady state (normalized) real wage, and 15
grid points on the lagged shadow policy rate on the interval between -8 and 10 annualized
percent. For the empirical model, we use 31 grid points for the Gaussian quadrature.

D Solution Accuracy

In this section, we report the accuracy of our numerical solutions for the stylized and em-
pirical models. Following Fern´andez-Villaverde, Gordon, Guerr´on-Quintana, and Rubio-Ram´ırez
(2015) and Maliar and Maliar (2015), we evaluate these residuals functions along a simulated
equilibrium path. The length of the simulation is 100,000.

D.1 Stylized model

For the stylized model, there are two residual functions, one associated with the consump-

tion Euler equation and the other associated with the sticky-price equilibrium condition.

R2,t =

(cid:12)
(cid:12)
(cid:12)
(cid:12)

R1,t =

1 − Cχc

t βδtRtEtC−χc

t+1 Π−1
t+1

Πt
¯Π

(cid:18)

− 1

(cid:19)

Πt
¯Π

−

h

(cid:12)
(cid:12)
(cid:12)
(1 − θ) + θ(1 − τ )wt
ϕ

−

Cχc
t
Yt

βδtEt

(cid:12)
(cid:12)
(cid:12)
Yt+1
Cχc

t+1 (cid:18)

Πt+1
¯Π

Πt+1
¯Π

− 1

(cid:19)

(D.1)

(D.2)

With our log-utility speciﬁcation, the residual R1,t measures the diﬀerence between the
chosen consumption today and today’s consumption consistent with the optimization behav-
ior of the household, as a percentage of the chosen consumption. The residual R2,t is given
by the diﬀerence between (Πt − ¯Π)Πt/ ¯Π and the sum of the term involving today’s real wages
and the term involving the expectations. Given that the standard deviation of Πt is about
0.0075 (∼= 31 basis points), Πt/ ¯Π is always close to one and thus the (Πt − ¯Π)Πt/ ¯Π is roughly
equal to (Πt − ¯Π). (Πt − ¯Π) is the deviation of inﬂation from the target rate of inﬂation.
Thus, the diﬀerence between this term and the sum of the term involving today’s real wages

i(cid:12)
(cid:12)
(cid:12)
(cid:12)

ECB Working Paper 1913, June 201641and the term involving the expectations measures how much the chosen inﬂation rate diﬀers
from the inﬂation rate consistent with the optimization behavior of ﬁrms.

Table D.1: Solution Accuracy

Mean[log10(Rk,t)]

95th-percentile of [log10(Rk,t)]

Stylized Model
k = 1: Euler equation error
k = 2: Sticky-price equation error

Empirical Model
k = 1: Euler equation error
k = 2: Sticky-price equation error
k = 3: Sticky-wage equation error

−6.5
−7.5

−4.3
−4.7
−4.5

−6.0
−6.9

−3.7
−4.2
−3.9

Reﬂecting the large number of grid points used to solve the stylized model, the approxima-
tion errors are very small. The average errors on the consumption Euler equation is 0.3 ∗ 10−4
percent (= 10−6.5) with the 95th percentile being 1.0 ∗ 10−4 percent (= 100 ∗ 10−6.0). The
average errors on the sticky price equation is 0.001 basis points (= 400 ∗ 100 ∗ 10−7.5) with
the 95th percentile being 0.005 basis points (= 400 ∗ 100 ∗ 10−6.9).

D.2 Empirical Model

For the empirical model, there are three residual functions of interest. As in the stylized
model, the ﬁrst and second residual functions are associated with the consumption Euler
equation and the sticky-price equilibrium condition, respectively. The third residual function
is associated with the sticky-price and sticky-wage equilibrium conditions.

