Pareto Improving Fiscal and Monetary Policies:
Samuelson in the New Keynesian Model∗
Mark Aguiar

Manuel Amador

Cristina Arellano

Princeton University,

University of Minnesota,

Federal Reserve Bankand NBER

Federal Reserve Bank ofof Minneapolis

Minneapolis, and NBER

July 2024

Abstract
This paper explores the normative and positive consequences of government bond issuances in a New Keynesian model with heterogeneous agents, focusing on how the stock ofgovernment bonds affects the cross-sectional allocation of resources in the spirit of Samuelson (1958). We characterize the Pareto optimal levels of government bonds and the associatedmonetary policy adjustments that should accompany Pareto-improving bond issuances. The
paper introduces a simple phase diagram to analyze the global equilibrium dynamics of inflation, interest rates, and consumption in response to changes in the stock of government
debt. It provides a tractable tool to explore the consequences of anticipated fiscal deficits; howgovernment indebtedness alters the economy’s response to cost-push and monetary policy
shocks; the use of fiscal policy to escape the Effective Lower Bound (ELB) on nominal interest
rates; and the resolution of the “forward guidance puzzle” with government debt.

∗ The paper benefited from comments by Adrien Auclert, Pierpaolo Benigno, Jordi Gali, Chen Lian, Ricardo

Reis, Matthew Rognlie, Silvana Tenreyro, Jaume Ventura, and Christian Wolf. Simeng Zeng provided excellent
research assistance. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Contact information: maguiar@princeton.edu;
amador.manuel@gmail.com; arellano.cristina@gmail.com

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1

Introduction

It is well known, going back to at least Samuelson (1958), that the introduction of outside assetscan (Pareto) improve the allocation of consumption in models with heterogeneity. The role ofgovernment bonds as a safe store of value, in particular, has played an important role in welfare
analyses of both overlapping generations (OLG) and Bewley-Huggett-Aiyagari incomplete market economies. We revisit this idea in a tractable OLG version of the standard New Keynesianmodel and focus on Pareto improvements for the welfare metric. In particular, we analyze thenormative and positive consequences of government bond issuances, and how these interact with
monetary policy. The main tool of analysis is a phase diagram that allows easy analysis of theeconomy’s response to a variety of shocks, without recourse to local approximations. The focus
on Pareto improvements complements the explicitly redistributional policies that have been thefocus of the recent policy analyses in heterogeneous agent New Keynesian (HANK) models.1
A common theme of our analysis is that government debt has profound consequences for the
impact of monetary policy and welfare. Expansions in debt can be beneficial when the economy
is at the ELB as it can be used to escape the ELB and disinflation. Further expansions of debt canbe welfare-improving, but only with appropriate monetary policy that eliminates undesirable
increases in inflation. Moreover, high levels of government debt reduce the effects of monetary
policy shocks, both on impact, and especially for future forward guidance announcements.
A few distinguishing characteristics of our approach should be flagged at the outset. The
analysis abstracts from the direct link between monetary policy and the government budget constraint studied in the classic paper of Sargent and Wallace (1981) and the more recent work on
the fiscal theory of the price level (e.g., Cochrane, 2023). To make the distinction from the latter
literature crystal clear, we model the government as issuing real bonds, although this is donefor expositional reasons rather than as a necessary component of the analysis. To contrast withthe work of Sargent and Wallace (1981), we explore a cashless economy with zero seigniorage
revenue.
We build on two standard platforms. To generate a link between government debt and the real
economy, we break Ricardian Equivalence using the perpetual youth framework of Blanchard(1985) and Yaari (1965), augmented to include endogenous labor supply but without physicalcapital. In this framework, we embed a textbook New Keynesian (NK) model of nominal rigidities,
as in Rotemberg (1982), Galı́ (2015), and Woodford (2004). This is a combination that has beenused several times in the literature, which we review below. It also generates heterogeneity in
1 Recent examples of optimal policy in HANK models using a utilitarian criteria include Bhandari et al. (2020),

Nuno and Thonmas (2021), Dávilla and Schaab (2022), Acharya, Challe, and Dogra (2020), Bilbiie and Ragot (2021),
and McKay and Wolf (2022) and Yang (2022). LeGrand, Martin-Baillon, and Ragot (2021) studies optimal policy using
a social welfare function with empirically derived weights. See also LeGrand and Ragot (2024).

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a tractable manner, providing clean insights that are applicable to richer heterogeneous agents
New Keynesian models now popular in the quantitative literature.
One useful assumption we make is that the individuals supplying labor (“workers”) save (inaggregate) in the government bond, while firm owners (“entrepreneurs”) save in equity. Thisdichotomy echoes the reality that many households whose primary income consists of labor
earnings do not own shares of firms. As a modeling choice, the key implication is that pricesetters and holders of government bonds value inter-temporal tradeoffs differently, as there may
be a wedge between the equilibrium return on government bonds and the return on equity. While
stark, this segmentation proves extremely convenient when analyzing dynamics.
From the workers’ problem, we obtain an aggregate Euler equation. The key distinction between this Euler equation and that of the standard model is the presence of government bonds,
reflecting that these are seen as wealth by non-Ricardian households. In the product market,
prices are set in a monopolistically competitive fashion subject to quadratic adjustment costs,
generating a standard NK Phillips curve that relates anticipated inflation, current inflation, andthe inverse markup, which in equilibrium is proportional to worker consumption. Finally, monetary policy is set through a nominal interest rate rule subject to an effective/zero lower bound(ELB). We will study the implications of a standard Taylor rule, where the nominal interest rate
is a linear function of deviations from the inflation target, with a slope coefficient strictly greaterthan one (i.e., the Taylor Principle holds), and an augmented rule where the nominal interest rate
also responds to the level of government debt.
Given these equilibrium restrictions, we characterize the economy as a system of two nonlinear ordinary differential equations (ODEs). We do so in terms of inflation and aggregate worker
consumption, using equilibrium conditions to substitute out the remaining macro aggregates. The
system of ODEs is amenable to analysis using a simple phase diagram, which we use to characterize global dynamics and the economy’s response to alternative fiscal and monetary policies.
Our benchmark experiment focuses on a debt-financed fiscal transfer. This exercise is of interest for three reasons. The first is that it is a common real world occurence; for example, US
Presidents Ronald Reagan, George W. Bush, Donald Trump, and Joseph Biden all implemented
either tax cuts and/or additional transfers financed by increased government borrowing. Anotherless successful attempt of a similar policy was UK Prime Minister Elizabeth Truss’ “mini-budget”
of September 2022.2 Second, as noted above, an expansion of government bonds has the potential
to be welfare-enhancing in a heterogeneous agent environment. Finally, the equilibrium dynamics that occur in anticipation of and simultaneously with a debt issuance depend crucially on
the monetary policy rule, making this an ideal laboratory to understand how rules designed for2 On the other hand, Presidents George H.W. Bush and William Clinton raised taxes and reduced debt, the mirror

image of the expansive policy.

