olivierloisel.comolivierloisel.com/research/PUZ_paper.pdfPegging the Interest Rate on Bank Reserves:...
89
✩ - ✩
Transcript of olivierloisel.comolivierloisel.com/research/PUZ_paper.pdfPegging the Interest Rate on Bank Reserves:...
Pegging the Interest Rate on Bank Reserves:
A Resolution of New Keynesian Puzzles and ParadoxesI
Behzad Dibaa, Olivier Loiselb
aDepartment of Economics, Georgetown University, 3700 O Street, N.W., Washington, D.C. 20057, U.S.A.bCREST (ENSAE), 5 avenue Henry Le Chatelier, 91120 Palaiseau, France
Abstract
Introducing bank reserves (or money) into the basic New Keynesian (NK) model o�ers a
resolution of NK puzzles and paradoxes. The resulting models deliver local-equilibrium de-
terminacy under an exogenous interest rate on reserves and an exogenous nominal stock of
reserves, even for an arbitrarily small monetary friction. This leads to a resolution of the
forward-guidance puzzle, the �scal-multiplier puzzle, and the paradox of �exibility. As the
monetary friction becomes vanishingly small, the models converge to the basic NK model
and serve to select a particular equilibrium of that model − which also o�ers a resolution of
the paradox of toil.
Keywords: Forward-guidance puzzle, �scal-multiplier puzzle, paradox of �exibility, paradox
of toil, interest rate on reserves
JEL Codes: E52, E58
IWe thank, for their helpful comments, Pierpaolo Benigno, Florin Bilbiie, James Bullard, Edouard Challe,Larry Christiano, John Cochrane, Davide Debortoli, Jordi Galí, Pedro Gete, Je�rey Huther, Chen Lian, DavidLópez-Salido, Pascal Michaillat, Kalin Nikolov, Ricardo Reis, Andreas Schabert, Harald Uhlig, Mirko Wieder-holt, Mike Woodford; our discussants Fernando Duarte, Marcus Hagedorn, Luigi Iovino, and Kevin Sheedy; ananonymous Referee, the Associate Editor Johannes Wieland, and the Editor Yuriy Gorodnichenko. We grate-fully acknowledge the �nancial support of the grant Investissements d'Avenir (ANR-11-IDEX-0003/LabexEcodec/ANR-11-LABX-0047).
Email addresses: [email protected] (Behzad Diba), [email protected] (Olivier Loisel)URL: sites.google.com/georgetown.edu/behzad-dibas-website (Behzad Diba),
olivierloisel.com (Olivier Loisel)
Preprint submitted to Journal of Monetary Economics October 5, 2020
1. Introduction
The Great Recession led central banks to peg their policy rates at the zero lower bound
(ZLB) and provide forward guidance about future policy rates. The recession also rekindled
interest in the use of discretionary �scal policy as a stabilization tool, and sparked debate
(in Europe) about implementing structural reforms. Standard New Keynesian (NK) models5
typically have stark implications for policy at the ZLB: forward guidance is powerful, �scal
multipliers are large, and structural reforms that increase potential output or enhance price
�exibility actually worsen the recession.
The models that lead to these stark policy implications, however, also lead to limit puzzles
that (we think) seem implausible. Asymptotically, the power of monetary forward guidance10
or a �scal expansion grows exponentially with the horizon of anticipated policy interventions.
Even small policy interventions will have arbitrarily large e�ects on current outcomes if they
are expected to take place in the su�ciently distant future. We call these limit implications
the forward-guidance puzzle and the �scal-multiplier puzzle (after related puzzles in the lit-
erature). Our limit puzzles, of course, do not highlight any empirically refutable implications15
of standard NK models, but we think they cast some doubt on the ZLB-policy implications
of these models away from the limit − i.e., when the policy-intervention horizon is �nite.
Similarly, we associate a limit puzzle (of a di�erent sort) with the so-called �paradox of
�exibility� that arises in standard NK models. The standard rendition of this paradox in
the literature is that greater price �exibility aggravates the contraction (and the de�ation)20
in a liquidity trap. Our limit rendition of the paradox is that the contractionary e�ect of the
liquidity trap grows without bound as the degree of price stickiness goes to zero, instead of
converging to its �nite �exible-price value.
In this paper, we show that a simple departure from the basic NK model (Woodford,
2003, Galí, 2008) o�ers a resolution of these limit puzzles, and substantially alters the ZLB-25
policy implications of the model away from the limits. This departure adds a (possibly small)
monetary friction to the basic NK model using, in turn, an ad-hoc money-demand function,
or a familiar money-in-utility (MIU) setup, or the model with a more explicit role for bank
reserves presented in Diba and Loisel (2020). Adding money − either interpreted or modeled
2
as bank reserves − to the NK model enables us to capture more precisely what central banks30
actually did during the Great Recession: they pegged the interest rate on bank reserves (the
IOR rate), and switched to balance-sheet policies that essentially set the nominal stock of
bank reserves.
In our monetary models, setting exogenously the two monetary-policy instruments, i.e.
the IOR rate and the nominal stock of bank reserves, delivers local-equilibrium determinacy35
for all structural-parameter values − in particular for any degree of price stickiness. This
determinacy property, as we explain in the text, is what makes the models solve the three
limit puzzles. In our models, when the IOR rate is pegged at the ZLB (or, more generally,
set exogenously), monetary- and �scal-policy interventions in the vanishingly distant future
have vanishingly small e�ects, instead of unboundedly large e�ects, on current outcomes;40
and, as the degree of price stickiness goes to zero, the contractionary e�ect of a liquidity trap
converges to its �nite �exible-price value (zero), instead of growing explosively. The ad-hoc
model and the MIU model enable us to make these points in simple and familiar frameworks;
for some structural-parameter values, however, they give rise to a �reversal puzzle� (recurrent
sign reversals in the e�ect of forward guidance on current outcomes as we change the guidance45
horizon). Our more structured model with banks, by contrast, never gives rise to this reversal
puzzle.
Our monetary models solve the three limit puzzles even for arbitrarily small monetary
frictions. In the limit, as we make the monetary frictions vanishingly small, these models
converge to the basic NK model and serve to select uniquely a particular equilibrium out of an50
in�nity of equilibria of that model under a permanently exogenous policy rate. The selected
equilibrium does not depend on how we add monetary frictions to the basic NK model, and we
argue that this robustness feature of our selected equilibrium would extend to other models
that deliver determinacy under an exogenous policy rate and converge to the basic NK model
as we shrink some friction. We show that our selected equilibrium exhibits neither the �scal-55
multiplier puzzle, nor the paradox of �exibility; and that it exhibits a distinctively weaker
form of forward-guidance puzzle. We compare our selected equilibrium with the equilibria
considered in Cochrane (2017) and Bilbiie (2019a), which are other equilibria of the basic NK
model under a permanently exogenous policy rate. One notable di�erence is that our selected
3
equilibrium never exhibits Neo-Fisherian e�ects: current and future interest-rate hikes are60
always de�ationary in our equilibrium.
We also compare our selected equilibrium with the standard equilibrium of the basic
NK model. The latter equilibrium, on which most of the literature focuses, is obtained
by assuming that the economy is back to steady state at a �nite date. The two equilibria
di�er substantially at the limits, since the limit puzzles are (fully or partially) solved in our65
selected equilibrium and are not in the standard equilibrium. But the two equilibria also di�er
substantially away from the limits. More speci�cally, in the standard equilibrium, anticipated
�scal expansions are expansionary, anticipated positive supply shocks are contractionary (the
so-called �paradox of toil�), and greater price �exibility worsens the recession in a liquidity
trap (the most common de�nition of the �paradox of �exibility�). Our selected equilibrium70
has exactly the opposite implications. Unlike the standard equilibrium, thus, our selected
equilibrium suggests that �scal policy has some limitations to �ght the recession at the ZLB,
and that structural reforms increasing potential output or enhancing price �exibility are not
counter-productive at the ZLB.
Our resolution of the NK puzzles and paradoxes rests on the assumption that demand for75
bank reserves is not fully satiated.1 Indeed, if it were fully satiated, then our monetary models
would exactly coincide with the basic NK model, and all the NK puzzles and paradoxes would
re-emerge. Our non-satiation assumption stands in contrast to views often expressed about
the US economy in recent years (e.g., Cochrane, 2014, 2018; Reis, 2016). We defend this
assumption in Diba and Loisel (2020). In the present paper, moreover, we show that we still80
solve the NK puzzles and paradoxes when demand for bank reserves is arbitrarily close to
satiation, and even when it is asymptotically satiated. In the latter case, indeed, the unique
equilibrium of our monetary models converges to our selected equilibrium of the basic NK
model. The reason is that going to satiation asymptotically alleviates the monetary friction
1Our non-satiation assumption only pertains to the prevailing (�nite) level of real reserves. A number ofpapers following Ono (2001) analyze the consequences of a stronger non-satiation assumption that imposesa positive lower bound on the marginal convenience yield of money, as real money balances tend to in�nity.Ono (2001) shows that if this lower bound is large enough, the MIU model can generate a permanent liquiditytrap, Keynesian aggregate demand shortages, and long-lasting stagnation. Ono and Ishida (2014) show thatthis setup can generate large �scal multipliers and the paradoxes of �exibility and toil, under some additionalassumptions.
4
and has the same e�ect as asymptotically removing the monetary friction from the model (as85
in our equilibrium-selection exercise).
Some brief remarks may serve to put our contribution in the context of the recent literature
on NK puzzles and paradoxes. The phrases �forward-guidance puzzle,� �paradox of �exibility,�
and �paradox of toil� were coined respectively by Del Negro et al. (2015), Eggertsson and
Krugman (2012), and Eggertsson (2010), while Farhi and Werning (2016) were the �rst to90
expose (what we call) the �scal-multiplier puzzle. Other early contributions related to at
least one of the NK puzzles and paradoxes include Christiano et al. (2011), Eggertsson
(2011, 2012), Eggertsson et al. (2014), Werning (2012), and Woodford (2011). Wieland
(2019) presents empirical evidence against the paradox of toil. Carlstrom et al. (2015) and
Cochrane (2017) clearly show the link between the forward-guidance and �scal-multiplier95
puzzles and indeterminacy under a permanently exogenous policy rate.
A number of contributions have proposed departures from the basic NK model in order
to solve or attenuate at least one of the NK puzzles and paradoxes (�attenuate� in the
sense of reducing the quantitative e�ects of policy interventions at the ZLB for a given
intervention horizon and a given degree of price stickiness). These departures may involve100
non-rational expectations (e.g. Farhi and Werning, 2019, Gabaix, 2020, García-Schmidt
and Woodford, 2019); information frictions (e.g. Angeletos and Lian, 2018, Kiley, 2016,
Wiederholt, 2015); incomplete markets (e.g. Bilbiie, 2019b, Hagedorn et al., 2019, McKay
et al., 2016); overlapping generations (Del Negro et al., 2015); non-Ricardian �scal policy
(Cochrane, 2017, 2018); and government bonds with a convenience yield (e.g. Bredemeier et105
al., 2018, Hagedorn, 2018, Michaillat and Saez, 2019).
Against this literature background, one distinguishing feature of our work is our focus on
money demand, and the role of balance-sheet policies that central banks introduced when
they switched to pegging the IOR rate.2 Another distinguishing feature of our work is the new
equilibrium-selection approach, which links our monetary models to a particular equilibrium110
of the basic NK model.
2In a recent contribution, Piazzesi et al. (2019) also show that setting exogenously the IOR rate and thenominal stock of bank reserves delivers determinacy in the MIU model and in a banking model of theirs.They do not formally study the implications for the NK puzzles and paradoxes.
5
Several contributions to the literature propose models that can solve the forward-guidance
puzzle by �discounting� the IS equation or the Phillips curve of the basic NK model, i.e. by
scaling down the coe�cients of their expectational terms. In the online appendix, we gen-
eralize Cochrane's (2016) comments on Gabaix (2020) to highlight two di�erences between115
these models and ours: (i) discounting models do not solve (our limit-puzzle rendition of) the
paradox of �exibility; (ii) their resolution of the forward-guidance puzzle overturns the stan-
dard Fisher e�ect (i.e. replaces the one-to-one long-term relationship between the in�ation
rate and the nominal interest rate with a negative relationship).
The rest of the paper is organized as follows. Section 2 exposes the puzzles and paradoxes120
in the basic NK model. Section 3 presents our main points in a simple setting that adds a
money-demand nexus to the basic NK model. Section 4 con�rms and strengthens our points
in the MIU model and a model with banks. Section 5 shows the robustness of our results
under asymptotic satiation of demand for reserves. The online appendix contains the proofs,
the comparison with discounting models, and some numerical illustrations.125
2. Puzzles and Paradoxes in the Basic NK Model
We start with a brief exposition of the puzzles and paradoxes in the basic NK model. The
log-linearized IS equation and Phillips curve of this model are
yt = Et {yt+1} −1
σ(it − Et {πt+1} − rt) + gt − Et {gt+1} (1)
πt = βEt {πt+1}+ κ (yt − δggt − δϕϕt) , (2)130
where yt denotes the output level, πt the in�ation rate, it the nominal interest rate on bonds,
rt a preference shock (stemming from variations in the discount factor), gt a government-
purchases shock, and ϕt a supply shock (stemming from shifts in the labor-disutility function,
labor-tax modi�cations, technology changes, or variations in the elasticity of substitution
between intermediate goods). All variables and shocks are expressed in log-deviation from135
their steady-state value, except gt (which is proportional to the log-deviation of government
purchases from steady state). The parameters satisfy β ∈ (0, 1), σ > 0, δg ∈ (0, 1), δϕ > 0,
κ > 0; and κ increases without bound as the degree of price stickiness goes to zero. Finally,
6
Et{.} denotes the rational-expectations operator at date t.
The monetary-policy instrument, in this model, is the interest rate it. Under a perma-140
nently exogenous policy rate (it = i∗t exogenous for all t ∈ Z), the IS equation (1) and the
Phillips curve (2) lead to the following dynamic equation relating πt to Et{πt+2}, Et{πt+1},
and exogenous terms:
Et{Pb(L−1
)πt}
= Zbt , (3)
where L denotes the lag operator, Pb(X) ≡ X2 − [1 + 1/β + κ/(βσ)]X + 1/β, and145
Zbt ≡−κβσ
(i∗t − rt) +(1− δg)κ
β(gt − Et {gt+1})−
δϕκ
β(ϕt − Et {ϕt+1})
(the subscript �b� and superscript �b� stand for �basic�). Since Pb(0) = 1/β > 0, Pb(1) =
−κ/(βσ) < 0, and limX→+∞Pb(X) = +∞ > 0, the roots of Pb(X) are two real numbers ρb
and ωb such that 0 < ρb < 1 < ωb. We can thus rewrite the dynamic equation (3) as
Et{(L−1 − ωb
) (L−1 − ρb
)πt}
= Zbt . (4)150
Because ρb ∈ (0, 1), however, we cannot iterate this equation forward to +∞ and get πt as a
bounded function of current and expected future exogenous shocks. Nor can we iterate the
equation backward to pin down a unique solution for πt, given the absence of a πt−1 term
in this equation. The dynamic system has one stable eigenvalue (ρb) and no predetermined
variable (no πt−1 term), Blanchard and Kahn's (1980) conditions are not satis�ed, and local-155
equilibrium indeterminacy arises under a permanently exogenous policy rate.
Under a temporarily exogenous policy rate, however, local-equilibrium determinacy can
be obtained by assuming that policy will switch in the future to a rule that sets a nominal
anchor (e.g., a Taylor rule like it = φπt with φ > 1). The literature about the NK puzzles and
paradoxes typically assumes that the economy is expected to be back to its steady state after160
a �nite date T : Et{rT+k} = Et{gT+k} = Et{ϕT+k} = Et{πT+k} = Et{yT+k} = Et{iT+k} = 0
for all t ≤ T and k ≥ 1. Under this assumption, we can use the method of partial fractions
7
to iterate the dynamic equation (4) forward until date T ; we get3
πt = Et{
Zbt
(L−1 − ωb) (L−1 − ρb)
}=
Etωb − ρb
{ρ−1b Zb
t
1− (ρbL)−1− ω−1b Zb
t
1− (ωbL)−1
}=
Etωb − ρb
{T−t∑k=0
(ρ−k−1b − ω−k−1b
)Zbt+k
}.165
Using the de�nition of Zbt , we then get
πt =κEt
β (ωb − ρb)
{−1
σ
T−t∑k=0
(ρ−k−1b − ω−k−1b
) (i∗t+k − rt+k
)+
T−t∑k=0
[(1− ρb) ρ−k−1b + (ωb − 1)ω−k−1b
][(1− δg) gt+k − δϕϕt+k]
}, (5)
where again i∗t+k denotes the exogenous value of the policy rate at date t + k. Finally,
using (5) to replace πt and πt+1 in the Phillips curve (2), and using also βρbωb = 1 and170
Pb(ρb) = Pb(ωb) = 0, we get
yt =κEt
βσ (ωb − ρb)
{−1
σ
T−t∑k=0
(ρ−kb
1− ρb+
ω−kbωb − 1
)(i∗t+k − rt+k
)+
T−t∑k=1
(ρ−kb − ω
−kb
)[(1− δg) gt+k − δϕϕt+k]
}+ gt. (6)
Equations (5) and (6) form what Cochrane (2017) calls the �standard equilibrium� of the
basic NK model. Because the stable eigenvalue ρb has been inverted to obtain these equations,175
the e�ects of expected future shocks on current in�ation and output grow essentially at rate
ρ−kb in the horizon k of the shocks. The farther away these shocks, the bigger their e�ects on
current variables. At the limit, shocks that are in�nitely distant in the future have in�nite
e�ects on current variables:
∀v ∈ {π, y}, ∀s ∈ {r, i∗, g, ϕ}, limk→+∞
| ∂vt∂st+k
| = +∞.180
3Our presentation in this section is a discrete-time version of the presentations in Werning (2012), Farhiand Werning (2016), and Cochrane (2017).
8
Moreover, the sign of ∂vt/∂st+k does not change with the horizon k in (5)-(6), for either
variable v and any shock s. So, the cumulative current e�ects of shocks that are expected to
persist for multiple periods in the future also grow explosively with the expected duration of
these shocks.
Consider, for instance, a negative preference shock expected to last for the next k periods,185
and suppose monetary policy cannot fully o�set this shock because of the binding ZLB
constraint: rt+j = r∗ < 0 and i∗t+j = i∗ < 0 for j ∈ {0, ..., k}, with r∗ < i∗. The net
e�ects on πt and yt are then negative and grow explosively with the horizon k. The central
bank can, however, easily o�set and even overturn these de�ationary and contractionary
pressures by promising to keep the policy rate at the ZLB for a su�ciently long time after190
the shock (it+j = i∗ < 0 for j ∈ {k + 1, ..., T − t}), because the power of forward guidance
grows exponentially with the duration T − t of the promised low-rate period. This arbitrarily
large e�ect of keeping the policy rate low for a su�ciently long time in the future is the
direct consequence of the arbitrarily large e�ect of setting the policy rate low at a single
date su�ciently distant in the future. So, we adopt the following de�nition for the forward-195
guidance puzzle:
De�nition 1 (Forward-Guidance Puzzle): When the policy rate is expected to be set
exogenously during at least the next k periods, the response of current in�ation and output to
an expected policy-rate cut k periods ahead goes to in�nity with k (i.e., limk→+∞ ∂vt/∂i∗t+k =
−∞ for v ∈ {π, y}).200
The government can also rely on a �scal expansion to o�set the de�ationary and contrac-
tionary e�ects of shocks during a ZLB episode. In our preceding example with a preference
shock, increasing government expenditures during the preference shock (gt+j = g∗ > 0 for
j ∈ {0, ..., k}) gives rise to a �scal multiplier that grows exponentially with the duration of
the shock (limk→+∞ ∂vt/∂g∗ = +∞ for v ∈ {π, y}). Again, the source of this arbitrarily large205
�scal multiplier for a su�ciently long �scal expansion is, ultimately, the arbitrarily large
e�ect of a one-o� �scal expansion su�ciently distant in the future. The latter e�ect seems
puzzling: conventional wisdom is that delaying a �scal expansion reduces its current impact.4
4We will come back to this issue in more detail in Subsection 3.2.
9
So, we will adopt the following de�nition for the �scal-multiplier puzzle:
De�nition 2 (Fiscal-Multiplier Puzzle): When the policy rate is expected to be set210
exogenously during at least the next k periods, the response of current in�ation and output to
an expected �scal expansion k periods ahead goes to in�nity with k (i.e., limk→+∞ ∂vt/∂gt+k =
+∞ for v ∈ {π, y}).
The inversion of the stable eigenvalue ρb, leading to the terms ρ−kb in (5)-(6), also gives
rise to another kind of puzzling implication. As the degree of price stickiness θ (i.e. the215
probability that a �rm cannot change its price at a given date) goes to zero, we have ρb →
0. This is because, as θ → 0, we have κ → +∞, and therefore ωb = [1 + β + κ/σ +√(1 + β + κ/σ)2 − 4β]/(2β) → +∞, and hence �nally ρb = Pb(0)/ωb = 1/(βωb) → 0. As
a consequence, the terms ρ−kb in (5) and (6) make current in�ation and output explode in
response to expected shocks k periods ahead, for any given horizon k ∈ {1, ..., T − t}, as220
prices are made more and more �exible:
∀v ∈ {π, y}, ∀s ∈ {r, i∗, g, ϕ}, ∀k ∈ {1, ..., T − t}, limθ→0| ∂vt∂st+k
| = +∞.
Moreover, the sign of ∂vt/∂st+k does not change with the horizon k in (5)-(6), for either
variable v and any shock s. So, the cumulative current e�ects of shocks that are expected to
persist for multiple periods in the future also grow explosively as the degree of price stickiness225
goes to zero.
Consider again our thought experiment about a temporary ZLB episode (rt+j = r∗ < 0
and it+j = i∗ < 0 for j ∈ {0, ..., k}, with r∗ < i∗). The contractionary e�ect on current
output, in (6), and the de�ationary pressure, in (5), will be stronger if our hypothetical
economy has more �exible prices, as we show in (online) Appendix A.3. Thus, greater price230
�exibility magni�es the e�ects of the ZLB episode. We do not view this implication of the
model as necessarily puzzling; nor do we think it is easily refutable. What seems puzzling to
us, however, is that the contractionary and de�ationary e�ects of the ZLB episode explode
as the degree of price stickiness θ goes to zero. This explosion implies, in particular, a stark
discontinuity at the limit of perfect price �exibility: current output drops by an arbitrarily235
large amount as θ gets arbitrarily close to zero, but it does not move at all when θ is exactly
10
zero.5 Since this arbitrarily large e�ect of the ZLB episode for a su�ciently low degree of
price stickiness is the direct consequence of the arbitrarily large e�ect of expected future
one-o� shocks, we will adopt the following de�nition for the paradox of �exibility:
De�nition 3 (Paradox of Flexibility): When the policy rate is expected to be set240
exogenously during at least the next k ≥ 1 periods, the response of current in�ation and
output to expected shocks k periods ahead goes to plus or minus in�nity as the degree of price
stickiness θ goes to zero (i.e., limθ→0 |∂vt/∂st+k| = +∞ for v ∈ {π, y} and s ∈ {r, i∗, g, ϕ}).