R2,t =

(cid:18)

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
R3,t =

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

(D.3)

Πw
t+1
¯Πw
(cid:12)
(cid:12)
i
(D.4)
(cid:12)
(cid:12)
(cid:12)

(D.5)

(cid:12)
(cid:12)
i
(cid:12)
(cid:12)
(cid:12)

R1,t =

1 −

β
aχc ˜λt

δtRtEt˜λt+1

−1

Πp

t+1

(cid:0)

(cid:1)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

+

Πw
t
¯Πw − 1
(cid:19)

Πw
t
¯Πw −

1 − θw
ϕw

h

θw
ϕw

N χn
t
˜λt ˜wt

+

˜λ−1
t
Nt ˜wt

β

aχc−1 δtEt

Nt+1 ˜wt+1
λ−1
t+1 (cid:18)

t+1

Πw
¯Πw − 1
(cid:19)

Πp
t
¯Πp − 1
(cid:19)

(cid:18)

Πp
t
¯Πp −

h

(1 − θp) + θp ˜wt
ϕp

+

˜λ−1
t
˜Yt

β

aχc−1 δtEt

˜Yt+1
˜λ−1
t+1 (cid:18)

t+1

Πp
¯Πp − 1
(cid:19)

Πp
t+1
¯Πp

Table D.1 shows the average and the 95th percentile of the residuals for the three equi-
librium conditions. The average size of the Euler equation errors is 0.005 percent (= 10−4.3)
with the 95th percentile of the errors being 0.02 percent (= 10−3.7). These are larger than
those in the stylized model, reﬂecting the coarser grid used in the solution of the empirical
model. However, this degree of accuracy fares well in comparison with what’s reported in
other studies such as Fern´andez-Villaverde, Gordon, Guerr´on-Quintana, and Rubio-Ram´ırez
(2015) and Maliar and Maliar (2015).

The average size of the sticky-price equation errors is 0.8 basis points (= 400∗100∗10−4.7)
with the 95th percentile being 2.7 basis points (= 400 ∗ 100 ∗ 10−4.2). These are again
larger than those in the stylized model, reﬂecting the coarser grid used in the solution of the
empirical model. While the size of the error is not ideal, it is acceptable as the approximation
errors of this magnitude do not materially aﬀect the main claim regarding the quantitative

ECB Working Paper 1913, June 201642importance of the deﬂationary bias. The instances of large approximation errors are only
observed in the area of the state space where the discount rate shock is high and the policy
rate is constrained by the eﬀective lower bound.

The sticky-wage equation errors are somewhat larger than the sticky-price equation errors.
The average size is 1.3 basis points (= 400 ∗ 100 ∗ 10−4.5) with the 95th percentile being 4.9
basis points (= 400 ∗ 100 ∗ 10−3.9). However, given that the standard deviation of the wage
inﬂation is more than two times as large as that of the price inﬂation (66 versus 31 basis
points), the sticky-wage equation errors are smaller in relative terms than the sticky-price
equation errors. As in the sticky-price equation error, the instances of larger approximation
errors are only observed in the area of the state space where the discount rate shock is high
and the policy rate is constrained by the eﬀective lower bound.

E Expected Time Until the Liftoﬀ

In this section, we present the survey-based measures of the expected time until the liftoﬀ
to support the claim that the market participants consistently underestimated the duration
of the lower bound episode since the federal funds rate hit the lower bound in late 2008. The
surveys we examine are (i) the Blue Chip Surveys, (ii) the Survey of Professional Forecasters,
and (iii) the Primary Dealers Survey.

The evidence from all three surveys is consistent with the claim that the market partici-
pants have consistently underestimated the duration of the lower bound episode. In particu-
lar, for the ﬁrst two years of the lower bound episode, the market participants expected that
the federal funds rate to stay at the ELB only for additional few quarters.28

E.1 Blue Chip Surveys

The Blue Chip Surveys consists of two monthly surveys, the Blue Chip Economic Indi-
cators Survey and the Blue Chip Financial Forecasts Survey. These two surveys ask their
participants (about 50 ﬁnancial institutions for each survey) their forecast paths of various
macroeconomic variables, including the 3-month Treasury Bill rate in the Economic Indi-
cators Survey and the federal funds rate in the Financial Forecasts Survey. The near-term
forecast horizon is up until the end of next calendar year and the frequency of the projection
is quarterly. Thus, the forecast path of the Treasury rate or the federal funds rate can tell
us the expected time until the liftoﬀ when the participants expect the ﬁrst liftoﬀ to occur
within two years.

