# Fiscal Stimulus of Last Resort

Alessandro Piergallini
University of Rome Tor Vergata Associate Professor of Economics, Department of
Economics and Finance, University of Rome Tor Vergata, Via Columbia 2, 00133
Rome, Italy. ORCID: 0000-0001-6458-0364. Phone: +390672595639. E-mail:
<EMAIL_ADDRESS>Homepage: https://sites.google.com/
view/alessandropiergallini. I wish to thank Paolo Canofari, Alessia Franzini,
Giovanni Piersanti, Michele Postigliola and Luca Vitali for very useful
comments. The usual disclaimers apply.

###### Abstract

I examine global dynamics in a monetary model with overlapping generations of
finite-horizon agents and a binding lower bound on nominal interest rates.
Debt targeting rules exacerbate the possibility of self-fulfilling liquidity
traps, for agents expect austerity following deflationary slumps. Conversely,
activist but sustainable fiscal policy regimes—implementing intertemporally
balanced tax cuts and/or transfer increases in response to disinflationary
trajectories—are capable of escaping liquidity traps and embarking inflation
into a globally stable path that converges to the target. Should fiscal
stimulus of last resort be overly aggressive, however, spiral dynamics around
the liquidity-trap steady state exist, causing global indeterminacy.

JEL Classification: E31; E62; E63.

Keywords: Monetary-Fiscal Policy Interactions; Overlapping Generations;
Intergenerational Wealth Effects; Multiple Equilibria; Global Dynamics.

##  1 Introduction

The recent observed plunge in global monetary policy rates towards zero or
below both as a result of the secular stagnation (Summers, 2016, 2018;
Krugman, 2020) and in the attempt to cope with the corona crisis—in a
macroeconomic environment featured by effective and expected inflation rates
systematically undershooting their targets for an extended period in most
major economies following the Great Recession (Bartsch et al., 2019)—poses
salient challenges for public policy design. A central question that arguably
urges for macroeconomic analysis is how to escape unintended deflationary
slumps associated to the existence of a liquidity-trap equilibrium when the
economy lies in the vicinity of the effective lower bound policy rate, whereby
conventional monetary policy reveals to be impotent in fostering aggregate
demand and therefore the level of prices, without triggering other sources of
macroeconomic instability—such as unsustainability of fiscal policy or a burst
of inflation. The contribution of the present paper is to analyze the issue of
aggregate stability in an overlapping generations monetary model that exhibits
multiple steady-state equilibria because of the existence of a binding lower
bound on nominal interest rates. By departing from the Ricardian debt
equivalence (Barro, 1974), due to a finite planning horizon of private agents
and to the absence of perfect intergenerational altruism as originally
proposed in the Yaari (1965) and Blanchard (1985)’s uncertain lifetime
approach, the setup developed in this study enables me to explore in an
analytically tractable way the effects of alternative fiscal policy regimes
and their interaction with monetary policy from both a local- and a global-
dynamics perspective.

The paper establishes policy-relevant results that would not appear in the
traditional infinitely-lived single representative agent framework. First I
show the dynamic consequences of a budgetary policy regime in which the fiscal
authority gradually adjusts the stock of real government liabilities relative
to the size of the economy in order to converge towards a target level in the
long run, as assumed by Minea and Villieu (2013), Maebayashi, Hori and
Futagami (2017), and Cheron et al. (2019) in non-monetary contexts with
endogenous growth to analyze the case of fiscal constraints in the spirit of
the Maastricht-Treaty framework prevailing in the European Union. I
demonstrate that such a regime of stringent fiscal discipline makes the
economic system prone to self-fulfilling deflationary (or disinflationary)
trajectories approaching to an unintended steady state that exhibits the
features of a liquidity trap, which prevents the monetary authority from
uniquely determining inflation along a saddle path passing through the target
rate.

In the present setup with dyachronous heterogeneity, deflationary dynamics
induce, per se, a redistribution of real financial wealth from future to
current generations, because the implied increase in the real value of
government liabilities rises the burden of future fiscal restrictions. Such an
intergenerational redistribution of wealth in favor of currently alive
generations would provide, ceteris paribus, a stimulus to aggregate demand and
thus to prices, tending as a consequence to counter-react to the initial
deflationary pressures. In other words, a stabilizing role of wealth effects,
potentially capable of making liquidity traps implausible along the lines of
the traditional theory of Pigou (1943, 1947) and Patinkin (1965), is operative
in my optimizing framework.

However, in a condition of general equilibrium that takes also into account
the design of fiscal policy, if the government is engaged in making the stock
of real government liabilities relative to the size of the economy converge to
a target level to comply constraints of the Maastricht-type, following
deflation primary surpluses must increase in order to avoid escalation of real
debt. Under these circumstances, the resulting fiscal austerity measures
generate a negative wealth effect on current cohorts’ consumption and turn to
reinforce the initial deflationary pressures, hence leading the economic
system to a liquidity-trap steady state, whereby monetary policy loses control
over the inflation rate. Global indeterminacy arises. Specifically, any
inflation trajectory originating below the saddle path passing through the
target steady state can be sustained as an equilibrium outcome, bringing about
macroeconomic instability.

Then I explore the issue of how to lift the economy out of liquidity traps. I
demonstrate that a large class of ‘activist’ but sustainable fiscal policy
regimes—decreasing taxes and/or increasing public transfers in response to
decelerating inflation under the respect of the government’s solvency
condition—enable the economic system to escape liquidity traps and, at the
same time, embark inflation into a globally stable path converging to the
target rate. My results indicate that to sustain global determinacy and thus
ensure macroeconomic stability, there is no need for the government to render
the liquidity-trap steady state fiscally unsustainable as predicted in the
context of the traditional Ramsey-type single representative agent paradigm.

Should deflationary spirals take place, a sequence of fiscal expansions
financed by issuance of bonds and future taxes net of transfers enlarges both
human and financial wealth for currently alive households, thereby sustaining
aggregate consumption and exerting a powerful stimulus to aggregate demand
despite the expected increase in real interest rates in the vicinity of the
trapping equilibrium, for government debt is net wealth for living
generations. This course of sustainable policy action, which partly shifts the
sequence of future net taxes to future generations thereby inducing current
generations to dissave, is shown to be capable of offsetting self-fulfilling
falls in prices, restoring in general equilibrium convergence of inflation
towards the intended steady state.

However, I demonstrate that not all types of expansionary budgetary policies
that are able to eradicate liquidity traps are compatible with global
determinacy. In particular, I find that when fiscal stimulus is overly
aggressive, (unstable) spiral dynamics around the liquidity-trap steady state
exist. In this case, even though convergence of inflation towards the target
rate along a saddle connection is ensured, global indeterminacy prevails. In
addition, the inflation rate may fluctuate for relatively long periods of time
in a region whereby monetary policy is necessarily ‘passive’—in the sense of
Leeper (1991)—because of the effective lower bound on the nominal interest
rate, away from the intended steady state. These results cast doubts on the
stabilizing properties of excessively lax fiscal boosts expected to be backed
by large future net taxes required to redeem the government debt eventually.

Two clear-cut policy implications stem from my analytical findings. First,
strict fiscal discipline does not support price stability, contrary to
conventional wisdom, at least since Sargent and Wallace (1981). Rather, it is
a potential source of macroeconomic imbalances, for it dampens aggregate
demand following deflationary slumps, thus exacerbating expectations-driven
instabilities in prices. A self-fulfilling ‘austerity-deflation’ nexus does
amplify the possibility of undesired liquidity traps. Second, enforcing an
intertemporally balanced and moderately aggressive bond-financed fiscal
stimulus at a time of out-of-equilibrium deflationary trajectories plays an
essential role, complementary to inflation-targeting-oriented interest rate
feedback rules (e.g., Taylor, 1999, 2012; Woodford, 2003; Galí, 2015; Walsh,
2017), in order to guarantee price and macroeconomic stability. Central banks
can determine equilibrium inflation globally under the rules-based approach to
monetary policy (Taylor, 2021), but only if supported by feedback budgetary
policy actions of ‘last resort’.

The paper is organized as follows. Section 2 points out the paper’s
connections with the literature and elucidates the novel contribution of the
present study. Section 3 develops the optimizing continuous time overlapping
generations model and specifies the monetary policy regime. Section 4 examines
global dynamics under a debt targeting fiscal policy regime. Section 5
examines global dynamics under an activist fiscal policy regime compatible
with the respect of the government’s intertemporal budget constraint. The
concluding Section 6 sums up the main results.

##  2 Related Literature

The macroeconomic framework set forth in this paper and the results that
emerge from the present analysis are markedly different than those prevailing
in most of the literature on ‘confidence-driven’ liquidity traps (see Bilbie,
2018, and Nakata and Schmidt, 2019). The seminal works by Benhabib, Schmitt-
Grohé and Uribe (2001, 2002) and Schmitt-Grohé and Uribe (2009) employ an
infinitely-lived single representative agent setup with real balance effects à
la Sidrauski (1967)-Brock (1974, 1975), thereby overlooking intergenerational
heterogeneity and the associated wealth effects on aggregate demand dynamics.
The analysis developed in this paper is an effort to fill this gap. In
particular, Benhabib, Schmitt-Grohé and Uribe (2002) show that fiscal policy
requires to be unsustainable at the steady state exhibiting deflation or
relatively low inflation in order to avoid the liquidity trap equilibrium, for
the agents’ transversality condition turns to be violated. In contrast, in the
present setting with overlapping generations of finitely-lived agents, which
brings the advantage of encompassing the Ramsey-type framework as a limiting
case, I demonstrate that fiscal boosts satisfying both the individuals’
transversality condition and the government’s solvency constraint do suffice
to escape liquidity traps and, at the same time, sustain global determinacy.
Stabilizing budgetary policies need not be intertemporally unbalanced. In
addition, however, I prove that fiscal stimulus of last resort should not be
overly aggressive, in order to avoid that the liquidity-trap steady state is
an unstable spiral point—which would give rise to global indeterminacy.

Performing a thorough numerical analysis of a nonlinear New Keynesian model
with distortionary taxation, Mertens and Ravn (2014) show that supply-side
policies such as cuts marginal labour tax rates—as opposed of demand-side
policies such as increases in government spending—are necessary to properly
offset expectations-driven liquidity traps. My paper differs in three relevant
dimensions. First, I present an alternative way to analyze the implications of
confidence-driven liquidity traps for fiscal policy design, since I employ an
overlapping generations model whereby demographic heterogeneity
matters—contrary to the standard New Keynesian setup. Taxes/transfers are lump
sum, prices are flexible and time is continuous, in order to make my analysis
directly comparable to Benhabib, Schmitt-Grohé and Uribe (2002) and reconsider
transparently the role of demand stimulating budgetary policies in a model
where private agents have finite horizons, without unnecessary complications.
Combining distortionary taxes and sticky prices with an overlapping
generations setting and analyzing the implied global dynamics are important
considerations for future research. The present modeling approach to monetary-
fiscal interactions in the presence of multiple steady-state equilibria could
then constitute a fruitful benchmark for more complex investigations along
these lines. Second, in my simple and tractable continuous-time model I am
enabled to explore analytically the scope for the existence—under a particular
type of fiscal policy—of a saddle connection among different steady states,
leading to global determinacy under the Taylor-rule monetary framework and
escaping the liquidity-trap equilibrium. Third, differently from Mertens and
Ravn (2014), I do find a positive role of demand-side-oriented fiscal stimulus
respecting the government’s intertemporal budget constraint in order to
restore an inflationary path uniquely converging to the central bank’s target
rate. The view that fiscal intervention stimulating aggregate demand, financed
by bond issues and future higher taxes and/or lower expenditures to avoid
public insolvency, is a powerful tool to reverse downward pressure on
inflation linked to the existence of a liquidity-trap equilibrium appears to
have sound microfoundations.