The forward-guidance and �scal-multiplier puzzles in De�nitions 1-2, and the paradox of
�exibility in De�nition 3, are �limit puzzles� in the sense that they arise only at the limit245
when the horizon of the (monetary- or �scal-)policy intervention goes to in�nity, or at the
limit when the degree of price stickiness goes to zero. In our view, however, the fact that
the standard equilibrium of the basic NK model exhibits these puzzles at the limits casts
some doubt on the policy implications of this equilibrium away from the limits − i.e. on
its implications for monetary policy, �scal policy, and structural reforms enhancing price250
�exibility, when the policy-intervention horizon is �nite and the degree of price stickiness is
not close to zero.
Finally, in addition to these three limit puzzles and for a related reason, the standard
equilibrium also exhibits the paradox of toil. This paradox says that, contrary to conventional
wisdom, current and expected future positive supply shocks − such as downward shifts in the255
labor-disutility function, labor-tax cuts, technology improvements, and reductions in market
power − are not expansionary:
De�nition 4 (Paradox of Toil): When the policy rate is expected to be set exogenously
during at least the next k ≥ 0 periods, an expected positive supply shock k periods ahead has
a zero or contractionary e�ect on current output (i.e., ∂yt/∂ϕt+k ≤ 0).260
In the standard equilibrium, indeed, current positive supply shocks are neutral (∂yt/∂ϕt =
0) and expected future ones are contractionary (∂yt/∂ϕt+k < 0 for any k ∈ {1, ..., T − t},
given the presence of ρ−kb > ω−kb in the second line of (6)).
5Current in�ation also drops by an arbitrarily large amount as θ gets arbitrarily close to zero. But wecannot talk of discontinuity (nor of continuity) because in�ation is indeterminate when θ is exactly zero.
11
3. Resolution of the Puzzles and Paradoxes in a Simple Model
We now add a money-demand nexus to the basic NK model, and show how setting265
exogenously the interest rate on money and the stock of money solves the NK puzzles and
paradoxes. The model is simple, but not micro-founded; we will consider two micro-founded
models in the next section.
3.1. Resolution of the Three Limit Puzzles
We consider a standard log-linear money-demand equation, of the kind estimated in the270
empirical literature, linking demand for real money balances mt positively to output yt and
negatively to the opportunity cost of holding money it − imt :
mt = χyyt − χi (it − imt ) , (7)
where imt denotes the interest rate on money, χy > 0, and χi > 0. Real money balances mt
are also linked to nominal money balancesMt and the price level pt through the identitymt =275
Mt − pt (where all variables are expressed in log-deviation from some constant value). The
monetary-policy instruments are, now, imt and Mt. When these instruments are permanently
exogenous (in particular imt = i∗t exogenous for all t ∈ Z), the IS equation (1), the Phillips
curve (2), the money-demand equation (7), and the identities mt = Mt−pt and πt = pt−pt−1lead to the following dynamic equation relating pt to Et{pt+2}, Et{pt+1}, pt−1, and exogenous280
terms:
Et{LP
(L−1
)pt}
= Zt, (8)
where P (X) ≡ X3 −(
2 +1
β+
κ
βσ+
χyσχi
)X2
+
[1 +
2
β+
(1 +
1
χi
)κ
βσ+
(1 +
1
β
)χyσχi
]X −
(1
β+
χyβσχi
),285
Zt ≡−κβσ
(i∗t − rt) +κ
βσχiMt +
[1−
(1 +
χyσχi
)δg
]κ
βgt
−(1− δg)κβ
Et {gt+1} −(
1 +χyσχi
)δϕκ
βϕt +
δϕκ
βEt {ϕt+1} .
12
We show in Appendix A.1 that the characteristic polynomial P(X) has one root inside the
unit circle (ρ ∈ (0, 1)) and two roots outside the unit circle (ω1 and ω2 with |ω1| ≤ |ω2|);
and that the latter roots are either positive real numbers or complex conjugates. We assume290
that they are distinct real numbers (so that ω2 > ω1 > 1), and postpone the discussion of
this assumption to the next section. Using the roots of P(X), we can rewrite the dynamic
equation (8) as
Et{(L−1 − ω1
) (L−1 − ω2
)(1− ρL) pt
}= Zt,
and we can use the method of partial fractions to solve this equation forward and get the295
unique bounded solution for pt − ρpt−1:6
pt − ρpt−1 = Et{
Zt(L−1 − ω1) (L−1 − ω2)
}=
Etω2 − ω1
{ω−11 Zt
1− (ω1L)−1− ω−12 Zt
1− (ω2L)−1
}=
Etω2 − ω1
{+∞∑k=0
(ω−k−11 − ω−k−12
)Zt+k
}. (9)
At any date t, pt−1 is known, and (9) pins down pt uniquely. If time starts at −∞, we can
iterate (9) backward and get pt as a unique bounded function of the exogenous forcing vari-300
ables Et−j{Zt−j+k} for all (j, k) ∈ N2. Thus, the model delivers local-equilibrium determinacy
under permanently exogenous monetary-policy instruments. The dynamic system still has
one stable eigenvalue (ρ), but this eigenvalue is now matched by a predetermined variable
(pt−1), so that Blanchard and Kahn's (1980) conditions are satis�ed.
This determinacy result can be interpreted as follows. Under exogenous monetary-policy305
instruments imt and Mt, the money-demand equation (7) makes the interest rate on bonds
it a strictly increasing function of output and, crucially, the price level: it = (χy/χi)yt +
(1/χi)pt + [imt − (1/χi)Mt]. If the price level (or output) rises, demand for nominal money
balances increases; and, given the exogenous policy instruments, the interest rate on bonds
must increase to clear the money market. Thus, our model with exogenous monetary-policy310
instruments is isomorphic to the basic NK model with a �Wicksellian rule� for the interest
rate it (which is the policy rate in the latter model); and Wicksellian rules are well known
6Con�ning our analysis to bounded solutions amounts to following the common practice to set aside theglobal indeterminacy inherent in models with �at money.
13
to ensure determinacy in the basic NK model (as shown in Woodford, 2003, Chapter 4). At
the ZLB, the central bank has to peg the IOR rate imt ; but it is as if it could set the interest
rate on bonds it according to this (shadow) Wicksellian rule ensuring determinacy.315
Using the price-level solution (9), the identity πt = pt − pt−1, and the Phillips curve (2),
we get
πt = − (1− ρ) pt−1 +Et
ω2 − ω1
{+∞∑k=0
(ω−k−11 − ω−k−12
)Zt+k
}, (10)
yt = −ϑpt−1 + δggt + δϕϕt −Et
(ω2 − ω1)κ
{+∞∑k=0
(ξ1ω
−k−11 − ξ2ω−k−12
)Zt+k
}, (11)
where ϑ ≡ (1 − ρ)(1 − βρ)/κ and ξj ≡ β(ωj + ρ − 1) − 1 for j ∈ {1, 2}. One key di�erence320
between our simple model's equilibrium (10)-(11) and the basic NK model's standard equi-
librium (5)-(6) is that the former involves only ω−k1 and ω−k2 terms with |ω1| > 1 and |ω2| > 1,
while the latter involves ρ−kb terms with ρb ∈ (0, 1), which are responsible for the limit puzzles
discussed in the previous section. As a consequence, the implications of our simple model for
the response of in�ation and output to anticipated future shocks are in sharp contrast to the325
corresponding implications of the basic NK model: the later shocks are expected to occur,
the bigger their current e�ects in the basic NK model, but the smaller their current e�ects
in our model, regardless of which type of shock (preference, monetary, �scal, or supply) we
consider. More speci�cally, in our model, shocks occurring at date t + k and announced at
date t do not a�ect pt−1; their e�ects on in�ation and output at date t decay at an exponential330
rate, converging to zero essentially like |ω1|−k. So, neither the forward-guidance puzzle nor
the �scal-multiplier puzzle can arise in our model. The central bank in our model can provide
forward guidance not only about low future policy rates (i∗t+k), but also about large future
balance sheets (Mt+k), in order to o�set the de�ationary pressures exerted by the negative
preference shock rt; but the e�ectiveness of both types of forward guidance decreases in the335
guidance horizon k. The following proposition formalizes these results:
Proposition 1 (Resolution of the Forward-Guidance and Fiscal-Multiplier Puz-
zles in the Simple Model): In the simple model, under exogenous monetary-policy in-
struments, the response of current in�ation and output to an expected policy-rate cut or
14
an expected �scal expansion k periods ahead converges to zero as k goes to in�nity (i.e.,340
limk→+∞ ∂vt/∂i∗t+k = limk→+∞ ∂vt/∂gt+k = 0 for v ∈ {π, y}).
Moreover, because we have 0 < ρ < 1 < |ω1| ≤ |ω2| for any degree of price stickiness
θ ∈ (0, 1) and in particular as θ → 0, the paradox of �exibility does not arise in our simple
model. In Appendix A.2, we show that the limits of πt and yt as θ → 0 take �nite values,
unlike their counterparts in the basic NK model, and that these values coincide with the values345
that πt and yt take under perfectly �exible prices (in particular, limθ→0 yt = δggt + δϕϕt). So,
the model involves no discontinuity at the θ = 0 point, in contrast to the basic NK model,
and we have the following proposition:
Proposition 2 (Resolution of the Paradox of Flexibility in the Simple Model):
In the simple model, under exogenous monetary-policy instruments, the responses of current350
in�ation and output to expected shocks k periods ahead converge to their �nite �exible-price
values as the degree of price stickiness goes to zero.
To illustrate graphically our resolution of the forward-guidance puzzle and the paradox
of �exibility, Figure 1 shows the e�ects of cutting the policy rate by 25 basis points (one
percentage point per annum) in Quarter t + k on the annualized in�ation rate in Quarter t355
(when the rate cut is announced). We start from a benchmark calibration borrowed from
Galí (2008, Chapter 3), which sets the degree of price stickiness θ to 2/3 (corresponding to
�3-quarter price rigidity�), and we then cut θ in half step by step to make prices more �exible.
The right panel in Figure 1 replicates the implausible implications of the basic NK model:
cutting the policy rate in a later quarter leads to an exponentially larger e�ect on current360
in�ation; and making prices more �exible accelerates these explosive e�ects. The left panel
shows the results for our simple model: with our benchmark value of θ = 2/3, the in�ationary
e�ects of the policy-rate cut are modest (about 10 basis points for a cut in one of the �rst
�ve quarters) and die o� relatively quickly with the horizon of the cut; as we make prices
more �exible, these in�ationary e�ects smoothly converge to the e�ects under perfect price365
�exibility (θ = 0). Figure 1 is, of course, only illustrative, as our simple model is not really
suitable for a quantitative assessment of the e�ects of forward guidance or other policies.7
7We provide a sensitivity analysis in Appendix G.2 − with a particular focus on the sensitivity to the
15
Our contribution here is to show that this model o�ers a qualitative resolution of the three
limit puzzles under any calibration.
3.2. Equilibrium Selection and Implications Away from the Limits370
In our simple model, we can asymptotically remove the monetary friction by making χi
go to in�nity. We then get a sequence of money-demand equations that converges to it = imt ,
and hence a sequence of models that converges to the basic NK model. The corresponding se-
quence of unique bounded solutions converges to a particular equilibrium (out of an in�nity of
equilibria) of the basic NK model under a permanently exogenous policy rate. To characterize375
the equilibrium that we select uniquely in this way, we �rst note that as χi → +∞, we have
P(X) → (X − 1)Pb(X) for any X ∈ R. Therefore, we get limχi→+∞(ρ, ω1, ω2) = (ρb, 1, ωb).
Using this result, βρbωb = 1, ρb + ωb = 1 + 1/β + κ/(βσ), and Pb(ρb) = 0, we then easily
obtain that the limits of (10) and (11) as χi → +∞ are
πt = − (1− ρb) pt−1 + Et
{− (1− ρb)
+∞∑k=0
(1− ω−k−1b
) (i∗t+k − rt+k
)380
+κρb
+∞∑k=0
ω−kb [(1− δg) gt+k − δϕϕt+k]
}, (12)
yt = −ρbσpt−1 −
ρbσEt
{+∞∑k=0
[1 + β (1− ρb)ω−kb
] (i∗t+k − rt+k
)+κ
+∞∑k=0
ω−kb [(1− δg) gt+k − δϕϕt+k]
}+ gt. (13)
We can highlight four main di�erences between our selected equilibrium (12)-(13) and the
standard equilibrium (5)-(6) of the basic NK model. The �rst di�erence is that there are no385
ρ−kb terms in (12)-(13), and as a consequence our selected equilibrium does not exhibit any
of the three limit puzzles − at least in the same form as the standard equilibrium. More
speci�cally, the e�ects of announcing at date t a �scal expansion at date t + k on in�ation
and output at date t decrease at rate ω−kb in the horizon k of the �scal expansion, and
value of the interest-rate semi-elasticity of money demand χi, given the wide range of estimates for χi in theempirical literature (Galí's value standing in the middle of the range). We also provide additional numericalillustrations under our benchmark calibration in Appendix G.1 (for the e�ects of forward guidance on output,and the e�ects of anticipated changes in �scal policy on in�ation and output).
16
asymptotically limk→+∞ ∂πt/∂gt+k = limk→+∞ ∂yt/∂gt+k = 0 (no �scal-multiplier puzzle).390
Moreover, given that limθ→0(ρb, ωb, κ/ωb) = (0,+∞, βσ) and ρbωb = 1/β, (12) and (13) imply
that limθ→0 πt and limθ→0 yt are �nite, and in particular that limθ→0 yt takes the same value
(i.e., δggt + δϕϕt) as yt under perfectly �exible prices (no paradox of �exibility).8 Finally, in
our selected equilibrium (12)-(13), the e�ects of announcing at date t a policy-rate cut at date
t+ k on in�ation and output at date t do not explode as the horizon k of the policy-rate cut395
goes to in�nity (unlike in the standard equilibrium). However, they do not converge to zero
either (unlike in our simple model outside its basic-NK-model limit). Instead, they converge
to non-zero �nite values: limk→+∞ ∂πt/∂i∗t+k = −(1 − ρb) and limk→+∞ ∂yt/∂i
∗t+k = −ρb/σ.
In this sense, the forward-guidance puzzle is not fully, but only partially solved in our selected
equilibrium, unlike the �scal-multiplier puzzle and the paradox of �exibility. We highlight400
this �rst di�erence between the two equilibria as follows:
Proposition 3 (Resolution of the Three Limit Puzzles in our Selected Equi-
librium of the Basic NK Model): In our selected equilibrium of the basic NK model,
unlike in the standard equilibrium of the basic NK model, the �scal-multiplier puzzle and
the paradox of �exibility are fully solved (i.e., limk→+∞ ∂vt/∂gt+k = 0 for v ∈ {π, y},405
limθ→0 |πt| < +∞, limθ→0 yt = δggt + δϕϕt), and the forward-guidance puzzle is partially
solved (i.e., 0 < limk→+∞ |∂vt/∂i∗t+k| < +∞ for v ∈ {π, y}).
We relate the partial resolution of the forward-guidance puzzle in our selected equilibrium
to price-level stationarity. More generally, we show that the forward-guidance puzzle cannot
be fully solved in any equilibrium of the basic NK model in which the policy rate is exogenous410
and the price level is stationary in response to temporary policy-rate shocks (which is the
case, in particular, of our selected equilibrium). To see this, note from (9) that our simple
model implies price-level stationarity in response to a one-o� policy-rate change (p∞ = p0
when i∗k = i∗ 6= 0 and i∗t = 0 for t ≥ 1 and t 6= k). Therefore, in our selected equilibrium
of the basic NK model, the price level is also stationary in response to a one-o� policy-rate415
change. Now, iterating the IS equation (1) forward to +∞ under this policy-rate change at
8We cannot compare limθ→0 πt with πt at θ = 0, since πt at θ = 0 is indeterminate in the basic NK model.
17
date k, using price-level stationarity, and using the terminal condition y∞ = 0, leads to
y1 =−i∗
σ− π1
σ.
This relationship is consistent with y1 and π1 converging towards non-zero �nite values as
k → +∞ (partial resolution of the puzzle), but inconsistent with y1 and π1 both converging420
towards zero (full resolution of the puzzle).
The second, third, and fourth di�erences between our selected equilibrium and the stan-
dard equilibrium pertain to the implications of these equilibria away from the limits, i.e.
for �nite horizons of the shocks and non-zero degrees of price stickiness. More speci�cally,
the second di�erence is that these equilibria have opposite implications about the sign of425
the e�ects of future government-purchases on current output. In the standard equilibrium
(6), anticipated �scal expansions are expansionary (∂yt/∂gt+k > 0 for k ≥ 1) − suggesting
that �scal policy is a potent stabilization tool at the ZLB. This implication of the standard
equilibrium arises from a feedback loop, �rst described in Farhi and Werning (2016), that
works back in time via the IS equation and the Phillips curve: given that πT+1 = yT+1 = 0,430
a �scal expansion at date T raises in�ation at date T , which lowers the real interest rate at
date T − 1, which raises output and in�ation at date T − 1, etc. This feedback loop is also
present in our selected equilibrium, but it is counteracted by the presence of the state variable
pt−1 in (12)-(13): the starting point of the loop, πT+1 and yT+1, now reacts endogenously
to prior developments. More speci�cally, (12) and the identity πt = pt − pt−1 imply that a435
pre-announced �scal expansion at date T (gT > 0) raises the price level at all dates starting
from the announcement date, and in particular at date T (pT > 0); in turn, (12) and (13)
then imply that πT+1 = −(1 − ρb)pT < 0 and yT+1 = −(ρb/σ)pT < 0. As a result, expected
future �scal expansions are now contractionary, re�ecting the familiar wealth e�ect present
in standard Real-Business-Cycle models (through a reduction in permanent income).440
The third di�erence between our selected equilibrium and the standard equilibrium is
about the e�ects of current and expected future supply shocks on current output. In the stan-
dard equilibrium (6), anticipated positive supply shocks are contractionary (∂yt/∂ϕt+k < 0
for k ≥ 1). In other words, the standard equilibrium exhibits the paradox of toil, and sug-
18
gests that structural reforms raising potential output may be best put on hold during a ZLB445
episode. This implication of the standard equilibrium arises from essentially the same feed-
back loop as the one at work for government-purchases shocks (described above). In our
selected equilibrium (13), this feedback loop is, again, counteracted by the presence of the
state variable pt−1. As a result, anticipated positive supply shocks are now expansionary
(∂yt/∂ϕt+k > 0 for k ≥ 1). Thus, our selected equilibrium does not exhibit the paradox of450
toil, and suggests that structural reforms raising potential output are not counter-productive
at the ZLB. Similarly, current positive supply shocks are expansionary in our selected equilib-
rium (∂yt/∂ϕt > 0). By contrast, they are neutral in the standard equilibrium (∂yt/∂ϕt = 0).
The mechanical reason for this neutrality is that the standard equilibrium is obtained by it-
erating the IS equation and the Phillips curve backward in time (from πT+1 = yT+1 = 0); so,455
the date-t + 1 variables πt+1 and yt+1 do not depend on the date-t shock ϕt; and the date-t
IS equation then implies that yt does not depend on ϕt either.
Finally, the fourth di�erence between the two equilibria is that they have opposite im-
plications about the sign of the e�ect of greater price �exibility on output during a ZLB
episode caused by a negative preference shock. In the standard equilibrium (6), greater price460
�exibility does not change the e�ects of current preference shocks on output (∂yt/∂rt) and
ampli�es the e�ects of expected future preference shocks on output (∂yt/∂rt+k for k ≥ 1), as
we show in Appendix A.3. As a result, greater price �exibility magni�es the contractionary
e�ect of a temporary ZLB episode caused by a negative preference shock (rt+j = r∗ < 0
and i∗t+j = i∗ < 0 for j ∈ {0, ..., k}, with r∗ < i∗). The standard equilibrium, thus, sug-465
gests that structural reforms enhancing price �exibility are counter-productive during a ZLB
episode. In our selected equilibrium (13), by contrast, greater price �exibility reduces the
e�ects of current and expected future preference shocks on output, as we show in Appendix
A.3. Therefore, our selected equilibrium suggests on the contrary that structural reforms
enhancing price �exibility help to �ght the recession during a ZLB episode.470
We summarize the second, third, and fourth di�erences between the two equilibria in the
following proposition:
Proposition 4 (Some Implications of our Selected Equilibrium Away From the
Limits, Including Resolution of the Paradox of Toil): In our selected equilibrium
19
of the basic NK model, unlike in the standard equilibrium of the basic NK model, expected475
future �scal expansions are contractionary (i.e., ∂yt/∂gt+k < 0 for k ≥ 1), current and
expected future positive supply shocks are expansionary (i.e., ∂yt/∂ϕt+k > 0 for k ≥ 0), and
greater price �exibility reduces the contraction caused by current and expected future negative
preference shocks (i.e., ∂2yt/∂θ∂rt+k > 0 for k ≥ 0).
3.3. Comparison With Other Equilibria in the Literature480
We now brie�y compare our selected equilibrium with some other equilibria studied in
the literature. Cochrane (2017) characterizes the set of all equilibrium paths, in the basic NK
model, with (it, rt) = (i∗, r∗) for 1 ≤ t ≤ T and (it, rt) = (0, 0) for t ≥ T+1. He shows that any
of these paths can be obtained as the unique local equilibrium under a temporary interest-rate
peg followed by a suitably designed interest-rate rule. He considers a �local-to-frictionless�485
equilibrium-selection criterion, which requires that equilibrium outcomes converge towards
�exible-price equilibrium outcomes as prices become more and more �exible. This criterion
does not select a unique equilibrium, but rules out some equilibria (including the standard
NK equilibrium). Our selected equilibrium satis�es this criterion, since it does not exhibit
the paradox of �exibility.490
Among the equilibria satisfying the local-to-frictionless criterion, Cochrane (2017) de-
scribes more speci�cally two particular equilibria that do not exhibit any of the NK puzzles
and paradoxes: (i) the �backward-stable� equilibrium, in which in�ation goes to zero back-
ward in time (limt→−∞ πt = 0) when the interest-rate peg between 1 and T is announced
at date −∞; and (ii) the �no-in�ation-jump� equilibrium, in which in�ation is zero at the495
start of the peg (π1 = 0). Two di�erences between these equilibria and our selected equi-
librium are worth emphasizing. First, the forward-guidance puzzle is fully solved in these
equilibria, but only partially solved in our equilibrium. Second, at the start of a liquidity
trap caused by r∗ < 0, in�ation is negative in our equilibrium (∂π1/∂r∗ > 0), while it is
positive in the backward-stable equilibrium (∂π1/∂r∗ < 0) and, by construction, zero in the500
no-in�ation-jump equilibrium (∂π1/∂r∗ = 0).9
9The result ∂π1/∂r∗ < 0 in the backward-stable equilibrium is straightforwardly obtained from Cochrane's
(2017) Equation (34) by setting C = 0 and t = Tl; the result ∂π1/∂r∗ > 0 in our equilibrium is straightfor-
20
The latter di�erence can be equivalently restated as follows. Unlike the backward-stable
equilibrium (and, to a lesser extent, the no-in�ation-jump equilibrium), our selected equilib-
rium does not imply any Neo-Fisherian e�ects: announcing a current or future interest-rate
hike, in our equilibrium, always reduces current in�ation (∂πt/∂i∗t+k < 0 for k ≥ 0), as fol-505
lows from (12). In the context of a monetary-policy normalization process, in particular,
our equilibrium thus highlights the de�ationary pressures that arise from expected future
interest-rate hikes. We highlight this di�erence in the following proposition:
Proposition 5 (Other Implication of our Selected Equilibrium Away From the
Limits: No Neo-Fisherian E�ects): In our selected equilibrium of the basic NK model,510
unlike in other equilibria of the basic NK model that do not exhibit any of the four puzzles and
paradoxes, current and expected future interest-rate hikes are de�ationary (i.e., ∂πt/∂i∗t+k < 0
for k ≥ 0).