Twice a year, the surveys ask longer-run projections of certain variables in the special
question section (March and October for the Economic Indicators and June and December
for the Financial Forecasts). The longer-run forecasts are in annual frequency for next 5 to 6
years. Towards the end of the lower bound episode, the Surveys also asked the participants
to provide the expected liftoﬀ date in the special questions section.

For each survey, we combine these various pieces of information in the following way
to construct a series for the expected period until the liftoﬀ. First, we use the average
probability distribution over the timing of the liftoﬀ to compute the expected time until the
liftoﬀ whenever that information is available. Second, if the probability distribution is not
available, we use the information from the near-term forecasts. The time of liftoﬀ is deﬁned

28While not shown, the expected duration of the lower bound episode based on the expected policy path

implied by the federal funds rate futures is also consistent with this claim.

ECB Working Paper 1913, June 201643to be the ﬁrst quarter when the median federal funds rate forecast exceeds 37.5 basis points.
Finally, when the policy rate is projected to stay at the ELB until the end of the near-term
forecast horizon, we use the information from the long-run projections if the Survey has that
information and leave the series blank when the Long-Range section is not available.

Figure E.1: Expected Time Until Liftoﬀ†

Blue Chip Economic Indicators

Blue Chip Financial Forecasts

12

10

8

6

4

2

)
s
r
e
t
r
a
u
q

n
i
(

n
o
i
t
a
r
u
D

12

10

8

6

4

2

)
s
r
e
t
r
a
u
q

n
i
(

n
o
i
t
a
r
u
D

0
2009 2010 2011 2012 2013 2014 2015 2016
Year

0
2009 2010 2011 2012 2013 2014 2015 2016
Year

Primary Dealers Survey

Survey of Professional Forecasters

12

10

8

6

4

2

)
s
r
e

t
r
a
u
q

n
i
(
n
o

i
t

a
r
u
D

12

10

8

6

4

2

)
s
r
e

t
r
a
u
q

n
i
(
n
o

i
t

a
r
u
D

0
2009 2010 2011 2012 2013 2014 2015 2016
Year

0
2009 2010 2011 2012 2013 2014 2015 2016
Year

† Data sourced from: Blue Chip Economic Indicators Survey (from January 2009 to August 2015); Blue Chip Financial
Forecasts Survey (from January 2009 to August 2015); Federal Reserve Bank of New York, Primary Dealer Survey,
accessed September 2015, https://www.newyorkfed.org/markets/primarydealer˙survey˙questions.html; Federal Reserve
Board, Survey of Professional Forecasters, accessed September 2015, https://www.philadelphiafed.org/research-and-
data/real-time-center/survey-of-professional-forecasters/

The top two panels in ﬁgure E.1 show the evolutions of the expected period until liftoﬀ
based on the Blue Chip Economic Indicators Survey and the Blue Chip Financial Forecasts
Survey. According to both panels, the market participants expected the lower bound episode
to be transitory in the early stage of the lower bound episode. The expectation shifted in
the second half of 2011, with the expectated duration of staying at the ELB exceeding 2
years. Starting in late 2012 or early 2013, the survey respondents gradually reduced their
expectation for the additional duration of the lower bound episode.

E.2 Primary Dealers Survey

The Primary Dealers Survey (the PD Survey in the remainder of the text), conducted by
the Federal Reserve Bank of New York, asks primary dealers about their policy expectations
eight times a year. The survey asks its participants their probability distribution over the
liftoﬀ timing (quarter or FOMC meeting). We compute the expected time until the liftoﬀ
using the average probability distribution over the liftoﬀ timing. The results of the PD Survey

ECB Working Paper 1913, June 201644 
 
 
 
 
 
 
 
are publicly available since January 2011.