Using a standard New Keynesian model à la Woodford (2003), Schmidt (2016)
shows that Ricardian government spending rules that prevent a decline in real
marginal costs following a confidence shock protect the economy from falling
into expectations-driven liquidity traps. My paper is complementary in three
important dimensions. First, my contribution is to demonstrate that
sustainable fiscal actions can be justified even in the most innocuous
monetary framework with flexible prices, once intergenerational heterogeneity
is accounted for. The stabilizing dynamic effect of Ricardian budgetary
policies does not require pricing frictions. Second, in the present model with
overlapping-generations demographics I show that out-of-target deflationary
slumps are typically escaped not by making the liquidity-trap steady state
incompatible with agents’ optimizing conditions in general equilibrium, as in
Schmidt (2016), but by making it locally unstable. Under these circumstances,
I prove that there exists a heteroclinic orbit connecting the unintended
steady state with the intended one, along which global determinacy of
equilibrium applies should fiscal policy design ensure that the undesired
steady state is an unstable node. Third, I analyze the dynamic implications of
a Ricardian experiment concerning an intertemporal reallocation of taxes and
transfers, in order to elucidate the consequences of intergenerational wealth
effects in an otherwise conventional model exhibiting the presence of
Ricardian equivalence and the existence of liquidity traps. Importantly, from
an empirical perspective, such a policy focus is justifiable from the fact
that fiscal stimulus packages in the aftermath of both the Great Recession and
the corona crisis have largely consisted of increases in public transfers
(e.g., Taylor, 2018; Bayer et al., 2020). Remarkably, the first fiscal plan
decided by the Biden administration in the United States—The American Rescue
Plan Act of 2021111https://www.congress.gov/bill/117th-congress/house-
bill/1319/text.—consists for approximately three quarters of transfers.

##  3 The Model

For the objectives of this work, I model intergenerational heterogeneity
within a monetary economy in continuous time by setting forth a modified
version of the Yaari (1965)-Blanchard (1985)-Weil (1989) overlapping
generations framework, whereby forward looking agents have finite horizons,
extended to incorporate money in the agents’ asset menu and embed the more
general case of non-separable preferences over consumption and real cash
balances. The resulting framework will permit me to remain close to the
literature on global perspectives of macroeconomic policy rules that
originates at least since the seminal contributions by Benhabib, Schmitt-Grohé
and Uribe (2001, 2002), based on the other hand upon the infinite-horizon
single representative agent setup.

###  3.1 The Individual’s Consumption Behavior

Agents face uncertainty about the duration of their lives. Each individual, in
particular, is assumed to be subject to a common and constant instantaneous
probability of death, $\mu>0$. At each instant of time $t$ a new generation is
born and total population is assumed to grow at a constant rate $n$.
Therefore, the birth rate is $\beta=n+\mu$. Denote by $N(t)$ total population
at time $t$, with $N\left(0\right)=1$ for simplicity. Hence, the size of the
generation born at time $t$ is $\beta N(t)=\beta e^{nt}$, while the size of
the surviving cohort born at time $s\leq t$ is $\beta
N\left(s\right)e^{-\mu(t-s)}=\beta e^{-\mu t}e^{\beta s}$. Population at time
$t$ is consequently given by $N(t)=\beta e^{-\mu t}\int_{-\infty}^{t}e^{\beta
s}ds$.

There is no intergenerational operative bequest motive, following Blanchard
(1985). Newly born agents have no assets. Individuals have identical
preferences and are assumed to supply one unit of labor inelastically, which
is transformed one-for-one into output, for analytical convenience. Each agent
belonging to the generation born at time $s\leq 0$ chooses the time path of
consumption, $\overline{c}(s,t)$, and real money balances,
$\overline{m}(s,t)$, in order to maximize the expected discounted value of the
utility function given by

$E_{0}\int_{0}^{\infty}\log\Upsilon\left(\overline{c}(s,t),\overline{m}(s,t)\right)e^{-\rho
t}dt,$ (1)

where $E_{0}$ denotes the expectation operator conditional on period $0$
information, $\rho>0$ is the pure rate of time preference, and the subutility
function $\Upsilon\left(\cdot\right)$ is strictly increasing, concave, and
linearly homogenous. According to Reis (2007), consumption and real money
balances are Edgeworth complements, that is, $\Upsilon_{cm}>0$. Following
Cushing (1999), the elasticity of substitution between the two is lower than
unity.

Using the fact that the probability at time $0$ of surviving at time $t\geq 0$
is $e^{-\mu t}$, the expected lifetime utility function (1) can be rewritten
as

$\int_{0}^{\infty}\log\Upsilon\left(\overline{c}(s,t),\overline{m}(s,t)\right)e^{-\left(\mu+\rho\right)t}dt,$
(2)

implying that the effective subjective discount rate with lifetime uncertainty
is $\mu+\rho$.

Individuals accumulate their real assets, $\overline{a}(s,t)$, in the form of
interest bearing public bonds, $\overline{b}(s,t)$, and real money balances,
so that $\overline{a}(s,t)=\overline{b}(s,t)+\overline{m}(s,t)$. The
instantaneous budget constraint in real terms takes the form

$\dot{\overline{a}}(s,t)=\left(R(t)-\pi(t)+\mu\right)\overline{a}(s,t)+\overline{y}(s,t)-\overline{\tau}(s,t)-\overline{c}(s,t)-R(t)\overline{m}(s,t),$
(3)

where $R(t)$ denotes the nominal interest rate, $\pi(t)$ the inflation rate,
$\overline{y}(s,t)$ output, and $\overline{\tau}(s,t)$ real lump-sum taxes net
of public transfers. Following Yaari (1965), the budget constraint
incorporates the hypothesis that in each period consumers of generation $s$
receive an actuarial fair premium, given by $\mu\overline{a}(s,t)$, from
perfectly competitive life insurance companies in exchange for their total
financial wealth at the time of death. The life insurance market avoids the
possibility for individuals of passing away leaving undesired bequests to
their heirs. As emphasized by Blanchard (1985), under the alternative
hypothesis of actuarial bonds issued by financial intermediaries, results
would be equivalent.

Agents are precluded from engaging in Ponzi’s games, implying

$\underset{t\rightarrow\infty}{\lim}\overline{a}(s,t)e^{-\int_{0}^{t}(R(j)-\pi(j)+\mu)dj}\geq
0.$ (4)

Let $\overline{x}(s,t)$ denote total consumption, defined as physical
consumption plus the interest forgone on real money holdings. That is,

$\overline{x}(s,t)\equiv\overline{c}(s,t)+R(t)\overline{m}(s,t).$ (5)

Hence, the agent’s optimizing problem can be solved using a two-stage
budgeting procedure (Deaton and Muellbauer, 1980; Marini and van der Ploeg,
1988).

In the first stage, consumers solve an intratemporal maximizing problem of
choosing the efficient allocation between consumption and real money balances,
in order to maximize the instantaneous subutility function
$\Upsilon\left(\cdot\right)$ for a given level of total consumption.
Optimality implies that the marginal rate of substitution between consumption
and real money balances must equal the nominal interest rate:

$\frac{\Upsilon_{m}\left(\overline{c}(s,t),\overline{m}(s,t)\right)}{\Upsilon_{c}\left(\overline{c}(s,t),\overline{m}(s,t)\right)}=R(t).$
(6)

Because preferences are linearly homogenous, condition (6) takes the form

$\overline{c}(s,t)=\Omega\left(R(t)\right)\overline{m}(s,t),$ (7)

where $\Omega^{\prime}\left(R(t)\right)>0$. This sign restriction follows from
$\Upsilon_{cc}-\Upsilon_{cm}\Upsilon_{c}/\Upsilon_{m}<0$ and
$\Upsilon_{mm}-\Upsilon_{cm}\Upsilon_{m}/\Upsilon_{c}<0$.

In the second stage, agents solve an intertemporal maximizing problem of
choosing the time path of total consumption, $\overline{x}(s,t)$, in order to
maximize their lifetime utility function (2) given the constraints (3)-(4) and
the optimal static condition (7). At optimum222See Appendix A for analytical
details.

$\dot{\overline{x}}(s,t)=\left(R(t)-\pi(t)-\rho\right)\overline{x}(s,t),$ (8)
$\underset{t\rightarrow\infty}{\lim}\overline{a}(s,t)e^{-\int_{0}^{t}(R(j)-\pi(j)+\mu)dj}=0.$
(9)

Integrating forward the instantaneous budget constraint (3) after using
definition (5), applying the transversality condition (9) and employing the
dynamic equation (8), total consumption can be expressed as a linear function
of total wealth:

$\overline{x}(s,t)=(\mu+\rho)\left(\overline{a}(s,t)+\overline{h}(s,t)\right),$
(10)

where

$\overline{h}(s,t)\equiv\int_{t}^{\infty}\left(\overline{y}(s,v)-\overline{\tau}(s,v)\right)e^{-\int_{t}^{v}(R(j)-\pi(j)+\mu)dj}dv$
(11)

quantifies human wealth, that is, the present discounted value of after-
tax/transfers labor income.

Using (5), (7), and (10), it also follows that individual phisical consumption
is a function of total wealth:

$\overline{c}(s,t)=\frac{(\mu+\rho)}{\Lambda(R(t))}\left(\overline{a}(s,t)+\overline{h}(s,t)\right).$
(12)

where $\Lambda\left(R(t)\right)\equiv 1+R(t)/\Omega\left(R(t)\right)$.
Combining (5), (7) and (8) yields the optimal time path of individual
consumption:

$\dot{\overline{c}}(s,t)=\left[\left(R(t)-\pi(t)-\rho\right)-\frac{\Lambda^{\prime}(R(t))}{\Lambda(R(t))}\dot{R}(t)\right]\overline{c}(s,t),$
(13)

where $\Lambda^{\prime}\left(R(t)\right)>0$. This sign restriction follows
from $\Omega\left(R(t)\right)-R(t)\Omega^{\prime}\left(R(t)\right)>0$, which
in turn depends on the elasticity of substitution between real money balances
and consumption,
$\Omega^{\prime}\left(R(t)\right)R(t)/\Omega\left(R(t)\right)$, assumed to be
lower than unity. Notice that, according to (13), the growth rate of optimal
consumption is identical across all generations, for it is independent of $s$.

###  3.2 Time Paths of Aggregate Variables

I can now derive the evolution of aggregate variables. The population
aggregate for a generic variable at individual level $\overline{z}(s,t)$ can
be obtained by integrating over all generations, so that

$\overline{Z}(t)\equiv\beta e^{-\mu
t}\int_{-\infty}^{t}\overline{z}(s,t)e^{\beta s}ds,$ (14)

where the upper case letter indicates the aggregate value at the population
level. The corresponding variable in per capita terms is denoted as

$\overline{z}(t)\equiv\overline{Z}(t)e^{-nt}=\beta\int_{-\infty}^{t}\overline{z}(s,t)e^{\beta\left(s-t\right)}ds.$
(15)

For analytical convenience, assume that each agent faces identical age-
independent income and net tax flows, so that
$\overline{y}(s,t)=\overline{y}(t)$ and
$\overline{\tau}(s,t)=\overline{\tau}(t)$, as in Blanchard (1985). Then, using
$\overline{a}(t,t)=0$ and consequently
$\overline{c}(t,t)=\left[\alpha(\mu+\rho)/\Lambda(R(t))\right]\overline{h}(t,t)$,
the budget constraint and the optimal time path of consumption expressed in
per capita terms are given by, respectively,333See Appendix B for analytical
details.