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Ricardian environments (mis)behave in non-Ricardian settings.
In particular, consider a surprise announcement at 𝑡 = 0 of a one-time debt-financed tax cut (ortransfer) that will take place in 𝑡 ′ > 0 periods. This is equivalent to a “helicopter” distribution of
government bonds to taxpayers at 𝑡 ′. For households to hold these bonds in equilibrium requires
some combination of a higher real interest rate and/or increased income.
First, consider a “standard Taylor rule” in which the monetary policy’s reaction function depends on inflation but does not depend directly on the amount of government debt outstanding,as would be appropriate in a model with Ricardian agents.3 We show that if 𝑡 ′ is not too distant, inflation, both the nominal and real interest rates, and worker consumption will jump up
on the announcement. After the initial impact, inflation, interest rates, and consumption will all
continue to increase, at an accelerating rate, reaching the new steady state at precisely 𝑡 = 𝑡 ′.4
At this new steady state, the economy has “moved along” the Phillips curve, and inflation, bothnominal and real interest rates, and workers’ consumption are all higher. The monetary policy
rule selects this particular combination of higher income and real rates to clear the bond market.
Anticipating this, as 𝑡 → 𝑡 ′ the economy experiences a prolonged boom with elevated inflation.
For the same fiscal experiment, the dynamics are markedly different if the monetary policy
rule explicitly responds to the increase in government debt. In this case, we assume monetary
policy increases the nominal rate simultaneously with the debt issuance; specifically, the nominal
rate is increased to the level associated with the change in the real rate that would occur in a
flexible price economy. Under this policy, the increase in the real rate alone clears the bondmarket without any increase in workers’ income. Anticipating this, there are no dynamics inanticipation of the debt issuance for 𝑡 < 𝑡 ′.
The analysis highlights the importance of adapting policy for non-Ricardian environments.
If the central bank were to follow the policy advice obtained from a standard Taylor rule, itwould induce unnecessary fluctuations in both consumption and inflation in response to fiscal
deficits. In particular, the central bank would appear to be conscientiously fighting the inflation
apparently caused by the lax fiscal authority, when in reality the rigidity of the monetary policy
rule is equally to blame for the adverse inflationary consequences.
Our phase diagram also provides clear implications and intuitions for the economy’s response
to a number of other shocks. For example, we show how the level of government indebtedness
matters for the economy’s response to a “demand shock” (modeled as a change in consumer
3 More precisely, in Ricardian models the long-run real interest rate is pinned down independently of debt, andhence bond issuances do not alter the long-run target nominal interest rate. This is separate from secular shifts inpreferences (discount factors), openness, or demographic changes that may play a role in long-run trends in the realinterest rate.
4 It is not obvious whether or why the economy must be at the new steady state at 𝑡 ′ , but we postpone a detailed
discussion of this to the body of the paper.

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discounting); a cost-push shock (modeled as a shift in the Phillips curve); and a shock to themonetary policy rule. In particular, whether an economy translates these shocks into changes
in inflation versus changes in consumption depends on the level of government debt. We findthat high indebtedness tends to dampen the consumption and inflation effects of monetary anddemand shocks while it exacerbates the inflation effect of a cost-push shock.
With the positive analysis of a fiscal expansion in hand, we turn to the welfare consequencesof changes in the stock of government debt. This question has been the focus of much work using
real models in the tradition of Aiyagari (1994) and also for the classic insights of Samuelson (1958)
and Balasko and Shell (1980). Building on our previous work (Aguiar, Amador, and Arellano,2022), we start our normative analysis by studying the feasibility of robust Pareto improvements(RPI), which are policies that induce changes in prices and taxes that expand the budget sets of
all agents. Our previous work established that a key condition in the feasibility of an RPI is thatsavers are willing to hold additional wealth without large increases in the real interest rate; thatis, the aggregate savings supply schedule is elastic. We show that the same condition holds in a
flexible price version of the current environment. With nominal rigidities, the monetary authority
can induce a greater savings elasticity with respect to the interest rate by exploiting the fact thathigher income implies the bond market can clear at a lower real rate. Unfortunately, this is not a
viable avenue to increase the feasibility of Pareto improvements, as higher income is associatedwith higher inflation via the Philips curve, which decreases entrepreneurs’ profits, both due tothe deadweight costs of price adjustment and the transfer to workers via higher wages. We showthat to implement an RPI, monetary policy needs to replicate the response of the flexible priceeconomy to the fiscal policy change.5 In this precise sense, monetary and fiscal policy must workin a complementary fashion to achieve the Pareto improvement.
While our focus is primarily on how and when government bonds improve allocations ineconomies with heterogeneity and nominal rigidities, our model also allows for a tractable analysis of two other questions that have been core topics in monetary economics. The first one
relates to the forward guidance puzzle of Del Negro, Giannoni, and Patterson (2023) and McKay,
Nakamura, and Steinsson (2016). This puzzle stems from the fact that in the standard representative agent New Keynesian model, an anticipated decline in interest rates far in the future hasa large impact on current demand. Our phase diagram shows clearly why, with positive government debt in a non-Ricardian environment, there is no such puzzle, providing an additional
insight into the resolution of the puzzle proposed by Del Negro, Giannoni, and Patterson (2023)
without relying on linear approximations. In a related paper, Farhi and Werning (2019) studies
a model with incomplete markets and argues that the forward guidance puzzle is not generally
5 That is, for an RPI, monetary policy should track the real interest rate that will prevail without nominal rigidities

but in the presence of the fiscal policy change (i.e., the appropriate Wicksellian interest rate, Woodford, 2004).

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resolved. We discuss the key difference in assumptions that lead to this result, and, in doing so,show precisely how alternative fiscal policies alter the effectiveness of forward guidance.
Another extension concerns a large transitory decline in the discount rate of savers, such that
the ELB binds and the economy enters a liquidity trap. In this case, we show that fiscal policy
provides an alternative to the traditional “forward guidance” of Krugman (1998), Eggertsson andWoodford (2003), and Werning (2007). In particular, the fiscal authority can reflate the economy
by issuing bonds and rebating the proceeds back to taxpayers. By doing so, it increases the flexprice real interest rate, allowing the economy to escape the ELB. The effectiveness of this policy,of course, leverages the non-Ricardian environment.

Related Literature
Our work builds on the literature that has integrated the Blanchard-Yaari perpetual youth model
into monetary models. Marini and Ploeg (1988) and Cushing (1999) identify monetary nonneutralities in this set-up in the context of flexible prices. Piergallini (2006) and Nistico (2016)
study optimal monetary policy in environments with sticky prices and find that strict inflation
targeting might no longer be optimal because of the additional financial wealth effects this framework contains. Galı́ (2021) and Piergallini (2023) focus on the case of low interest rates, 𝑅 < 𝐺,
and study the implications for asset pricing bubbles and liquidity trap equilibria.6 Relative tothis work, our contribution focuses on the interactions between monetary and fiscal policy, as
in the recent work of Angeletos, Lian, and Wolf (2023). These authors explore how and whenfiscal deficits can be “self-financing,” either because they generate a boom in output that raises(proportional) taxes, or (with nominal bonds) because they generate inflation. In recent work, Kaplan, Nikolakoudis, and Violante (2023) explores the mechanics of the fiscal theory of the pricelevel in a heterogeneous agent economy. Our focus is not on how deficits can be self-financing
or the fiscal theory of the price level, but rather on the interplay of fiscal and monetary policy
in improving the cross-sectional allocation of output via government bonds. Moreover, we study
global non-linear solutions which contrast with the first-order approximation approach used inthese papers.
Michaillat and Saez (2021) study a New Keynesian framework where the natural rate of interest depends on financial factors. They introduce preferences where households obtain utility
from their relative wealth with respect to the population. Our phase diagram is similar to thephase diagram they use to analyze several of the anomalies present in the standard New Keyne6 See also the contributions of Lepetit (2022), Leith and Wren-Lewis (2000), Nisticò (2012).

We have alreadymentioned the contributions of Del Negro, Giannoni, and Patterson (2023) and Farhi and Werning (2019) that are aso
based on a monetary version of the Blanchard-Yaari model.