The bottom-left panel of ﬁgure E.1 shows the evolution of the expected period until the
liftoﬀ based on the PD Survey. Consistent with the measures based on the Blue Chip survey,
the expected duration of the additional period of the lower bound episode increase markedly
in the second half of 2011. The expected duration hovers around 10 quarters during 2012,
and has declined steadily since then.

E.3 Survey of Professional Forecasters

The Survey of Professional Forecasters (SPF) is a quarterly survey of about 40 individ-
uals in academia, ﬁnancial industries, and policy institutions, administered by the Federal
Reserve Bank of Philadelphia. Like the Blue Chip Surveys, the SPF asks its participants
their projections of various macroeconomic variables, including the 3-month Treasury rate.
For the near-term projection that extends to the end of the next calendar year, the forecasts
are available in quarterly frequency. For the longer horizon, the forecast is available in annual
frequency.

The bottom-right panel of ﬁgure E.1 shows the evolution of the expected period until
liftoﬀ based on the SPF. Consistent with the Blue Chip Surveys and the Primary Dealers
Survey, the SPF shows that the market anticipated the lower bound episode to last for only
about one additional year until the second half of 2011. The expected duration averages
about 9 quarters in 2012 and 2013. The expected duration started declining in the second
half of 2013 and has come down to 2 quarters in February 2015.

F Low inﬂation prior to the ELB episodes

Figure F.1 shows the evolution of inﬂation and policy rate past two decades in six select
economies (U.S., Canada, Euro Area, U.K., Sweden, and Switzerland). We exclude Japan
because the policy rate in Japan has been at the ELB for most of the past two decades and
thus the Japanese economy is better described by the deﬂationary equilibrium that ﬂuctuates
around a deﬂationary steady state in the context of sticky-price models.29 In all of the six
economies, the policy rate became constrained at the ELB during the recent global recession
for the ﬁrst time in the post WWII era and the central bank has an explicit inﬂation target
of 2 percent.

Figure F.2 shows the average inﬂation in these six economies over the period when the
policy rate was above the ELB. Since the sample period is short, the ﬁgure shows how the
conditional average depends on the starting date of the sample as well as the conﬁdence band
for plus/minus two standard errors.

According to the ﬁgure, inﬂation averaged below the inﬂation target while the policy rate
was away from ELB the all six countries. In the U.S. and Canada, inﬂation averaged about 20
and 30 basis point below the 2 percent target while the ELB was not binding. The conditional
averages in these two countries are closer to 2 percent and the conﬁdence band includes 2
percent when the starting date of the sample is around 2000. Thus, we cannot completely
rule out the possibility that the observed undershooting of the inﬂation target is statistically
insigniﬁcant. However, the ﬁgures for the U.S. and Canada demonstrate the overall tendency
for the central banks to undershoot the long-run target by non-trivial amount in these two
countries. In the Euro Area, the United Kingdom, Sweden, and Switzerland, the conditional

29Aruoba, Cuba-Borda, and Schorfheide (2014) provide an empirical support for this view.

ECB Working Paper 1913, June 201645Figure F.1: Policy Rates and Inﬂation in Six Economies†

United States

Canada

4

3

2

1

0
1995
1995

4

3

2

1

0
1995
1995

6

4

2

0

Core Inflation Rate
Policy Rate

2000
2000

2005
2005

2010
2010

2015
2015

Euro area (19 countries)

2000
2000

2005
2005

2010
2010

2015
2015

Sweden

8

6

4

2

0

8

6

4

2

0

12

8

4

0

4

3

2

1

0
1995
1995

4

3

2

1

0
1995
1995

4

2

0

2000
2000

2005
2005

2010
2010

2015
2015

United Kingdom

2000
2000

2005
2005

2010
2010

2015
2015

Switzerland

8

6

4

2

0

8

6

4

2

0

4

2

0

−2
1995
1995

2000
2000

2005
2005
Year

2010
2010

−4

2015
2015

−2
1995
1995

2000
2000

2010
2010

−2

2015
2015

2005
2005
Year

†Shaded regions mark the ELB era. Red horizontal lines represent inﬂation target rate.

average inﬂation away from the ELB are substantially below the target rate of 2 percent,
regardless of the starting date of the sample.