$\dot{\overline{a}}(t)=\left(R(t)-\pi(t)-n\right)\overline{a}(t)+\overline{y}(t)-\overline{\tau}(t)-\overline{c}(t)-R(t)\overline{m}(t),$
(16)
$\dot{\overline{c}}(t)=\left[\left(R(t)-\pi(t)-\rho\right)-\frac{\Lambda^{\prime}(R(t))}{\Lambda(R(t))}\dot{R}(t)\right]\overline{c}(t)-\frac{\beta(\rho+\mu)}{\Lambda(R(t))}\overline{a}(t),$
(17)

Equation (17) stipulates that evolution over time of per capita consumption
also depends upon the level of per capita real financial wealth
$\overline{a}(t)$. A higher real wealth brings about a stimulus to current
consumption at expense of future consumption. This is because the absence of
perfect intergenerational altruism implies that future cohorts’ consumption is
not valued by agents alive today. Government liabilities are net wealth for
living generations, for agents may not be alive to pay future taxes required
to guarantee public solvency. Intergenerational heterogeneity prevails: older
generations are wealthier than younger generations, and therefore consume more
and save less. Only in the special case in which the birth rate $\beta$ is
equal to zero, the law of motion of per capita consumption collapses to the
traditional Euler equation characterizing the infinitely-lived single
representative agent monetary framework (e.g., Benhabib, Schmitt-Grohé and
Uribe, 2002), whereby solely interest-rate movements affect consumption
dynamics.

###  3.3 The Public Sector

The monetary authority controls the nominal interest rate $R(t)$ and is
subject to the zero lower bound on policy rates, which prevents to set a
negative value of $R(t)$. As pointed out by Buiter (2020), the effective lower
bound that the monetary authority has to satisfy must equal the zero nominal
interest rate on currency minus the carry cost of currency, due for instance
to storage or insurance. In my model, I have abstracted from the carry cost of
currency, for analytical convenience. So the effective lower bound must be
zero. Modifying the model to account for the possibility of a negative nominal
interest rate, which is consistent with the recent behavior of some central
banks, would not alter the results of the present analysis in any essential
dimension. All that is necessary for my findings to hold is that there exists
some binding lower bound on policy rates. Whether such a bound is negative,
zero, or positive is inconsequential.

Having said this, let me assume that monetary policy is described by a
feedback policy rule of the form

$R(t)=\Psi(\pi(t)),$ (18)

where $\Psi(\cdot)$ is a continuous, positive, increasing and strictly convex
function. The assumption of a strictly positive nominal interest rate in the
present model is made to avoid discontinuity in money demand when $R(t)=0$.
Denoting by $\pi^{\ast}>0$ the target level of steady state inflation, I
assume $\Psi^{\prime}(\pi^{\ast})>1$. That is, monetary policy is ‘active’ by
satisfying the Taylor principle (see Taylor, 1999, 2012, 2021, Woodford, 2003,
Galí, 2015, and Walsh, 2017). This constraint is meant to guarantee that
whenever monetary policy makers detect symptoms of inflationary
(disinflationary) pressure, they will tighten (ease) policy sufficiently to
ensure an increase (decrease) in the real interest rate.

Public transfer payments and interest payments on government bonds are
financed by lump-sum taxation, seignorage revenues and issuance of new bonds.
Without loss of generality for the present analysis, I set government
purchases equal to zero. Hence, the flow budget constraint of the public
sector in per capita terms is given by

$\dot{\overline{b}}(t)+\dot{\overline{m}}(t)=\left(R(t)-\pi(t)-n\right)\overline{b}(t)-\overline{\tau}(t)-\left(\pi(t)-n\right)\overline{m}(t).$
(19)

Following Benhabib, Schmitt-Grohé and Uribe (2001, 2002) and Canzoneri, Cumby
and Diba (2010), equation (19) can be written as

$\dot{\overline{a}}(t)=\left(R(t)-\pi(t)-n\right)\overline{a}(t)-\overline{s}\left(t\right),$
(20)

where $\overline{a}(t)=\overline{b}(t)+\overline{m}(t)$ are total government
liabilities and
$\overline{s}\left(t\right)=\overline{\tau}(t)+R(t)\overline{m}(t)$ is the
primary surplus inclusive of interest savings from the issuance of money.

For my purposes, throughout the paper fiscal policies are assumed to be
Ricardian in the sense of Benhabib, Schmitt-Grohé and Uribe (2001, 2002). That
is, the government must respect the terminal boundary condition precluding
Ponzi’s games and requiring that the present discounted value of total
government liabilities in per capita terms converges to zero,

$\underset{t\rightarrow\infty}{\lim}\overline{a}(t)e^{-\int_{0}^{t}(R(j)-\pi(j)-n)dj}=0,$
(21)

for all possible, equilibrium and off-equilibrium, time paths of the remaining
endogenous variables. Integrating equation (20) forward, given condition (21)
that ensures public solvency, yields the intertemporal budget constraint of
the public sector:

$\overline{a}(0)=\int_{0}^{\infty}\overline{s}\left(t\right)e^{-\int_{0}^{t}(R(j)-\pi(j)-n)dj}dt.$
(22)

###  3.4 Equilibrium Inflation Dynamics

Equilibrium in the goods’ market requires that
$\overline{y}\left(t\right)=\overline{c}(t)$. Equilibrium in the money market
implies $\overline{m}(t)=\overline{c}(t)/\Omega(R(t))$. Total output
$\overline{Y}(t)$ is assumed to grow at the rate of population growth $n$,
without loss of generality. Per capita output is thus constant,
$\overline{y}(t)=\overline{y}$.

For a generic variable at the population level $\overline{Z}(t)$, let now

$z\left(t\right)\equiv\frac{\overline{Z}\left(t\right)}{\overline{Y}\left(t\right)}=\frac{\overline{z}\left(t\right)}{\overline{y}}$
(23)

indicate the corresponding variable relative to the size of the economy
$\overline{Y}\left(t\right)$. It then follows

$m(t)=\frac{1}{\Omega(R(t))}.$ (24)

Furthermore, from the law of motion of per capita consumption (17), the real
interest rate that guarantees equilibrium in the goods’ market is increasing
in the real value of government liabilities relative to output:

$R(t)-\pi(t)=\rho+\frac{\Lambda^{\prime}(R(t))}{\Lambda(R(t))}\dot{R}(t)+\frac{\beta(\rho+\mu)}{\Lambda(R(t))}a\left(t\right).$
(25)

Using the monetary policy rule (18) into equation (25), one obtains that
equilibrium dynamics of inflation must obey

$\dot{\pi}(t)=\frac{\Lambda(\Psi(\pi(t)))}{\Lambda^{\prime}(\Psi(\pi(t)))\Psi^{\prime}(\pi(t))}\left(\Psi(\pi(t))-\pi(t)-\rho\right)-\frac{\beta(\rho+\mu)}{\Lambda^{\prime}(\Psi(\pi(t)))\Psi^{\prime}(\pi(t))}a\left(t\right).$
(26)

From (26), the time path of inflation is influenced by the real financial
wealth-to-output ratio, except in the limiting case in which the birth rate
$\beta$ is equal to zero. Therefore, in order to close the model and
investigate the implied macroeconomic dynamics, one needs to specify the
policy rule describing the behavior of the fiscal authority.

##  4 Strict Fiscal Discipline and Liquidity Traps

Let me first pay attention to the dynamic consequences of a budgetary policy
regime in which the fiscal authority gradually adjusts the stock of real
government liabilities relative to the size of the economy in order to
converge towards a given target level $a^{\ast}>0$ in the long run. Thus,
consistently with Minea and Villieu (2013), Maebayashi, Hori and Futagami
(2017) and Cheron et al. (2019), the fiscal adjustment rule takes the form

$\dot{a}(t)=-\phi\left(a(t)-a^{\ast}\right),$ (27)

where $a^{\ast}$ is to be interpreted as a government policy parameter and
$\phi>0$ captures the pace of the consolidation in the case in which $a(t)$ is
larger than $a^{\ast}$.

Given the supply of money that in equilibrium endogenously adjusts to the
demand of money according to condition (24)—since in my model the monetary
authority controls the nominal interest rate $R(t)$ on the basis of rule
(18)—equation (27) consequently sets down the equilibrium issuance of
government bonds, given by

$b\left(t\right)=a(t)-\frac{1}{\Omega(\Psi(\pi(t)))}.$ (28)

Combining (20) expressed in per unit of output and (27), it also follows that
to implement the fiscal policy targeting rule the government must adjust the
primary surplus according to

$s\left(t\right)=\left(\Psi(\pi(t))+\phi-\pi(t)-n\right)a(t)-\phi a^{\ast}.$
(29)

Setting $\dot{\pi}(t)=0$ and $\dot{a}(t)=0$ in equations (26) and (27) yields
the following ‘modified Fisher equation’:

$\Psi(\pi)=\rho+\frac{\beta(\rho+\mu)}{\Lambda(\Psi(\pi))}a^{\ast}+\pi.$ (30)

Because the monetary policy reaction function $\Psi(\cdot)$ is
positive—respecting the zero lower bound on nominal interest rates—and
satisfies $\Psi^{\prime}(\cdot),\Psi^{\prime\prime}(\cdot)>0$, the steady-
state relation (30) has two solutions, $\pi^{\ast}$ and $\pi^{L}$. In
addition, since I have assumed that $\pi^{\ast}>0$ is the target inflation
rate at which $\Psi^{\prime}(\pi^{\ast})>1$, the alternative steady-state
value $\pi^{L}$ must obey $\pi^{L}<\pi^{\ast}$, is possibly negative, and
necessarily does not verify the Taylor principle, that is,
$\Psi^{\prime}(\pi^{L})<1$, so that monetary policy is ‘passive’. Thus, the
Taylor principle cannot prevail globally.