This property also distinguishes our equilibrium from the one considered by Bilbiie (2019a).
Bilbiie (2019a) uses McCallum's (1999) Minimal-State-Variable (MSV) criterion to select an515
equilibrium of the basic NK model under an exogenous AR(1) stochastic process for the
interest rate. He �nds that this MSV equilibrium may imply Neo-Fisherian e�ects − like
Cochrane's (2017) equilibria, and unlike ours. We suspect, based on Wieland's (2018) analy-
sis of �scal multipliers, that the MSV criterion actually selects Cochrane's (2017) backward-
stable equilibrium for the parameter values that give rise to Neo-Fisherian e�ects.10520
Another interesting parallel is between our selected equilibrium of the basic NK model
and the equilibrium of Mankiw and Reis's (2002) sticky-information model. Carlstrom et al.
(2015) and Kiley (2016) show that Mankiw and Reis's (2002) model fully solves the �scal-
multiplier puzzle, the paradox of �exibility, and the paradox of toil, and partially solves the
forward-guidance puzzle − exactly like the basic NK model with our selected equilibrium.11525
wardly obtained from (12).10Wieland (2018) shows that the MSV criterion selects either the backward-stable equilibrium with a
negative �scal multiplier, or the standard NK equilibrium with a positive �scal multiplier, depending on thepersistence of the �scal shock. Since the model is linear, it seems likely that this insight also explains whyNeo-Fisherian e�ects may (or may not) arise in the MSV equilibrium, depending on the persistence of theinterest-rate shock.
11Of course, the paradox of �exibility solved by Mankiw and Reis's (2002) model is about the e�ects ofinformation �exibility, not price �exibility.
21
Thus, our selected equilibrium brings the canonical sticky-price model at par with its sticky-
information cousin in terms of their ability to fully or partially solve all four NK puzzles
and paradoxes. Kiley (2016) also points out that Mankiw and Reis's (2002) model implies
price-level stationarity when the central bank follows a non-inertial interest-rate rule after
the temporary interest-rate peg; in this case, we can attribute the inability of that model to530
fully solve the forward-guidance puzzle to price-level stationarity, for the same reason as in
the basic NK model with our selected equilibrium.
4. Resolution of the Puzzles and Paradoxes in Other Models
We have so far presented our main points in the context of a simple but ad hoc setup.
It remains to show that these points can be made in more structured models generating a535
demand for money. In this section, we consider in turn a standard MIU model, with money
interpreted as interest-bearing reserve balances at the central bank, and a model with a more
explicit role for banks and bank reserves that we develop in Diba and Loisel (2020).
Throughout the section, for convenience, we keep the same notations as in the previous
section for the characteristic polynomial (P(X)), the roots of this polynomial (ρ, ω1, ω2),540
and the exogenous driving term in the dynamic equation (Zt), although all of them are in
fact model-speci�c.
4.1. MIU Model
We consider essentially the same MIU model as in Woodford (2003, Chapter 4).12 In
Appendix B, for completeness, we present this model, derive the necessary and su�cient545
condition for steady-state existence and uniqueness under exogenous monetary-policy instru-
ments, and log-linearize the equilibrium conditions around the unique steady state. In the
case in which the utility function is separable in consumption and money, we obtain the same
IS equation (1) and Phillips curve (2) as in the two models considered so far (the basic NK
model in Section 2 and our simple model in Section 3). In the alternative case in which550
the utility function is not separable in consumption and money, we obtain an IS equation
12The main di�erence is that he considers di�erentiated types of labor, while we consider, for simplicity, asingle type of labor.
22
and a Phillips curve that involve real-money terms (because the marginal utility of consump-
tion depends on real money). In both cases, we also obtain a money-demand equation that
is isomorphic to its counterpart (7) in our simple model, except that it also involves the
government-purchases shock (because money demand now depends on consumption, which555
we eliminate using the goods-market-clearing condition).
In Appendix C, we conduct the same analysis with our MIU model as in the previous
section with our simple model. More speci�cally, we show that the log-linearized equilibrium
conditions of the MIU model, under permanently exogenous monetary-policy instruments imt
and Mt, lead to a dynamic equation of type (8), where the characteristic polynomial P(X)560
is again of degree 3. We show that P(X) has again one root inside the unit circle (ρ ∈ (0, 1))
and two roots outside the unit circle (ω1 and ω2 with |ω1| ≤ |ω2|), the latter roots being either
positive real numbers or complex conjugates. We assume again that ω1 and ω2 are distinct real
numbers, and postpone the discussion of this assumption to the end of the subsection. With
one eigenvalue inside the unit circle (ρ) for one predetermined variable (pt−1), our MIU model565
satis�es Blanchard and Kahn's (1980) conditions and delivers determinacy for all structural-
parameter values, and in particular for any degree of price stickiness. Therefore, our MIU
model solves the three limit puzzles (forward-guidance puzzle, �scal-multiplier puzzle, and
paradox of �exibility) in the same way as our simple model in the previous section. We
determine the unique local equilibrium of our MIU model in the same way as previously, and570
obtain in particular that in�ation in this equilibrium is again characterized by (10) − keeping
in mind, though, that the roots ρ, ω1, ω2, and the exogenous driving term Zt have changed.
We also show in Appendix C that, as with the simple model of the previous section,
we can asymptotically remove the monetary friction from (some speci�cations of) our MIU
model for equilibrium-selection purposes. In the separable-utility case, for instance, we can575
make the scale parameter of the money-utility function go to zero. In the case of utility
over a constant-elasticity-of-substitution (CES) aggregator of money and consumption, we
can make the quasi-share parameter on money go to zero. Making this scale or quasi-share
parameter γ go to zero removes asymptotically the marginal bene�t of holding money. To
prevent real money balances from shrinking to zero, we need to concomitantly remove the580
opportunity cost of holding money, i.e. to make the steady-state IOR rate Im go to the
23
steady-state interest rate on bonds 1/β. We show that as Im goes to 1/β at the same speed
as γ goes to zero, the steady state and reduced form of our MIU model with separable utility
or with utility over a CES aggregator converge to the steady state and reduced form of the
basic NK model, with real money balances bounded away from zero and in�nity along the585
way.
Therefore, as we asymptotically remove the monetary friction from our MIU model, the
characteristic polynomial P(X) goes to (X− 1)Pb(X); its roots ρ, ω1, and ω2 go respectively
to ρb, 1, and ωb; and the exogenous driving term Zt in the dynamic equation goes to Zbt .
Using these limit results, we get that in�ation (10) in the unique local equilibrium of our590
MIU model converges to (12); and, therefore, that output converges to (13). Thus, our MIU
model serves to select the same equilibrium of the basic NK model under a permanently
exogenous policy rate as our simple model in the previous section. This selected equilibrium,
in particular, does not exhibit the paradox of toil.
We can thus summarize the results obtained with the MIU model as follows:595
Proposition 6 (Robustness of the Results With the MIU Model): The MIU
model solves the three limit puzzles, like the simple model (i.e. Propositions 1-2 still hold when
�simple� is replaced with �MIU�); and the MIU model serves to select the same equilibrium of
the basic NK model (characterized by Propositions 3-5) as the simple model.
We have so far assumed that the roots ω1 and ω2 are real numbers− both in this subsection600
in the context of the MIU model, and in the previous section in the context of the simple
model. These roots, however, can be complex (non-real) numbers in either model. In the
case of complex roots, all the equations that we have derived are still valid, and all the
results that we have obtained still hold. In particular, these two models still solve the three
limit puzzles, and still serve to select the equilibrium (12)-(13) of the basic NK model, which605
solves the paradox of toil. However, the case of complex roots is awkward for our purposes
because it implies recurrent sign reversals in the e�ects of future shocks on current in�ation
and output, as we change the horizon of the shocks. We illustrate this implication with the
e�ect of forward-guidance policy on in�ation. When ω1 and ω2 are complex, we can write
them as ω1 = Reiφ and ω2 = Re−iφ, where R > 1 and φ ∈ (0, 2π). Therefore, using (10)610
24
with ∂Zt/∂i∗t = −κ/(βσ) and ∂Zt/∂i
∗t+k = 0 for k 6= 0, we can write the e�ect of a future
policy-rate change i∗t+k = i∗ 6= 0 on current in�ation πt as
πt =
(ω−k−12 − ω−k−11
ω2 − ω1
)κi∗
βσ=−κR−k−2 sin [(k + 1)φ] i∗
βσ sin (φ).
The sign of this e�ect changes an in�nity of times as the horizon k of the policy-rate change
moves to +∞. We call this implication the �reversal puzzle:�13615
De�nition 5 (Reversal Puzzle): Under exogenous monetary-policy instruments, the
sign of the e�ect of an expected policy-rate change k periods ahead on current in�ation changes
recurrently as the horizon k goes to +∞ (i.e., ∀k ∈ N, ∃k′ ∈ N such that k′ > k and
(∂πt/∂i∗t+k)(∂πt/∂i
∗t+k′) < 0).
The simple model (in the previous section) and the MIU model (in the present subsection)620
can give rise to this reversal puzzle because their characteristic polynomial has complex roots
under some calibrations. In the next subsection, we will consider a model that cannot give
rise to the reversal puzzle, because its characteristic polynomial has real roots under all
calibrations.
4.2. Model with Banks625
We now turn to a model in which money is explicitly made of bank reserves. In this model,
which we present in detail in Diba and Loisel (2020), �rms must borrow the wage bill (or some
fraction of it) from banks. Banks incur costs making loans, and holding reserves mitigates
these costs. The model makes weak standard assumptions about utility and production
functions, like monotonicity and concavity, without specifying any functional form. The key630
log-linearized equilibrium conditions are, again, an IS equation, a Phillips curve, and a money-
demand equation. The IS equation is the same as the IS equation (1) of the basic NK model.
Like its counterpart in the MIU model, the Phillips curve involves real reserves (because real
reserves now reduce banking costs, which in turn lowers the borrowing costs of �rms and
hence their marginal cost of production). The money-demand equation is isomorphic to its635
13Carlstrom et al. (2015) show that the basic NK model augmented with in�ation indexation gives rise toa more severe form of �reversal puzzle.�
25
counterpart (7) in our simple model, except that it also involves the government-purchases
shock (like the money-demand equation of the MIU model) and the supply shock (because
the demand for reserves now depends on the volume of loans, which in turn depends on �rms'
wage bill, which in turn depends on the supply shock for a given output level).
In Appendix D, we conduct the same analysis with this model as with our previous two640
models. More speci�cally, we show that the log-linearized equilibrium conditions of our
model with banks, under permanently exogenous monetary-policy instruments imt and Mt,
lead to a dynamic equation of type (8), where the characteristic polynomial P(X) is again
of degree 3. We show that the roots of P(X) are three real numbers ρ, ω1, and ω2 such that
0 < ρ < 1 < ω1 < ω2. With one eigenvalue inside the unit circle (ρ) for one predetermined645
variable (pt−1), thus, our model with banks satis�es Blanchard and Kahn's (1980) conditions,
delivers determinacy, and solves the three limit puzzles in the same way as our previous two
models. We determine the unique local equilibrium of our model with banks in the same way
as previously, and obtain in particular that in�ation in this equilibrium is again characterized
by (10).650
In addition, as in our previous two models, we can asymptotically remove the monetary
friction from our model with banks for equilibrium-selection purposes. As we elaborate
in Appendix D, as the scale parameter of banking costs and the steady-state interest-rate
spread are shrunk to zero (at suitable rates, to keep a positive and �nite level of steady-state
real reserve balances in the limit), the steady state and reduced form of our model with655
banks converge to the steady state and reduced form of the basic NK model. Therefore, as
previously, the characteristic polynomial P(X) goes to (X − 1)Pb(X); its roots ρ, ω1, and ω2
go respectively to ρb, 1, and ωb; and the exogenous driving term Zt goes to Zbt . Using these
limit results, we get again that in�ation (10) in the unique local equilibrium of our model
with banks converges to (12); and, therefore, that output converges to (13). Thus, our model660
with banks serves to select the same equilibrium of the basic NK model under a permanently
exogenous policy rate as our previous two models − an equilibrium that, in particular, does
not exhibit the paradox of toil.
We can thus summarize the results obtained with the model with banks as follows:
Proposition 7 (Robustness of the Results With the Model With Banks): The665
26
model with banks solves the three limit puzzles, like the simple model and the MIU model (i.e.
Propositions 1-2 still hold when �simple model� is replaced with �model with banks�); and the
model with banks serves to select the same equilibrium of the basic NK model (characterized
by Propositions 3-5) as the simple model and the MIU model.
Unlike our previous two models, however, our model with banks solves the three limit670
puzzles without ever giving rise to the reversal puzzle de�ned in the previous subsection,
since ω1 and ω2 are always positive real numbers in this model. Using ω1 < ω2 in (10) with
∂Zt/∂i∗t = −κ/(βσ) and ∂Zt/∂i
∗t+k = 0 for k 6= 0, we straightforwardly get the following
proposition:
Proposition 8 (Resolution of the Reversal Puzzle, and No Neo-Fisherian Ef-675
fects, in the Model With Banks): In the model with banks, under exogenous monetary-
policy instruments, the sign of the e�ect of an expected policy-rate change k periods ahead on
current in�ation does not depend on the horizon k and is negative (i.e., ∂πt/∂i∗t+k < 0 for all
k ∈ N).
4.3. Robustness of the Selected Equilibrium680
The basic NK model has an in�nity of (local) equilibria under a permanently exogenous
policy rate. The equilibrium-selection mechanism that we have proposed consists in adding
a monetary friction to the basic NK model, getting a unique equilibrium under exogenous
monetary-policy instruments in the resulting monetary model, and considering the limit of
this equilibrium as the monetary friction is gradually shrunk to zero. Despite their di�erences,685
all the three monetary models that we have considered (the simple model, the MIU model,
and the model with banks) have served to select the same equilibrium of the basic NK model.
This robustness feature of our selected equilibrium, we argue, extends to other models
that deliver determinacy under an exogenous policy rate and smoothly converge to the basic
NK model as we gradually shrink some friction. At least, any model that has these two690
properties and whose dynamic equation under an exogenous policy rate relates pt to Et{pt+2},
Et{pt+1}, pt−1, and exogenous terms (like our three monetary models) will lead to our selected
equilibrium. Indeed, any model of this kind will have in�ation in the unique equilibrium given
27
by (10), although the roots ρ, ω1, ω2, and the exogenous driving term Zt in (10) will be model-
speci�c. As the friction is shrunk, the characteristic polynomial will go to (X − 1)Pb(X), so695
its roots ρ, ω1, and ω2 will converge to ρb, 1, and ωb. Similarly, the exogenous driving term
Zt will converge to Zbt . Therefore, in�ation in the unique equilibrium of the model, (10), will
converge to in�ation in our selected equilibrium of the basic NK model, (12). Our conjecture
is that models with the same properties but dynamic systems of higher orders will also lead
to our selected equilibrium.700
There are, of course, models that are not amenable to our equilibrium-selection exercise,
either (i) because we cannot gradually shrink a friction in these models and make them
smoothly converge to the basic NK model, or (ii) because they no longer deliver determinacy
under an exogenous policy rate as we gradually shrink the friction. An example of Case (i)
is the cash-in-advance (CIA) model with leisure implicitly serving as credit good, as we show705
in Appendix E. Examples of Case (ii) can be found in the discounting models used in the
literature to solve the forward-guidance puzzle (as we mention in the Introduction). There
is typically a friction that can be gradually shrunk to zero in these models, and gradually
shrinking it to zero typically makes the models smoothly converge to the basic NK model.
However, as we show in Appendix F.4, some of these models cease to deliver determinacy710
under an exogenous policy rate as they approach the basic NK model; so, we cannot use
them for our equilibrium-selection exercise.
What we need for our equilibrium-selection exercise, in essence, is a model that smoothly
converges to the basic NK model as some friction is gradually shrunk to zero, and delivers
determinacy under an exogenous policy rate along the way. Our conjecture is that any model715
of this kind will lead to our selected equilibrium.
5. Resolution of the Puzzles and Paradoxes Under Asymptotic Satiation
Our resolution of the NK puzzles and paradoxes rests on the assumption that demand for
reserves is not fully satiated. In our monetary models, demand for reserves would be fully
satiated if the marginal bene�t of holding reserves were exactly zero − which would make720
the IS equations and the Phillips curves of these models exactly the same as those of the
basic NK model. In that case, the opportunity cost of holding reserves would also have to be
28
exactly zero − implying that the money-demand equations of these models would collapse to
it = imt . Thus, our monetary models would then exactly coincide with the basic NK model,
and all the NK puzzles and paradoxes would re-emerge.725
Our non-satiation assumption stands in contrast to views often expressed about the US
economy in recent years (e.g., Cochrane, 2014, 2018; Reis, 2016). In Diba and Loisel (2020),
we present two formal arguments in defense of this assumption; for convenience, we summa-
rize these arguments in Appendix D.5. In the present section, moreover, we show that the
assumption we need to make is only that demand for reserves not be fully satiated. More730
speci�cally, we show that even though we no longer solve the NK puzzles and paradoxes
when demand for reserves is fully satiated, we still solve them when it is arbitrarily close to
satiation, and even when it is asymptotically satiated.
To establish this result, we make the steady-state IOR rate go to the steady-state interest
rate on bonds 1/β in the MIU model and the model with banks.14 This policy experiment735
asymptotically removes the opportunity cost of holding reserves. This time, unlike previously,
we do not concomitantly remove the marginal bene�t of holding reserves (by shrinking a scale
or quasi-share parameter). Therefore, the steady-state real stock of reserves goes to in�nity,
and demand for reserves is asymptotically satiated.
As we show in Appendices C.5 and D.4, the steady states and reduced forms of our MIU740
model with separable utility and of our model with banks then converge to the steady state
and reduced form of the basic NK model − under a condition that is met, in particular, for
isoelastic money-utility and banking-cost functions. Therefore, as previously, the character-
istic polynomial P(X) goes to (X − 1)Pb(X); its roots ρ, ω1, and ω2 go respectively to ρb, 1,
and ωb; the exogenous driving term Zt goes to Zbt ; and, again, the unique local equilibrium745
converges to our selected equilibrium of the basic NK model. Thus, at the limit, when de-
mand for reserves is asymptotically satiated, we fully solve the �scal-multiplier puzzle, the
paradox of �exibility, and the paradox of toil; and we partially solve the forward-guidance
puzzle. More generally, we get the following proposition:
14We cannot conduct this policy experiment in the simple model, because there is no steady-state IORrate in this ad-hoc model.
29
Proposition 9 (Robustness of the Results Under Asymptotic Satiation): Under750
asymptotic satiation of demand for reserves, the unique equilibrium of the MIU model and the
model with banks coincides with our selected equilibrium of the basic NK model (characterized
by Propositions 3-5).
The policy experiment we are conducting in this section involves a sequence of di�erent
policies in a given model, and the corresponding sequence of unique equilibrium outcomes755
converging to satiation. By contrast, our previous equilibrium-selection exercises involved a
sequence of di�erent policies in di�erent models (with smaller and smaller monetary frictions),
and the corresponding sequence of unique equilibrium outcomes staying away from satiation.
Despite this di�erence, however, the two sequences of unique equilibrium outcomes converge
to the same limit equilibrium outcome (namely, our selected equilibrium of the basic NK760
model). The reason is that going to satiation asymptotically alleviates the monetary friction
and has the same e�ect as asymptotically removing the monetary friction from the model.
6. Conclusion
In this paper, we have proposed a resolution of the puzzles and paradoxes that arise
in the basic NK model at the ZLB. In reality, it is the IOR rate, not the interest rate on765
bonds, that the Great Recession forced central banks to peg near zero. Central banks also
conducted balance-sheet policies. To capture what they actually did, we analyze models of
monetary policy in which the central bank sets exogenously the IOR rate and the nominal
stock of reserves. The three models we consider − a model with an ad-hoc money-demand
function, a familiar MIU setup, and a model with a more explicit role for bank reserves −770
lead essentially to the same conclusions. By delivering local-equilibrium determinacy under
exogenous monetary-policy instruments, for any degree of price stickiness, they o�er a full
resolution of the three limit puzzles − the forward-guidance puzzle, the �scal-multiplier
puzzle, and the paradox of �exibility. The more structured model with banks, however, has
the advantage over the other two models of never giving rise to what we call the �reversal775
puzzle.�
As we shrink the monetary friction to zero, or as we asymptotically satiate the demand
for reserves, these models converge to the basic NK model and serve to select a particular
30
equilibrium of that model under a permanently exogenous policy rate. The equilibrium
that we select in this way does not depend on the model we start from (neither one of the780
three monetary models we consider, nor, as we argue, any other suitable model). Unlike the
standard equilibrium of the basic NK model, our selected equilibrium does not exhibit the
three limit puzzles, nor the paradox of toil. Compared to the standard equilibrium, it has
opposite implications for the e�ects of expected �scal expansions, expected supply shocks,
and greater price �exibility, on current output in a liquidity trap. Unlike other equilibria of785
the basic NK model considered in the literature, it does not imply any Neo-Fisherian e�ect.
Our results have important policy implications at the ZLB: �scal policy and monetary
forward guidance may not be such powerful stabilization tools, and the pursuit of structural
reforms increasing potential output or enhancing price �exibility is not counterproductive. In
this paper, we have made these points analytically in tractable models. We think it would be790
useful to explore these points quantitatively by adding a money-demand nexus to larger-scale
models of monetary policy.
References
[1] Angeletos, G.-M., and Lian, C. (2018), �Forward Guidance Without Common Knowl-
edge,� American Economic Review, 108(9), 2477-2512.795
[2] Bilbiie, F.O. (2019a), �Neo-Fisherian Policies and Liquidity Traps,� mimeo.
[3] Bilbiie, F.O. (2019b), �Monetary Policy and Heterogeneity: An Analytical Framework,�
mimeo.
[4] Blanchard, O.J., and Kahn, C.M. (1980), �The Solution of Linear Di�erence Models
under Rational Expectations,� Econometrica, 48(5), 1305-1311.800
[5] Bredemeier, C., Kaufmann, C., and Schabert, A. (2018), �Interest-Rate Spreads and
Forward Guidance,� European Central Bank Working Paper No. 2186.