G More Sensitivity Analyses

G.1 Price Indexation

In this subsection, we examine the implications of price indexation for the risky steady
state. Figures G.1 and G.2 show how the key features of the model vary with the degree of
price indexation. Note that, as we vary the degree of price indexation, we adjust the standard
deviation of the discount rate shock so that the standard deviation of the policy rate in the
model remains unchanged at 2.34.

As shown in Figure G.1, the volatility of inﬂation increases, and the volatility of the
output gap decreases, with the degree of price indexation. As the volatility of the policy rate
stays constant, the expected duration of the ELB episode increases as the degree of price
indexation increases. According to Figure G.2, inﬂation, output, and the policy rate at the
risky steady state increases, decreases, and increases with the degree of price indexation,
respectively. However, the magnitudes are small. Thus, price-setters do not need to be fully
forward-looking for the ELB risk to reduce the risky steady state inﬂation by non-trivial
amount.

ECB Working Paper 1913, June 201646 
 
Figure F.2: Average Inﬂation Rates Away From the Lower Bound
with Alternative Starting Dates†

United States

Canada

2.1

2

1.9

1.8

1.7

1.6

1.5

1995

2.5

2

1.5

1
1995

2.5

2

1.5

1

0.5

1996

1997

1998

1999

2000

2001

Euro area (19 countries)

1996

1997

1998

1999

2000

2001

Sweden

2.1

2

1.9

1.8

1.7

1.6

1.5

1995

2.5

2

1.5

1
1995

2.5

2

1.5

1

0.5

1996

1997

1998

1999

2000

2001

United Kingdom

1996

1997

1998

1999

2000

2001

Switzerland

0
1995

1996

1997

1998
Start Date

1999

2000

2001

0
1995

1996

1997

1998
Start Date

1999

2000

2001

†Horizontal lines mark the inﬂation target rate. Dashed lines represent the 95% conﬁdence interval (2 standard errors)
of the average inﬂation, which is shown by the solid black lines.

Figure G.1: Moments with
Alternative Degrees of Price Indexation

Inflation

Output Gap

−3.1

−3.3

−3.5

−3.7

0.8
0.8

0.6
0.6
0.4
0.4
0.2
0.2
Price Indexation

E[ELB Duration]

Unconditional Stdev
E[•|ELB]

0.6
0.6
0.4
0.4
0.2
0.2
Price Indexation

Policy Rate

1.3

3.1

1.2

1.1

1

3

2.9

0.9

0.8
0.8

2.8
0
0

0.14

9.6

0.13

s
r
e

t
r
a
u
Q

9.4

9.2

9

0.6
0.6
0.4
0.4
0.2
0.2
Price Indexation

0.12

0.8
0.8

8.8
0

0.2
0.6
0.4
Price Indexation

0.8

0.5

0.45

0.4

0.35

0.3
0
0

2.4

2.3

2.2
0
0

ECB Working Paper 1913, June 201647 
 
Figure G.2: Risky Steady State with
Alternative Degrees of Price Indexation

RSS Inflation

RSS Output Gap

0.34

0.33

0.32

0.31

0.3

0.8

0.29
0

0.2
0.6
0.4
Price Indexation

0.8

15.5

15

−3

x 10
7

d

6.7

σ

0.2
0.6
0.4
Price Indexation

RSS Policy Rate

0.2
0.6
0.4
Price Indexation

0.8

14.5
0
0

ELB Frequency
Stdev[Demand Shocks]