Examine next local equilibrium dynamics. Linearizing the dynamic equation (26)
around a steady-state point $\left(a^{\ast},\pi\right)$ and using (27), one
obtains the system

$\left(\begin{array}[]{c}\dot{a}(t)\\\
\dot{\pi}(t)\end{array}\right)=J^{\left(a^{\ast},\pi\right)}\left(\begin{array}[]{c}a(t)-a^{\ast}\\\
\pi(t)-\pi\end{array}\right),$ (31)

where

$J^{\left(a^{\ast},\pi\right)}=\left(\begin{array}[]{cc}-\phi&0\\\
-\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi)\right)\Psi^{\prime}(\pi)}&J_{22}^{\left(a^{\ast},\pi\right)}\end{array}\right),$
(32)

with

$J_{22}^{\left(a^{\ast},\pi\right)}=\frac{\left(\Psi^{\prime}(\pi)-1\right)\Lambda\left(\Psi(\pi)\right)}{\Lambda^{\prime}\left(\Psi(\pi)\right)\Psi^{\prime}(\pi)}+\frac{\beta(\rho+\mu)}{\Lambda\left(\Psi(\pi)\right)}a^{\ast}.$

The two eigenvalues of the Jacobian matrix $J^{\left(a^{\ast},\pi\right)}$ are
$-\phi<0$ and $J_{22}^{\left(a^{\ast},\pi\right)}$. Observe that
$J_{22}^{\left(a^{\ast},\pi^{\ast}\right)}>0$ since
$\Psi^{\prime}(\pi^{\ast})>1$, so that one eigenvalue is positive and one
eigenvalue is negative when evaluated at the target-inflation steady state
$\left(a^{\ast},\pi^{\ast}\right)$. Because $\pi(t)$ is a jump variable with a
free initial condition and $a\left(t\right)$ is a state variable,444The stock
of real government liabilities relative to output should be counted as a state
variable of the system because its value cannot jump independently of the
inflation rate. To see this, denote by $A\left(0\right)$ and $M\left(0\right)$
the initial stocks of nominal government liabilities and nominal money,
respectively, whose values are predetermined. Hence, the ratio
$A\left(0\right)/M\left(0\right)=a\left(0\right)/m\left(0\right)=a\left(0\right)\Omega(\Psi(\pi(0)))$
cannot jump, because $A\left(0\right)/M\left(0\right)$ is predetermined. It
follows that only $\pi(0)$ can jump freely in system (31), and the Blanchard-
Kahn conditions guarantee that a steady state is locally determined if one
root of the Jacobian matrix is positive and one root is negative. local
determinacy of equilibrium prevails in the neighborhood of the intended steady
state. That is, around $\left(a^{\ast},\pi^{\ast}\right)$ there exists a
unique equilibrium converging asymptotically to that steady state.
Specifically, the only trajectory of $\left(a\left(t\right),\pi(t)\right)$
converging asymptotically to $\left(a^{\ast},\pi^{\ast}\right)$ is given by
the following saddle-path solution:

$\pi\left(t\right)=\pi^{\ast}+\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\phi+J_{22}^{\left(\pi^{\ast},a^{\ast}\right)}\right)}\left(a\left(t\right)-a^{\ast}\right),$
(33) $a\left(t\right)=a^{\ast}+\left(a\left(0\right)-a^{\ast}\right)e^{-\phi
t}.$ (34)

where equation (33) describes the stable arm of the saddle path, which is
positively sloped—reflecting the fact that real government liabilities
positively affect the inflation rate via the wealth effect on consumption.
Observe, on the other hand, that
$J_{22}^{\left(a^{\ast},\pi^{L}\right)}\gtrless 0$ if and only if
$\left[\beta(\rho+\mu)/\Lambda\left(\Psi(\pi)\right)\right]a^{\ast}\gtrless\left(1-\Psi^{\prime}(\pi^{L})\right)\Lambda\left(\Psi(\pi^{L})\right)/\left[\Lambda^{\prime}\left(\Psi(\pi^{L})\right)\Psi^{\prime}(\pi^{L})\right]$,
since $\Psi^{\prime}(\pi^{L})<1$. Appendix C shows, in particular, that the
case $J_{22}^{\left(a^{\ast},\pi^{L}\right)}<0$ proves to largely apply for
any empirically plausible model’s parameterization. So both eigenvalues are
robustly negative when evaluated at the low-inflation steady state
$\left(a^{\ast},\pi^{L}\right)$. This implies that local indeterminacy of
equilibrium prevails in the neighborhood of the unintended steady state. That
is, around $\left(a^{\ast},\pi^{L}\right)$ there exists a continuum of
equilibrium paths of $\left(a\left(t\right),\pi(t)\right)$ converging
asymptotically to that steady state.

Now, one might arguably appeal to the equilibrium determinacy result around
the intended steady state to persuade readers about the stabilizing effects of
strict fiscal constraints of the type given by the adjustment rule (27), in
particular by advocating three properties of the local solution. First, under
the debt targeting fiscal regime equations (33) and (34) imply that, for
$a\left(0\right)\neq a^{\ast}$, inflation uniquely converges towards the
central bank’s target rate. Second, the higher the speed of adjustment of
government liabilities towards the long-run objective $a^{\ast}$, measured by
the parameter $\phi$, the faster inflation approaches to the target. Third,
equations (30) and (33)-(34) yield the following long-run and impact effects
of a government policy change in the target level $a^{\ast}$ on the inflation
rate:

$\frac{d\pi^{\ast}}{da^{\ast}}=\frac{\beta(\rho+\mu)}{\Lambda(\Psi(\pi^{\ast}))\left(\Psi^{\prime}(\pi^{\ast})-1\right)+\beta(\rho+\mu)\Lambda^{\prime}(\Psi(\pi^{\ast}))\Psi^{\prime}(\pi^{\ast})/\Lambda(\Psi(\pi^{\ast}))}>0,$
(35)
$\frac{d\pi\left(0\right)^{+}}{da^{\ast}}=\frac{d\pi^{\ast}}{da^{\ast}}+\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\phi+J_{22}\right)}\left(\frac{da\left(0\right)^{+}}{d\pi\left(0\right)^{+}}\frac{d\pi\left(0\right)^{+}}{da^{\ast}}-1\right),$
(36)

so that, using the fact that
$A\left(0\right)/M\left(0\right)=a\left(0\right)/m\left(0\right)=a\left(0\right)\Omega(\Psi(\pi(0)))$,
where $A\left(0\right)$ and $M\left(0\right)$ denote the initial predetermined
stocks of nominal government liabilities and nominal money, respectively,
which implies
$da\left(0\right)^{+}/d\pi\left(0\right)^{+}=-\Omega^{\prime}(\Psi(\pi(0)))\Psi^{\prime}(\pi(0)/$
$\left[\left(A\left(0\right)/M\left(0\right)\right)\Omega(\Psi(\pi(0)))^{2}\right]$,

$\displaystyle\frac{d\pi\left(0\right)^{+}}{da^{\ast}}$ $\displaystyle=$
$\displaystyle\frac{\phi\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})J_{22}\left(\phi+J_{22}\right)}$
(37)
$\displaystyle\times\left(1+\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\phi+J_{22}\right)}\frac{\Omega^{\prime}(\Psi(\pi(0)))\Psi^{\prime}(\pi(0)}{\frac{A\left(0\right)}{M\left(0\right)}\Omega(\Psi(\pi(0)))^{2}}\right)^{-1}$
$\displaystyle>$ $\displaystyle 0.$

Thus, a reduction in the target level of real government liabilities relative
to the size of the economy enables the monetary authority to set a lower
inflation target, for it induces a fall in the steady-state real interest rate
via the modified Fisher equation. This reflects the fact that the fiscal
consolidation brings about a redistribution of resources from current to
future generations, since it lowers the future tax burden required to finance
the reduced interest payments on government bonds. On impact inflation jumps
downwards from $\pi(0)$ to $\pi\left(0\right)^{+}$ onto the new stable arm
described by equation (33), undershooting the new long-run equilibrium, since
$d\pi\left(0\right)^{+}/da^{\ast}<d\pi^{\ast}/da^{\ast}$. Notice that only in
the special case of a zero birth rate, $\beta=0$, inflation would be
independent of fiscal variables, since the model would collapse into the
infinitely-lived single representative agent paradigm, exhibiting the
Ricardian debt equivalence and no longer describing an economy with
intergenerational heterogeneity.

Summing up, should the economy be expected to permanently remain in the
neighborhood of the inflation target, the above three implications obtained in
the general case $\beta>0$ do support the consensus view that an appropriate
fiscal commitment and a correct pace at which a fiscal consolidation plan is
implemented sustain the central bank’s objective of maintaining price
stability. The point, however, is that the overall story is radically
different if one does not limit to local analysis and look instead at the
global behavior of the system.

To this purpose, first observe that from equation (27), the
$\dot{a}(t)=0$–locus is given by $a\left(t\right)=a^{\ast}$, which in the
phase plane $\left(a\left(t\right),\pi(t)\right)$ is vertical. On the other
hand, from equation (26), the $\dot{\pi}(t)=0$–locus is implicitly given by

$\left(\Psi(\pi(t))-\pi(t)-\rho\right)\Lambda(\Psi(\pi(t)))=\beta(\rho+\mu)a\left(t\right).$
(38)

Because

$\left.\frac{d\pi(t)}{da\left(t\right)}\right|_{\dot{\pi}(t)=0}=\frac{\beta(\rho+\mu)}{\left(\Psi^{\prime}(\pi(t))-1\right)\Lambda(\Psi(\pi(t)))+\left(\Psi(\pi(t))-\pi(t)-\rho\right)\Lambda^{\prime}\left(\Psi(\pi(t))\right)\Psi^{\prime}(\pi(t))},$
(39)

which is positive at $\left(a^{\ast},\pi^{\ast}\right)$, negative at
$\left(a^{\ast},\pi^{L}\right)$ (see Appendix C), and infinity when
$\left(1-\Psi^{\prime}(\pi(t))\right)\Lambda(\Psi(\pi(t)))=\left(\Psi(\pi(t))-\pi(t)-\rho\right)\Lambda^{\prime}\left(\Psi(\pi(t))\right)\Psi^{\prime}(\pi(t))$,
in the phase plane $\left(a\left(t\right),\pi(t)\right)$ the
$\dot{\pi}(t)=0$–locus is horizontally U-shaped. The two loci intersect at the
steady states $\left(a^{\ast},\pi^{\ast}\right)$ and
$\left(a^{\ast},\pi^{L}\right)$. Then, from (27) and (26), I have
$\dot{a}(t)>\left(<\right)0$ if $a<\left(>\right)a^{\ast}$ and
$\dot{\pi}(t)>\left(<\right)0$ if
$\left(\Psi(\pi(t))-\pi(t)-\rho\right)\Lambda(\Psi(\pi(t)))>\left(<\right)\beta(\rho+\mu)a\left(t\right)$.

The resulting global dynamics are characterized in Figure 1.

Figure 1: Dynamic behavior of $\left(a\left(t\right),\pi(t)\right)$ under
strict fiscal discipline

The stable arm of the saddle point passing through the steady state
$\left(a^{\ast},\pi^{\ast}\right)$, has a slope given by
$\beta(\rho+\mu)/\left[\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\phi+J_{22}^{\left(\pi^{\ast},a^{\ast}\right)}\right)\right]>0$
that is lower than the slope of the $\dot{\pi}(t)=0$–locus evaluated at
$\left(a^{\ast},\pi^{\ast}\right)$, given by
$\beta(\rho+\mu)/\left[\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})J_{22}^{\left(\pi^{\ast},a^{\ast}\right)}\right]>0$.
It clearly emerges that, in the neighborhood of the intended steady state
$\left(a^{\ast},\pi^{\ast}\right)$, there exists an infinite number of initial
values for the inflation rate featured by $\pi\left(0\right)<\pi^{\ast}+$
$\left\\{\beta(\rho+\mu)/\left[\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\phi+J_{11}^{\left(\pi^{\ast},a^{\ast}\right)}\right)\right]\right\\}\left(a\left(0\right)-a^{\ast}\right)$
such that $\left(a\left(t\right),\pi(t)\right)$ will spiral into deflationary
trajectories converging to the trapping steady state
$\left(a^{\ast},\pi^{L}\right)$.555Notice that global inflation dynamics
originating around $\left(a^{\ast},\pi^{\ast}\right)$ follow a (non-)monotonic
behavior if they start to the right (left) of the $\dot{\pi}(t)=0$–locus. The
saddle manifold associated with $\left(a^{\ast},\pi^{\ast}\right)$ is
precisely the boundary of the basin of attraction of
$\left(a^{\ast},\pi^{L}\right)$.