6sian model. Their model however features Ricardian equivalence, and thus it cannot address theeffect of changes in the level of government debt, which is our focus.
Our paper is to a large degree motivated by the growing literature on Heterogeneous Agents
New Keynesian (HANK) models, which highlights the interaction between fiscal and monetary
policies. In HANK models, government debt-financed transfers are non-Ricardian because richer
households, who hold a larger share of the debt, have a lower marginal propensity to consume(MPC) than poorer households. Kaplan, Moll, and Violante (2018), for example, shows that theeffectiveness of monetary policy depends crucially on the fiscal response. Specifically, monetary
expansions are more potent when the government transfers the savings from the interest rate
expense back to households. Auclert, Rognlie, and Straub (2018) study fiscal policy multipliers
and highlight the importance of the aggregate intertetemporal MPCs, for the magnitude of themultipliers. Other studies, such as McKay, Nakamura, and Steinsson (2016), have examined theimpact of forward guidance and the ELB, and found that precautionary savings motives can also
temper the power of forward guidance. Our simpler OLG model shares some of the propertiesof these richer HANK models as in both frameworks households’ wealth matters for the determination of interest rates. Our theoretical analysis complements the existing quantitative workand provides insights that can be useful in other applications.
In our framework, government debt matters for the interest rate directly, beyond its effect on
aggregate consumption, because the holdings of government debt differ across generations. Thisis related to findings in Krishnamurthy and Vissing-Jorgensen (2012), that illustrate empiricallythat government debt carries a convenience yield, which tends to fall with more debt, and thatprovides a framework to rationalize these findings, where a representative agent values government bonds in the utility function. Eggertsson, Mehrotra, and Robbins (2019), while studying
an OLG model with nominal ridigities, demonstrate how an expansion of government debt canhelp an economy escape a permanently binding ELB, an exercise we replicate and discuss later
using our model. Also related is Mian, Straub, and Sufi (2022) who study how the restrictions
from the ELB on monetary policy interact with fiscal policy. Similar to our perpetual youth environment, they find that a low level of government debt can increase the likelihood of a binding
ELB equilibrium. However, their emphasis is on the impact of this configuration on the fiscal
space of the government rather than on inflation, which never exceeds the target level. Bassetto
and Sargent (2020) shares with us the focus on monetary/fiscal interactions. In particular, their
example in Section 4.1 showcases the use of the debt Laffer curve, and the result that an optimal
policy may stop issuing debt before reaching 𝑟 = 𝑔 + 𝑛 because of heterogeneity and distributional concerns. Their case fits nicely within our RPI criteria. Finally, a series of papers Caballero,
Farhi, and Gourinchas (2017), Caballero and Farhi (2017), and Caballero, Farhi, and Gourinchas(2021) explore the role of government bonds in improving economic outcomes when monetary7policy is constrained by the ELB. Our analysis of fiscal policy at the ELB echoes their emphasis
on how a shortage of government debt can generate slumps at the ELB and how bonds become
an aggregate “demand shifter.” We extend the analysis to the impact of debt on global dynamics
away from the ELB and the distribution of resources across heterogeneous agents.

2

Environment

The environment builds closely on the canonical perpetual youth model of Blanchard and Yaari,embedded in the textbook New Keynesian paradigm, as in Galı́ (2021). Time is continuous andthere is no aggregate uncertainty. All announcements will be zero probability “MIT” shocks. A
measure of workers supply labor (with a potential lifecycle of earnings), save in a government
bond, and are subject to a constant hazard of death, which they insure via annuities. The crucial element for the non-Ricardian aspect of the model is that a new cohort of workers is born
every period. On the production side, firms produce differentiated intermediate goods, competemonopolistically, and face quadratic price adjustment costs as in Rotemberg (1982). Workers andthe owners of firms (entrepreneurs) are segmented in the sense that workers cannot own sharesin firms. This will be useful to separate the return on government bonds from the internal rateof return to private equity. Finally, the government conducts fiscal and monetary policy. In thefollowing subsections, we fill in the details on each block of the model and then characterize theequilibrium.

2.1

Workers

The worker sector closely follows Blanchard (1985) (as well as Buiter, 1988 for the extension to
population and technological growth). At any point in time 𝑠, a cohort of workers of size (𝜆 +𝑛)𝑒 𝑛𝑠
is born, where 𝑛 denotes population growth. Each worker faces a constant hazard rate of dying,
given by 𝜆 > 0. The expected lifespan of a worker is therefore 1/𝜆. We require 𝜆 + 𝑛 > 0.
Letting 𝜙(𝑠, 𝑡) denote the time-𝑡 size of the cohort born at 𝑠 ≤ 𝑡, we have𝜙(𝑠, 𝑡) = (𝜆 + 𝑛)𝑒 𝑛𝑠 𝑒 −𝜆(𝑡−𝑠) .
The size of the total worker population at time 𝑡, denoted 𝑚𝑡 , is thenZ𝑡𝜙(𝑠, 𝑡)𝑑𝑠 = 𝑒 𝑛𝑡 .

𝑚𝑡 =
−∞

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A (representative) worker born in period 𝑠 and alive at 𝑡 ≥ 𝑠 has preferences given by:Z∞𝑒 −(𝜌+𝜆)(𝜏−𝑡)𝑢(𝑐(𝑠, 𝜏), 𝑙(𝑠, 𝜏))𝑑𝜏,(1)𝑡where 𝜌 > 0 is the subjective discount factor; 𝑐(𝑠, 𝑡) is consumption of the final good; and 𝑙(𝑠, 𝑡)
is the amount of labor supplied. Workers effectively discount the future with the sum of the
discount factor and the probability of dying. It will be useful to consider the following functionalform:𝑢(𝑐, 𝑛) = ln 𝑐 + 𝜓 ln (1 − 𝑙) ,with 𝑐 ≥ 0 and 𝑙 ≤ 1.7
A worker’s productivity 𝑧 changes over the life cycle. Specifically, define 𝑧 0 ≡ (𝜆+𝛼 +𝑛)/(𝜆+𝑛).
The productivity of a worker born at time 𝑠 and alive at time 𝑡 ≥ 𝑠 is given by:8𝑧(𝑠, 𝑡) = 𝑧 0𝑒 𝑔𝑡 𝑒 −𝛼(𝑡−𝑠),so 𝑧 declines exponentially with age at rate 𝛼 ≥ 0, and grows with time at rate 𝑔, where 𝑔 captures
technological growth. Aggregate productivity at time 𝑡 is thenZ𝑡
𝑍 (𝑡) ≡

𝜙(𝑠, 𝑡)𝑧(𝑠, 𝑡)𝑑𝑠 = 𝑒 (𝑔+𝑛)𝑡 .

−∞As in Blanchard, workers can perfectly insure their survival risk in spot annuity markets. Let𝑖(𝑡) denote the nominal return on government bonds. For each nominal unit (“dollar”) held by aworker in the annuity, they receive (𝑖(𝑡) + 𝜆)𝑑𝑡 if they survive the next 𝑑𝑡 → 0 periods. If they die,the insurance intermediaries receive the asset. As 𝜆𝑑𝑡 workers die, the insurance sector breaks
even with probability one.
7 Note that we do not restrict 𝑙 ≥ 0. A negative 𝑙 is equivalent to the worker hiring another individual to assist

them in “daily living,” removing that unit of time from the production sector, a result that we think is reasonable.
Ascari and Rankin (2007) highlighted the possibility of negative labor supply in the perpetual youth framework andproposed using preferences without wealth effects on labor supply to eliminate the possibility.
8 We assume that all cohorts enjoy productivity growth as they age. Instead of modeling growth as a “time effect,”
we could have assumed that growth is across cohorts:𝑧(𝑠, 𝑡) = 𝑧 0𝑒 𝑔𝑠 𝑒 −𝛼(𝑡 −𝑠) .
As is well known due to the fact that age, cohort, and time have a linear relationship, the two approaches have asimple correspondence. In particular, let 𝛼˜ ≡ 𝛼 − 𝑔, we have˜ −𝑠)𝑧(𝑠, 𝑡) = 𝑧 0𝑒 𝑔𝑡 𝑒 −𝛼(𝑡 −𝑠) = 𝑧 0𝑒 𝑔𝑠 𝑒 −𝛼(𝑡.