0.6
0.6
0.4
0.4
0.2
0.2
Price Indexation

6.4

0.8
0.8

1.74

1.73

1.72

1.71

1.7
0

3.25

3.24

3.23

3.22

3.21

3.2

3.19

3.18
0

G.2 Wage Indexation

In this subsection, we examine the implications of wage indexation for the risky steady
state. Figures G.3 and G.4 show how the key features of the model vary with the degree of
wage indexation. Note that, as we vary the degree of wage indexation, we adjust the standard
deviation of the discount rate shock so that the standard deviation of the policy rate in the
model remains unchanged at 2.34.

As shown in Figure G.3, the volatilities of inﬂation, the output gap, and the policy rate as
well as the expected duration of the ELB episode increase with the degree of wage indexation.
According to Figure G.2, inﬂation, output, and the policy rate at the risky steady state
decreases, increases, and decreases with the degree of wage indexation, respectively. The
eﬀects of wage indexation on the risky steady state is quantitatively large. For example,
inﬂation is about 1.03 percentage point and the policy rate is about 2.17 percent at the risky
steady state when the degree of wage indexation is 0.4. Thus, even when wage-setters are
not fully forward-looking, the wedge between the risky and deterministic steady states can
be large.

ECB Working Paper 1913, June 201648 
 
Figure G.3: Moments with
Alternative Degrees of Wage Indexation

Inflation

Output Gap

Unconditional Stdev
E[•|ELB]

1.2

4

1

3.75

0.8

3.5

0.6

3.25

0.1
0.1

0.2
0.2
Wage Indexation

0.3
0.3

0.4

0.4
0.4

3

0
0

0.1
0.1

0.2
0.2
Wage Indexation

0.3
0.3

Policy Rate

E[ELB Duration]

−2

−2.5

−3

−3.5

−4

0.4
0.4

1

0.8

0.6

0.4

0.2

0
0.4
0.4

12

11

10

s
r
e

t
r
a
u
Q

9

8

0

0.1
0.1

0.2
0.2
Wage Indexation

0.3
0.3

0.1

0.2
Wage Indexation

0.3

0.4

Figure G.4: Risky Steady State with
Alternative Degrees of Wage Indexation

RSS Inflation

RSS Output Gap

0.1

0.2
Wage Indexation

0.3

0.4

RSS Policy Rate

1.5

1

0.5

0

0

30

25

20

15

0.1

0.2
Wage Indexation

0.3

0.4

ELB Frequency
Stdev[Discount Factor Shocks]

−3

x 10
8

7.5

7

d

σ

6.5

6
0.4
0.4

0.5

0.45

0.4

0.35

0.3

0
0

3

2.8

2.6

2.4

2.2

2

0
0

2

1.8

1.6

1.4

1.2

1

0

3.5

3

2.5

2

0

0.1

0.2
Wage Indexation

0.3

0.4

10

0
0

0.1
0.1

0.2
0.2
Wage Indexation

0.3
0.3

G.3 A Model with Monetary Policy Shocks

In this subsection, we introduce monetary policy shocks into the baseline empirical model.
The standard deviation of the monetary policy shock is set to the standard deviation of the
residuals in the interest-rate feedback rule computed using the U.S. data before the federal
funds rate hit the ELB (σr = 0.14
100 ). The standard deviation of the discount rate shock is
adjusted so that the standard deviation of the policy rate in the model remains unchanged
at 2.34.

ECB Working Paper 1913, June 201649 
 
 
 
As shown in Table G.1, The moments and risky steady states of this model are similar
to those of the baseline empirical model. This reﬂects the fact that the magnitude of the
monetary policy shock is very small, accounting for only less than 5 percent of the total
variation in the policy rate.