[6] Carlstrom, C.T., Fuerst, T.S., and Paustian, M. (2015), `In�ation and Output in New
Keynesian Models With a Transient Interest-Rate Peg,� Journal of Monetary Economics,
76, 230-243.805
31
[7] Christiano, L., Eichenbaum, M., and Rebelo, S. (2011), �When Is the Government Spend-
ing Multiplier Large?,� Journal of Political Economy, 119(1), 78-121.
[8] Cochrane, J.H. (2014), �Monetary Policy With Interest on Reserves,� Journal of Eco-
nomic Dynamics and Control, 49, 74-108.
[9] Cochrane, J.H. (2016), �Comments on `A Behavioral New Keynesian Model' by Xavier810
Gabaix,� mimeo.
[10] Cochrane, J.H. (2017), �The New-Keynesian Liquidity Trap,� Journal of Monetary Eco-
nomics, 92, 47-63.
[11] Cochrane, J.H. (2018), �Michelson-Morley, Fisher, and Occam: The Radical Implications
of Stable Quiet In�ation at the Zero Bound,� in Eichenbaum, M., and Parker, J.A. (eds)815
NBER Macroeconomics Annual 2017, University of Chicago Press, 32, 113-226.
[12] Del Negro, M., Giannoni, M.P., and Patterson, C. (2015), �The Forward-Guidance Puz-
zle,� mimeo.
[13] Diba, B.T., and Loisel, O. (2020), �A Model of Post-2008 Monetary Policy,� mimeo.
[14] Eggertsson, G.B. (2010), �The Paradox of Toil,� Federal Reserve Bank of New York Sta�820
Report No. 433.
[15] Eggertsson, G.B. (2011), �What Fiscal Policy is E�ective at Zero Interest Rates?,� in
Acemoglu, D., and Woodford, M. (eds) NBER Macroeconomics Annual 2010, University
of Chicago Press, 25, 59-112.
[16] Eggertsson, G.B. (2012), �Was the New Deal Contractionary?,� American Economic825
Review, 102(1), 524-555.
[17] Eggertsson, G.B., and Krugman, P. (2012), �Debt, Deleveraging, and the Liquidity Trap:
A Fisher-Minsky-Koo Approach,� Quarterly Journal of Economics, 127(3), 1469-1513.
[18] Eggertsson, G.B., Ferrero, A., and Ra�o, A. (2014), �Can Structural Reforms Help
Europe?,� Journal of Monetary Economics, 61, 2-22.830
[19] Farhi, E., and Werning, I. (2016), �Fiscal Multipliers: Liquidity Traps and Currency
Unions,� in Taylor, J.B., and Uhlig, H. (eds) Handbook of Macroeconomics, Amsterdam:
Elsevier Science, 2, 2417-2492.
32
[20] Farhi, E., and Werning, I. (2019), �Monetary Policy, Bounded Rationality, and Incom-
plete Markets,� American Economic Review, 109(11), 3887-3928.835
[21] Gabaix, X. (2020), �A Behavioral New Keynesian Model,� American Economic Review,
110(8), 2271-2327.
[22] Galí, J. (2008), Monetary Policy, In�ation, and the Business Cycle: An Introduction to
the New Keynesian Framework, Princeton: Princeton University Press.
[23] García-Schmidt, M., and Woodford, M. (2019), �Are Low Interest Rates De�ationary?840
A Paradox of Perfect-Foresight Analysis,� American Economic Review, 109(1), 86-120.
[24] Hagedorn, M. (2018), �Prices and In�ation when Government Bonds are Net Wealth,�
mimeo.
[25] Hagedorn, M., Luo, J., Manovskii, I., and Mitman, K. (2019), �Forward Guidance,�
Journal of Monetary Economics, 102, 1-23.845
[26] Kiley, M.T. (2016), �Policy Paradoxes in the New Keynesian Model,� Review of Economic
Dynamics, 21, 1-15.
[27] Mankiw, G.N., and Reis, R. (2002), �Sticky Information Versus Sticky Prices: A Proposal
to Replace the New Keynesian Phillips Curve,� Quarterly Journal of Economics, 117(4),
1295-1328.850
[28] McCallum, B.T. (1999), �Role of the Minimal State Variable Criterion in Rational Ex-
pectations Models,� International Tax and Public Finance, 6(4), 621-639.
[29] McKay, A., Nakamura, E., and Steinsson, J. (2016), �The Power of Forward Guidance
Revisited,� American Economic Review, 106(10), 3133-3158.
[30] Michaillat, P., and Saez, E. (2019), �Resolving New Keynesian Anomalies with Wealth855
in the Utility Function,� forthcoming in the Review of Economics and Statistics.
[31] Ono, Y. (2001), �A Reinterpretation of Chapter 17 of Keynes' General Theory: E�ec-
tive Demand Shortage Under Dynamic Optimization,� International Economic Review,
42(1), 207-236.
[32] Ono, Y., and Ishida, J. (2014), �On Persistent Demand Shortages: A Behavioural Ap-860
proach,� Japanese Economic Review, 65(1), 42-69.
33
[33] Piazzesi, M., Rogers, C., and Schneider, M. (2019), �Money and Banking in a New
Keynesian Model,� mimeo.
[34] Reis, R. (2016), �Funding Quantitative Easing to Target In�ation,� in Designing Resilient
Monetary Policy Frameworks for the Future, Federal Reserve Bank of Kansas City, 423-865
478.
[35] Werning, I. (2012), �Managing a Liquidity Trap: Monetary and Fiscal Policy,� mimeo.
[36] Wiederholt, M. (2015), �Empirical Properties of In�ation Expectations and the Zero
Lower Bound,� mimeo.
[37] Wieland, J.F. (2018), �Zero Lower Bound Government Spending Multipliers and Equi-870
librium Selection,� mimeo.
[38] Wieland, J.F. (2019), �Are Negative Supply Shocks Expansionary at the Zero Lower
Bound?,� Journal of Political Economy, 127(3), 973-1007.
[39] Woodford, M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy,
Princeton: Princeton University Press.875
[40] Woodford, M. (2011), �Simple Analytics of the Government Expenditure Multiplier,�
American Economic Journal: Macroeconomics, 3(1), 1-35.
34
Figure 1: E�ect of a policy-rate cut at date t+ k on in�ation at date t
0 5 10 15 200
0.2
0.4
0.6
0.8
0 5 10 15 200
50
100
150
200
250
Note: The �gure displays the e�ect on πt of announcing at date t a one-percentage-point-per-annumcut in imt+k (for the simple model) or it+k (for the basic NK model), as a function of k ∈ {0, ..., 20}.Benchmark parameter values are set as in Galí (2008, Chapter 3): β = 0.99, σ = 1, χy = 1, χi = 4,and κ = λ[(1− θ)(1− βθ)/θ] = 0.13, where λ = 3/4 and θ = θ∗ ≡ 2/3. As θ takes the values θ∗/2,θ∗/4, θ∗/8, and θ∗/16, κ takes respectively the values 1.00, 3.13, 7.57, and 16.54.
35
Online Appendix to
�Pegging the Interest Rate on Bank Reserves:A Resolution of New Keynesian Puzzles and Paradoxes�880
Behzad Diba and Olivier Loisel
Appendix A: Simple Model (Analytical Results)
A.1. Root Analysis
We �rst show that 0 < ρ < 1 < |ω1| ≤ |ω2|. Since P(0) = −1/β − χy/(βσχi) < 0 and
P(1) = κ/(βσχi) > 0, P(X) has either one or three real roots inside (0, 1). Moreover, since885
P(X) < 0 for all X < 0, P(X) has no negative real roots. Therefore, P(X) has at least
one real root inside (0, 1), which we denote by ρ, and its other two roots, which we denote
by ω1 and ω2 with |ω1| ≤ |ω2|, must be (i) both real and inside (0, 1), or (ii) both real and
larger than 1, or (iii) both complex and conjugates of each other. Now, given that P(X) is
of type X3− a2X2 + a1X − a0, we have ρ+ω1 +ω2 = a2 ≡ 2 + 1/β+κ/(βσ) +χy/(σχi) > 3.890
Therefore, Case (i) is impossible, and in Case (iii) the common real part of ω1 and ω2 is larger
than 1. As a consequence, in the remaining two possible cases, namely Cases (ii) and (iii),
ω1 and ω2 lie outside the unit circle.
We now show that ω1 and ω2 can be real numbers, and that they can also be complex
(non-real) numbers. Suppose, for instance, that χy and χi go to 0, with χy/χi constant.895
Then, a1 ≡ 1 + 2/β + (1 + 1/χi)κ/(βσ) + (1 + 1/β)χy/(σχi) goes to +∞, while a2 ≡
2 + 1/β + κ/(βσ) + χy/(σχi) and a0 ≡ 1/β + χy/(βσχi) remain constant. Therefore, for
su�ciently small values of χy and χi, P(X) = X3 − a2X2 + a1X − a0 is positive for all
X ≥ 1, so that Case (ii) is impossible and ω1 and ω2 are complex numbers. By contrast,
suppose now that χy and χi go to +∞, with χy/χi constant. Then, P [1 + χy/(σχi)] goes to900
−[1 + χy/(σχi)]κχy/(βσ2χi), which is negative. Therefore, for su�ciently large values of χy
and χi, we have P [1 + χy/(σχi)] < 0, which, together with P(1) > 0, implies that ω1 and ω2
are positive real numbers.
Online Appendix − 1
A.2. Resolution of the Paradox of Flexibility
Using the de�nition of Zt, and after some simple algebra, we can rewrite (10) and (11) as905
πt = − (1− ρ) pt−1 +κ
β (ω2 − ω1)Et
{− 1
σ
+∞∑k=0
(ω−k−11 − ω−k−12
)(i∗t+k − rt+k −
Mt+k
χi
)
−+∞∑k=0
(ξg1ω
−k−11 − ξg2ω−k−12
)gt+k +
+∞∑k=0
(ξϕ1 ω
−k−11 − ξϕ2 ω−k−12
)δϕϕt+k
}, (A.1)
yt = −ϑpt−1 + gt +Et
β (ω2 − ω1)
{1
σ
+∞∑k=0
(ξ1ω
−k−11 − ξ2ω−k−12
)(i∗t+k − rt+k −
Mt+k
χi
)
++∞∑k=0
(ξ1ξ
g1ω−k−11 − ξ2ξg2ω−k−12
)gt+k −
+∞∑k=0
(ξ1ξ
ϕ1 ω−k−11 − ξ2ξϕ2 ω−k−12
)ϕt+k
}, (A.2)910
where ϑ ≡ (1− ρ)(1− βρ)/κ and
ξj ≡ β (ωj + ρ− 1)− 1,
ξgj ≡ (1− δg) (ωj − 1) +δgχyσχi
,
ξϕj ≡ δϕ (ωj − 1)− δϕχyσχi
for j ∈ {1, 2}.915
The only parameter that depends on the degree of price stickiness θ in the structural
equations (1), (2), and (7) is the slope κ of the Phillips curve (2). We have limθ→0 κ = +∞
and therefore
−βσ limθ→0
[P (X)
κ
]= X (X − ωn1 )
for any X ∈ R, where ωn1 ≡ (1 + χi)/χi > 1, which implies in turn that920
limθ→0
ρ = 0, limθ→0
ω1 = ωn1 , and limθ→0
ω2 = +∞. (A.3)
Using (A.3) and
(1− ρ) (ω1 − 1) (ω2 − 1) = P (1) =κ
βσχi,
Online Appendix − 2
we also get that
limθ→0
κ
ω2
= βσ. (A.4)925
Using (A.3) and (A.4), we can easily determine the limits of (A.1) and (A.2) as θ → 0:
limθ→0
πt = −pt−1 − Et
{+∞∑k=0
(ωn1 )−k−1{i∗t+k − rt+k −
Mt+k
χi+
[σ (1− δg) + χyδg
χi
]gt+k
+
[(χy − σ) δϕ
χi
]ϕt+k
}}+ σ (1− δg) gt − σδϕϕt, (A.5)
limθ→0
yt = δggt + δϕϕt. (A.6)
These limits are �nite, unlike their counterparts in the basic NK model.930
We now show that the right-hand sides of (A.5) and (A.6) coincide with the values taken
by πt and yt when prices are perfectly �exible (θ = 0). The �exible-price value of yt is
straightforwardly obtained by setting to zero the last term in the Phillips curve (2), which is
proportional to (the log-deviation of) �rms' marginal cost of production:
yt = δggt + δϕϕt. (A.7)935
This value is identical to the right-hand side of (A.6). Using the IS equation (1), the money-
demand equation (7), the identity mt = Mt − pt, the exogenous policy-rate setting imt = i∗t ,
and the solution for �exible-price output (A.7), we get the following dynamic equation under
�exible prices:
pt = (ωn1 )−1 Et {pt+1} − (ωn1 )−1{i∗t − rt −
Mt
χi−[σ (1− δg)−
χyδgχi
]gt940
+σ (1− δg)Et {gt+1}+
(σ +
χyχi
)δϕϕt − σδϕEt {ϕt+1}
}.
Iterating this equation forward to +∞ leads to the following value for the price level pt in
Online Appendix − 3
our simple model under �exible prices:
pt = −Et
{+∞∑k=0
(ωn1 )−k−1{i∗t+k − rt+k −
Mt+k
χi+
[σ (1− δg) + χyδg
χi
]gt+k
+
[(χy − σ) δϕ
χi
]ϕt+k
}}+ σ (1− δg) gt − σδϕϕt,945
which implies in turn that the value of πt ≡ pt−pt−1 in our simple model under �exible prices
coincides with the right-hand side of (A.5). Thus, our simple model solves the paradox of
�exibility: the limits of πt and yt as θ → 0 are �nite and coincide with the values of πt and
yt when θ = 0.
A.3. E�ects of Greater Price Flexibility950
We measure the degree of price �exibility by the reduced-form parameter κ (which is
inversely related to the degree of price stickiness θ). We show that: (i) ∂2πt/∂κ∂rt+k > 0 for
k ≥ 0 in both our selected equilibrium and the standard equilibrium of the basic NK model;
and (ii) ∂2yt/∂κ∂rt+k < 0 for k ≥ 0 in our selected equilibrium, while ∂2yt/∂κ∂rt = 0 and
∂2yt/∂κ∂rt+k > 0 for k ≥ 1 in the standard equilibrium.955
We start with (i). In our selected equilibrium, we have ∂πt/∂rt+k = (1− ρb)(1− ω−k−1b ).
Using ωb = [1 + β + κ/σ +√
(1 + β + κ/σ)2 − 4β]/(2β), we get ∂ωb/∂κ > 0. In turn, using
this result and ρbωb = 1/β, we get ∂ρb/∂κ < 0. We conclude that ∂2πt/∂κ∂rt+k > 0. In the
standard equilibrium, now, we have
∂πt∂rt+k
=κ(ρ−k−1b − ω−k−1b
)βσ (ωb − ρb)
=βkκ
σ
(ωk+1b − ρk+1
b
ωb − ρb
)=βkκ
σ
k∑j=0
ωk−jb ρjb =βkκ
σ
k∑j=0
β−jωk−2jb ,960
where the second and fourth equalities follow from ρbωb = 1/β. For k = 0, thus, we
straightforwardly get ∂2πt/∂κ∂rt = 1/σ > 0. For k ≥ 1, we introduce the function
x 7→ fk(x) ≡∑k
j=0 β−jxk−2j, and we write ∂πt/∂rt+k = βkκfk(ωb)/σ. The �rst derivative of
fk(x) is
f ′k (x) =1
x
k∑j=0
(k − 2j) β−jxk−2j =1
x
b(k−1)/2c∑j=0
(k − 2j) β−j
[xk−2j − 1
(βx)k−2j
],965
Online Appendix − 4
where b.c denotes the �oor operator. Using ρbωb = 1/β, we then get
f ′k (ωb) =1
ωb
b(k−1)/2c∑j=0
(k − 2j) β−j(ωk−2jb − ρk−2jb
)> 0.
So, we have∂2πt
∂κ∂rt+k=βk
σ
[fk (ωb) + κf ′k (ωb)
∂ωb∂κ
]> 0.
We now turn to (ii). In our selected equilibrium, we have970
∂yt∂rt+k
=ρbσ
[1 + β (1− ρb)ω−kb
]=
1
σ
(ρb + βk+1ρk+1
b − βk+1ρk+2b
),
where the second equality follows from ρbωb = 1/β. Therefore, we get
∂2yt∂κ∂rt+k
=1
σ
[1 + (k + 1) βk+1ρkb − (k + 2) βk+1ρk+1
b
] ∂ρb∂κ
=1
σ
[(1− βk+1ρk+1
b
)+ (k + 1) βk+1ρkb (1− ρb)
] ∂ρb∂κ
< 0.975
In the standard equilibrium, we have
∂yt∂rt+k
=κ
βσ2 (ωb − ρb)
(ρ−kb
1− ρb+
ω−kbωb − 1
)=
1
σ (ωb − ρb)[(ωb − 1) ρ−kb + (1− ρb)ω−kb
]=
βk
σ
[(ωb − 1)
(ωkb − ρkbωb − ρb
)+ ρkb
],
where the second equality follows from ρb + ωb = 1 + 1/β + κ/(βσ) and ρbωb = 1/β, and the
third one from ρbωb = 1/β. For k = 0, we straightforwardly get ∂yt/∂rt = 1/σ and hence980
∂2yt/∂κ∂rt = 0. For k = 1, we get ∂yt/∂rt+1 = (β/σ)(ρb + ωb − 1) = (1/σ)(1 + κ/σ) and
hence ∂2yt/∂κ∂rt+1 = σ−2 > 0. Finally, for k ≥ 2, we get
∂yt∂rt+k
=βk
σ
[(ωb − 1) fk−1 (ωb) + ρkb
]=βk
σ
[(ωb − 1) fk−1 (ωb) + β−kω−kb
],
where the function fk−1(.) is de�ned above and where the second equality follows from ρbωb =
Online Appendix − 5
1/β. So,985
∂2yt∂κ∂rt+k
=βk
σ
[fk−1 (ωb) + (ωb − 1) f ′k−1 (ωb)− kβ−kω−k−1b
] ∂ωb∂κ
=βk
σ
[(ωb − 1) f ′k−1 (ωb) +
1
ωb
(k−1∑j=0
ωk−jb ρjb − kρkb
)]∂ωb∂κ
> 0,
where the second equality follows from ρbωb = 1/β.
Appendix B: MIU Model (Presentation and Log-Linearization)990
In this appendix (and the following ones), to lighten up the notation, we sometimes omit
function arguments when no ambiguity results.
B.1. Households
Households get utility from consumption (ct) and real money (mt), and disutility from
labor (ht). Their intertemporal utility is995
Ut = Et
{∞∑k=0
βkζt+k
[u (ct+k,mt+k)−
v (ht+k)
ϕ1,t+k
]},
where β ∈ (0, 1). The utility function u, de�ned over the set of pairs of positive real numbers
R2>0, is twice di�erentiable, strictly increasing (uc > 0, um > 0), strictly concave (ucc < 0,
umm < 0, uccumm− (ucm)2 > 0), with ucm ≥ 0, and it satis�es the standard Inada conditions
limct→0 uc (ct,mt) = +∞, limct→+∞ uc (ct,mt) = 0,
limmt→0 um (ct,mt) = +∞, limmt→+∞ um (ct,mt) = 0.
The labor-disutility function v, de�ned over the set of non-negative real numbers R≥0, is twice1000
di�erentiable, strictly increasing (v′ > 0), and weakly convex (v′′ ≥ 0). The intertemporal
utility Ut is a�ected by two stochastic exogenous shocks of mean one: the discount-factor
shock ζt, and the labor-disutility shock ϕ1,t. The latter shock is the �rst of the four alternative
supply shocks that we consider.
Online Appendix − 6
Households choose ct, ht, mt, and real bonds bt to maximize their utility function subject1005
to their budget constraint
ct + bt +mt ≤It−1Πt
bt−1 +Imt−1Πt
mt−1 + wtht + τt, (B.1)
where It denotes the gross nominal interest rate on bonds, Imt the gross nominal interest rate
on money, Πt ≡ Pt/Pt−1 the gross in�ation rate (with Pt the price level), wt the real wage,
and τt captures �rm pro�ts and the government's lump-sum taxes or transfers. Let λt denote1010
the Lagrange multiplier on the period-t budget constraint. The �rst-order conditions of this
maximization problem are
λt = ζtuc (ct,mt) , (B.2)
1
It= βEt
{λt+1
λtΠt+1
}, (B.3)1015
λtwt =ζtv′ (ht)
ϕ1,t
, (B.4)
ζtum (ct,mt) = λt − βImt Et{λt+1
Πt+1
}.
Using (B.2) and (B.3), we can rewrite the last condition as1020
ImtIt
= 1− um (ct,mt)
uc (ct,mt). (B.5)
B.2. Firms
There is a continuum of monopolistically competitive �rms owned by households and
indexed by i ∈ [0, 1]. Each �rm i uses ht(i) units of labor to produce
yt (i) = ϕ2,tf [ht (i)] (B.6)1025
units of output. The production function f , de�ned over R≥0, is twice di�erentiable, strictly
increasing (f ′ > 0), weakly concave (f ′′ ≤ 0), and such that f(0) = 0. The stochastic
exogenous technology shock ϕ2,t, of mean one, is the second of the four alternative supply
shocks that we consider. The third supply shock that we consider, ϕ3,t, also of mean one,
Online Appendix − 7
captures a labor subsidy received by �rms (when ϕ3,t > 1) or labor tax paid by �rms (when1030
ϕ3,t < 1): if Wt denotes the pre-subsidy or pre-tax nominal wage, then the after-subsidy or
after-tax nominal wage paid by �rms is Wt/ϕ3,t.
Following Calvo (1983), we assume that at any date, each �rm, whatever its history, has
the probability θ ∈ [0, 1) not to be allowed to reset its price. If allowed to reset its price at
date t, �rm i chooses its new price P ∗t (i) to maximize the present value of the pro�ts that1035
this price will generate:
Et
{+∞∑k=0
(βθ)kλt+k
λtΠt,t+k
[P ∗t (i) yt+k (i)− Wt+kht+k (i)
ϕ3,t+k
]},
subject to the production function (B.6) and the demand schedule
yt+k (i) =
[P ∗t (i)
Pt+k
]−εϕ4,t+k
yt+k, (B.7)
where Πt,t+k ≡ Pt+k/Pt for any k ∈ N, ε > 0 denotes the steady-state elasticity of sub-1040
stitution between di�erentiated goods, and yt ≡ [∫ 1
0yt(i)
(εϕ4,t−1)/(εϕ4,t)di]εϕ4,t/(εϕ4,t−1). The
stochastic exogenous shock ϕ4,t, of mean one, a�ecting the elasticity of substitution between
di�erentiated goods, is the last of the four alternative supply shocks that we consider.
Using (B.6), we can rewrite the present value of the pro�ts generated by P ∗t (i) as
Et
{+∞∑k=0
(βθ)kλt+k
λtΠt,t+k
[P ∗t (i) yt+k (i)− Wt+k
ϕ3,t+k
f−1[yt+k (i)
ϕ2,t+k
]]}.1045
Choosing P ∗t (i) to maximize this present value subject to (B.7) leads to the �rst-order con-
dition
Et
{+∞∑k=0
(βθ)kλt+k (εϕ4,t+k − 1)
λtΠt,t+k
[P ∗t (i)−
(εϕ4,t+k
εϕ4,t+k − 1
)Wt+k
ϕ2,t+kϕ3,t+kf ′ [ht+k (i)]
]yt+k (i)
}= 0.