In other words, once a global perspective is accounted for, inflation no
longer needs to jump on the saddle path leading to the target rate to
guarantee dynamic stability. All initial values of inflation bounded above by
the saddle path are equilibrium values for they make
$\left(a\left(t\right),\pi(t)\right)$ approach to the alternative steady state
$\left(a^{\ast},\pi^{L}\right)$. This implies that the economic system is
affected by global indeterminacy.

In the present framework with dyachronous heterogeneity, deflationary paths
generate, per se, a redistribution of real financial wealth from future to
current generations, since the associated rise in the stock of government
liabilities in real terms causes the fiscal burden borne by future cohorts’ to
increase in order to avoid public insolvency. This intergenerational
reallocation of resources beneficial to current generations would exert,
ceteris paribus, a stimulus to aggregate consumption and as a consequence to
prices, thereby potentially offsetting the initial deflationary perturbations,
along lines analogous to the traditional wealth-effect channel emphasized in
the static models of Pigou (1943, 1947) and Patinkin (1965). Nevertheless, if
the fiscal policy regime forces the government to adjust gradually the real
amount of government liabilities relative to the size of the economy at a
certain level—similarly, say, to the permanent commitment rules in the
Maastricht Treaty and the Fiscal Compact reforming the Stability and the
Growth Pact in the European Union666The Fiscal Compact—formally ‘Treaty on
Stability, Coordination and Governance in the Economic and Monetary Union’—is
in force since 2013, although temporarily suspended from 2020 to 2023 because
of the pandemic crisis, and features a well-defined debt reduction benchmark
rule. Specifically, the rule establishes that Member States whose debt-to-GDP
ratio exceeds the 60% threshold are required to reduce their ratios to the
reference value at an average rate of one-twentieth per year.—following out-
of-fundamentals deflationary slumps primary surpluses are expected to expand
in order to choke off a potential escalation of real debt. The resulting
fiscal austerity actions bring about a negative wealth effect on current
cohorts’ consumption and consequently strengthen the deflationary pressures
over time, hence validating the initial arbitrary revision in agents’
expectations and making the economy descend in a self-fulfilling manner into a
steady state that displays the characteristics of a liquidity trap: monetary
policy turns to be powerless in positively affecting aggregate demand and thus
prices, losing control over inflation and incurring sunspot fluctuations to
the extent that the interest-rate policy stance is passive in the vicinity of
the effective lower bound—so that the Taylor principle is no longer satisfied.

In synthesis, the stabilizing effects of fiscal consolidation strategies that
result from a local-dynamics perspective break down, and are even reversed,
from a global-dynamics perspective. Unlike conventional wisdom, strict fiscal
discipline does not support price stability for, on the contrary, is likely to
give rise to a self-fulfilling austerity-decelerating inflation nexus that
amplifies the possibility of expectations-driven liquidity traps.

##  5 Escaping Liquidity Traps through Activist Fiscal Policy Regimes

Consider now the dynamic implications of a policy regime in which the fiscal
authorities are assumed to react actively—in a complementary way with respect
to monetary policy—in the event that off-target decelerating inflation paths
leading to a liquidity trap materialize, provided that intertemporal budget
constraint of the government is always satisfied.

Applying (23) to (20) yields the flow budget constraint of the public sector
in terms of ratios to output:

$\dot{a}(t)=\left(R(t)-\pi(t)-n\right)a(t)-s\left(t\right).$ (40)

The solvency condition thus requires that

$\underset{t\rightarrow\infty}{\lim}a(t)e^{-\int_{0}^{t}(R(j)-\pi(j)-n)dj}=0,$
(41)

or, equivalently,

$a(0)=\int_{0}^{\infty}s\left(t\right)e^{-\int_{0}^{t}(R(j)-\pi(j)-n)dj}dt.$
(42)

Suppose, in particular, that the feedback budgetary policy rule takes the form

$s\left(t\right)=\Theta\left(\Psi(\pi(t))-\pi(t)\right)a\left(t\right)+\Gamma\left(\Psi(\pi(t))-\pi(t)\right),$
(43)

where $\Theta(\cdot)$ and $\Gamma(\cdot)$ are continuous and increasing
functions. The idea behind rule (43) is that, whenever fiscal policy makers
observe symptoms of decelerating inflation—with the central bank that
decreases the real interest rate according to the Taylor principle in the
attempt to reverse dynamics, they will ease budgetary policy by ensuring a
decline in primary surpluses or a rise in primary deficits through tax cuts
and/or transfer increases. The fiscal expansion complementary to the Taylor-
type feedback rule is assumed to occur both directly via $\Gamma(\cdot)$ and
indirectly via the reduction of the sensitivity of the primary surplus with
respect to total government liabilities, measured by $\Theta(\cdot)$.
Functions $\Theta(\cdot)$ and $\Gamma(\cdot)$ are such that the solvency
condition (41) is globally satisfied. Specifically, using the resulting law of
motion of $a(t)$,

$\dot{a}(t)=\left(\Psi(\pi(t))-\pi(t)-n-\Theta\left(\Psi(\pi(t))-\pi(t)\right)\right)a(t)-\Gamma\left(\Psi(\pi(t))-\pi(t)\right),$
(44)

this occurs when, should government liabilities embark on potentially
explosive paths, beyond a certain arbitrarily chosen threshold level for
$a(t)$ function $\Theta(\cdot)$ becomes permanently positive (see, e.g., Bohn,
1991, 1998), so that

$\underset{t\rightarrow\infty}{\lim}\int_{0}^{t}\Theta\left(\Psi(\pi(v))-\pi(v)\right)dv=+\infty,$
(45)

and, in addition,

$\underset{t\rightarrow\infty}{\lim}-e^{-\int_{0}^{t}\Theta\left(\Psi(\pi(v))-\pi(v)\right)dv}\int_{0}^{t}\Gamma\left(\Psi(\pi(v))-\pi(v)\right)e^{-\int_{0}^{v}(R(j)-\pi(j)-n-\Theta\left(\Psi(\pi(j))-\pi(j)\right))dj}dv=0.$
(46)

Under (45), condition (46) is surely verified when the present discounted
value of future primary deficits associated to function $\Gamma(\cdot)$ in
(43) is lower than $+\infty$—an arguably realistic assumption.

Setting $\dot{a}(t)=0$ and $\dot{\pi}(t)=0$ in the dynamic equations (44) and
(26) yields

$a=\frac{\Gamma\left(\Psi(\pi)-\pi\right)}{\left(\Psi(\pi)-\pi-n-\Theta\left(\Psi(\pi)-\pi\right)\right)},$
(47) $\Psi(\pi)=\rho+\frac{\beta(\rho+\mu)}{\Lambda(\Psi(\pi))}a+\pi.$ (48)

Close inspection of (47) and (48) reveals that there exist at least two steady
states, $\left(a^{\ast},\pi^{\ast}\right)$ and $\left(a^{L},\pi^{L}\right)$,
obeying $\pi^{\ast}>\pi^{L}$, $\Psi^{\prime}(\pi^{\ast})>1$,
$\Psi^{\prime}(\pi^{L})<1$, and $a^{\ast}\gtreqless a^{L}$. This reflects the
fact that, for a given $a$, the modified Fisher equation (48) continues to
have two solutions for $\pi$, one in which monetary policy is active and one
in which monetary policy is passive, analogously to the previous section.

Linearizing equations (44) and (26) in the neighborhood of any steady-state
point $\left(a,\pi\right)$, one obtains the system

$\left(\begin{array}[]{c}\dot{a}(t)\\\
\dot{\pi}(t)\end{array}\right)=K^{\left(a,\pi\right)}\left(\begin{array}[]{c}a(t)-a\\\
\pi(t)-\pi\end{array}\right),$ (49)

where

$K^{\left(a,\pi\right)}=\left(\begin{array}[]{cc}\Psi(\pi)-\pi-n-\Theta\left(\Psi(\pi)-\pi\right)&\left(\Psi^{\prime}(\pi)-1\right)\left(\begin{array}[]{c}a-\Theta^{\prime}\left(\Psi(\pi)-\pi\right)a\\\
-\Gamma^{\prime}\left(\Psi(\pi)-\pi\right)\end{array}\right)\\\
-\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi)\right)\Psi^{\prime}(\pi)}&K_{22}^{\left(a,\pi\right)}\end{array}\right),$
(50)

with

$K_{22}^{\left(a,\pi\right)}=\frac{\left(\Psi^{\prime}(\pi)-1\right)\Lambda\left(\Psi(\pi)\right)}{\Lambda^{\prime}\left(\Psi(\pi)\right)\Psi^{\prime}(\pi)}+\frac{\beta(\rho+\mu)}{\Lambda\left(\Psi(\pi)\right)}a.$

The determinant and the trace of the Jacobian matrix $K^{\left(a,\pi\right)}$
are

$\displaystyle\det\text{ }K^{\left(a,\pi\right)}$ $\displaystyle=$
$\displaystyle\left(\Psi(\pi)-\pi-n-\Theta\left(\Psi(\pi)-\pi\right)\right)K_{22}^{\left(a,\pi\right)}$
(53)
$\displaystyle+\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi)\right)\Psi^{\prime}(\pi)}\left(\Psi^{\prime}(\pi)-1\right)\left(\begin{array}[]{c}a-\Theta^{\prime}\left(\Psi(\pi)-\pi\right)a\\\
-\Gamma^{\prime}\left(\Psi(\pi)-\pi\right)\end{array}\right),$ $\text{tr
}K^{\left(a,\pi\right)}=\Psi(\pi)-\pi-n-\Theta\left(\Psi(\pi)-\pi\right)+K_{22}^{\left(a,\pi\right)}.$
(54)

Observe that $\det$ $K^{\left(a^{\ast},\pi^{\ast}\right)}<0$ if

$\displaystyle\Gamma^{\prime}\left(\Psi(\pi^{\ast})-\pi^{\ast}\right)+\Theta^{\prime}\left(\Psi(\pi^{\ast})-\pi^{\ast}\right)a^{\ast}$
(55) $\displaystyle>$ $\displaystyle
a^{\ast}+\frac{\left(\Psi(\pi^{\ast})-\pi^{\ast}-n-\Theta\left(\Psi(\pi^{\ast})-\pi^{\ast}\right)\right)\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})K_{22}^{\left(a^{\ast},\pi^{\ast}\right)}}{\beta(\rho+\mu)\left(\Psi^{\prime}(\pi^{\ast})-1\right)}.$

That is, local determinacy of equilibrium applies in the neighborhood of the
intended steady state $\left(a^{\ast},\pi^{\ast}\right)$, where
$\Psi^{\prime}(\pi^{\ast})>1$, provided that fiscal policy is sufficiently
reactive to potential disinflationary pressures. In particular, the locally
unique trajectory of $\left(a\left(t\right),\pi(t)\right)$ converging
asymptotically to $\left(a^{\ast},\pi^{\ast}\right)$ is given by the saddle-
path solution expressed by