Thus whether technological growth affects all cohorts equally over time or just cohorts at birth (or any linear combination of the two) is covered by the representation in the text.

9Let 𝑃(𝑡) be the price of the final good at time 𝑡 and let 𝑊 (𝑡) denote the nominal wage per
efficiency unit of labor. Workers of cohort 𝑠 at time 𝑡 pay non-distortionary taxes 𝑃(𝑡)𝑇 (𝑠, 𝑡). We
let
𝑇 (𝑠, 𝑡) = 𝑧(𝑠, 𝑡)𝑇 (𝑡),(2)where 𝑇 is the tax burden per effective unit of labor. Specifically, we have indexed taxes by cohort
in order to allow the tax burden to decline with productivity.9 Aggregate tax revenue at 𝑡 is:Z𝑡
𝑇 (𝑡) ≡

𝜙(𝑠, 𝑡)𝑇 (𝑠, 𝑡)𝑑𝑠 = 𝑍 (𝑡)𝑇 (𝑡)−∞Let 𝑃(𝑡)𝑎(𝑠, 𝑡) denote the nominal asset position of the representative agent from cohort 𝑠 attime 𝑡. The flow budget constraint for cohort 𝑠 is given by:𝑑¤[𝑃(𝑡)𝑎(𝑠, 𝑡)] = 𝑃(𝑡)𝑎(𝑠,𝑡) + 𝑎(𝑠,¤ 𝑡)𝑃(𝑡)𝑑𝑡= (𝑖(𝑡) + 𝜆)𝑃(𝑡)𝑎(𝑠, 𝑡) + 𝑊 (𝑡)𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡) − 𝑃(𝑡)𝑐(𝑠, 𝑡) − 𝑃(𝑡)𝑇 (𝑠, 𝑡),where a “dot” indicates the derivative with respect to time. Dividing through by 𝑃(𝑡), we have𝑎(𝑠,¤ 𝑡) = (𝑟 (𝑡) + 𝜆)𝑎(𝑠, 𝑡) + 𝑤(𝑡)𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡) − 𝑐(𝑠, 𝑡) − 𝑇 (𝑠, 𝑡),(3)where 𝑤(𝑡) ≡ 𝑊 (𝑡)/𝑃(𝑡) is the real wage, 𝑟 (𝑡) ≡ 𝑖(𝑡) − 𝜋(𝑡) is the real interest rate, and 𝜋(𝑡) ≡¤𝑃(𝑡)/𝑃(𝑡)is the rate of inflation.10 Households are subject to the natural borrowing limit, 𝑎(𝑠, 𝑡) ≥𝑎(𝑠, 𝑡), which, combined with the log preferences, ensures an interior consumption sequence at
an optimum. LettingR𝜏𝑅(𝑡, 𝜏) ≡ 𝑒 − 𝑡 (𝑟 (𝑚)+𝜆)𝑑𝑚
we can integrate the flow budget constraint forward to obtain:
Z∞
𝑅(𝑡, 𝜏) [𝑐(𝑠, 𝜏) + 𝑇 (𝑠, 𝜏) − 𝑤(𝜏)𝑧(𝑠, 𝜏)𝑙(𝑠, 𝜏)] 𝑑𝜏 .

𝑎(𝑠, 𝑡) =(4)

𝑡Workers are born with zero wealth; that is, 𝑎(𝑠, 𝑠) = 0 for all cohorts 𝑠.
Given a sequence of aggregate taxes 𝑇 (𝑡) and prices {𝑤(𝑡), 𝑟 (𝑡)}, a worker born at time 𝑠chooses sequences {𝑐(𝑠, 𝑡), 𝑙(𝑠, 𝑡)}𝑡 ≥𝑠 to maximize (1) subject to (4), with 𝑎(𝑠, 𝑠) = 0, as well as
the constraints 𝑐(𝑠, 𝑡) ≥ 0, 𝑙(𝑠, 𝑡) ≤ 1, and 𝑎(𝑠, 𝑡) ≥ 𝑎(𝑠, 𝑡) for all 𝑡 ≥ 𝑠, where the natural borrowing
9 We do this for simplicity as it facilitates aggregation. Note that the tax (or transfer if negative) remains lump-sum

and it is not a function of the labor supply choice.
10 For some policy experiments, we may want to consider an unanticipated lump-sum “helicopter” drop of assets(government) bonds to various cohorts at a fixed time 𝑡 0 . We will be more explicit about this in Section 2.3, but tostreamline the exposition we will suppress this from the notation until needed.

10limit isZ∞
𝑅(𝑡, 𝜏)𝑧(𝑡, 𝜏) (𝑤(𝑡) − 𝑇 (𝑡)) 𝑑𝜏,𝑎(𝑠, 𝑡) ≡ −(5)𝑡which is the present value of maximal labor earnings (i.e., 𝑙 = 1) net of taxes. Equilibrium will
require that 𝑎(𝑠, 𝑡) be finite. Note also that 𝑎(𝑡, 𝑡) must be negative, as newborns have no wealth,𝑎(𝑡, 𝑡) = 0.
The solution to the workers’ problem is characterized as follows:
Lemma 1. Suppose that 𝑎(𝑠, 𝑡) is finite for all (𝑠, 𝑡). The following conditions characterize theoptimal consumption and labor plan of a worker born at 𝑠 evaluated at 𝑡 ≥ 𝑠:(i) The Euler equation:
𝑐(𝑠,
¤ 𝑡)
= 𝑟 (𝑡) − 𝜌;𝑐(𝑠, 𝑡)(6)(ii) The static labor-consumption condition:
𝜓𝑐(𝑠, 𝑡) = 𝑤(𝑡)𝑧(𝑠, 𝑡) (1 − 𝑙(𝑠, 𝑡)) .

(7)(iii) and a consumption function:



𝜌 +𝜆
(𝑎(𝑠, 𝑡) + ℎ(𝑠, 𝑡) − T (𝑠, 𝑡)) ,𝑐(𝑠, 𝑡) =
1 +𝜓
where

(8)

Z∞
ℎ(𝑠, 𝑡) ≡𝑅(𝑡, 𝜏)𝑧(𝑠, 𝜏)𝑤(𝜏)𝑑𝜏𝑡represents potential “human wealth” andZ∞
T (𝑠, 𝑡) ≡𝑅(𝑡, 𝜏)𝑧(𝑠, 𝜏)𝑇 (𝜏)𝑑𝜏 .

(9)𝑡

represents the present value tax burden.
In the next subsection, we discuss aggregation. We flag a few elements of the individual
worker’s problem that will be useful. One is that all cohorts that are alive have consumption thatgrows at the same rate (𝑟 (𝑡) − 𝜌) and the level of consumption is linear in total wealth net of taxes.
A second feature is that given wages, labor supply is linear in consumption, with labor income
equal to potential income 𝑤(𝑡)𝑧(𝑠, 𝑡) minus a fraction 𝜓 of consumption. This feature allows us to11express consumption as a fraction of financial wealth and potential human wealth.
2.1.1

Aggregation

We now characterize the aggregate behavior of the workers, integrating over the various cohorts.
Given that the size of a cohort 𝑠 at time 𝑡 is 𝜙(𝑠, 𝑡), and letting capital letters indicate aggregate
quantities, we can integrate over 𝑠 to define the aggregates:
Z𝑡𝑤