(B.8)
In the limit case of perfectly �exible prices (θ = 0), and in a symmetric equilibrium (P ∗t (i) =
Online Appendix − 8
Pt and ht(i) = ht), this �rst-order condition becomes1050
Pt =
(εϕ4,t
εϕ4,t − 1
)Wt
ϕ2,tϕ3,tf ′ (ht). (B.9)
B.3. Government
The government consists of a �scal authority and a monetary authority. The �scal au-
thority consumes an exogenous quantity gt ≥ 0 of goods, does not issue bonds, and sets
lump-sum taxes on households so as to balance its budget (making �scal policy Ricardian).1055
We assume for simplicity that government purchases gt are wasted, but the results would be
unchanged if they entered households' utility function in a separable way.
The monetary authority − i.e., the central bank − has two independent instruments: the
nominal stock of money Mt > 0, or equivalently its (gross) growth rate µt ≡ Mt/Mt−1 >
0, and the (gross) nominal interest rate on money Imt ≥ 0. We assume that the central1060
bank injects reserves via lump-sum transfers.1 The consolidated budget constraint of the
government is thus
Mt = Imt−1Mt−1 + Ptgt − Tt, (B.10)
where Tt denotes the net lump-sum tax imposed by the government (the �scal authority's
tax minus the monetary authority's transfer).1065
To capture a lower bound on Imt in a simple and stark way, we assume that cash (with
no interest payments) is a perfect substitute for deposits at the central bank in terms of
providing liquidity services to households. This introduces a zero lower bound (ZLB) for the
net nominal IOR rate Imt − 1 in our model. In an equilibrium with Imt > 1, households will
hold no cash. In an equilibrium with Imt = 1, the decomposition of money into reserves and1070
cash will be indeterminate, but also inconsequential.
1It would be straightforward to modify our model and allow changes in money balances to be matchedby changes in the monetary authority's holdings of bonds issued by households or the �scal authority; suchfeatures, however, would not play a role in our analysis.
Online Appendix − 9
B.4. Market-Clearing Conditions
The bond-market-clearing condition is
bt = 0,
the money-market-clearing condition is1075
mt =Mt
Pt, (B.11)
and the goods-market-clearing condition is
ct + gt = yt. (B.12)
B.5. Steady-State Existence and Uniqueness
We consider steady-state values of policy-instruments such that Im ≥ 1, µ = 1, and g ≥ 0.1080
Since µ = 1, the set of steady states is the same under sticky prices (θ > 0) as under �exible
prices (θ = 0), so that we can use the �rst-order condition of �rms' optimization problem
under �exible prices (B.9) to characterize this set. We �rst use (B.2), (B.4), (B.6), (B.9),
and (B.12) to get
uc [f (h)− g,m] =
(ε
ε− 1
)v′ (h)
f ′ (h). (B.13)1085
We then consider two alternative cases in turn, separable utility (ucm = 0) and non-separable
utility (ucm > 0). We show that in both cases, the necessary and su�cient condition for
steady-state existence and uniqueness is Im < 1/β.
In the separable-utility case, the left-hand side of (B.13) does not depend on m and
decreases from +∞ to 0 as h increases from h ≡ f−1(g) to +∞. The right-hand side of1090
(B.13) increases as h increases from h to +∞. Therefore, there is a unique value of h in
(h,+∞) that satis�es (B.13). Moreover, (B.3), (B.5), and (B.13) imply
(ε− 1
ε
)f ′ (h)
v′ (h)um [f (h)− g,m] = 1− βIm. (B.14)
The left-hand side of (B.14) decreases from +∞ to 0 asm increases from 0 to +∞. Therefore,
Online Appendix − 10
there is a unique value of m that satis�es (B.14) if and only if the right-hand side of (B.14)1095
is positive. In other words, there exists a unique steady state if and only if Im < 1/β.
In the non-separable-utility case, (B.13) implicitly and uniquely de�nes a function M
such that
m =M (h) . (B.15)
This function is de�ned over (h,+∞), and it is strictly increasing (M′ > 0). We then use1100
(B.3), (B.5), (B.13), and (B.15) to get
(ε− 1
ε
)f ′ (h)
v′ (h)um [f (h)− g,M (h)] = 1− βIm. (B.16)
The function z(h) ≡ um [f (h)− g,M (h)] is strictly decreasing in h. The reason is that
(B.13) implies that uc [f (h)− g,M (h)] is strictly increasing in h, i.e. that
ucc [f (h)− g,M (h)] f ′ (h) + ucm [f (h)− g,M (h)]M′ (h) > 0,1105
which implies in turn that
z′ (h) = ucmf′ (h) + ummM′ (h) <
−f ′ (h)
ucm
(uccumm − u2cm
)≤ 0,
where the functions ucc, umm, and ucm are evaluated at [f (h)− g,M (h)]. Since z′(h) < 0,
the left-hand side of (B.16) decreases from +∞ to 0 as h increases from h to +∞. Therefore,
there is a unique value of h that satis�es this equation if and only if its right-hand side is1110
positive. In other words, there exists a unique steady state if and only if Im < 1/β.
B.6. Log-Linearization
We assume that Im < 1/β and log-linearize the equilibrium conditions of the model
around its unique steady state. To derive the Phillips curve (C.2), we log-linearize �rms'
�rst-order condition (B.8), and use the de�nition of the real wage wt ≡ Wt/Pt, to get1115
P ∗t = (1− βθ)Et
{+∞∑k=0
(βθ)k[wt+k + Pt+k − mp t+k|t − ϕ3,t+k −
ϕ4,t+k
ε− 1
]}, (B.17)
Online Appendix − 11
where variables with hats denote log-deviations from steady-state values, it ≡ It, and mp t+k|t
denotes the marginal productivity in period t+k for a �rm whose price was last set in period
t. Log-linearizing the production function (B.6) gives
ht =f
f ′h(yt − ϕ2,t) , (B.18)1120
so that we can rewrite mp t+k|t as
mp t+k|t = ϕ2,t +f ′′h
f ′h t+k|t = mpt+k +
f ′′h
f ′
(h t+k|t − ht+k
)= mpt+k +
ff ′′
(f ′)2(y t+k|t − yt+k
)= mpt+k −
εff ′′
(f ′)2
(P ∗t − Pt+k
), (B.19)
where mpt+k denotes the average marginal productivity in period t+k. Using this result and
πt ≡ log (Πt) = (1− θ)(P ∗t − Pt−1
),1125
and following the same steps as in, e.g., Galí (2008, Chapter 3), we can rewrite (B.17) as
πt = βEt {πt+1}+(1− θ) (1− βθ)
θ[1− εff ′′
(f ′)2
] (wt − mpt − ϕ3,t −
ϕ4,t
ε− 1
). (B.20)
Now, log-linearizing the goods-market-clearing condition (B.12) gives
ct + gt = yt, (B.21)
where ct ≡ (c/y)ct and gt ≡ (g/y)gt. Log-linearizing the �rst-order condition (B.4), and1130
using (B.18) and (B.21), gives
wt =
(−uccy
uc+v′′h
v′f
f ′h
)yt −
ucmm
ucmt +
uccy
ucgt − ϕ1,t −
v′′h
v′f
f ′hϕ2,t. (B.22)
Moreover, we have
mpt = ϕ2,t +ff ′′
(f ′)2(yt − ϕ2,t) . (B.23)
Online Appendix − 12
Using (B.22) and (B.23), we can then rewrite (B.20) as the Phillips curve1135
πt = βEt {πt+1}+ κ (yt − δmmt − δggt − δϕϕt) (B.24)
with
κ ≡ (1− θ) (1− βθ)
θ[1− εff ′′
(f ′)2
] ψ > 0,
δm ≡(ucmm
uc
)ψ−1 ≥ 0,
δg ≡(−uccyuc
)ψ−1 ∈ (0, 1),1140
δϕ ≡{1ϕt=ϕ1,t +
[1 +
v′′h
v′f
f ′h− ff ′′
(f ′)2
]1ϕt=ϕ2,t + 1ϕt=ϕ3,t +
(1
ε− 1
)1ϕt=ϕ4,t
}ψ−1 > 0,
where
ψ ≡ −uccyuc
+v′′h
v′f
f ′h− ff ′′
(f ′)2> 0.
Note that, to write this Phillips curve in a compact way, we have considered a single supply
shock ϕt ∈ {ϕ1,t, ϕ2,t, ϕ3,t, ϕ4,t} and used indicator functions in the de�nition of δϕ: for any1145
k ∈ {1, 2, 3, 4}, 1ϕt=ϕk,t takes the value one if ϕt = ϕk,t and the value zero otherwise.
Log-linearizing the �rst-order condition (B.5) and using (B.21) gives the money-demand
equation
mt = χy (yt − gt)− χi (it − imt ) , (B.25)
where imt ≡ Imt and1150
χy ≡(ucmm
uc− ummm
um
)−1(ucmy
um− uccy
uc
)> 0,
χi ≡(ucmm
uc− ummm
um
)−1(βIm
1− βIm
)> 0.
Finally, log-linearizing the �rst-order condition (B.3) and using (B.21) gives the IS equa-
Online Appendix − 13
tion
yt = Et {yt+1} −1
σ(it − Et {πt+1} − rt)− ηEt {∆mt+1} − Et {∆gt+1} , (B.26)1155
where ∆ ≡ 1− L denotes the �rst-di�erence operator, rt ≡ −Et{∆ζt+1}, and
σ ≡ −uccyuc
> 0,
η ≡(−uccyuc
)−1ucmm
uc≥ 0.
Appendix C: MIU Model (Log-Linearized Version)
This appendix proves Proposition 6 (stated in the main text), which essentially says that1160
our MIU model delivers the same results as our simple model. The �rst subsection provides
an outline of the proof, following the same steps as in Section 3 for our simple model. The
following subsections prove some speci�c claims made in the �rst subsection.
For convenience, we keep the same notations as in our simple model in Section 3 for the
reduced-form parameters (σ, κ, δg, δϕ, χy, χi), the characteristic polynomial (P(X)), the1165
roots of this polynomial (ρ, ω1, ω2), and the exogenous driving term in the dynamic equation
(Zt), although all of them are in fact model-speci�c.
C.1. Outline of the Proof of Proposition 6
We start from the log-linearized reduced form of our MIU model, made of the IS equa-
tion (B.26), the Phillips curve (B.24), and the money-demand equation (B.25) derived in1170
Appendix B.6. For simplicity, we replace the notations yt, mt, gt, and ϕt with the notations
yt, mt, gt, and ϕt (as everywhere in the main text), and we thus write these equations as
yt = Et {yt+1} −1
σ(it − Et {πt+1} − rt) + η (mt − Et {mt+1}) + gt − Et {gt+1} , (C.1)
πt = βEt {πt+1}+ κ (yt − δmmt − δggt − δϕϕt) , (C.2)
mt = χy (yt − gt)− χi (it − imt ) , (C.3)1175
where β ∈ (0, 1), η ≥ 0, δm ≥ 0, δg ∈ (0, 1), and all the other parameters are positive.
Online Appendix − 14
In the case in which the utility function is not separable in consumption and money, we
have η > 0 and δm > 0. In this case, the IS equation (C.1) involves real-money terms (in
mt and Et{mt+1}) because the marginal utility of consumption in the consumption Euler
equation depends on real money. Similarly, the Phillips curve (C.2) involves a real-money1180
term (in mt) because real money increases the marginal utility of consumption, which in
turn decreases the real wage and hence the marginal cost of production of �rms. In the
alternative case in which the utility function is separable in consumption and money, we
have η = δm = 0, and these two equations become identical to the IS equation (1) and the
Phillips curve (2) of the two models considered so far (the basic NK model in Section 21185
and our simple model in Section 3). The money-demand equation (C.3) is isomorphic to its
counterpart (7) in our simple model, except for the presence of the government-purchases
shock gt. This shock appears in (C.3) because money demand now depends on consumption,
which we have eliminated using the goods-market-clearing condition.
Our MIU model implies, in particular, the following two restrictions on the reduced-form1190
parameters:
η =δmδg
, (C.4)
δmχy < 1, (C.5)
as we show in Appendix C.2. These restrictions will play an important role in our determinacy1195
result below (as we will see). The equality (C.4) says that the weight of mt relative to gt
(and Et{mt+1} relative to Et{gt+1}) in the IS equation, η, is identical to the weight of mt
relative to gt in the Phillips curve, δm/δg. The reason is that mt and gt come exclusively from
the marginal utility of consumption in both equations. The marginal utility of consumption
depends negatively on consumption, and therefore positively on gt for a given yt (through the1200
goods-market-clearing condition); and it depends non-negatively on mt, with a weight of mt
relative to gt equal to η = δm/δg. The inequality (C.5) re�ects how holding money mitigates
changes in �rms' marginal cost of production (through the real wage). For a given spread
it − imt , a rise in output yt has two opposite e�ects on �rms' marginal cost of production
(i.e., on the term in factor of κ in the Phillips curve): a standard positive direct e�ect (with1205
Online Appendix − 15
elasticity 1), and a negative indirect e�ect via the implied rise in money mt (with elasticity
δmχy). The inequality states that the direct e�ect dominates the indirect one (i.e., δmχy < 1).
Under permanently exogenous monetary-policy instruments imt and Mt (in particular
imt = i∗t exogenous for all t ∈ Z), the IS equation (C.1), the Phillips curve (C.2), the money-
demand equation (C.3), and the identities mt = Mt − pt and πt = pt − pt−1 lead to the1210
following dynamic equation relating pt to Et{pt+2}, Et{pt+1}, pt−1, and exogenous terms:
Et{LP
(L−1
)pt}
= Zt
with P (X) ≡ X3 −[2 +
1
β+
κ
βσ+
(1− δg) δmκβδg
+χyσχi
]X2 +
[1 +
2
β+
κ
βσ
+(1− δg) δmκ
βδg+
(1 + β)χyβσχi
+(1− δmχy)κ
βσχi
]X −
(1
β+
χyβσχi
),1215
Zt ≡−κβσ
(i∗t − rt) +
[(1− δg) δm
δg+
1− δmχyσχi
]κ
βMt −
(1− δg) δmκβδg
Et {Mt+1}
+
(1 +
χyσχi
)(1− δg)κ
βgt −
(1− δg)κβ
Et {gt+1} −(
1 +χyσχi
)δϕκ
βϕt +
δϕκ
βEt {ϕt+1} ,
where we have used the equality (C.4) to replace η by δm/δg. Using the inequality (C.5), we
show in Appendix C.3 that, as in our simple model of Section 3, the characteristic polynomial1220
P(X) has one root inside the unit circle (ρ ∈ (0, 1)) and two roots outside the unit circle (ω1
and ω2 with |ω1| ≤ |ω2|). With one eigenvalue inside the unit circle (ρ) for one predetermined
variable (pt−1), thus, our MIU model satis�es Blanchard and Kahn's (1980) conditions and
has a unique bounded solution under permanently exogenous monetary-policy instruments.
In the MIU model, as in the simple model of Section 3, setting exogenously imt and Mt1225
amounts to following a �shadowWicksellian rule� for it. Indeed, if the price level rises (making
real money fall, given that nominal money is �xed), or if output rises, then the marginal
utility of real money increases. Since the IOR rate is �xed, the interest rate on bonds has
then to increase for private agents to remain indi�erent between holding money and holding
bonds. What is di�erent from Section 3, however, is that existing results for Wicksellian1230
Online Appendix − 16
rules in the basic NK model (e.g., Woodford, 2003, Chapter 4) do not apply to the MIU
model with non-separable utility (i.e. with η = δm/δg > 0). In fact, not all Wicksellian rules
would ensure determinacy in this model. What our determinacy result says, thus, is that the
speci�c shadow Wicksellian rule that arises under permanently exogenous monetary-policy
instruments, given the restriction (C.5) that the model imposes on its coe�cients, always1235
delivers determinacy.
We solve the dynamic equation forward in the same way as in Section 3, and obtain that
in�ation in the unique bounded solution is again characterized by (10) − keeping in mind,
though, that the roots ρ, ω1, ω2, and the exogenous driving term Zt have changed. Using
the solution for in�ation (10), the Phillips curve (C.2), and the identities mt = Mt − pt and1240
πt = pt − pt−1, we then get the solution for output:
yt = −ϑpt−1 + δmMt + δggt + δϕϕt −Et
(ω2 − ω1)κ
{+∞∑k=0
(ξ1ω
−k−11 − ξ2ω−k−12
)Zt+k
}, (C.6)
where now ϑ ≡ (1 − ρ)(1 − βρ)/κ + δmρ and ξj ≡ β(ωj + ρ − 1) + κδm − 1 for j ∈ {1, 2}.
Like our simple model's equilibrium (10)-(11), and unlike the basic NK model's standard
equilibrium (5)-(6), the MIU model's equilibrium (10) and (C.6) involves only ω−k1 and ω−k21245
terms with ω1 > 1 and ω2 > 1. Therefore, the longer the horizon k, the smaller the e�ects of
shocks occurring at date t+k on in�ation and output at date t in the MIU model, regardless
of which type of shock (preference, monetary, �scal, or supply) we consider. In particular,
neither the forward-guidance puzzle nor the �scal-multiplier puzzle can arise in the MIU
model. Moreover, because determinacy obtains for any degree of price stickiness θ ∈ (0, 1)1250
and in particular as θ → 0, the paradox of �exibility does not arise either. In Appendix C.4,
we show that the limits of πt and yt as θ → 0 take �nite values, and that these values coincide
with the values that πt and yt take under perfectly �exible prices.
In Appendix C.5, we show that we can asymptotically remove the monetary friction from
our MIU model in (at least) two cases: the case with separable utility, and the case of utility1255
over a constant-elasticity-of-substitution (CES) aggregator of money and consumption. In
either case, as we remove the monetary friction at the same speed as we shrink the steady-
state spread between the interest rate on bonds and the IOR rate, the steady state and
Online Appendix − 17
reduced form of our MIU model converge to the steady state and reduced form of the basic
NK model, with real money balances bounded away from zero and in�nity along the way. In1260
particular, the reduced-form parameters σ, κ, δg, and δϕ converge to their counterparts in the
basic NK model, while η, δm, 1/χi, and χy/χi converge to zero. As a result, the characteristic
polynomial P(X) goes to (X − 1)Pb(X); its roots ρ, ω1, and ω2 go respectively to ρb, 1, and
ωb; and the exogenous driving term Zt goes to Zbt . Using these limit results, we get that the
unique local equilibrium of our MIU model (10) and (C.6) converges to (12)-(13). Thus, our1265
MIU model serves to select the same equilibrium of the basic NK model under a permanently
exogenous policy rate as our simple model in the previous section.
Proposition 6 follows.
C.2. Restrictions on the Reduced-Form Parameters
The reduced-form parameters η, δm, and δg, which are de�ned in Appendix B.6, are1270
straightforwardly linked to each other through the equality (C.4). The reduced-form pa-
rameters δm and χy, which are also de�ned in Appendix B.6, satisfy the inequality (C.5)
because
1− δmχy = 1−[−uccyuc
+v′′h
v′f
f ′h− ff ′′
(f ′)2
]−1(ucmm
uc− ummm
um
)−1(ucmc
um− uccy
uc
)ucmm
uc
=
[−uccyuc
+v′′h
v′f
f ′h− ff ′′
(f ′)2
]−1(ucmm
uc− ummm
um
)−1{[v′′h
v′f
f ′h− ff ′′
(f ′)2
]1275 (
ucmm
uc− ummm
um
)+
(y − c)muccummucum
+cm
ucum
(uccumm − u2cm
)}> 0.
C.3. Root Analysis
We �rst show that 0 < ρ < 1 < |ω1| ≤ |ω2|. To that aim, we write the polynomial P(X)
as1280
P (X) = X3 − a2X2 + a1X − a0
Online Appendix − 18
with
a2 ≡ 2 +1
β+
κ
βσ+
(1− δg) δmκβδg
+χyσχi
> 3,
a1 ≡ 1 +2
β+
κ
βσ+
(1− δg) δmκβδg
+(1 + β)χyβσχi
+(1− δmχy)κ
βσχi> 0,
a0 ≡1
β+
χyβσχi
> 0,1285
where the inequality a2 > 3 comes from β ∈ (0, 1) and δg ∈ (0, 1), and the inequality a1 > 0
from δg ∈ (0, 1) and (C.5).
We have P(0) = −a0 < 0 and P(1) = (1− δmχy)κ/(βσχi) > 0, where the last inequality
comes from (C.5). Therefore, P(X) has either one or three real roots inside (0, 1). Moreover,
the inequalities a2 > 0, a1 > 0, and a0 > 0 imply that P(X) < 0 for all X < 0, so that P(X)1290
has no negative real roots. Therefore, P(X) has at least one real root inside (0, 1), which we
denote by ρ, and its other two roots, which we denote by ω1 and ω2 with |ω1| ≤ |ω2|, must
be (i) both real and inside (0, 1), or (ii) both real and larger than 1, or (iii) both complex
and conjugates of each other. Now, we have ρ + ω1 + ω2 = a2 > 3. Therefore, Case (i)
is impossible, and in Case (iii) the common real part of ω1 and ω2 is larger than 1. As a1295
consequence, in the remaining two possible cases, namely Cases (ii) and (iii), ω1 and ω2 lie
outside the unit circle.
We now show that ω1 and ω2 can be real numbers, and that they can also be complex
(non-real) numbers. Consider, for example, the separable and iso-elastic speci�cation
u(ct,mt) =c1−σct − 1
1− σc+m1−σmt − 1
1− σm,1300
where σc > 0 and σm > 0. Under this speci�cation, σ, κ, δm, δg, and χy/χi do not depend
on σm, but χi and χy do. Therefore, a2 and a0 do not depend on σm, but a1 does. Since
limσm→+∞ χi = 0, we have limσm→+∞ a1 = +∞. As a consequence, for su�ciently large
values of σm, P(X) = X3 − a2X2 + a1X − a0 is positive for all X ≥ 1, so that Case (ii) is
Online Appendix − 19
impossible and ω1 and ω2 are complex numbers. Moreover, since limσm→0 χi = +∞, we have1305
limσm→0
P(
1 +χyσχi
)= −
(1 +
χyσχi
)χyκ
βσ2χi< 0.
Therefore, for su�ciently small values of σm, we have P [1 + χy/(σχi)] < 0, which, together
with P(1) > 0, implies that ω1 and ω2 are positive real numbers.
By continuity, there also exist non-separable speci�cations of u that can make ω1 and
ω2 real or complex depending on the calibration. Consider, for instance, the iso-elastic1310
speci�cation
u(ct,mt) =c1−σct − 1
1− σc+m1−σmt − 1
1− σm+ εcνtm
1−νt ,
where ν ∈ (0, 1) and ε > 0. If ε is su�ciently small, then, as above, ω1 and ω2 will be real
for su�ciently small values of σm and complex for su�ciently large values of σm.