$\pi\left(t\right)=\pi^{\ast}-\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\varepsilon_{1}-K_{22}^{\left(\pi^{\ast},a^{\ast}\right)}\right)}\left(a\left(t\right)-a^{\ast}\right),$
(56)
$a\left(t\right)=a^{\ast}+\left(a\left(0\right)-a^{\ast}\right)e^{\varepsilon_{1}t}.$
(57)

where $\varepsilon_{1}$ is the negative eigenvalue associated to
$K^{\left(a^{\ast},\pi^{\ast}\right)}$ and equation (56) is the stable arm of
the saddle path, which is positively sloped since
$K_{22}^{\left(\pi^{\ast},a^{\ast}\right)}>0$—reflecting again the positive
comovement between agents’ wealth and inflation. On the other hand, the
unintended steady state $\left(a^{L},\pi^{L}\right)$ with
$\Psi^{\prime}(\pi^{L})<1$, exhibiting deflation or disinflation, may no
longer be a sink—differently from the case of a strict fiscal discipline
examined in Section 4. Indeed, observe that $\det$
$K^{\left(a^{L},\pi^{L}\right)}>0$ if

$\displaystyle\Gamma^{\prime}\left(\Psi(\pi^{L})-\pi^{L}\right)+\Theta^{\prime}\left(\Psi(\pi^{L})-\pi^{L}\right)a^{L}$
(58) $\displaystyle>$ $\displaystyle
a^{L}+\frac{\left(\Psi(\pi^{L})-\pi^{L}-n-\Theta\left(\Psi(\pi^{L})-\pi^{L}\right)\right)\Lambda^{\prime}\left(\Psi(\pi^{L})\right)\Psi^{\prime}(\pi^{L})\left(-K_{22}^{\left(a^{L},\pi^{L}\right)}\right)}{\beta(\rho+\mu)\left(1-\Psi^{\prime}(\pi^{L})\right)},\text{
\ \ \ }$

where $K_{22}^{\left(a^{L},\pi^{L}\right)}<0$ on the basis of any empirically
plausible model’s calibration (see Appendix C), and tr
$K^{\left(a^{L},\pi^{L}\right)}>0$ if

$\Theta\left(\Psi(\pi^{L})-\pi^{L}\right)<\rho-n+\frac{\beta(\rho+\mu)}{\Lambda(\Psi(\pi^{L}))}a+K_{22}^{\left(a^{L},\pi^{L}\right)}.$
(59)

That is, the steady state $\left(a^{L},\pi^{L}\right)$ becomes unstable if
fiscal policy is sufficiently activist and the sensitivity of the primary
surpluses with respect to government liabilities is relatively low. For what
will follow, in particular, it is worth pointing out that $\left(\text{tr
}K^{\left(a^{L},\pi^{L}\right)}\right)^{2}-4\det$
$K^{\left(a^{\ast},\pi^{\ast}\right)}>\left(<\right)0$ if

$\displaystyle\Gamma^{\prime}\left(\Psi(\pi^{L})-\pi^{L}\right)+\Theta^{\prime}\left(\Psi(\pi^{L})-\pi^{L}\right)a^{L}$
(60) $\displaystyle<$
$\displaystyle\left(>\right)a^{L}+\left[\frac{\left(\Psi(\pi^{L})-\pi^{L}-n-\Theta\left(\Psi(\pi^{L})-\pi^{L}\right)\right)\Lambda^{\prime}\left(\Psi(\pi^{L})\right)\Psi^{\prime}(\pi^{L})\left(-K_{22}^{\left(a^{L},\pi^{L}\right)}\right)}{\beta(\rho+\mu)\left(1-\Psi^{\prime}(\pi^{L})\right)}\right]$
$\displaystyle+\frac{\Lambda^{\prime}\left(\Psi(\pi^{L})\right)\Psi^{\prime}(\pi^{L})\left(\Psi(\pi^{L})-\pi^{L}-n-\Theta\left(\Psi(\pi^{L})-\pi^{L}\right)+K_{22}^{\left(a^{L},\pi^{L}\right)}\right)^{2}}{4\beta(\rho+\mu)\left(1-\Psi^{\prime}(\pi^{L})\right)}.$

As a result, provided that fiscal policy is not overly aggressive, the roots
associated to the Jacobian matrix $K^{\left(a^{L},\pi^{L}\right)}$ are real,
implying that $\left(a^{L},\pi^{L}\right)$ is an unstable node. In this case,
there exists a continuum of curved paths that are tangent to the negatively-
slopped unstable branch given by

$\pi\left(t\right)=\pi^{L}-\frac{\beta(\rho+\mu)}{\Lambda^{\prime}\left(\Psi(\pi^{L})\right)\Psi^{\prime}(\pi^{L})\left(\eta_{1}-K_{22}^{\left(\pi^{L},a^{L}\right)}\right)}\left(a\left(t\right)-a^{L}\right),$
(61)

where $\eta_{1}>0$ is the non-dominant eigenvalue of
$K^{\left(a^{L},\pi^{L}\right)}$. By contrast, should fiscal policy be
excessively aggressive, roots are complex, implying that
$\left(a^{L},\pi^{L}\right)$ is an unstable spiral point.

I am now ready to elucidate the consequences for global dynamics under
activist fiscal policy regimes satisfying conditions (55), (58) and (59). From
equation (44), the $\dot{a}(t)=0$–locus is given by

$\frac{\Gamma\left(\Psi(\pi(t))-\pi(t)\right)}{\Psi(\pi(t))-\pi(t)-n-\Theta\left(\Psi(\pi(t))-\pi(t)\right)}=a(t),$
(62)

from which

$\left.\frac{d\pi(t)}{da\left(t\right)}\right|_{\dot{a}(t)=0}=\frac{\Gamma\left(\Psi(\pi(t))-\pi(t)\right)}{\left(\Psi^{\prime}(\pi(t))-1\right)\left(\Gamma^{\prime}\left(\Psi(\pi(t))-\pi(t)\right)+\Theta^{\prime}\left(\Psi(\pi(t))-\pi(t)\right)a(t)-a(t)\right)a(t)}.$
(63)

In what follows, I shall assume the restrictions
$\left.\Gamma\left(\Psi(\pi(t))-\pi(t)\right)\right|_{\dot{a}(t)=0}>0$ and
$\left.\Gamma^{\prime}\left(\Psi(\pi(t))-\pi(t)\right)+\Theta^{\prime}\left(\Psi(\pi(t))-\pi(t)\right)a(t)-a(t)\right|_{\dot{a}(t)=0}>0$,
according to which fiscal policy avoids the existence of more than two steady
states. Hence, the slope of the $\dot{a}(t)=0$–locus is positive at
$\left(\pi^{\ast},a^{\ast}\right)$, negative at $\left(a^{L},\pi^{L},\right)$,
and infinity when $\Psi^{\prime}(\pi(t))=1$. Therefore, in the phase plane
$\left(a\left(t\right),\pi(t)\right)$ the $\dot{a}(t)=0$–locus is horizontally
U-shaped. As in the previous section, the $\dot{\pi}(t)=0$–locus implicitily
given by (38) is also horizontally U-shaped. The two loci intersect at the
steady states $\left(a^{\ast},\pi^{\ast}\right)$ and
$\left(a^{L},\pi^{L}\right)$. From the properties of the Jacobian matrix, it
emerges that the slope of the $\dot{a}(t)=0$–locus is lower than the slope of
the $\dot{\pi}(t)=0$–locus if evaluated at $\left(a^{\ast},\pi^{\ast}\right)$
and the opposite occurs if the slopes are evaluated at
$\left(a^{L},\pi^{L}\right)$. From (44) and (26), I have that
$\dot{a}(t)>\left(<\right)0$ if
$\left(\Psi(\pi(t))-\pi(t)-n-\Theta\left(\Psi(\pi(t))-\pi(t)\right)\right)a(t)>\left(<\right)\Gamma\left(\Psi(\pi(t))-\pi(t)\right)$
and $\dot{\pi}(t)>\left(<\right)0$ if
$\left(\Psi(\pi(t))-\pi(t)-\rho\right)\Lambda(\Psi(\pi(t)))>\left(<\right)\beta(\rho+\mu)a\left(t\right)$.

In the case of real roots associated to $K^{\left(a^{L},\pi^{L}\right)}$,
which require that fiscal policy is not overly aggressive at the liquidity-
trap steady state, the resulting global dynamics are characterized in Figure
2.

Figure 2: Dynamic behavior of $\left(a\left(t\right),\pi(t)\right)$ under
moderately activist fiscal policies Figure 3: Dynamic behavior of
$\left(a\left(t\right),\pi(t)\right)$ under overly activist fiscal policies

The stable arm of the saddle point passing through
$\left(a^{\ast},\pi^{\ast}\right)$ has a positive slope given by
$-\beta(\rho+\mu)/\left[\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})\left(\varepsilon_{1}-K_{22}^{\left(\pi^{\ast},a^{\ast}\right)}\right)\right]>0$,
which is higher than the slope of the $\dot{a}(t)=0$–locus evaluated at
$\left(a^{\ast},\pi^{\ast}\right)$, given by (63) at
$\left(a^{\ast},\pi^{\ast}\right)$, and lower than the slope of the
$\dot{\pi}(t)=0$–locus evaluated at $\left(a^{\ast},\pi^{\ast}\right)$, given
by
$\beta(\rho+\mu)/\left[\Lambda^{\prime}\left(\Psi(\pi^{\ast})\right)\Psi^{\prime}(\pi^{\ast})K_{22}^{\left(\pi^{\ast},a^{\ast}\right)}\right]>0$.
Since the steady state $\left(a^{L},\pi^{L}\right)$ is an unstable node, there
exists one trajectory—the heteroclinic orbit $H$—originating in the
neighborhood of the steady state $\left(a^{L},\pi^{L}\right)$, negatively
slopped around $\left(a^{L},\pi^{L}\right)$ because tangent to the negatively
slopped eigenspace related to the nondominant eigenvalue $\eta_{1}$ expressed
by equation (61), and converging asymptotically to the steady state
$\left(a^{\ast},\pi^{\ast}\right)$—locally along the associated saddle path
whose stable arm is given by equation (56).777As the heteroclinic orbit $H$ is
tangent to the eigenspace associated to $\eta_{1}$ in the neighborhood
$\left(a^{L},\pi^{L}\right)$, it lies between the $\dot{a}(t)=0$–locus and
that eigenspace and never crosses this line. As a result, the $H$–trajectory
is enclosed by a ‘trapping region’ whose sides are given by the
$\dot{\pi}(t)=0$–locus between the two steady-state equilibria, a line passing
through the steady state $\left(a^{L},\pi^{L}\right)$ whose slope is given by
the eigenvector associated to the non-dominant eigenvalue $\eta_{1}$, and a
line passing through the steady state $\left(a^{\ast},\pi^{\ast}\right)$ whose
slope is given by the slope of the associated saddle path, between
$\left(a^{\ast},\pi^{\ast}\right)$ and the previous line. All the trajectories
starting inside the trapping area escape from it, with the exception of those
starting at any point along the heteroclinic orbit. In other words, only the
orbit does not hit the boundaries of the trapping area and follows a non-
monotonous path—in terms of dynamic behavior of $a(t)$—that changes direction
when $\dot{\pi}(t)=0$. The existence of such a saddle connection guarantees
global determinacy of equilibrium, according to which even if the economy lies
in the neighborhood of the liquidity-trap steady state at which monetary
policy is passive, now inflation and real government liabilities will uniquely
escape from $\left(a^{L},\pi^{L}\right)$ and converge towards the target
steady state at which monetary policy is active. The reason why liquidity
traps are eradicated with no commitment to trigger unsustainable budget
deficits is that a sequence of tax cuts and transfer increases financed by
bond issues and future primary surpluses expands both human and financial
wealth for currently alive households, thus spuring aggregate consumption,
since government debt is net wealth for living generations. This course of
sustainable fiscal policy measures, which partly shift the sequence of future
net taxes to future generations thereby leading current generations to
dissave, is capable of reflating the economy due to the induced excess demand
in the goods’ market and restoring in general equilibrium dynamic convergence
of inflation towards the intended steady state.