𝐶 (𝑡) ≡

𝜙(𝑠, 𝑡)𝑐(𝑠, 𝑡)𝑑𝑠
Z −∞𝑡

𝑁 (𝑡) ≡

𝜙(𝑠, 𝑡)𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡)𝑑𝑠
Z −∞𝑡

𝐴(𝑡) ≡

𝜙(𝑠, 𝑡)𝑎(𝑠, 𝑡)𝑑𝑠
Z −∞𝑡

𝐻 (𝑡) ≡

𝜙(𝑠, 𝑡)ℎ(𝑠, 𝑡)𝑑𝑠
Z −∞𝑡

T (𝑡) ≡

𝜙(𝑠, 𝑡)T (𝑠, 𝑡)𝑑𝑠.
−∞

The following characterizes the aggregate behavior of workers:
Lemma 2. Given a path {𝑤(𝑡), 𝑟 (𝑡),𝑇 (𝑡)}, worker optimization implies:(i) An aggregate Euler equation:(𝜌 + 𝜆)(𝛼 + 𝜆 + 𝑛)𝐴(𝑡);𝐶¤𝑤 (𝑡) = (𝑟 (𝑡) − 𝜌 + 𝛼 + 𝑛)𝐶 𝑤 (𝑡) −
1 +𝜓

(10)

(ii) An aggregate labor supply:𝜓𝐶 𝑤 (𝑡) = 𝑤(𝑡)𝑍 (𝑡) − 𝑤(𝑡)𝑁 (𝑡);(11)(iii) An aggregate consumption function:



𝜌 +𝜆
[𝐴(𝑡) + 𝐻 (𝑡) − T (𝑡)] ;𝐶 (𝑡) =
1 +𝜓
𝑤

(12)

(iv) And an aggregate evolution of financial wealth:¤ = 𝑟 (𝑡)𝐴(𝑡) + 𝑤(𝑡)𝑍 (𝑡) − 𝑇 (𝑡) − (1 + 𝜓 )𝐶 𝑤 (𝑡).
𝐴(𝑡)

12

(13)

The aggregate labor supply and consumption function, conditions (11) and (12), follow immediately from integrating across cohorts their static decisions. The aggregate dynamics of consumption and financial wealth, however, also have to consider that fraction 𝜆 of the population
dies every period and a new cohort is born. Those dying in aggregate have average financial
wealth 𝐴(𝑡) while those being born have zero. The difference in wealth between those dying
and those being born matters for the aggregate Euler equation (10) and the evolution of financial
wealth (13). Financial wealth 𝐴(𝑡) shows up in (10) because richer agents are replaced by poorer
agents; the growth of aggregate consumption is lower with high aggregate wealth. The aggregate
Euler equation illustrates that the level of household financial wealth 𝐴𝑡 matters for aggregate
consumption dynamics, in addition to the interest rate, discount rate, and the age profile of productivity. From (13), aggregate worker wealth evolves “as if” all cohorts inelastically supply 𝑍 (𝑡)efficiency units of labor while at the same time spending an extra 𝜓 on consumption. This reflectsthat for each individual, any increase in consumption reduces labor income at the linear rate 𝜓
via the income effect on labor supply. Note also that Lemma 2 holds for an arbitrary distributionof individual financial assets among surviving cohorts.
∞ , equations (11), (12), and (13) completely
Given an initial 𝐴(0) and a path for {𝑤(𝑡), 𝑟 (𝑡),𝑇 (𝑡)}𝑡=0∞ .
characterize the aggregates of the household sector, {𝐴(𝑡), 𝐶 𝑤 (𝑡), 𝑁 (𝑡)}𝑡=0

2.2

Entrepreneurs

The technology side of the model is familiar from the standard textbook New Keynesian model.11
There is a measure-one continuum of entrepreneurs, each of whom operates a firm that producesa unique intermediate input 𝑗 ∈ [0, 1]. Intermediate firm technology is given by 𝑦 𝑗 (𝑡) = ℓ 𝑗 (𝑡),where ℓ 𝑗 are efficiency units of labor. Firms hire these units in a competitive labor market at real
wage 𝑤(𝑡).
Entrepreneurs sell their output to a competitive final goods sector that combines inputs using
a constant-elasticity-of-substitution technology:
Z 1
𝑌 (𝑡) =𝑦 𝑗 (𝑡)

𝜂−1
𝜂

𝜂
 𝜂−1

𝑑𝑗

.

0

Given the constant-returns-to-scale technology and competitive behavior, there is no value added
generated by this sector and hence no need to detail the ownership of final-good firms. The price11 In particular, this part of the model follows Kaplan, Moll, and Violante (2018) closely.

reproduces their Lemma 1.

13

Our Lemma 3 below

index of the final good is given by
Z 1
𝑃(𝑡) =𝑝 𝑗 (𝑡)1−𝜂 𝑑 𝑗

1
 1−𝜂

,

0

where 𝑝 𝑗 (𝑡) is the price of intermediate 𝑗 ∈ [0, 1] .
For simplicity, and not crucial for what follows, we assume that there are no demographic
dynamics for the entrepreneurs: entrepreneurs live forever. They have linear utility and discount
˜ Entrepreneurial wealth consists of shares of its firm. Shares in firms are (potentially)at the rate 𝜌.
traded among entrepreneurs, but as mentioned above are not available to workers. Moreover,entrepreneurs do not hold government bonds. This is stated as an assumption but is consistent
with any equilibrium in which 𝑟 (𝑡) < 𝜌˜ for all 𝑡.
The entrepreneur’s consumption/savings problem is given by:
Z∞
max

∞
{𝑐(𝜏)}𝜏=𝑡𝑒˜−𝜌(𝜏−𝑡)𝑐(𝜏)𝑑𝜏

Z∞
s.t.

𝑡

˜
𝑒 −𝜌(𝜏−𝑡)𝑐(𝜏)𝑑𝜏 ≤ 𝑉 (𝑡),𝑡where 𝑉 is the value of equity held by the entrepreneur at time 𝑡. We already impose in the
˜ which follows
representative entrepreneur’s problem that the internal rate of return to equity is 𝜌,from the linearity of preferences and the requirement that consumption be interior in equilibrium.
In a symmetric equilibrium, entrepreneurs do not actively trade shares among themselves, andthe value of an individual entrepreneur’s shares will be equal to the value of their firm, denoted
𝑄.
Entrepreneurs compete monopolistically and choose a sequence of prices to maximize thevalue of their firm. We restrict attention to symmetric equilibria in which all firms pursue an
identical policy.
Intermediate good firms face a nominal friction when setting prices. Let 𝑝(𝑡) be the nominal
price of an individual variety, where we drop the 𝑗 index. Intermediate good firms choose the¤rate of change in their nominal price, 𝑥(𝑡) ≡ 𝑝(𝑡)/𝑝(𝑡)and pay a cost 𝑓 (𝑥)𝑌 , where:𝜑
𝜑𝜋𝑥 − 2 𝜋 2
𝑓 (𝑥) = 𝜑2 𝑥 2𝜑𝜋𝑥 − 𝜑2 𝜋 2

if 𝑥 < 𝜋if 𝑥 ∈ [𝜋, 𝜋]
if 𝑥 > 𝜋 .