Table G.1: A Model with Monetary Policy Shocks

(a) Moments

Moment

Variable

St.Dev.(·)

E(X|ELB)

Output gap
Inﬂation
Policy rate
Output gap
Inﬂation
Policy rate

ELB

·

Frequency
Expected/Actual Duration
100σǫ,δ

Model
(original)
3.0
0.31
2.34
−3.7
1.21
0.13
13.8%
8.6 quarters
0.69

Model
(w/ MP shocks)

2.9
0.29
2.34
−3.3
1.28
0.13

13.5%
6.3 quarters
0.67

Data
(1995Q3–2015Q2)
2.9
0.52
2.34
−5.2
1.48
0.13
36%
26 quarters

·

(b) Risky Steady State†

Inﬂation Output Gap Policy Rate

DSS
RSS (w/ σr = 0.14
(Wedge)
RSS (original)
(Wedge)

100 )

2
1.75
(−0.25)
1.74
(−0.26)

0
0.27
(0.27)
0.30
(0.30)

3.75
3.28
(−0.47)
3.26
(−0.49)

†DSS stands for ‘Deterministic Steady State’ while RSS stands for ‘Risky Steady State.’

G.4 A Model with TFP Shocks

In this subsection, we introduce TFP shocks into the baseline empirical model in such
a way that the standard deviations of output, inﬂation, and the policy rate are roughly
unchanged from the baseline empirical model. To do so, we adjust a few parameters (ϕp = 700
and φπ = 3.5) and choose the size of the standard deviation of the TFP shock so that the
volatility of output explained by the TFP shock accounts for 10 percent, as opposed to 25
percent, of the total output volatility. This model is used in the non-technical summary of
this paper.

As shown in Table G.2, the moments and risky steady states of this model is broadly

similar to those of the baseline empirical model.

ECB Working Paper 1913, June 201650Table G.2: A Model with TFP Shocks

Moment

Variable

St.Dev.(·)

E(X|ELB)

Output gap
Inﬂation
Policy rate
Output gap
Inﬂation
Policy rate

ELB

·

Frequency
Expected/Actual Duration
100σǫ,δ

(a) Moments

Model
(original)
3.0
0.31
2.34
−3.7
1.21
0.13
13.8%
8.6 quarters
0.69

Model
(w/ TFP shocks)

3.1
0.42
2.34
−3.4
1.18
0.13

15.9%
9.0 quarters
0.66

Data
(1995Q3–2015Q2)
2.9
0.52
2.34
−4.2
1.48
0.13
36%
26 quarters

·

(b) Risky Steady State†

Inﬂation Output Gap Policy Rate

DSS
RSS (w/ TFP shocks)
(Wedge)
RSS (original)
(Wedge)

2
1.71
(−0.29)
1.74
(−0.26)

0
0.32
(0.32)
0.30
(0.30)

3.75
3.04
(−0.71)
3.26
(−0.49)

†DSS stands for ‘Deterministic Steady State’ while RSS stands for ‘Risky Steady State.’

H Distribution of Inﬂation in a Perfect-Foresight Version of

the Stylized Model

Figure H.1 shows the unconditional distribution of inﬂation in the perfect-foresight version
of the model. The distribution is positively skewed, reﬂecting the fact that the nonlinearity
associated with the ELB is dominated by the other nonlinear features of the model. As
a result, the unconditional average of inﬂation is slightly above 2 percent. The standard
deviation of the discount rate shock is the same as the one for the stochastic model. With
this standard deviation, the probability of being at the ELB is x percent in the perfect-
foresight model.

Figure H.2 shows how the unconditional average of inﬂation and the conditional averages
of inﬂation at and away from the ELB vary with the probability of being at the ELB in the
model with the ELB constraint solved under the assumption of perfect-foresight.