On the other hand, in the case of complex roots associated to
$K^{\left(a^{L},\pi^{L}\right)}$, which occur when fiscal policy is overly
aggressive at the unintended steady state, the implied global dynamics become
those displayed in Figure 3. Even though fiscal expansions are able to escape
liquidity traps along a heteroclinic orbit leading to
$\left(a^{\ast},\pi^{\ast}\right)$ as in the previous case, now global
indeterminacy prevails because the steady state $\left(a^{L},\pi^{L}\right)$
is an unstable spiral point. Hence, if the economy lies in the vicinity of the
liquidity-trap steady state, there exists a large class of initial values for
the inflation rate compatible with a globally stable perfect foresight
equilibrium. Furthermore, the inflation rate may fluctuate for relatively long
periods of time in a region whereby monetary policy is necessarily passive
because of the effective lower bound on the nominal interest rate, away from
the intended steady state. Taken together, these results give analytical
support to the view that excessively lax fiscal stimuli, although expected to
be repaid by large future net taxes needed to redeem the government debt
eventually, are likely to be a quite severe source of macroeconomic
instability.

In synthesis, should the latter case of macroeconomic instability induced by
too aggressive fiscal boosts be ruled out by fiscal policy makers, the
theoretical findings demonstrated in my analysis clearly lead to the
conclusion that moderately activist demand side oriented budgetary
interventions, responding to deflationary pressures under the respect of the
government’s intertemporal budget constraint, do constitute an essential tool
of ‘last resort’ in order to avert liquidity traps and, at the same time,
preserve the stabilizing properties of the rules-based approach to monetary
policy.

##  6 Conclusions

As the nominal interest rates set by central banks in most major economies
have turned close to zero and even negative as a consequence of the secular
stagnation and in order to counter the ongoing pandemic crisis—within a
prolonged period of below-target inflation—the issue of how to escape
undesirable deflationary paths associated to a situation of liquidity trap
without creating space for other forms of macroeconomic instability, such as
fiscal unsustainability or a burst of inflation, is a pressing concern for
policy makers. The present paper explores the scope for aggregate stability in
a monetary model with overlapping generations of finitely-lived agents that
displays multiple steady-state equilibria due to the existence of a binding
lower bound on nominal interest rates. First, I depart from the standard
literature on confidence-driven liquidity traps based upon the infinitely-
lived single representative agent theoretical paradigm, because in the present
framework wealth effects on aggregate demand dynamics emerging from
intergenerational heterogeneity critically affect both the monetary and fiscal
policy transmission mechanism. Hence, the model I present is a natural
extension of the seminal work by Benhabib, Schmitt-Grohé and Uribe (2002).
Second, I depart from the standard literature on fiscal policy multipliers
based upon local approximations of dynamic stochastic general equilibrium
models, because I take into account non-linearities and potential connections
between multiple steady-state equilibria resulting from a global-dynamics
perspective. Hence, my framework relaxing the Ricardian equivalence due to
disconnectedness across generations is a natural alternative to the prominent
work by Mertens and Ravn (2014), grounded on a nonlinear New Keynesian model
with distortionary taxation. Third, I concentrate on the stabilizing role of
intertemporal reallocations of taxes and transfers in an otherwise standard
monetary model with flexible prices. Hence, the model I employ is a useful
complement the theoretical analysis developed by Schmidt (2016), based on the
implications of Ricardian government spending rules in the context a typical
New Keynesian model.

My analysis leads to three conclusions. First, a rigid degree of fiscal
discipline of the type prescribed by the Maastricht-Treaty and the Fiscal-
Compact frameworks in the European Union, in which the government is required
to gradually adjust the stock of real government liabilities relative to the
size of the economy in order to guarantee convergence towards a target level
in the long run, is not a precondition to price stability, as commonly
believed. On the contrary, it is a potentially severe source of macroeconomic
imbalances, because it typically makes the economy exposed to the emergence of
unintended liquidity traps, which preclude the central bank from uniquely
pinning down inflation at the target rate. The central reason is that, under
this fiscal policy regime, following deflationary dynamics driven by arbitrary
revisions in private agents’ expectations the government is expected to rise
primary surpluses in order to rule out escalation of debt in real terms. Thus
austerity actions produce a negative wealth effect on current cohorts’
consumption, amplifying the initial deflationary perturbations over time.
Global indeterminacy emerges, since any inflation trajectory bounded above by
the saddle path passing through the target steady state can be validated as an
equilibrium outcome.

Second, implementing an intertemporally balanced fiscal stimulus in response
to out-of-equilibrium deflationary paths—with no need to make the liquidity-
trap steady state fiscally unsustainable as argued within the classical
Ramsey-type single representative agent paradigm—exerts an decisive role,
complementary to inflation-targeting-oriented interest rate feedback rules of
the Taylor-type, in ensuring macroeconomic stability. Sustainable bond-
financed fiscal boosts, decreasing taxes and/or increasing public transfers in
reaction to deflationary pressures under the respect of the public solvency
condition, are capable of eradicating liquidity traps and, at the same time,
drive inflation into a globally stable trajectory converging to the target
rate. The reason is twofold. On the one hand, under a fiscal policy regime
preventing the applications of fiscal austerity measures following
deflationary slumps, a redistribution of real financial wealth from future to
current generations occurs, since deflation increases the real value of
government liabilities, thereby enhancing the burden of future fiscal
retrenchment. On the other hand, fiscal expansions financed by bond issues and
future taxes net of transfers enlarge both human and non-human wealth for
living generations. The two effects induce currently alive individuals to
dissave, spurring aggregate consumption despite the increases in real interest
rates in the vicinity of the trapping equilibrium. Following the resulting
excess demand in the goods’ market, price increases are necessary to absorb
the disequilibrium.

Third, global determinacy is not guaranteed by all types of expansionary
budgetary policies that seek to fight liquidity traps. Overly aggressive
fiscal stimulus, in particular, induces spiral dynamics around the liquidity-
trap steady state. In this event, although liquidity traps are escaped along a
saddle connection from the steady state at which monetary policy is passive to
the steady state at which monetary policy is active and convergence of
inflation to the target rate is ensured, global indeterminacy applies, thus
warning against the enforcement of excessively pronounced fiscal boosts
expected to be paid back by large future primary surpluses.

Should the latter case be avoided, however, the foregoing results provide
sound analytical microfoundations to the view that demand side oriented fiscal
intervention respecting the government’s intertemporal budget constraint,
because of the existence of a binding lower bound on nominal interest rates
that severely limit the power of the central bank, constitutes a tool of ‘last
resort’ for liquidity traps to be aptly averted and, at the same time, for
aggregate stability to be preserved.

## Appendix A

Using the definition of total consumption (5) and the optimal intratemporal
condition (7), the instantaneous utility function takes the form

$\log\Upsilon\left(\overline{c}(s,t),\overline{m}(s,t)\right)=\log
q(t)+\log\overline{x}(s,t),$ (64)

where
$q(t)\equiv\Upsilon\left(\frac{\Omega(R(t))}{\Omega(R(t))+R(t)},\frac{1}{\Omega(R(t))+R(t)}\right)$
is identical across all generations and can be interpreted as the utility-
based cost of living index of the basket of physical goods and real balances.
Thus, the intertemporal optimization problem can be expressed as follows:

$\underset{\\{\overline{x}(s,t)\\}}{\max}\int_{0}^{\infty}\log
q(t)+\log\overline{x}(s,t)e^{-\left(\mu+\rho\right)t}dt,$ (65)

subject to

$\dot{\overline{a}}(s,t)=\left(R(t)-\pi(t)+\mu\right)\overline{a}(s,t)+\overline{y}(s,t)-\overline{\tau}(s,t)-\overline{x}(s,t),$
(66)

the no-Ponzi game condition (4), and given $\overline{a}(s,0)$. Hence,
optimality implies the Euler equation in terms of total consumption (8) and
the transversality condition (9). Integrating forward (66) and employing both
the transversality condition (9) and the law of motion of total consumption
(8) yield the level total consumption expressed as a linear function of total
wealth, that is, equation (10). From (7), I have

$\overline{x}(s,t)=\Lambda(R(t))\overline{c}(s,t),$ (67)

where $\Lambda(R(t))\equiv 1+R(t)/\Omega\left(R(t)\right)$. Time-
differentiating (67) gives

$\dot{\overline{x}}(s,t)=\Lambda^{\prime}(R(t))\overline{c}(s,t)\dot{R}(t)+\Lambda(R(t))\dot{\overline{c}}(s,t).$
(68)

Substituting (67) and (68) into (8) results in the law of motion for
individual consumption (13).

## Appendix B

Aggregate wealth in per capita terms is, by definition, given by

$\overline{a}(t)=\beta\int_{-\infty}^{t}\overline{a}(s,t)e^{\beta\left(s-t\right)}ds.$
(69)

Differentiating with respect to time gives

$\dot{\overline{a}}(t)=\beta\overline{a}(t,t)-\beta\overline{a}(t)+\beta\int_{-\infty}^{t}\dot{\overline{a}}(s,t)e^{\beta\left(s-t\right)}ds,$
(70)

Since $\overline{a}(t,t)$ is equal to zero, by assumption, using (3) into (70)
yields

$\displaystyle\dot{\overline{a}}(t)$ $\displaystyle=$
$\displaystyle-\beta\overline{a}(t)+\mu\overline{a}(t)+\left(R(t)-\pi(t)\right)\overline{a}(t)+\overline{y}(t)-\overline{\tau}(t)-\overline{c}(t)-R(t)\overline{m}(t)$
(71) $\displaystyle=$
$\displaystyle\left(R(t)-\pi(t)-n\right)\overline{a}(t)+\overline{y}(t)-\overline{\tau}(t)-\overline{c}(t)-R(t)\overline{m}(t).$

From (12), the per capita aggregate consumption is given by

$\overline{c}(t)=\frac{(\mu+\rho)}{\Lambda(R(t))}\left(\overline{a}(t)+\overline{h}(t)\right),$
(72)

Differentiating with respect to time the definition of per capita aggregate
consumption gives

$\dot{\overline{c}}(t)=\beta\overline{c}(t,t)-\beta\overline{c}(t)+\beta\int_{-\infty}^{t}\dot{\overline{c}}(s,t)e^{\beta\left(s-t\right)}ds,$
(73)

where $\overline{c}(t,t)$ represents consumption of the newborn generation.
Because $\overline{a}(t,t)=0$ and $\overline{h}(t,t)=\overline{h}(t)$, (12)
implies

$\overline{c}(t,t)=\frac{(\mu+\rho)}{\Lambda(R(t))}\overline{h}(t).$ (74)

Substituting (13), (72) and (74) into (73) yields the time path of per capita
aggregate consumption, given by (17).