The costs of price adjustment are weakly convex, continuous, and continuously differentiable.
For intermediate inflation rates 𝑥 ∈ [𝜋, 𝜋], for some 𝜋 < 0 < 𝜋, adjustment costs are quadratic,
as in Rotemberg (1982). For extreme rates of change, costs are linear. In the spirit of Nakamura
and Steinsson (2010), this captures that for high inflation environments, price setting is different
14than at moderate inflation rates. We shall see that this induces a vertical Phillips curve at extreme
inflation.
¤ to maximize the value of the firm:
The entrepreneur chooses a path of 𝑝 via control 𝑥 = 𝑝/𝑝Z∞
𝑄(𝑡) = sup{𝑥(𝜏 }𝜏 ≥𝑡˜[Π(𝑝(𝜏), 𝜏) − 𝑓 (𝑥)𝑌 (𝜏)] 𝑑𝑡𝑒 −𝜌(𝜏−𝑡)𝑡¤ = 𝑥(𝑡)𝑝(𝑡),subject to: 𝑝(𝑡)where Π(𝑝, 𝑡) are the real flow profits gross of adjustment costs of a firm charging price 𝑝 at time
𝑡:



𝑝 −𝜂
𝑝
− 𝑤(𝑡)
𝑌 (𝑡).
Π(𝑝, 𝑡) =𝑃(𝑡)𝑃(𝑡)Note that we assume the government does not tax (or subsidize) entrepreneurs. This rules out
implicit transfers to workers through the taxation of entrepreneurs to pay interest on the debt
held by workers.12
The solution to the optimal pricing plan generates the following Phillips curve:
Lemma 3. Let 𝑔𝑌 (𝑡) ≡ 𝑌¤ (𝑡)/𝑌 (𝑡) denote the real growth rate; 𝑤 ★ = (𝜂 − 1)/𝜂 denote theflexible price optimal inverse markup; and 𝜅˜ ≡ 𝜂/𝜑. In a symmetric equilibrium with a path¤
for aggregate for real wages {𝑤(𝑡)}, aggregate inflation 𝜋(𝑡) ≡ 𝑃(𝑡)/𝑃(𝑡)satisfies:𝜋(𝑡)¤ = (𝜌˜ − 𝑔𝑌 (𝑡))𝜋(𝑡) + 𝜅˜ 𝑤 ★ − 𝑤(𝑡)if 𝜋(𝑡) ∈ [𝜋, 𝜋];(14)

and for 𝜋(𝑡) ∈/ [𝜋, 𝜋], we have:

if 𝜋(𝑡) < 𝜋★

if 𝜋(𝑡) > 𝜋 .

(𝜌˜ − 𝑔𝑌 (𝑡)) 𝜋 = 𝜅˜ 𝑤(𝑡) − 𝑤 ★
(𝜌˜ − 𝑔𝑌 (𝑡)) 𝜋 = 𝜅˜ 𝑤(𝑡) − 𝑤

For interior inflation, 𝜋(𝑡) ∈ [𝜋, 𝜋], the last term in (14) represents the deviation from theflex-price markup, with a positive value indicating that the markup is higher than the flex-pricemarkup. At the extreme rates of inflation, the real wage is uniquely pinned down for any 𝜋,
generating a “vertical” Phillips curve in 𝜋 × 𝑤 space. Let 𝑤 and 𝑤 denote the associated low andhigh real wages, respectively.13
One loose end is that the firm always has the option to shut down production. Flow profits
12 This assumption is made for simplicity, and allows for a simple equilibrium value for aggregate output, as we

discuss below.
13 Specifically, 𝑤 ≡ 𝜌𝜋/˜ 𝜅˜ − 𝑤 ★ and 𝑤 ≡ 𝜌𝜋/˜ 𝜅˜ − 𝑤 ★.

15

are negative if 𝑓 (𝜋(𝑡)) > 1 − 𝑤(𝑡). We rule out equilibria that violate this condition. In particular,
this rules out equilibria in which inflation explodes in either direction.
R1
Let 𝐶 𝑒 (𝑡) ≡ 0 𝑐 𝑗 (𝑡)𝑑 𝑗 denote the consumption of the entrepreneurial sector. Given paths∞ , we say {𝐶 𝑒 (𝑡), 𝑌 (𝑡), 𝑉 (𝑡), 𝑄(𝑡)} ∞ and {𝜋(𝑡)} ∞ that solve the entrepreneurs’ problem{𝑤(𝑡), 𝜌˜ }𝑡=0
𝑡=0
𝑡=0
and satisfy Lemma 3 characterize the entrepreneurs’ sector.

2.3

Government

The government sets fiscal and monetary policies under full commitment. Fiscal policy consistsof a sequence of non-distortionary aggregate taxes, 𝑇 (𝑡), and real debt 𝐵(𝑡), subject to the budget
constraint¤ = 𝑟 (𝑡)𝐵(𝑡) − 𝑇 (𝑡).
𝐵(𝑡)

(15)

We assume the government borrows in real bonds promising a real return to differentiate ouranalysis from the fiscal theory of the price level.14 We assume a vanishing small amount of
nominal bonds that carry the nominal rate 𝑖(𝑡) in order to ensure the Fisher equation 𝑖 = 𝑟 + 𝜋
holds in equilibrium.
Our policy experiments involve a discrete change to the stock of government debt. For example, suppose at time 𝑡 0 the government expands government debt by a discrete amount. The
fiscal authority rebates the proceeds to workers. Note that this involves a sale of bonds offset
by an aggregate transfer of equal amount. We could circumvent this sale-transfer by assuming
the government simply “helicopter drops” the new bonds to households. Specifically, let 𝜉(𝑠, 𝑡)denote the cumulative bond transfers to cohort 𝑠 as of time 𝑡 ≥ 𝑠. Let 𝑑𝜉(𝑠, 𝑡) = 𝜉(𝑠, 𝑡)−lim𝜏↑𝑡 𝜉(𝑠, 𝜏)denote the amount of new bonds transferred to workers of cohort 𝑠 at time 𝑡. Our experiments
assume 𝜉(𝑠, 𝑡) = 0 for 𝑡 < 𝑡 0 , and is constant thereafter. Note that 𝜉 can be negative as well as
positive, with negative numbers representing an expropriation of assets. LetZ𝑡𝑑𝜉(𝑡) ≡𝑑𝜉(𝑠, 𝑡)𝑑𝑠−∞

denote the aggregate change to worker assets.
14 Although in our current model in continuous time with instantaneous bonds and a price level process that cannot
jump (given the adjustment costs), the fiscal theory of the price level will remain inoperative even if the government
were to issue nominal bonds.

16

We then augment (3) to incorporate the helicopter drop to write:𝑑𝑎(𝑠, 𝑡) = [(𝑟 (𝑡) + 𝜆)𝑎(𝑠, 𝑡) + 𝑤(𝑡)𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡) − 𝑐(𝑠, 𝑡) − 𝑇 (𝑠, 𝑡)] 𝑑𝑡 + 𝑑𝜉(𝑠, 𝑡).
We augment (15) similarly,ih𝑑𝐵(𝑡) = 𝑟 (𝑡)𝐵(𝑡) − 𝑇 (𝑡) 𝑑𝑡 + 𝑑𝜉(𝑡).
As noted already, the evolution of aggregate worker wealth evolves independently of the idiosyncratic distribution of wealth, given a path of prices and taxes (see equation 13). Thus, we focus
on the aggregate change in government debt, 𝑑𝜉(𝑡 0 ), rather than the specifics of the distribution.
Monetary policy is set via an “augmented” interest rate rule: 
𝐵(𝑡) 𝐵(0)𝑖(𝑡) = max 𝜄¯ + 𝜃 𝜋 𝜋(𝑡) + 𝜃𝑏−
,0 ,𝑍 (𝑡) 𝑍 (0)(16)

where 0 is the ELB on nominal interest rates. When 𝜃𝑏 = 0, monetary policy responds only to
inflation, as in a standard Taylor rule, and we impose (unless otherwise noted) that the “Taylor
principle” 𝜃 𝜋 > 1 holds. In this case, the intercept 𝜄¯ is a constant that in the baseline equilibrium
corresponds to the level of the real interest rate that is realized when inflation is zero. In our
analysis, we also consider a monetary policy rule that can change in response to fiscal policy,
hence 𝜃𝑏 . As we will see, this term is useful as fiscal policy affects the real rate in the flexible
price allocation.15