C.4. Resolution of the Paradox of Flexibility1315
Using the de�nition of Zt, and after some simple algebra, we can rewrite (10) and (C.6)
as
πt = − (1− ρ) pt−1 +κ
β (ω2 − ω1)Et
{− 1
σ
+∞∑k=0
(ω−k−11 − ω−k−12
) (i∗t+k − rt+k
)+
+∞∑k=0
(ξM1 ω
−k−11 − ξM2 ω−k−12
)Mt+k −
+∞∑k=0
(ξg1ω
−k−11 − ξg2ω−k−12
)gt+k
++∞∑k=0
(ξϕ1 ω
−k−11 − ξϕ2 ω−k−12
)ϕt+k
}, (C.7)1320
yt = −ϑpt−1 +δmδgMt + gt +
Etβ (ω2 − ω1)
{1
σ
+∞∑k=0
(ξ1ω
−k−11 − ξ2ω−k−12
) (i∗t+k − rt+k
)−
+∞∑k=0
(ξ1ξ
M1 ω
−k−11 − ξ2ξM2 ω−k−12
)Mt+k +
+∞∑k=0
(ξ1ξ
g1ω−k−11 − ξ2ξg2ω−k−12
)gt+k
−+∞∑k=0
(ξ1ξ
ϕ1 ω−k−11 − ξ2ξϕ2 ω−k−12
)ϕt+k
}, (C.8)
Online Appendix − 20
where ϑ ≡ (1− ρ)(1− βρ)/κ+ δmρ and
ξj ≡ β (ωj + ρ− 1) + κδm − 1,1325
ξMj ≡ 1− δmχyσχi
− (1− δg) (ωj − 1) δmδg
,
ξgj ≡(ωj − 1− χy
σχi
)(1− δg) ,
ξϕj ≡(ωj − 1− χy
σχi
)δϕ
for j ∈ {1, 2}.
The only parameter that depends on the degree of price stickiness θ in the structural1330
equations (C.1), (C.2), and (C.3) is the slope κ of the Phillips curve (C.2). We have limθ→0 κ =
+∞ and hence [−βσδg
δg + (1− δg)σδm
]limθ→0
[P (X)
κ
]= X (X − ωn1 )
for any X ∈ R, where
ωn1 ≡ 1 +
[1− δmχy
δg + (1− δg)σδm
]δgχi
> 1,1335
where in turn the inequality follows from δg ∈ (0, 1) and (C.5). Therefore, we get
limθ→0
ρ = 0, limθ→0
ω1 = ωn1 , and limθ→0
ω2 = +∞. (C.9)
Using (C.9) and
(1− ρ) (ω1 − 1) (ω2 − 1) = P (1) =(1− δmχy)κ
βσχi,
we also get that1340
limθ→0
κ
ω2
=βσδg
δg + (1− δg)σδm. (C.10)
Online Appendix − 21
Using (C.9) and (C.10), we can easily determine the limits of (C.7) and (C.8) as θ → 0:
limθ→0
πt = −pt−1 +δg
δg + (1− δg)σδmEt
{+∞∑k=0
(ωn1 )−k−1{−(i∗t+k − rt+k
)+ (ωn1 − 1)Mt+k +
[χyχi− σ (ωn1 − 1)
][(1− δg) gt+k − δϕϕt+k]
}}+
σδgδg + (1− δg)σδm
[(1− δg) δm
δgMt + (1− δg) gt − δϕϕt
], (C.11)1345
limθ→0
yt =δgδm
δg + (1− δg)σδmEt
{+∞∑k=0
(ωn1 )−k−1{i∗t+k − rt+k − (ωn1 − 1)Mt+k
+
[σ (ωn1 − 1)− χy
χi
][(1− δg) gt+k − δϕϕt+k]
}}+
δgδg + (1− δg)σδm
[δmMt + δggt +
(1 +
σδmδg
)δϕϕt
]. (C.12)
These limits are �nite, unlike their counterparts in the basic NK model.
We now show that the right-hand sides of (C.11) and (C.12) coincide with the values1350
taken by πt and yt when prices are perfectly �exible (θ = 0). To determine these values, we
�rst log-linearize the �rst-order condition of �rms' optimization problem under �exible prices
(B.9), and use (B.18), to get
wt =ff ′′
(f ′)2yt +
[1− ff ′′
(f ′)2
]ϕ2,t + ϕ3,t +
ϕ4,t
ε− 1. (C.13)
Using (B.22) and (C.13), considering a single supply shock ϕt ∈ {ϕ1,t, ϕ2,t, ϕ3,t, ϕ4,t}, and1355
replacing the notations yt, mt, gt, and ϕt by the notations yt, mt, gt, and ϕt (for simplicity
and consistency with the main text), we then get
yt = δmmt + δggt + δϕϕt. (C.14)
Finally, using the IS equation (C.1), the money-demand equation (C.3), the identity mt =
Mt − pt, the exogenous policy-rate setting imt = i∗t , and the solution for �exible-price output1360
Online Appendix − 22
(C.14), we get the following dynamic equation under �exible prices:
pt = (ωn1 )−1 Et {pt+1}+δg (ωn1 )−1
δg + (1− δg)σδm
{− (i∗t − rt) +
[1− δmχy
χi+
(1− δg)σδmδg
]Mt
−(1− δg)σδmδg
Et {Mt+1}+
(σ +
χyχi
)(1− δg) gt − σ (1− δg)Et {gt+1}
−(σ +
χyχi
)δϕϕt + σδϕEt {ϕt+1}
},
where we have used the equality (C.4) to replace η by δm/δg. Iterating this equation forward1365
to +∞ leads to the following value for the price level pt in our MIU model under �exible
prices:
pt =δg
δg + (1− δg)σδmEt
{+∞∑k=0
(ωn1 )−k−1{−(i∗t+k − rt+k
)+ (ωn1 − 1)Mt+k
+
[χyχi− σ (ωn1 − 1)
][(1− δg) gt+k − δϕϕt+k]
}}+
σδgδg + (1− δg)σδm
[(1− δg) δm
δgMt + (1− δg) gt − δϕϕt
], (C.15)1370
which implies in turn that the value of πt ≡ pt− pt−1 in our MIU model under �exible prices
coincides with the right-hand side of (C.11). In turn, using (C.14), (C.15), and the identity
mt = Mt − pt, we get that the value of yt in our MIU model under �exible prices coincides
with the right-hand side of (C.12). Thus, our MIU model solves the paradox of �exibility:
the limits of πt and yt as θ → 0 are �nite and coincide with the values of πt and yt when1375
θ = 0.
C.5. Convergence to the Basic NK Model
We start with the separable speci�cation
u(ct,mt) = u1 (ct) + γu2 (mt) ,
where γ > 0 is a scale parameter. Under this speci�cation, the steady-state value h of ht,1380
given by (B.13) with uc[f(h)− g,m] = u′1[f(h)− g], is identical to the steady-state value of
ht in the basic NK model (with consumption-utility function u1). The IS equation (C.1) and
Online Appendix − 23
the Phillips curve (C.2) are also identical to the IS equation (1) and the Phillips curve (2) of
the basic NK model, in the sense that their reduced-form parameters take the same values
(in particular η = 0 and δm = 0). The steady-state value m of mt is given by (B.14), which1385
can be rewritten as
u′2 (m) =
(1− βIm
γ
)(ε
ε− 1
)v′ (h)
f ′ (h). (C.16)
If (Im, γ) goes to (1/β, 0) with (1 − βIm)/γ bounded away from zero and in�nity, as in the
thought experiment of Subsection 4.1, then m is bounded away from zero and in�nity. In
this case, χy = (−u′′2m/u′2)−1(−u′′1y/u′1) is also bounded away from zero and in�nity, while1390
χi = (−u′′2m/u′2)−1βIm/(1 − βIm) goes to in�nity. Therefore, 1/χi and χy/χi converge to
zero, and the money-demand equation (C.3) converges to it = imt . Alternatively, if Im goes
to 1/β holding γ constant, as in the policy experiment of Section 5, then m goes to in�nity
(asymptotic satiation). In that case, we still have 1/χi and χy/χi converging to zero, and
(C.3) converging to it = imt , if the elasticity −u′′2m/u′2 is bounded from above − a condition1395
that is met, in particular, for isoelastic u2 functions.
We now turn to the CES-based speci�cation
u(ct,mt) = U{
[(1− γ) cαt + γmαt ]1/α
},
where α ∈ (−∞, 1), γ ∈ (0, 1), and the function U , de�ned over the set of positive real
numbers R>0, is twice di�erentiable, strictly increasing (U ′ > 0), and strictly concave (U ′′ <1400
0). In addition, we impose that U ′′(x)x/U ′(x) ≤ 1− α for any x > 0, which is the necessary
and su�cient condition for ucm ≥ 0. Under this speci�cation, (B.5) at the steady state can
be rewritten as m = φc = φ[f(h)− g], where
φ ≡[
γ
(1− γ) (1− βIm)
] 11−α
,
which implies that (B.13) can in turn be rewritten as1405
(1− γ) [(1− γ) + γφα]1−αα U ′
{[(1− γ) + γφα]
1α [f (h)− g]
}=
(ε
ε− 1
)v′ (h)
f ′ (h). (C.17)
Online Appendix − 24
If (Im, γ) goes to (1/β, 0) with (1 − βIm)/γ bounded away from zero and in�nity, as in the
thought experiment of Subsection 4.1, then φ is bounded away from zero and in�nity, and
(C.17) converges to
U ′ [f (h)− g] =
(ε
ε− 1
)v′ (h)
f ′ (h).1410
This last equation, which characterizes the limit value of h, is the same as the equation
implicitly and uniquely de�ning the steady-state value h? of ht in the basic NK model (with
consumption-utility function U). Therefore, h, y, and c converge respectively to h?, y? ≡
f(h?), and c? ≡ f(h?)− g, while m = φ[f(h)− g] is bounded away from zero and in�nity. As
a consequence, we have C ≡ [(1− γ)cα + γmα]1/α → c? and1415
−uccyuc
=( cC
)α yc
{(1− α) γφα + (1− γ)
[−U ′′ (C)C
U ′ (C)
]}−→ −U ′′ (c?) y?
U ′ (c?),
ucmm
uc= γφα
( cC
)α{(1− α)−
[−U ′′ (C)C
U ′ (C)
]}−→ 0,
−ummmum
=( cC
)α{(1− α) (1− γ) + γφα
[−U ′′ (C)C
U ′ (C)
]}−→ 1− α,
ucmy
um= (1− γ)
( cC
)α yc
{(1− α)−
[−U ′′ (C)C
U ′ (C)
]}−→ (1− α)
y?
c?−[−U ′′ (c?) y?
U ′ (c?)
].
Using these limit results, we get that the reduced-form parameters σ, κ, δg, and δϕ converge1420
to their counterparts in the basic NK model, while η, δm, 1/χi, and χy/χi converge to zero.
We conclude that the steady state and reduced form of our MIU model, under the CES-based
speci�cation, converge to the steady state and reduced form of the basic NK model.
Appendix D: Model With Banks
This appendix proves Proposition 7 (stated in the main text), which essentially says that1425
our model with banks delivers the same results as our simple model and our MIU model.
The �rst subsection provides an outline of the proof, following the same steps as in Section 3
for our simple model and Appendix C.1 for our MIU model. The following subsections prove
some speci�c claims made in the �rst subsection.
Online Appendix − 25
D.1. Outline of the Proof of Proposition 71430
As we show in Diba and Loisel (2020), the IS equation of our model with banks is the same
as the IS equation (1) of the basic NK model, while the Phillips curve and the money-demand
equation of our model with banks are
πt = βEt {πt+1}+ κ (yt − δmmt − δggt − δϕϕt) , (D.1)
mt = χyyt − χi (it − imt )− χggt − χϕϕt, (D.2)1435
where β ∈ (0, 1), δg ∈ (0, 1), χϕ ≥ 0, and all the other parameters are positive. The Phillips
curve (D.1) is isomorphic to its counterpart (C.2) in the MIU model − but not identical to it,
since the reduced-form parameters κ, δm, δg, and δϕ have changed (even though we keep, for
convenience, the same notation). The reason why real reservesmt appear in the Phillips curve
(D.1) is that they reduce banking costs, which in turn lowers the borrowing costs of �rms and1440
hence their marginal cost of production. Like its counterpart (C.3) in the MIU model, the
money-demand equation (D.2) involves the government-purchases shock gt because money
demand depends on consumption, which we have eliminated using the goods-market-clearing
condition.2 Unlike (C.3), however, it also involves the supply shock ϕt, because the demand
for reserves now depends on the volume of loans, which in turn depends on �rms' wage bill,1445
which in turn depends on the supply shock for a given output level.3
Our model with banks implies, in particular, the following restriction on the reduced-form
parameters:
σ < χy <1
δm, (D.3)
as we show in Diba and Loisel (2020). This double inequality will play a key role in our1450
results below (as we will see). The �rst inequality in (D.3) arises from the fact that bank
loans serve to �nance the wage bill (or some fraction of it). If output yt increases by 1%
for given government purchases gt, the marginal utility of consumption decreases by σ%;
2We no longer have χg = χy, though, because money demand now depends on yt not only throughconsumption (via the goods-market-clearing condition), but also through loans (which are proportional tothe wage bill).
3The only exception is when the supply shock is a markup shock − in which case χϕ = 0.
Online Appendix − 26
so, the wage, the wage bill, and loans all increase by more than σ%; and, in turn, so does
the demand for reserves mt for a given spread it − imt (i.e., χy > σ). The second inequality1455
in (D.3) is similar to the inequality (C.5) in the MIU model. Here, it re�ects how holding
reserves mitigates changes in banking costs. For a given spread it − imt , a rise in output yt
has two opposite e�ects on �rms' marginal cost of production (i.e., on the term in factor of
κ in the Phillips curve): a standard positive direct e�ect (with elasticity 1), and a negative
indirect e�ect via the implied rise in reserves mt (with elasticity δmχy). The inequality states1460
that the direct e�ect dominates the indirect one (i.e., δmχy < 1).
Under permanently exogenous monetary-policy instruments imt and Mt (in particular
imt = i∗t exogenous for all t ∈ Z), the IS equation (1), the Phillips curve (D.1), the money-
demand equation (D.2), and the identities mt = Mt − pt and πt = pt − pt−1 lead to the
following dynamic equation relating pt to Et{pt+2}, Et{pt+1}, pt−1, and exogenous terms:1465
Et{LP
(L−1
)pt}
= Zt,
where P (X) ≡ X3 −[2 +
1
β+
χyσχi
+
(1
σ− δm
)κ
β
]X2 +
[1 +
2
β+
(1 +
1
β
)χyσχi
+
(1
σ− δm
)κ
β+ (1− δmχy)
κ
βσχi
]X −
(1
β+
χyβσχi
),
Zt ≡−κβσ
(i∗t − rt) +
[1
σχi−(
1 +χyσχi
)δm
]κ
βMt +
δmκ
βEt {Mt+1}1470
+
[(1 +
χgσχi
)−(
1 +χyσχi
)δg
]κ
βgt −
(1− δg)κβ
Et {gt+1}
+
[χϕσχi−(
1 +χyσχi
)δϕ
]κ
βϕt +
δϕκ
βEt {ϕt+1} .
Using the double inequality (D.3), we show in Appendix D.2 that the roots of the character-
istic polynomial P(X) are three real numbers ρ, ω1, and ω2 such that 0 < ρ < 1 < ω1 < ω2.
With one eigenvalue inside the unit circle (ρ) for one predetermined variable (pt−1), thus,1475
our model with banks satis�es Blanchard and Kahn's (1980) conditions and has a unique
bounded solution under permanently exogenous monetary-policy instruments.
In our model with banks, setting exogenously imt and Mt also amounts to following a
Online Appendix − 27
�shadow Wicksellian rule� for it, as in the previous two models (the simple model of Section
3 and the MIU model of Subsection 4.1). Existing results for Wicksellian rules in the basic1480
NK model do not apply to our model with banks, and not all Wicksellian rules would ensure
determinacy in this model. What our determinacy result says is that the speci�c shadow
Wicksellian rule that arises under permanently exogenous monetary-policy instruments, given
the restriction (D.3) that the model imposes on its coe�cients, always delivers determinacy.
We determine the unique local equilibrium of our model with banks in the same way as1485
the unique local equilibrium of our MIU model in Appendix C.1. We obtain that in�ation
and output in this equilibrium are again characterized by (10) and (C.6) − keeping in mind,
though, that the roots ρ, ω1, ω2, the reduced-form parameters κ, δm, δg, δϕ, and the exogenous
driving term Zt have changed. Since (10) and (C.6) involve only ω−k1 and ω−k2 terms with
ω1 > 1 and ω2 > 1, neither the forward-guidance puzzle nor the �scal-multiplier puzzle can1490
arise in our model with banks. Moreover, because determinacy obtains for any degree of
price stickiness θ ∈ (0, 1) and in particular as θ → 0, the paradox of �exibility does not arise
either in this model, as we formally show in Appendix D.3.
As we elaborate in Appendix D.4, as the scale parameter of banking costs and the steady-
state interest-rate spread are shrunk to zero (at suitable rates, to keep a positive and �nite1495
level of steady-state real reserve balances in the limit), the steady state and reduced form of
our model with banks converge to the steady state and reduced form of the basic NK model.
Therefore, as previously, the characteristic polynomial P(X) goes to (X − 1)Pb(X); its roots
ρ, ω1, and ω2 go respectively to ρb, 1, and ωb; and the exogenous driving term Zt goes to Zbt .
Using these limit results, we get again that the unique local equilibrium of our model with1500
banks (10) and (C.6) converges to (12)-(13). Thus, our model with banks serves to select the
same equilibrium of the basic NK model under a permanently exogenous policy rate as our
previous two models.
Proposition 7 follows.
Online Appendix − 28
D.2. Root Analysis1505
We show that 0 < ρ < 1 < ω1 < ω2. The polynomial P(X) can be rewritten as
P (X) = X3 −(
1 + 2β + βΘ1 + Θ2
β
)X2 +
[2 + β + (1 + β) Θ1 + Θ2 + Θ3
β
]X −
(1 + Θ1
β
)= (X − 1−Θ1)
[X2 −
(1 + β + Θ2
β
)X +
1
β
]−(
Θ1Θ2 −Θ3
β
)X,
where Θ1 ≡ χy/(σχi) > 0, Θ2 ≡ (1/σ − δm)κ, and Θ3 ≡ (1 − δmχy)κ/(σχi). The double
inequality (D.3) implies Θ2 > 0, Θ3 > 0, and Θ1Θ2−Θ3 = (χy − σ)κ/(σ2χi) > 0. Therefore,1510
we get P(0) = −(1+Θ1)/β < 0, P(1) = Θ3/β > 0, P(1+Θ1) = −(Θ1Θ2−Θ3)(1+Θ1)/β < 0,
and limX→+∞P(X) = +∞ > 0. As a consequence, the roots of P (X) are three real numbers
ρ, ω1, and ω2 such that 0 < ρ < 1 < ω1 < 1 + Θ1 < ω2.
D.3. Resolution of the Paradox of Flexibility
Using the de�nition of Zt, and after some simple algebra, we can rewrite (10) and (C.6)1515
as
πt = − (1− ρ) pt−1 +κ
β (ω2 − ω1)Et
{− 1
σ
+∞∑k=0
(ω−k−11 − ω−k−12
) (i∗t+k − rt+k
)+
+∞∑k=0
(ξM1 ω
−k−11 − ξM2 ω−k−12
)Mt+k −
+∞∑k=0
(ξg1ω
−k−11 − ξg2ω−k−12
)gt+k
++∞∑k=0
(ξϕ1 ω
−k−11 − ξϕ2 ω−k−12
)ϕt+k
}, (D.4)
1520
yt = −ϑpt−1 + gt +Et
β (ω2 − ω1)
{1
σ
+∞∑k=0
(ξ1ω
−k−11 − ξ2ω−k−12
) (i∗t+k − rt+k
)−
+∞∑k=0
(ξ1ξ
M1 ω
−k−11 − ξ2ξM2 ω−k−12
)Mt+k +
+∞∑k=0
(ξ1ξ
g1ω−k−11 − ξ2ξg2ω−k−12
)gt+k
−+∞∑k=0
(ξ1ξ
ϕ1 ω−k−11 − ξ2ξϕ2 ω−k−12
)ϕt+k
}, (D.5)
Online Appendix − 29
where ϑ ≡ (1− ρ)(1− βρ)/κ+ δmρ and
ξj ≡ β (ωj + ρ− 1) + κδm − 1,1525
ξMj ≡ δm (ωj − 1) +1− δmχyσχi
,
ξgj ≡ (1− δg) (ωj − 1) +δgχy − χg
σχi,
ξϕj ≡ δϕ (ωj − 1) +χϕ − δϕχy
σχi
for j ∈ {1, 2}.
The only parameter that depends on the degree of price stickiness θ in the structural1530
equations (1), (D.1), and (D.2) is the slope κ of the Phillips curve (D.1). We have limθ→0 κ =
+∞ and hence (−βσ
1− σδm
)limθ→0
[P (X)
κ
]= X (X − ωn1 )
for any X ∈ R, where
ωn1 ≡ 1 +1− δmχy
(1− σδm)χi> 1,1535
where in turn the inequality follows from (D.3). Therefore, we get
limθ→0
ρ = 0, limθ→0
ω1 = ωn1 , and limθ→0
ω2 = +∞. (D.6)
Using (D.6) and
(1− ρ) (ω1 − 1) (ω2 − 1) = P (1) =(1− δmχy)κ
βσχi,
we also get that1540
limθ→0
κ
ω2
=βσ
1− σδm. (D.7)
Using (D.6) and (D.7), we can easily determine the limits of (D.4) and (D.5) as θ → 0:
limθ→0
πt = −pt−1 +1
1− σδmEt
{+∞∑k=0
(ωn1 )−k−1{−(i∗t+k − rt+k
)+ (ωn1 − 1)Mt+k
−[σ (1− δg) (ωn1 − 1) +
δgχy − χgχi
]gt+k +
[σδϕ (ωn1 − 1) +
χϕ − δϕχyχi
]ϕt+k
}}+
σ
1− σδm[−δmMt + (1− δg) gt − δϕϕt] , (D.8)1545
Online Appendix − 30
limθ→0
yt =δm
1− σδmEt
{+∞∑k=0
(ωn1 )−k−1{i∗t+k − rt+k − (ωn1 − 1)Mt+k
+
[σ (1− δg) (ωn1 − 1) +
δgχy − χgχi
]gt+k −
[σδϕ (ωn1 − 1) +
χϕ − δϕχyχi
]ϕt+k
}}+
1
1− σδm[δmMt + (δg − σδm) gt + δϕϕt] . (D.9)
These limits are �nite, unlike their counterparts in the basic NK model.