When the probability of being at the ZLB is suﬃciently small, the unconditional average
of inﬂation is slightly positive, reﬂecting the positive skewness implied by other nonlinear
features of the model besides the ELB. However, when the probability of being at the ELB
is suﬃciently high, the distribution becomes negatively skewed and the average inﬂation
becomes below 2 percent. The conditional average of inﬂation away from the ELB monoton-
ically increases with the probability of being at the ELB. In the perfect-foresight version of
the model with the ELB, the median of inﬂation is always 2 percent. As the ELB frequency
increases, there are more masses on the areas of the distribution where the ELB binds and
the conditional average of inﬂation in those areas decrease with the probability of being at
the ELB. The conditional average of inﬂation away from the ELB is by deﬁnition the average
inﬂation in the region that excludes those areas with low inﬂation. Thus, as the probability
of the ELB frequency increases, the conditional average of inﬂation at the ELB increases.

ECB Working Paper 1913, June 201651Figure H.1: Unconditional Distribution of Inﬂation
in a Perfect-Foresight Version of the Stylized Model

Inflation Target: 2
Average Inflation: 2.03

Skewness: 0.1283

ELB Binds

1.5

2

2.5

−6

−4

−2

0

2

4

6

8

Figure H.2: Conditional and Unconditional Averages of Inﬂation
in a Perfect-Foresight Version of the Stylized Model

)

%
d
e
z

i
l

a
u
n
n
A

(

n
o
i
t
a
l
f
n
I

2.5

2.4

2.3

2.2

2.1

2

1.9
0

E[Π|R
 = 1]
t

E[Π]
E[Π|R
 > 1]
t

−0.6

−0.8

−1

−1.2

−1.4

−1.6

−1.8

)

%
d
e
z

i
l

a
u
n
n
A

(

n
o
i
t
a
l
f
n
I

2

6
4
 = 1] (in %)
Prob[R
t

8

10

12

−2
0

2

6
4
 = 1] (in %)
Prob[R
t

8

10

12

As discussed in the main text, the increase is non-monotonic in the stochastic model with
the ELB; The conditional average of inﬂation away from the ELB initially increases, but later
declines, with the ELB frequency. Unlike in the perfect-foresight version of the model, an
increase in the probability of being at the ELB shifts down the distribution of inﬂation, as the
forward-looking price-setters take into account the ELB risk in their pricing decisions. This
force dominates the other force to increase the conditional average of inﬂation away from the
ELB, and lowers the conditional average of inﬂation away from the ELB. Provided that the
ELB frequency is suﬃciently high, the conditional average of inﬂation away from the ELB is
below 2 percent in the stochastic model where price-setters take the ELB risk into account,

ECB Working Paper 1913, June 201652 
 
 
 
 
 
 
 
 
 
while it is above 2 percent in the perfect-foresight version of the model that abstracts from
the ELB risk. As a result, the fact that the conditional average fell below 2 percent when
the policy rate is away from the ELB can be seen as the manifestation of the ELB risk, an
idea that is pursued in Section 5.

ECB Working Paper 1913, June 201653Acknowledgements 
We would like to thank Lena Boneva, Oliver de Groot, Michael Kiley, Jean-Philippe LaForte, John Roberts and seminar participants at 
the University of Tokyo for useful comments. Paul Yoo provided excellent research assistance. The views expressed in this paper, and 
all errors and omissions, should be regarded as those solely of the authors, and are not necessarily those of the Federal Reserve Board 
of Governors, the Federal Reserve System, or the European Central Bank. 

Timothy Hills 
Stern School of Business, New York University, New York, USA; email: hills.timoteo@gmail.com 

Taisuke Nakata 
Division of Research and Statistics, Federal Reserve Board, Washington, D.C., USA; email: taisuke.nakata@frb.gov 

Sebastian Schmidt 
European Central Bank, Monetary Policy Research Division, Frankfurt am Main, Germany; email: sebastian.schmidt@ecb.int 

© European Central Bank, 2016 

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This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library at 
or from RePEc: Research Papers in Economics.  

Information on all of the papers published in the ECB Working Paper Series can be found on the ECB’s website. 

ISSN 
ISBN 
DOI 
EU catalogue No 

1725-2806 (online) 
978-92-899-2161-9 
10.2866/615932 
QB-AR-16-030-EN-N