2.4

Definition of Equilibrium

Given a path for fiscal policy {𝐵(𝑡),𝑇 (𝑡)} and monetary policy rule (16), an equilibrium is a¤path of prices {𝜋(𝑡) = 𝑃(𝑡)/𝑃(𝑡)}starting from 𝑃(0) = 1; real interest rates {𝑟 (𝑡)} for government bonds; nominal interest rates {𝑖(𝑡)}; real wages {𝑤(𝑡)}; firm values {𝑄(𝑡)}; and quantities{𝑌 (𝑡), 𝐶 𝑤 (𝑡), 𝐶 𝑒 (𝑡), 𝑁 (𝑡), 𝐴(𝑡)} such that:(i) 𝐶 𝑤 (𝑡), 𝑁 (𝑡), and 𝐴(𝑡) solve workers’ problem given 𝑤, 𝑟 , and 𝑇 and where 𝑎(𝑠, 𝑡) in (5) is finite
for all (𝑠, 𝑡);(ii) 𝐶 𝑒 (𝑡), 𝑌 (𝑡), 𝑄(𝑡), and 𝜋(𝑡) solve the entrepreneurs’ problem given 𝑤;(iii) the bond market clears 𝐴(𝑡) = 𝐵(𝑡);15 There are several papers that study adjustments to the Taylor rule in cases where Ricardian equivalence fails. Seefor example Curdia and Woodford (2010). Closer to our case, Nisticò (2012) studies a Blanchard-Yaari environment
where the “Wicksellian” interest rate deviates (due to preference/demand shocks) from the one of the representative
agent.

17

(iv) the resource condition is satisfied 𝐶 𝑤 (𝑡) + 𝐶 𝑒 (𝑡) = (1 − 𝑓 (𝜋)) 𝑌 (𝑡) = (1 − 𝑓 (𝜋)) 𝑁 (𝑡);(v) arbitrage between nominal and real returns by households implies that the Fisher equation
is satisfied, 𝑖(𝑡) = 𝑟 (𝑡) + 𝜋(𝑡); and(vi) the monetary policy rule and the government budget constraint are satisfied.

3

Characterizing Equilibrium Dynamics

In this section, we introduce a phase diagram that will be the main tool to analyze equilibrium
dynamics. As a first step, we note a possibly surprising aspect of the equilibrium; namely, effective
aggregate labor supply, 𝑁 (𝑡), and output, 𝑌 (𝑡), both grow at an exogenous constant rate 𝑔 + 𝑛.
To see this, let us use the aggregate evolution of financial wealth (13), together with the budget
constraint of the government, and the equilibrium condition 𝐴(𝑡) = 𝐵(𝑡), and we obtain:
𝑤(𝑡)𝑍 (𝑡) = (1 + 𝜓 )𝐶 𝑤 (𝑡).
The static labor-consumption condition (11) states that 𝜓𝐶 𝑤 (𝑡) = 𝑤(𝑡)𝑍 (𝑡) − 𝑤(𝑡)𝑁 (𝑡). Hence:
𝑁 (𝑡) =

𝑍 (𝑡)
,
1 +𝜓

and 𝑌 (𝑡) = 𝑁 (𝑡). In equilibrium, the aggregate labor supply curve is “vertical”; that is, independent of other equilibrium outcomes. This result stems from the balanced growth preferences ofworkers plus the segmentation of bond markets and taxation. In this model, therefore, fiscal andmonetary policies can change the share of output consumed, due to their effect on adjustment
costs and inflation, and the share of consumption going to workers versus entrepreneurs, but not
the total amount of output produced.
From the workers’ side, recall the aggregate Euler equation (10). Substituting the asset marketclearing condition 𝐴(𝑡) = 𝐵(𝑡), we obtain(𝜌 + 𝜆)(𝛼 + 𝜆 + 𝑛)
𝐵(𝑡).
𝐶¤𝑤 (𝑡) = (𝑟 (𝑡) − 𝜌 + 𝛼 + 𝑛)𝐶 𝑤 (𝑡) −1 +𝜓
As 𝐵(𝑡) increases, in order to clear the bond market we need some combination of a higher 𝑟 (𝑡),higher 𝐶 𝑤 (𝑡), or lower growth rate of 𝐶 𝑤 (𝑡).
As usual, it is convenient to rewrite aggregates in terms of total effective units of labor. In

18

particular, letc(𝑡) ≡𝐵(𝑡)𝐶 𝑤 (𝑡), and b(𝑡) ≡
.
𝑍 (𝑡)
𝑍 (𝑡)

As 𝑍 (𝑡) grows at rate 𝑔 + 𝑛, we have:¤ = (𝑟 (𝑡) − 𝜌 − 𝑔 + 𝛼) c(𝑡) − 𝜇b(𝑡),c(𝑡)¤ = (𝑟 (𝑡) − 𝑔 − 𝑛) b(𝑡) − 𝑇 (𝑡).
b(𝑡)
where
𝜇≡

(17)
(18)

(𝜌 + 𝜆)(𝛼 + 𝜆 + 𝑛)
1 +𝜓Finally, (1 + 𝜓 )c(𝑡) = 𝑤(𝑡), and letting c★ ≡ 𝑤 ★/(1 + 𝜓 ), we can rewrite the Phillips curve as:


ˆ
𝜋(𝑡)
¤ = 𝜌𝜋(𝑡)+ 𝜅 c★ − 𝑐(𝑡) ,(19)

˜ We postpone the discussion offor 𝜋(𝑡) ∈ [𝜋, 𝜋], and where 𝜌ˆ ≡ 𝜌˜ − (𝑔 + 𝑛) and 𝜅 ≡ (1 + 𝜓 )𝜅.
dynamics for 𝜋(𝑡) ∈/ [𝜋, 𝜋] to below. We will assume that 𝜌˜ > (𝑔 + 𝑛), a restriction that guarantees
that the firm’s pricing problem is well defined.

3.1

Balanced Growth Paths

It will be useful to describe a few properties of a balanced growth path (BGP). In a BGP, allquantities grow at the rate of output, and the real interest rate is constant.
Setting c¤ = 0, and letting b, c, and 𝑟 denote the corresponding BGP values, we have:b (𝑟 − 𝜌 − 𝑔 + 𝛼)
=
.
c
𝜇

(20)

Note that the right-hand side is linearly increasing in 𝑟 . The interest rate in financial autarky,which occurs when b = 0, is given by 𝜌 + 𝑔 − 𝛼, and for higher levels of financial assets we have𝑟 > 𝜌 + 𝑔 − 𝛼. In a BGP, as debt relative to worker consumption increases, the real interest rate
must also increase for the bond market to clear.
Setting b¤ = 0, we get from the government sequential budget constraint:
𝑇 = (𝑟 − 𝑔 − 𝑛)b.
In a BGP, 𝑤 > 𝑇 for the borrowing limit, 𝑎(𝑡, 𝑡) to be negative. That is, (𝑟 − 𝑔 − 𝑛)b = 𝑇 < (1 +𝜓 )c,19

which requires(𝑟 − 𝑔 − 𝑛)(𝑟 − 𝜌 − 𝑔 + 𝛼)
< 1 ⇒ 𝑟 < 𝜌 + 𝑔 + 𝑛 + 𝜆,
(1 + 𝜓 )𝜇where we used that 𝑟 > 𝜌 +𝑔−𝛼. For 𝑎(𝑠, 𝑡) to be finite for all (𝑠, 𝑡), we also require that 𝑟 > 𝑔−𝛼 −𝜆,but this is satisfied given 𝜌 > 0 and 𝜆 ≥ 0 and 𝑟 ≥ 𝜌 + 𝑔 − 𝛼. Thus, in a BGP, the interest rate
must satisfy:
𝜌 + 𝑔 − 𝛼 ≤ 𝑟 < 𝜌 + 𝑔 + 𝑛 + 𝜆.