We now show that the right-hand sides of (D.8) and (D.9) coincide with the values taken1550
by πt and yt when prices are perfectly �exible (θ = 0). The �exible-price value of yt is
straightforwardly obtained by setting to zero the last term in the Phillips curve (D.1), which
is proportional to (the log-deviation of) �rms' marginal cost of production:
yt = δmmt + δggt + δϕϕt. (D.10)
Using the IS equation (1), the money-demand equation (D.2), the identity mt = Mt− pt, the1555
exogenous policy-rate setting imt = i∗t , and the solution for �exible-price output (D.10), we
get the following dynamic equation under �exible prices:
pt = (ωn1 )−1 Et {pt+1}+(ωn1 )−1
1− σδm
{− (i∗t − rt) +
(1− δmχy
χi− σδm
)Mt + σδmEt {Mt+1}
+
[χg − δgχy
χi+ σ (1− δg)
]gt − σ (1− δg)Et {gt+1}
+
(χϕ − δϕχy
χi− σδϕ
)ϕt + σδϕEt {ϕt+1}
}.1560
Iterating this equation forward to +∞ leads to the following value for the price level pt in
our model with banks under �exible prices:
pt =1
1− σδmEt
{+∞∑k=0
(ωn1 )−k−1{−(i∗t+k − rt+k
)+ (ωn1 − 1)Mt+k
−[σ (1− δg) (ωn1 − 1) +
δgχy − χgχi
]gt+k +
[σδϕ (ωn1 − 1) +
χϕ − δϕχyχi
]ϕt+k
}}+
σ
1− σδm[−δmMt + (1− δg) gt − δϕϕt] , (D.11)1565
Online Appendix − 31
which implies in turn that the value of πt ≡ pt− pt−1 in our model with banks under �exible
prices coincides with the right-hand side of (D.8). In turn, using (D.10), (D.11), and the
identity mt = Mt − pt, we get that the value of yt in our model with banks under �exible
prices coincides with the right-hand side of (D.9). Thus, our model with banks solves the
paradox of �exibility: the limits of πt and yt as θ → 0 are �nite and coincide with the values1570
of πt and yt when θ = 0.
D.4. Convergence to the Basic NK Model
In a previous version of this paper (Diba and Loisel, 2019), we show that the steady state
and reduced form of our model with banks converge to the steady state and reduced form
of the basic NK model, with the steady-state stock of real reserves m bounded away from1575
zero and in�nity, as the scale parameter of banking costs γ and the steady-state interest-rate
spread 1/β − Im are shrunk to zero at the same speed (as in the thought experiment of
Subsection 4.2).
More speci�cally, we prove this result in three steps. First, we show that the steady-state
values of all endogenous variables in our model with banks converge, as (Im, γ) → (1/β, 0),1580
to their counterparts in the corresponding basic NK model − with the exception of the
steady-state value of real reserves m, which does not exist in the basic NK model. Second,
we show that m remains bounded away from zero and in�nity along the way, provided that
(1/β − Im)/γ is itself bounded away from zero and in�nity, i.e. provided that 1/β − Im and
γ are shrunk to zero at the same speed. Third, we build on the �rst two steps to show that1585
the reduced form of our model with banks converges to the reduced form of the basic NK
model as (Im, γ)→ (1/β, 0) with (1/β − Im)/γ bounded away from zero and in�nity.
In essence, making the steady-state IOR rate Im go to the steady-state interest rate on
bonds I = 1/β asymptotically removes the steady-state opportunity cost of holding reserves.
Making the banking-cost-scale parameter γ go to zero asymptotically removes the steady-1590
state marginal banking cost (provided that m is bounded away from zero) and the steady-
state marginal bene�t of holding reserves (even when m is bounded from above). Imposing
that (1/β − Im)/γ be bounded away from zero and in�nity ensures that the steady-state
opportunity cost and marginal bene�t of holding reserves go hand in hand to zero, so that
Online Appendix − 32
m is itself bounded away from zero and in�nity. Asymptotically, given that all steady-state1595
costs related to banking and reserve holding are removed, the steady state and reduced form
of the model converge to the steady state and reduced form of the basic NK model.
It is straightforward to check that we still get this steady-state and reduced-form conver-
gence, this time with m going to in�nity (asymptotic satiation), as we hold γ constant and
shrink only 1/β − Im to zero (as in the policy experiment of Section 5), provided that two1600
conditions are met. The �rst condition is that the marginal banking cost should go to zero
as the stock of real reserves goes to in�nity (limmt→+∞ Γ`(`t,mt) = 0, where `t denotes real
loans and Γ the banking-cost function). The second condition is that banking-cost elasticities
(Γ```/Γ`, Γmmm/Γm, Γ`m`/Γm, and Γ`mm/Γ`) should be bounded from above. These two con-
ditions are met, in particular, when the banking-cost function Γ is isoelastic, which happens1605
when the loan-production and banker-labor-disutility functions are themselves isoelastic.
D.5. Defense of the Non-Satiation Assumption
In Diba and Loisel (2020), we present in detail our model with banks and show in partic-
ular that this model can account, in qualitative terms, for three key features of US in�ation
during the Great Recession: no signi�cant de�ation, little in�ation volatility, and no signif-1610
icant in�ation following quantitative-easing (QE) policies. These results, like our resolution
of NK puzzles and paradoxes in the present paper, rest on the assumption that demand for
bank reserves was not fully satiated in the US. For this reason, in Diba and Loisel (2020),
we address in detail two types of arguments that go against our non-satiation view.
First, some observers may make a case for satiation noting that the federal-funds rate1615
and Treasury-bill (T-bill) returns were below the IOR rate for several years in the aftermath
of the crisis. We do not think this contradicts our claim that reserves still had a positive
marginal convenience yield during this period. Most of the trading activity in the federal-
funds market over this period involved banks borrowing funds from entities that do not
have direct access to the IOR rate (particularly from Federal Home Loan Banks). Given1620
the presence of such eager lenders, the federal-funds rate had to be below the IOR rate to
incentivize the borrowers (banks with direct access to the IOR rate). As to T-bill returns, the
low rates could re�ect strong demand by non-bank entities − using T-bills as, e.g., collateral
Online Appendix − 33
or international reserve asset. We formalize this counter-argument in Diba and Loisel (2020)
by introducing government bonds providing liquidity services into our model with banks, and1625
showing that the resulting model reconciles the observed negative spread between T-bill and
IOR rates with our non-satiation assumption.
The second argument making a case for satiation of demand for reserves is the fact
that large increases in reserve balances during the second and third rounds of quantitative
easing (QE2 and QE3) had no apparent e�ect on expected in�ation, as Reis (2016) points1630
out. Our counter-argument is that this evidence may also be consistent with demand for
reserves being close to satiation, rather than fully satiated. More speci�cally, we show in
Diba and Loisel (2020) that in our model with banks, large increases in the money supply
(say, doubling the stock of reserves) can have very small in�ationary e�ects (around twenty
basis points) if the demand for reserves is close to satiation and the monetary expansion is1635
perceived as temporary (say, balance-sheet normalization is expected to occur in about �ve
years). Distinguishing between the two possibilities (arbitrarily close to satiation versus fully
satiated demand) may be di�cult in practice. In fact, in contrast to Reis's (2016) evidence
about expected in�ation, Krishnamurthy and Lustig (2019) �nd statistically signi�cant e�ects
of monetary policy, during QE2 and QE3, on the convenience yield of US Treasury bills and1640
the foreign-exchange value of the dollar.
Appendix E: CIA Model
In this appendix, we consider a sticky-price CIA model with leisure serving as the credit
good. We show that the log-linearized reduced form of this model cannot smoothly converge
to the log-linearized reduced form of the basic NK model, neither by gradually removing the1645
monetary friction, nor by gradually satiating the demand for money. As a result, we cannot
use this model to select an equilibrium of the basic NK model under an exogenous policy
rate.
Whenever possible, we use the same notation as with the MIU model in Appendix B.
For simplicity, we abstract from preference, supply, and government-purchases shocks (i.e.1650
ζt = ϕ1,t = ϕ2,t = ϕ3,t = ϕ4,t = 1 and gt = g ≥ 0), as we do not need them to make our point.
Online Appendix − 34
In our CIA model, households choose bt, ct, ht, and mt to maximize
Ut = Et
{∞∑k=0
βk [u (ct+k)− v (ht+k)]
}
subject to their budget constraint
bt +mt ≤It−1Πt
bt−1 +Imt−1Πt
(mt−1 − ct−1) +wt−1Πt
ht−1 + τt
and their cash-in-advance constraint
ct ≤ mt,
taking all prices as given. The monetary friction is that households need to use cash to buy
goods in the goods exchange, and they can acquire cash only in the �nancial exchange that
takes place before the goods exchange within the same period. Since all of consumption is
subject to the CIA constraint, the model has no parameter that we can shrink to gradually1655
remove the monetary friction. Therefore, we cannot make the model converge smoothly to
the basic NK model by gradually removing the monetary friction.
In fact, the only parameters of the model that do not exist in the basic NK model are
the steady-state interest rate on money Im and the steady-state gross growth rate of money
µ. So, the only possibility for the model to smoothly converge to the basic NK model would1660
be to gradually shrink the spread between the steady-state interest rate on money Im and
the steady-state interest rate on bonds µ/β, and as a result gradually satiate the demand for
money. In what follows, we show that the steady state of the model would then smoothly
converge to the steady state of the basic NK model, but its reduced form would not smoothly
converge to the reduced form of the basic NK model − essentially because the CIA constraint1665
remains binding along the way for su�ciently small shocks (since Im remains below µ/β along
the way).
To do so, we start by deriving the �rst-order conditions of households' maximization
problem (stated above):
u′(ct) = βImt Et{λt+1
Πt+1
}+ λt,
Online Appendix − 35
λt = βImt Et{λt+1
Πt+1
}+ λt,
λt = βItEt{λt+1
Πt+1
},
v′ (ht) = βwtEt{λt+1
Πt+1
},
where λt and λt denote the Lagrange multipliers associated with the budget and cash-in-
advance constraints respectively. We then turn to the other equilibrium conditions of the
model. Firms are subject to Calvo's (1983) constraints on the frequency at which they can
change their prices. They receive cash from consumers and hoard it until the next period.
Thus, in the speci�c case of perfectly �exible prices (θ = 0), �rm i chooses Pt(i) to maximize
Et{βλt+1
λtΠt+1
}[Imt
Pt(i)yt(i)
Pt− wtht(i)
]
subject to the production function
yt (i) = f [ht (i)]
and the demand schedule
yt (i) =
[Pt(i)
Pt
]−εyt.
Using the Euler equation above, and the symmetry between �rms, we can write the �rst-order
condition of this �exible-price maximization problem as
(ε
ε− 1
)wt = Imt f
′(ht).
Finally, the bond-market-clearing condition is
bt = 0,
Online Appendix − 36
the money-market-clearing condition is
mt =Mt
Pt,
and the goods-market-clearing condition is
ct + g = yt.
As with our MIU model, we set µ to one, so that the set of steady states is the same under
sticky prices (θ > 0) as under �exible prices (θ = 0). So, we can use the �rst-order condition
of �rms' optimization problem under �exible prices (above) to characterize this set. When1670
all variables are constant over time (in particular h ≡ ht = ht+1), the equilibrium conditions
above implyv′(h)
f ′(h)= βIm
(ε− 1
ε
)u′[f(h)− g]. (E.1)
Under standard assumptions on u, v and h, the left-hand side of (E.1) is increasing in h, from
0 (as h = 0) to +∞ (as h→ +∞), while its right-hand side is decreasing in h, from +∞ (as
h→ h, where h is de�ned by f(h) = g) to 0 (as h→ +∞). Therefore, there exists a unique
solution in h to (E.1), and hence a unique steady state of the model. As Im gradually goes
to 1/β, (E.1) smoothly converges to
v′(h)
f ′(h)=
(ε− 1
ε
)u′[f(h)− g],
which is the equation that the steady-state value of labor h satis�es in the basic NK model.
Therefore, the steady state of our CIA model smoothly converges to the steady state of the
basic NK model as Im gradually goes to 1/β.1675
Since Im goes to 1/β without ever reaching 1/β, the CIA constraint remains binding along
the way (for su�ciently small shocks). Therefore, log-linearizing the equilibrium conditions
Online Appendix − 37
around the unique steady state leads to the following reduced form:
yt = Et {yt+1} −1
σ(it − Et {πt+1}) ,
πt = βEt {πt+1}+ κ [yt + δi (it − imt )] ,
yt = νmt,
where mt = Mt − Pt, πt = Pt − Pt−1, and all parameters are positive:
σ ≡ −u′′yu′
> 0,
κ ≡ (1− θ) (1− βθ)
θ[1− εff ′′
(f ′)2
] [−u
′′y
u′+v′′h
v′f
f ′h− ff ′′
(f ′)2
]> 0,
δi ≡[−u
′′y
u′+v′′h
v′f
f ′h− ff ′′
(f ′)2
]−1> 0,
ν ≡ c
y∈ (0, 1).
This reduced form does not converge to the reduced form of the basic NK model as Im1680
gradually goes to 1/β. Thus, the reduced form of our CIA model does not smoothly converge
to the reduced form of the basic NK model.
Appendix F: Discounting Models
�Discounting models� provide an alternative to our approach for solving the forward-
guidance puzzle. In this appendix, we establish three points about a class of discounting1685
models. The �rst two points generalize Cochrane's (2016) comments on Gabaix (2020). More
speci�cally, the �rst point is that discounting models do not solve our paradox of �exibility
(De�nition 3). While we have no evidence against (nor in favor of) their implication that
greater price �exibility magni�es the contraction and the de�ation at the ZLB, we think
that their implication about a discontinuity at the �exible-price limit − highlighted in our1690
De�nition 3 − seems implausible. As we will show, this discontinuity comes from the fact
that discounting models do not deliver determinacy under an exogenous interest rate when
prices are �exible.
Online Appendix − 38
The second point is that discounting models cannot solve the forward-guidance puzzle
without generating a negative long-term relationship between the in�ation rate and the in-1695
terest rate on bonds. To our knowledge, existing empirical evidence does not support the
precise one-to-one long-term relationship implied by our monetary models. Nonetheless, the
existence of a positive long-term relationship between the in�ation rate and nominal interest
rates is a standard presumption of our textbooks, and is broadly re�ected in cross-country
data.1700
Our third point illustrates a limitation of our equilibrium-selection argument in Sections 3-
4. Although discounting models converge to the basic NK model as we shrink the underlying
friction, we cannot use them to uniquely select the equilibrium presented in Subsection 3.2
(nor any other equilibrium). This is because discounting models do not deliver determinacy
under an exogenous interest rate beyond some point, as we approach the basic-NK-model1705
limit. This illustrates more broadly that our equilibrium-selection argument must start with
a model that has higher-order dynamics than the second-order dynamics of the basic NK
model.
F.1. A Class of Discounting Models
We consider a class of models whose reduced form, in the absence of shocks other than1710
interest-rate shocks, is made of an IS equation and a Phillips curve of type
yt = ξ1Et {yt+1} −ξ2σEt {it − πt+1} , (F.1)
πt = βξ3 (θ)Et {πt+1}+ κ (θ) [yt − ξ4 (θ)Et {yt+1}] , (F.2)
where β ∈ (0, 1), σ > 0, ξ1 > 0, ξ2 > 0, and, for all θ ∈ (0, 1), ξ3(θ) ≥ 0, ξ4(θ) ∈ [0, 1),
and κ(θ) > 0, with limθ→0 ξ3(θ) < +∞ and limθ→0 κ(θ) = +∞.4 This class of reduced forms1715
nests the reduced form of the basic NK model as a special case in which ξ1 = ξ2 = ξ3(θ) = 1
and ξ4(θ) = 0. More generally, this class allows the coe�cients of Et{yt+1} and Et{πt+1} to
4We focus on discrete-time discounting models for the sake of comparability with our monetary models, butwe have no reason to expect that continuous-time discounting models behave di�erently. Indeed, Michaillatand Saez (2019) show that their continuous-time discounting model has the same three properties as the oneslisted above.
Online Appendix − 39
be smaller (�positive discounting�) or larger (�negative discounting�) than in the basic NK
model, and also allows for a Et{yt+1} term in the Phillips curve. In particular, this class
encompasses the reduced forms of three models that have been shown to be able to solve the1720
forward-guidance puzzle: (i) Gabaix's (2020) benchmark model, in which (ξ1, ξ3(θ)) ∈ (0, 1)2
and ξ4(θ) = 0; (ii) Angeletos and Lian's (2018) model, in which (ξ1, ξ2, ξ3(θ), ξ4(θ)) ∈ (0, 1)4;
and (iii) Bilbiie's (2019) model with external price-adjustment costs, in which (ξ1, ξ2) ∈ (0, 1)2
and ξ3(θ) = ξ4(θ) = 0. In addition, it also encompasses the reduced forms of: (iv) Bilbiie's
(2019) model with internal price-adjustment costs, in which (ξ1, ξ2) ∈ (0, 1)2, ξ3(θ) = 1, and1725
ξ4(θ) = 0; (v) McKay et al.'s (2017) model, in which also (ξ1, ξ2) ∈ (0, 1)2, ξ3(θ) = 1, and
ξ4(θ) = 0; (vi) Ravn and Sterk's (2018) model with risk-neutral equity investors, in which
ξ3(θ) = 1 and ξ4(θ) ∈ (0, 1); and (vii) Woodford's (2019) model with exponentially distributed
planning horizons and no learning, in which ξ1 = ξ2 = ξ3(θ) ∈ (0, 1) and ξ4(θ) = 0.5
F.2. Paradox of Flexibility1730
Like the basic NK model, and unlike our monetary models, discounting models exhibit
the paradox of �exibility (as stated in De�nition 3).6 They make in�ation and output explode
in response to future shocks as the degree of price stickiness θ goes to zero. To establish this
result, we assume that the interest rate is set exogenously from date 1 to some date T ≥ 2,
and that the economy is at its steady state at date T + 1; and we show that the responses of1735
|π1| and |y1| to an interest-rate change at any date k ∈ {2, ..., T} go to in�nity as θ → 0:
Proposition 10: Discounting models exhibit the paradox of �exibility: if ik = i∗ 6= 0,
it = 0 for all t ∈ {1, ..., T}r{k}, and yT+1 = πT+1 = 0, then limθ→0 |π1| = limθ→0 |y1| = +∞.
Proof: We start with the case in which ξ3(θ) > 0 or ξ4(θ) > 0. In this case, the system
made of the IS equation (F.1) and the Phillips curve (F.2) can be rewritten as1740
Et
yt+1
πt+1
= P
yt
πt
+ Zit (F.3)
5However, it does not encompass the reduced forms of McKay et al.'s (2016) and Del Negro et al.'s (2015)models, which involve some discounting too but are more complex.
6They may attenuate this paradox, though, as Angeletos and Lian (2018) show in the context of theirdiscounting model.
Online Appendix − 40
with
P ≡ 1
ϕ (θ)
κ (θ) ξ2 + βσξ3 (θ) −ξ2κ (θ)σ [ξ4 (θ)− ξ1] σξ1
and Z ≡ ξ2ϕ (θ)
βξ3 (θ)
κ (θ) ξ4 (θ)
,where ϕ(θ) ≡ βσξ1ξ3(θ) + κ(θ)ξ2ξ4(θ) > 0. The characteristic polynomial of P is
P (X) ≡ X2 − σξ1 + κ (θ) ξ2 + βσξ3 (θ)
ϕ (θ)X +
σ
ϕ (θ).1745
Since P(0) 6= 0, P is invertible. Iterating the dynamic equation (F.3) forward to date T , and
using the terminal condition yT+1 = πT+1 = 0 and the invertibility of P, we get y1
π1
= −P−(k−1)Zi∗.
For any X ∈ R, we have
1
ξ2limθ→0
[ϕ (θ)P (X)
κ (θ)
]=[limθ→0
ξ4 (θ)]X2 −X.1750
One root of the polynomial on the right-hand side of this equation is zero. Therefore, one root
of P(X) converges towards zero as θ → 0, which implies in turn that limθ→0 ||P−1|| = +∞.
Using the fact that ||Z|| is bounded away from zero as θ → 0, we conclude that limθ→0 |y1| =
limθ→0 |π1| = +∞.
In the alternative case in which ξ3(θ) = ξ4(θ) = 0, the system made of the IS equation1755
(F.1) and the Phillips curve (F.2) implies the following dynamic equation in in�ation:
[ξ1 +
κ (θ) ξ2σ
]Et {πt+1} = πt +
κ (θ) ξ2σ
it. (F.4)
Iterating this dynamic equation forward to date T , and using the terminal condition πT+1 = 0,
we get
π1 = −[ξ1 +
κ (θ) ξ2σ
]k−1κ (θ) ξ2i
∗
σ,1760
so that limθ→0 |π1| = +∞. Using the Phillips curve (F.2) with ξ3(θ) = ξ4(θ) = 0, we then get
Online Appendix − 41
limθ→0 |y1| = +∞. �
The preceding proof is based on two properties of discounting models: (i) these models
generate indeterminacy under a permanently exogenous policy rate when prices are su�-
ciently �exible, as their dynamic system then has one stable eigenvalue not matched by any1765
predetermined variable, and (ii) this stable eigenvalue goes to zero as prices are made more
and more �exible. As in the basic NK model in Section 2, this stable eigenvalue magni�es the
e�ects of future conditions (at date k) on initial outcomes (at date 1), and these e�ects grow
explosively as this eigenvalue goes to zero − thus giving rise to the paradox of �exibility.
Indeterminacy under su�ciently �exible prices, in turn, follows by continuity from inde-1770
terminacy under perfectly �exible prices. Under perfectly �exible prices, the Phillips curve
(F.2) collapses to the dynamic equation yt = [limθ→0 ξ4(θ)]Et{yt+1}, which pins down yt
uniquely if limθ→0 ξ4(θ) 6= 1. Under an exogenous policy rate it, the IS equation (F.1) then
pins down expected future in�ation Et{πt+1}, but not current in�ation πt. Thus, discounting
models may deliver determinacy under a permanently exogenous policy rate for some degrees1775
of price stickiness, but cannot do it for su�ciently small degrees of price stickiness.
In our monetary models, by contrast, the interest rate pegged at the ZLB is the IOR rate
imt , not the interest rate on bonds it. Setting exogenously the IOR rate and the nominal
stock of reserves − two monetary-policy instruments under the direct control of central
banks − makes the (market-determined) interest rate on bonds evolve according to a shadow1780
Wicksellian rule, as we have explained. This shadow Wicksellian rule ensures determinacy
for any degree of price stickiness, and in particular for perfectly �exible prices − thus solving
the paradox of �exibility.
F.3. Fisher E�ect
Discounting models cannot deliver determinacy under a permanently exogenous policy1785
rate without making the in�ation rate and the interest rate negatively related to each other
in the long term. Therefore, they cannot both solve the forward-guidance puzzle and imply
a long-term relationship consistent in sign (let alone in size) with the standard Fisher e�ect.7
7Gabaix (2020), however, adds price indexation and �in�ation guidance� to his benchmark discountingmodel and shows that the resulting model (which does not belong to the class of discounting models we
Online Appendix − 42
The following proposition formalizes this point:
Proposition 11: In discounting models, if setting the policy rate exogenously delivers1790
local-equilibrium determinacy, then a permanent increase in the policy rate leads to a perma-
nent decrease in the in�ation rate.