## Appendix C

Suppose that the subutility function
$\Upsilon\left(\overline{c}(s,t),\overline{m}(s,t)\right)$ is of the CES-type,
in line with Galí (2015) and Walsh (2017):

$\Upsilon\left(\overline{c}(s,t),\overline{m}(s,t)\right)=\left[\delta\overline{c}(s,t)^{\frac{\varepsilon-1}{\varepsilon}}+\left(1-\delta\right)\overline{m}(s,t)^{\frac{\varepsilon-1}{\varepsilon}}\right]^{\frac{\varepsilon}{\varepsilon-1}},$
(75)

with $0<\delta,\varepsilon<1$, where $\varepsilon$ represents the elasticity
of substitution between consumption and real money holdings. It then follows

$\displaystyle\frac{\overline{m}(s,t)}{\overline{c}(s,t)}$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega(R\left(t\right))}$ (76) $\displaystyle=$
$\displaystyle\left(\frac{\delta}{1-\delta}\right)^{-\varepsilon}R\left(t\right)^{-\varepsilon},$
$\Lambda\left(R\left(t\right)\right)=1+\left(\frac{\delta}{1-\delta}\right)^{-\varepsilon}R\left(t\right)^{1-\varepsilon},$
(77) $\displaystyle\Lambda^{\prime}(R\left(t\right))$ $\displaystyle=$
$\displaystyle\left(1-\varepsilon\right)\left(\frac{\delta}{1-\delta}\right)^{-\varepsilon}R\left(t\right)^{-\varepsilon}$
(78) $\displaystyle=$
$\displaystyle\left(1-\varepsilon\right)\frac{\overline{m}(s,t)}{\overline{c}(s,t)}.$

For advanced economies’ annual data, in the monetary policy literature it is
common to set $\rho=0.04$ and $\Psi(\pi^{\ast})=0.06$, (see, e.g., Woodford,
2003, and Benhabib, Schmitt-Grohé and Uribe, 2001). I initially set
$\varepsilon=0.5$, implying a log-log interest elasticity of money demand of
$-0.5$, consistently with the seminal paper by Lucas (2000). Using (76), the
parameter $\delta$ is initially chosen so that the implied annual consumption
velocity of money is equal to unity, which is plausible value when a broad
monetary aggregate such as $M3$ is employed. So I obtain $\delta=0.9434$. In
the baseline calibration, I set $\Psi(\pi^{L})=0.001$ and
$\Psi^{\prime}(\pi^{L})=0.1$ and $a^{\ast}=0.6$. In line with United Nations
World Population Prospects 2019 for 2015-2020 with reference to high-income
countries,888https://population.un.org/wpp/. I set $n=0.0047$ and
$\mu=0.012366$, implying a life expectancy at birth of $80.87$ years and a
birth rate of $0.017066$. Therefore, I have

$\displaystyle J_{22}^{\left(a^{\ast},\pi^{L}\right)}$ $\displaystyle=$
$\displaystyle\frac{\left(\Psi^{\prime}(\pi^{L})-1\right)\left[1+\left(\frac{\delta}{1-\delta}\right)^{-\varepsilon}\Psi(\pi^{L})^{1-\varepsilon}\right]}{\Psi^{\prime}(\pi^{L})\left[\left(1-\varepsilon\right)\left(\frac{\delta}{1-\delta}\right)^{-\varepsilon}\Psi(\pi^{L})^{-\varepsilon}\right]}+\frac{\beta(\rho+\mu)}{1+\left(\frac{\delta}{1-\delta}\right)^{-\varepsilon}\Psi(\pi^{L})^{1-\varepsilon}}a^{\ast}$
$\displaystyle=$ $\displaystyle-1.951.$

The result $J_{22}^{\left(a^{\ast},\pi^{L}\right)}<0$ is robust to large
variations in parameter values. In particular,
$J_{22}^{\left(a^{\ast},\pi^{L}\right)}$ continues to be negative if: (a)
$\varepsilon$, the elasticity of substitution between consumption and real
money holdings, is lowered from the baseline value of $0.5$ to a value in the
range $\left(0.05,0.5\right)$, implying a diminished log-log interest
elasticity of money demand consistently with the empirical results obtained by
Ireland (2009); (b) the annual consumption velocity of money is increased from
$1$ to a value in the range $\left(1,20\right)$, should $M0$, $M1$, or $M2$ be
used as monetary aggregates; (c) $\Psi(\pi^{L})$, the nominal interest rate at
the liquidity trap steady-state, is increased from $0.001$ to a value in the
range $\left(0.001,0.1\right)$; (d) $\Psi^{\prime}(\pi^{L})$, the monetary
policy feedback reaction to inflation at the liquidity-trap steady-state, is
decreased from $0.1$ to a value in the range $\left(0.1,0.01\right)$; (e)
$a^{\ast}$, the target level for the government liabilities to output ratio,
is increased from $0.6$ to a value in the range $\left(0.6,2\right)$.

## References

Barro, R. J. (1974), “Are Government Bonds Net Wealth?”, Journal of Political
Economy 82, 1095-1117.

Bartsch, E., J. Boivin, S. Fischer and P. Hildebrand (2019), “Dealing with the
Next Downturn: From Unconventional Monetary Policy to Unprecedented Policy
Coordination”, SUERF Policy Note 105.

Bayer, C., B. Born, R. Luetticke and G. Müller (2020), “The Coronavirus
Stimulus Package: How Large is the Transfer Multiplier?”, CEPR Discussion
Papers 14600.

Benhabib, J., S. Schmitt-Grohé and M. Uribe (2001), “The Perils of Taylor
Rules”, Journal of Economic Theory 96, 40-69.

Benhabib, J., S. Schmitt-Grohé and M. Uribe (2002), “Avoiding Liquidity
Traps”, Journal of Political Economy 110, 535-563.

Bilbiie, F. (2018), “Neo-Fisherian Policies and Liquidity Traps”, CEPR
Discussion Papers 13334.

Blanchard, O. J. (1985), “Debt, Deficits, and Finite Horizons”, Journal of
Political Economy 93, 223-247.

Brock, W. A. (1974), “Money and Growth: The Case of Long-Run Perfect
Foresight”, International Economic Review 15, 750-777.

Brock, W. A. (1975), “A Simple Perfect Foresight Monetary Model”, Journal of
Monetary Economics 1, 133-150.

Buiter, W. H. (2020), “Three Strikes Against the Fed”, VoxEU.org, 3 April.

Canzoneri, M., R. Cumby and B. Diba (2010), “The Interaction Between Monetary
and Fiscal Policy”, in B. M. Friedman and M. Woodford (Eds.), Handbook of
Monetary Economics 3, Amsterdam/Boston: North-Holland/Elsevier, 935-999.

Cheron, A., K. Nishimura, C. Nourry, T. Seegmuller and A. Venditti (2019),
“Growth and Public Debt: What Are the Relevant Trade-Offs?”, Journal of Money,
Credit and Banking 51, 655-682.

Cushing, M. J. (1999), “The Indeterminacy of Prices under Interest Rate
Pegging: The Non-Ricardian Case”, Journal of Monetary Economics 44, 131-148.

Deaton, A. and J. Muellbauer (1980), Economics and Consumer Behavior, Oxford:
Oxford University Press.

Galí, J. (2015), Monetary Policy, Inflation and the Business Cycle, Princeton:
Princeton University Press.

Ireland, P. N. (2009), “On the Welfare Cost of Inflation and the Recent
Behavior of Money Demand”, American Economic Review 99, 1040-1052.

Krugman, P. (2020), “The Case for Permanent Stimulus”, in R. Baldwin and B.
Weder di Mauro (Eds.), Mitigating the COVID Economic Crisis: Act Fast and Do
Whatever It Takes 3, London: CEPR Press, 213-219.

Leeper, E. M. (1991), “Equilibria under ’Active’ and ’Passive’ Monetary and
Fiscal Policies”, Journal of Monetary Economics 27, 129-147.

Lucas, R. E. (2000), “Inflation and Welfare”, Econometrica 68, 247-274.

Maebayashi, N., T. Hori and K. Futagami (2017), “Dynamic Analysis of
Reductions in Public Debt in an Endogenous Growth Model with Public Capital”,
Macroeconomic Dynamics 21, 1454-1483.

Marini, G. and F. Van der Ploeg (1988), “Monetary and Fiscal Policy in an
Optimizing Model with Capital Accumulation and Finite Lives”, Economic Journal
98, 772-786.

Mertens, K. and Ravn, M. (2014), “Fiscal Policy in an Expectations-Driven
Liquidity Trap”, Review of Economic Studies 81, 1637-1667.

Minea, A. and P. Villieu (2013), “Debt Policy Rule, Productive Government
Spending, and Multiple Growth Paths: A Note”, Macroeconomic Dynamics 17,
947-954.

Nakata, T. and Schmidt, S. (2019), “Expectations-Driven Liquidity Traps:
Implications for Monetary and Fiscal Policy”, ECB Working Paper Series 2304.

Patinkin, D. (1965), Money, Interest, and Prices, New York: Harper and Row.

Pigou, A. C. (1943), “The Classical Stationary State”, The Economic Journal
53, 343-351.

Pigou, A. C. (1947), “Economic Progress in a Stable Environment”, Economica
14, 180-188.

Reis, R. (2007), “The Analytics of Monetary Non-Neutrality in the Sidrauski
Model”, Economics Letters 94, 129-135.

Sargent, T. J. and N. Wallace (1981), “Some Unpleasant Monetarist Arithmetic”,
Federal Reserve Bank of Minneapolis Quarterly Review 5, 1-17.

Schmidt, S. (2016), “Lack of Confidence, the Zero Lower Bound, and the Virtue
of Fiscal Rules”, Journal of Economic Dynamics and Control 70, 36-53.

Schmitt-Grohé, S. and Uribe, M. (2009), “Liquidity Traps with Global Taylor
Rules”, International Journal of Economic Theory, 85-106.

Sidrauski, M. (1967), “Rational Choice and Patterns of Growth in a Monetary
Economy”, American Economic Review 57, 534-544.

Summers, L. H. (2016), “Secular Stagnation and Monetary Policy”, Federal
Reserve Bank of St. Louis Review 98, 93-110.

Summers, L. H. (2018), “Secular Stagnation and Macroeconomic Policy”, IMF
Economic Review 66, 226-250.

Taylor, J. B. (Ed.) (1999), Monetary Policy Rules, Chicago: University of
Chicago Press.

Taylor, J. B. (2012), “Monetary Policy Rules Work and Discretion Doesn’t: A
Tale of Two Eras”, Journal of Money Credit and Banking 44, 1017-1032.

Taylor, J. B. (2018), “Fiscal Stimulus Programs During the Great Recession”,
Economics Working Paper 18117, Hoover Institution.

Taylor, J. B. (2021), “Simple Monetary Rules: Many Strengths and Few
Weakenesses”, European Journal of Law and Economics, forthcoming.

Walsh, C. E. (2017), Monetary Theory and Policy, Cambridge: The MIT Press.

Weil, P. (1989), “Overlapping Families of Infinitely-Lived Agents”, Journal of
Public Economics 38, 183-198.

Woodford, M. (2003), Interest and Prices, Princeton: Princeton University
Press.

Yaari, M. E. (1965), “Uncertain Lifetime, Life Insurance, and the Theory of
the Consumer”, The Review of Economic Studies 32, 137-150.