Proof: We start with the case in which ξ3(θ) > 0 or ξ4(θ) > 0. In this case, under a
permanent peg it = i∗, the system made of the IS equation (F.1) and the Phillips curve (F.2)
can be rewritten as (F.3) with it = i∗. If the peg ensures local-equilibrium determinacy,1795
then P (X), the characteristic polynomial of P (derived in Appendix F.2), must have no root
inside the unit circle, because the system has no predetermined variable. In particular, P (X)
must have no root inside the real-number interval [0, 1], which requires that P (0)P (1) > 0,
i.e. equivalently
σ (1− ξ1) [1− βξ3 (θ)]− κ (θ) ξ2 [1− ξ4 (θ)] > 0. (F.5)1800
In the unique local equilibrium, the (constant) in�ation rate is easily obtained as
πt = π∗ ≡ −κ (θ) ξ2 [1− ξ4 (θ)] i∗
σ (1− ξ1) [1− βξ3 (θ)]− κ (θ) ξ2 [1− ξ4 (θ)].
Given (F.5), π∗ is negatively related to i∗.
In the alternative case in which ξ3(θ) = ξ4(θ) = 0, under a permanent peg it = i∗,
the system made of the IS equation (F.1) and the Phillips curve (F.2) implies the dynamic1805
equation (F.4) with it = i∗. Therefore, for the peg to ensure determinacy, we need
σ (1− ξ1)− κ (θ) ξ2 > 0. (F.6)
In the unique local equilibrium, the (constant) in�ation rate is easily obtained as
πt = π∗ ≡ −κ (θ) ξ2i∗
σ (1− ξ1)− κ (θ) ξ2.
Given (F.6), π∗ is negatively related to i∗. �1810
consider) can both solve the forward-guidance puzzle and make in�ation respond positively to the nominalinterest rate in the long term.
Online Appendix − 43
The preceding proof is simple, but mechanical. In what follows, we o�er an interpretation
of this result that involves a shadow interest-rate rule and the Taylor principle. The question
(negatively) answered by Proposition 11 is whether the system made of the modi�ed IS
equation (F.1), the modi�ed Phillips curve (F.2), and the permanent peg it = i∗ can have
a unique stationary solution and make in�ation, in this unique stationary solution, depend1815
positively on i∗. This question will receive exactly the same answer if that system is replaced
by the system made of the standard IS equation (1), the modi�ed Phillips curve (F.2), and
the shadow interest-rate rule
it = ξ2i∗ + σ (1− ξ1)Et {yt+1}+ (1− ξ2)Et {πt+1} . (F.7)
Indeed, the two systems have exactly the same implications for local-equilibrium determinacy1820
and the dynamics of in�ation and output (they di�er only in terms of the implied dynamics
for it). So consider the latter system. The Taylor principle (as de�ned by Woodford, 2003,
Chapter 4) states that a necessary condition for local-equilibrium determinacy is that the
modi�ed Phillips curve (F.2) and the shadow interest-rate rule (F.7) should make the interest
rate react more than one-to-one to the in�ation rate in the long term, that is to say1825
ζ ≡ σ (1− ξ1) [1− βξ3 (θ)]
κ (θ) [1− ξ4 (θ)]+ (1− ξ2) > 1. (F.8)
In the unique local equilibrium, the (constant) interest rate i and the (constant) in�ation
rate π are therefore linked to each other by the relationship i = ξ2i∗+ ζπ, where ζ > 1. Now,
the standard IS equation (1) implies that they should be equal to each other: i = π. As a
consequence, we get1830
π =−ξ2i∗
ζ − 1.
Thus, the necessary condition for local-equilibrium determinacy (F.8) imposed by the Taylor
principle requires that π be negatively related to i∗.
This con�ict between the Taylor principle and the Fisher e�ect does not arise in our
monetary models. First, the interest rate pegged at the ZLB in these models is the (directly1835
controlled) IOR rate imt , not the (market-determined) interest rate on bonds it. Under
Online Appendix − 44
exogenous monetary-policy instruments, the interest rate on bonds evolves according to a
shadow Wicksellian rule that always ensures determinacy. Second, these models generate the
standard Fisher e�ect, i.e. a one-to-one long-term relationship between the in�ation rate
and the interest rate on bonds. Indeed, the money-demand equations (7), (C.3), and (D.2)1840
imply that a permanent change in nominal-money growth Mt−Mt−1 = δ∗ leads to the same
permanent change in in�ation πt = δ∗. In turn, the IS equations (1) and (C.1) imply that
Mt − Mt−1 = πt = δ∗ leads to the same permanent change in the interest rate on bonds
it = δ∗.8
F.4. Convergence to the Basic NK Model1845
Although discounting models converge to the basic NK model as we shrink the underlying
friction (e.g., the degree of bounded rationality in Gabaix, 2020, information frictions in
Angeletos and Lian, 2018, market incompleteness in Bilbiie, 2019), we cannot use them to
uniquely select the equilibrium presented in Subsection 3.2 (nor any other equilibrium). The
reason is that discounting models no longer deliver determinacy under an exogenous interest1850
rate as they approach the basic NK model:
Proposition 12: Discounting models generate indeterminacy under a permanently ex-
ogenous policy rate when (ξ1, ξ2, ξ3(θ), ξ4(θ)) is su�ciently close to (1, 1, 1, 0).
Proof: We focus on the case in which ξ3(θ) is su�ciently close to 1 for ξ3(θ) > 0. In this
case, the system made of the IS equation (F.1) and the Phillips curve (F.2) can be rewritten1855
as (F.3), and its characteristic polynomial under a permanently exogenous policy rate is
P(X) (as de�ned in Appendix F.2). It is straightforward to check that as (ξ1, ξ2, ξ3(θ), ξ4(θ))
goes to (1, 1, 1, 0), P(X) converges to Pb(X) for any X ∈ R, and therefore the two roots of
P(X) converge to the two roots ρb ∈ (0, 1) and ωb > 1 of Pb(X). For (ξ1, ξ2, ξ3(θ), ξ4(θ))
su�ciently close to (1, 1, 1, 0), thus, only one root of P(X) lies outside the unit circle. With1860
only one eigenvalue outside the unit circle for two non-predetermined variables (Et{yt+1} and
Et{πt+1}), discounting models then generate indeterminacy. �
8Under the assumption that non-optimized prices are indexed to steady-state in�ation, the Phillips curves(2), (C.2), and (D.1) remain valid and residually determine the permanent change in output.
Online Appendix − 45
The basic reason for this result is that the reduced form (F.1)-(F.2) of discounting models
involves in�ation and output, but not the price level, like the reduced form (1)-(2) of the
basic NK model. As a result, under a permanently exogenous policy rate, the characteristic1865
polynomial P(X) of their dynamic system is of degree two, like the characteristic polynomial
Pb(X) of the dynamic equation of the basic NK model.
The reduced forms of our monetary models, by contrast, involve not only in�ation and
output, but also the price level pt through the real stock of reserves mt = Mt − pt in the
money-demand equations (7), (C.3), and (D.2). As a result, under permanently exogenous1870
monetary-policy instruments, their dynamic equation cannot be written in terms of in�ation
and involve only Et{πt+2}, Et{πt+1}, and πt. Instead, it has to be written in terms of the price
level, and to involve Et{pt+2}, Et{pt+1}, pt, and pt−1. As a consequence, the characteristic
polynomial P(X) of our monetary models' dynamic equation is of degree three.
As our monetary models converge to the basic NK model, the roots ρ, ω1, and ω2 of1875
their characteristic polynomial converge respectively to ρb ∈ (0, 1), 1, and ωb > 1, where
the limit value 1 of ω1 simply re�ects the identity πt = pt − pt−1. As long as our monetary
models do not exactly coincide with the basic NK model, ω1 remains outside the unit circle.
With two eigenvalues outside the unit circle (ω1 and ω2) for two non-predetermined variables
(Et{pt+2} and Et{pt+1}), therefore, our monetary models ensure determinacy even when they1880
are arbitrarily close to the basic NK model.
Put di�erently, the price-level term in our shadow Wicksellian rules − which directly
comes from the price-level term in the money-demand equations − acts as an error-correction
term that makes the price level stationary and hence determinate in our monetary models.
As these models converge to the basic NK model, the coe�cient of the price level in the1885
shadow Wicksellian rules goes to zero. But as long as our monetary models do not exactly
coincide with the basic NK model, this coe�cient remains positive, and both stationarity
and determinacy of the price level ensue.
Online Appendix − 46
Appendix G: Simple Model (Numerical Illustrations)
G.1. Additional Numerical Illustrations under our Benchmark Calibration1890
In Section 3 of the main text, Figure 1 illustrates numerically the e�ects of forward
guidance (i.e. future policy-rate cuts) on in�ation in our simple model, and compares these
e�ects to the implications of the standard NK equilibrium. In this appendix, we illustrate and
discuss the e�ects of forward guidance on output, and the e�ects of anticipated changes in
�scal policy on in�ation and output − both in the equilibrium of our simple model and in the1895
standard equilibrium of the basic NK model. We continue to use our benchmark calibration
taken from Galí (2008, Chapter 3), which sets θ = 2/3 (corresponding to �3-quarter price
rigidity�); we also report the e�ects of cutting θ in half, step by step, to make prices more
�exible.
Figure G.1 shows the e�ects of forward guidance on output in the two models. As before,1900
our policy experiment is to cut the policy rate by 25 basis points (one percentage point per
annum) in Quarter t + k, and we display the e�ects on output in Quarter t (when the rate
cut is announced). The right panel in Figure G.1 replicates the implausible implications of
the basic NK model. The left panel shows that our model does not share these implications.
The rate cut has small e�ects on output (less than 0.2 percent of steady-state output to begin1905
with), and the e�ects die o� quickly as we delay the rate cut. Moreover, these e�ects decline
smoothly as we make prices more �exible; they converge to the �exible-price (θ = 0) e�ects.
To analyze the e�ects of �scal policy, we add government purchases to Galí's calibration.
We set the share of government purchases in output to 0.3 in the steady state. We follow Galí's
calibration for the structural parameters (like the intertemporal elasticity of substitution) and1910
adjust the reduced-form parameters (like the coe�cient 1/σ on the real interest rate in the IS
equation) to re�ect the introduction of government purchases. Our policy experiment is an
increase in government purchases, amounting to one percent of steady-state output, occurring
(only) in Quarter t+ k and announced in Quarter t. Figures G.2 and G.3 display the e�ects
on in�ation and output in Quarter t. Once again, the comparison between the left and the1915
right panels shows that our model's equilibrium does not share the puzzling implications of
the basic NK model's standard equilibrium: the e�ects of anticipated �scal policy die out as
Online Appendix − 47
we delay the policy intervention, and they converge to the �exible-price values as we make
prices more and more �exible.
Another notable di�erence between our model's equilibrium and the basic NK model's1920
standard equilibrium under our benchmark calibration is that anticipated �scal expansions
have a contractionary e�ect on output in our model.9 Several contributions (e.g., Christiano
et al., 2011) suggest that anticipated �scal expansions can have large positive output mul-
tipliers at the ZLB according to the basic NK model. The right-hand panel of Figure G.3
con�rms this implication of the basic NK model. This implication arises from a feedback loop1925
�rst described in Farhi and Werning (2016). As we explain in Subsection 3.2 of the main text,
this feedback loop works back in time via the IS equation and the Phillips curve: given that
πT+1 = yT+1 = 0, a �scal expansion at date T raises in�ation at date T , which lowers the real
interest rate at date T − 1, which raises output and in�ation at date T − 1, and so on. This
feedback loop is also present in our model, but πT+1 and yT+1 are endogenously determined1930
when the �scal expansion is announced. As a result, expected future �scal expansions can
reduce current output in our model (as is the case under the calibration we use for Figure
G.3). Intuitively, these contractionary e�ects of anticipated �scal expansions may come from
wealth e�ects that also arise in standard Real-Business-Cycle models: consumers realize that
the future �scal expansion reduces their permanent income, and they respond by lowering1935
current consumption.
The e�ects of anticipated �scal expansions on in�ation may be dominated either by the
wealth e�ect we mention above (which is de�ationary) or by an in�ationary e�ect that we
can trace back to staggered price setting. The latter e�ect arises because the �scal expansion
is expected to raise prices in the future, and this motivates current price setters to set higher1940
prices too. Under our benchmark calibration, the e�ects of anticipated �scal expansions on
in�ation (displayed in the left panel of Figure G.2, for various horizons k and price-stickiness
degrees θ) are small, and mostly negative.
9Our analytical derivations in the main text show that this is always the case when we use our model togo to the basic-NK-model limit. Figure G.3 makes the point numerically under our benchmark calibration,without taking the model to the basic-NK-model limit.
Online Appendix − 48
Figure G.1: E�ect of a policy-rate cut at date t+ k on output at date t
0 5 10 15 200
0.05
0.1
0.15
0.2
0 5 10 15 200
50
100
150
Note: The �gure displays the e�ect on yt of announcing at date t a one-percentage-point-per-annum
cut in imt+k (for the simple model) or it+k (for the basic NK model), as a function of k ∈ {0, ..., 20}.Parameter values are the same as for Figure 1 in the main text. More speci�cally, benchmark
parameter values are set as in Galí (2008, Chapter 3): β = 0.99, σ = 1, χy = 1, χi = 4, andκ = λ[(1− θ)(1− βθ)/θ] = 0.13, where λ = 3/4 and θ = θ∗ ≡ 2/3. As θ takes the values θ∗/2, θ∗/4,θ∗/8, and θ∗/16, κ takes respectively the values 1.00, 3.13, 7.57, and 16.54.
Figure G.2: E�ect of government purchases at date t+ k on in�ation at date t
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0 5 10 15 200
20
40
60
80
100
Note: The �gure displays the e�ect on πt of announcing at date t a one-percent-of-steady-state-
output increase in gt+k, as a function of k ∈ {1, ..., 20}. The steady-state share of government
purchases in output is set to 0.3, and benchmark structural-parameter values are set as in Galí (2008,
Chapter 3), implying β = 0.99, σ = 1.43, δg = 0.42, χy = 1, χi = 4, and κ = λ[(1− θ)(1− βθ)/θ] =0.15, where λ = 0.86 and θ = θ∗ ≡ 2/3. As θ takes the values θ∗/2, θ∗/4, θ∗/8, and θ∗/16, κ takes
respectively the values 1.15, 3.58, 8.65, and 18.90.
Online Appendix − 49
Figure G.3: E�ect of government purchases at date t+ k on output at date t
0 5 10 15 20-0.15
-0.1
-0.05
0
0 5 10 15 200
10
20
30
40
Note: The �gure displays the e�ect on yt of announcing at date t a one-percent-of-steady-state-
output increase in gt+k, as a function of k ∈ {1, ..., 20}. Parameter values are the same as for Figure
G.2 above.
G.2. Numerical Sensitivity Analysis
The quantitative impressions conveyed by Figure 1 in the main text and Figures G.1-G.31945
in the previous appendix are not particularly sensitive to Galí's (2008, Chapter 3) choices
about the parameters of the basic NK model, nor to his (standard) assumption of a unitary
income elasticity of money demand. The value taken by the interest semi-elasticity of money
demand χi, however, does matter for the quantitative impression conveyed by our results
about the e�ects of forward guidance. The value of χi a�ects both the magnitude and the1950
persistence of the e�ects of future changes in the IOR rate on current in�ation and output.
Our choice of χi = 4, following Galí, represents a middle-of-the-range value compared to
estimates that we could take from the empirical literature on money demand.
Semi-log speci�cations of money demand typically yield small estimates of χi based on
US data. The estimates in Stock and Watson (1993) and Cochrane (2018), for example,1955
suggest semi-elasticities close to −0.1 on an annual basis.10 Given the quarterly frequency of
our model, these estimates correspond to χi = 0.4 (one order of magnitude smaller than the
value we use for Figure 1). By contrast, log-log speci�cations of money demand, estimated
on US or cross-country data, suggest interest elasticities around −1/4 (e.g., Teles and Zhou,
10Ball's (2001) estimate of −0.05 is even closer to zero.
Online Appendix − 50
2005) or −1/3 (e.g., Teles et al., 2016). If we set the opportunity cost of holding money to1960
one percent per quarter, an elasticity of −1/3 implies χi = 33 (one order of magnitude larger
than the value we use for Figure 1). Figure G.4 shows how the quantitative e�ects of forward
guidance on in�ation vary when we set χi to 0.4 or 33.
The policy experiment and the parameter values (other than the value of χi) used for
Figure G.4 are the same as earlier for Figures 1 and G.1. The right panel in Figure G.41965
replicates the implausible implications of the basic NK model. The left panel shows the
results for our simple model with χi = 0.4, and the middle panel shows the results with
χi = 33. The left panel suggests that the in�ationary e�ects of anticipated IOR-rate cuts are
tiny (below 3 basis points to begin with, and dying o� quickly). The middle panel suggests
that forward guidance has a sizable and more persistent e�ect on in�ation (announcing that1970
the IOR rate will be cut by one percentage point in 20 quarters raises current in�ation by 17
basis points).
Beyond this quantitative di�erence, however, both the left and middle panels of Figure
G.4 also illustrate the analytical results that are the main focus of our paper: the in�ationary
e�ects go to zero as we cut the IOR rate in the more distant future (k → +∞), and they1975
converge to the �exible-price e�ects as we make prices more �exible (θ → 0). Whatever the
calibration, our simple model exhibits neither the forward-guidance puzzle nor the paradox
of �exibility.
Figure G.5 shows the e�ects of forward guidance on output (under the same policy ex-
periment and parameter values we describe above). Again, the quantitative impressions we1980
get are sensitive to the value of the semi-elasticity χi. The e�ects are tiny if we set χi = 0.4,
but more noteworthy and persistent if we set χi = 33.
Of course, our simple model is not really suitable for a quantitative assessment of the
e�ects that one may associate with forward-guidance policies. Nonetheless, we suspect that
the sensitivity of quantitative results to the speci�cation of money demand may also be1985
present in richer (larger-scale) models. So, we suspect that the unsettled state of empirical
research on money demand may hinder sharp answers to interesting policy questions in this
context.
Online Appendix − 51
Figure G.4: E�ect of a policy-rate cut at date t+ k on in�ation at date tfor alternative values of χi
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 200
0.2
0.4
0.6
0.8
1
0 5 10 15 200
50
100
150
200
250
Note: The �gure displays the e�ect on πt of announcing at date t a one-percentage-point-per-annumcut in imt+k (for the simple model) or it+k (for the basic NK model), as a function of k ∈ {0, ..., 20}.Parameter values (except the value of χi) are the same as for Figure 1 in the main text and Figure
G.1 above.
Figure G.5: E�ect of a policy-rate cut at date t+ k on output at date tfor alternative values of χi
0 5 10 15 20-0.02
0
0.02
0.04
0.06
0.08
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0 5 10 15 200
50
100
150
Note: The �gure displays the e�ect on yt of announcing at date t a one-percentage-point-per-annum
cut in imt+k (for the simple model) or it+k (for the basic NK model), as a function of k ∈ {0, ..., 20}.Parameter values (except the value of χi) are the same as for Figure 1 in the main text and Figures
G.1 and G.4 above.
Online Appendix − 52
References
[1] Angeletos, G.-M., and Lian, C. (2018), �Forward Guidance Without Common Knowl-1990
edge,� American Economic Review, 108(9), 2477-2512.
[2] Ball, L. (2001), �Another Look at Long-run Money Demand,� Journal of Monetary
Economics, 47(1), 31-44.
[3] Bilbiie, F.O. (2019), �Monetary Policy and Heterogeneity: An Analytical Framework,�
mimeo.1995
[4] Calvo, G.A. (1983), �Staggered Prices in a Utility-Maximising Framework,� Journal of
Monetary Economics, 12(3), 383-398.
[5] Christiano, L., Eichenbaum, M., and Rebelo, S. (2011), �When Is the Government Spend-
ing Multiplier Large?,� Journal of Political Economy, 119(1), 78-121.
[6] Cochrane, J.H. (2016), �Comments on `A Behavioral New Keynesian Model' by Xavier2000
Gabaix,� mimeo.
[7] Cochrane, J.H. (2018), �Michelson-Morley, Fisher, and Occam: The Radical Implications
of Stable Quiet In�ation at the Zero Bound,� in Eichenbaum, M., and Parker, J.A. (eds)
NBER Macroeconomics Annual 2017, University of Chicago Press, 32, 113-226.
[8] Del Negro, M., Giannoni, M.P., and Patterson, C. (2015), �The Forward-Guidance Puz-2005
zle,� mimeo.
[9] Diba, B.T., and Loisel, O. (2019), �Pegging the Interest Rate on Bank Reserves: A Res-
olution of New Keynesian Puzzles and Paradoxes,� Georgetown University Department
of Economics Working Paper No. 19-05.
[10] Diba, B.T., and Loisel, O. (2020), �A Model of Post-2008 Monetary Policy,� mimeo.2010
[11] Farhi, E., and Werning, I. (2016), �Fiscal Multipliers: Liquidity Traps and Currency
Unions,� in Taylor, J.B., and Uhlig, H. (eds) Handbook of Macroeconomics, Amsterdam:
Elsevier Science, 2, 2417-2492.
[12] Gabaix, X. (2020), �A Behavioral New Keynesian Model,� American Economic Review,
110(8), 2271-2327.2015
Online Appendix − 53
[13] Galí, J. (2008), Monetary Policy, In�ation, and the Business Cycle: An Introduction to
the New Keynesian Framework, Princeton: Princeton University Press.
[14] Krishnamurthy, A., and Lustig, H. (2019), �Mind the Gap in Sovereign Debt Markets:
The U.S. Treasury Basis and the Dollar Risk Factor,� prepared for the 2019 Jackson
Hole Symposium of the Federal Reserve Bank of Kansas City.2020
[15] McKay, A., Nakamura, E., and Steinsson, J. (2016), �The Power of Forward Guidance
Revisited,� American Economic Review, 106(10), 3133-3158.
[16] McKay, A., Nakamura, E., and Steinsson, J. (2017), �The Discounted Euler Equation:
A Note,� Economica, 84(336), 820-831.
[17] Michaillat, P., and Saez, E. (2019), �Resolving New Keynesian Anomalies with Wealth2025
in the Utility Function,� forthcoming in the Review of Economics and Statistics.
[18] Ravn, M.O., and Sterk, V. (2018), �Macroeconomic Fluctuations with HANK and SAM:
An Analytical Approach,� mimeo.
[19] Reis, R. (2016), �Funding Quantitative Easing to Target In�ation,� in Designing Resilient
Monetary Policy Frameworks for the Future, Federal Reserve Bank of Kansas City, 423-2030
478.
[20] Stock, J.H., and Watson, M.W. (1993), �A Simple Estimator of Cointegrating Vectors
in Higher Order Integrated Systems,� Econometrica, 61(4), 783-820.
[21] Teles, P., Uhlig, H., and Valle e Azevedo, J. (2016), �Is Quantity Theory Still Alive?,�
The Economic Journal, 126(591), 442-464.2035
[22] Teles, P., and Zhou, R. (2005), �A Stable Money Demand: Looking for the Right Mon-
etary Aggregate,� Federal Reserve Bank of Chicago Economic Perspectives, 1Q/2005,
50-63.
[23] Woodford, M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy,
Princeton: Princeton University Press.2040
[24] Woodford, M. (2019), �Monetary Policy Analysis When Planning Horizons Are Finite,�
in Eichenbaum, M., and Parker, J.A. (eds) NBER Macroeconomics Annual 2018, Uni-
versity of Chicago Press, 33, 1-50.
Online Appendix − 54