(21)This condition summarizes the equilibrium restrictions that, in a BGP, the interest rate is higher
than the autarky level and that the borrowing constraint is non-negative and finite.
Flexible price interest rates. It will be useful to define the BGP real interest rate associatedwith the flexible price markup, given debt, 𝑟★(b) as well as its inverse with the following:b𝑟★(b) ≡ 𝜌 + 𝑔 − 𝛼 + 𝜇 ★
cc★
b★(𝑟 ) ≡ (𝑟 − 𝜌 − 𝑔 + 𝛼) .
𝜇

(22)
(23)

The equilibrium dynamics of the real interest rate 𝑟 (𝑡) will depend on the monetary policy rule.
When the monetary rule (16) responds to debt with parameter 𝜃𝑏 = 𝜃𝑏★ = 𝜇/c★ and its intercept is𝜄 = 𝑟★(b𝑜 ), then it can implement the interest rate associated with the flex price equilibrium. We
turn next to various configurations of fiscal policy and monetary rules and their impact on theeconomy’s dynamics. We first characterize the case in which monetary policy is not bound by
the ELB, and then discuss dynamics in the binding-ELB region of the state space in Section 5.2.

3.2

Dynamics away from the ELB

Let us first consider a fiscal policy that maintains a constant b(𝑡) = b𝑜 for all 𝑡 ≥ 0. When the ELB
is not binding, monetary rule (16) becomes𝑖(𝑡) = 𝜄¯ + 𝜃 𝜋 𝜋(𝑡).
Here, we assume that 𝜄¯ = 𝑟★(b𝑜 ) consistent with a target inflation rate of zero. For the ELB not tobind we require 𝑖(𝑡) > 0, which implies 𝜋(𝑡) > −¯𝜄 /𝜃 𝜋 . This will be the range of inflation relevant
for this subsection.
The monetary rule, combined with the Fisher equation implies 𝑟 (𝑡) = 𝑖(𝑡)−𝜋(𝑡) = 𝜄¯+(𝜃 𝜋 −1)𝜋(𝑡).
20Figure 1: Phase Diagram: Dynamics Away From the ELB𝜋𝜋¤ = 0

𝜋
c★, 0
0



c
c¤ = 0Substituting into (17), we have
¤ = (¯𝜄 + (𝜃 𝜋 − 1)𝜋(𝑡) − 𝜌 − 𝑔 + 𝛼) c(𝑡) − 𝜇b𝑜 .
c(𝑡)

(24)

Equations (19) and (24) are two ordinary differential equations (ODEs) in 𝜋(𝑡) and c(𝑡). Thissystem of two equations, combined with the condition that inflation is bounded and c ≥ 0 in
equilibrium, characterizes all possible equilibria in which the ELB does not bind.
To analyze the dynamic system, we use the phase diagram in Figure 1. The curve labelled“𝜋¤ = 0” sets 𝜋¤ in (19) to zero:𝜋=𝜅c − c★
𝜌ˆfor 𝜋 ∈ [𝜋, 𝜋].

(25)

As c increases above c★, firms would like to raise their markup, which is counterbalanced by thecosts of adjusting prices faster. The stationary point trades off higher 𝜋 against higher c.
Along the 𝜋¤ = 0 curve inflation is constant, at a value that increases with c. Above this locus,𝜋¤ > 0, and below we have 𝜋¤ < 0. These dynamics are represented by the arrows pointing up anddown in the phase diagram.
Outside of [𝜋, 𝜋], the lower terms in (14) imply that the Phillips curve is vertical. There is a
subtlety when it comes to dynamics for 𝜋 ∈/ [𝜋, 𝜋]. Along the vertical portion of the 𝜋¤ = 0 locus,
constant inflation is consistent with profit maximization, but so are movements along the verticalsection, as firms are indifferent about the choice of 𝜋.

21

The curve labelled “c¤ = 0” is the locus of points at which c¤ = 0. From (24), we have:
 
 𝑜
𝜇
𝜄¯ − 𝜌 − 𝑔 + 𝛼
b
+
.
𝜋 =−
𝜃𝜋 − 1𝜃𝜋 − 1 c
(26)

As c increases, the bond market requires a lower real interest rate to clear. Given that 𝜃 𝜋 > 1,
this implies a lower rate of inflation and an even lower nominal interest rate. This generates anegative relationship between c and 𝜋 in order to keep c¤ = 0.
From (24), we see that as 𝜋 increases for a given c relative to the c¤ = 0 locus, c¤ > 0. These
dynamics are depicted by the horizontal arrows in Figure 1. Note that when b𝑜 = 0, the c¤ = 0
curve would be horizontal with inflation equal to the first term in (26).
Again, care must be taken for 𝜋 ∈/ [𝜋, 𝜋]. For inflation outside this interval, there are uniquelevels of consumption, c and c, that are consistent with firm optimization. Thus, there are no
consumption dynamics in equilibrium outside [𝜋, 𝜋]; that is, if 𝜋 ∈/ [𝜋, 𝜋] is part of an equilibrium
trajectory, c is constant while the economy moves along the relevant vertical portion of the 𝜋¤ = 0
curve.
The intersection of the two curves, denoted by (c★, 0), is the zero-inflation steady state whichis the target of the monetary policy. This requires the monetary rule intercept 𝜄¯ to be set to𝜄¯ = 𝑟★(b𝑜 ), so that the bond market clears at the zero-inflation steady state.16 Note that achieving
the zero inflation outcome requires an intercept that depends on the level of debt, which reflects
the non-Ricardian environment, and anticipates our discussions of fiscal and monetary policy
coordination. The zero inflation steady state is unstable. In particular, the eigenvalues of thelinearized system evaluated at the steady state both have real parts strictly greater than zero.
It is possible that in our model the ELB binds. In the next section, we will however abstractfrom this possibility and postpone the discussion of the ELB and the implications for fiscal andmonetary policy interactions until Section 5.2.

4

Positive and Normative Implications of Government Debt

This section considers how changes in the stock of government debt affect the economy, both froma positive and normative viewpoint, under alternative monetary policy rules. We also discuss how
debt impacts the economy’s response to demand, monetary, and cost-push shocks.
16 Note that 𝜄¯ = 𝑟𝑜𝑜★(b ) > 𝜌 + 𝑔 − 𝛼 for b > 0, and thus the c¤ = 0 lines crosses the horizontal axes. In addition, theplot as drawn requires that the ELB does not bind at the intersection, that is, 𝜄¯ = 𝑟★(b𝑜 ) > 0. We will discuss the casewhere the ELB binds at the zero-inflation steady state later in Section 5.2.

22

4.1

The Economic Dynamics of Anticipated Deficits

Our main policy experiment involves an anticipated debt-financed tax cut (or transfer increase).
Many politicians run on such a platform, including US Presidents Reagan and George W. Bush, aswell as, most recently, the mini-budget of UK Prime Minister Elizabeth Truss. Presidents Trump
and Biden also implemented some combination of tax cuts and large transfer programs, particularly after the onset of the COVID pandemic. We show that anticipated deficits may increaseinflation and may increase or decrease consumption on impact, depending on the horizon andthe response of monetary policy.
We initialize the current period as 𝑡 = 0 and assume we are at the zero inflation steady statewith some level of debt b𝑜 for 𝑡 < 0. At 𝑡 = 0, there is an unanticipated announcement that attime 𝑡 ′ > 0 the government will increase debt to b′ > b𝑜 and rebate the proceeds to workers.
We trace out the path of inflation and worker consumption until the economy reaches the newsteady state. In doing so, we focus on the dynamics away from the ELB.
In Figure 2, we replicate the phase diagram from Figure 1 and we first study the case when𝜃𝑏 = 0. At 𝑡 ′, the “c¤ = 0” shifts out due to the change in b, which is the dashed line labeled c¤ = 0.
The size of this shift for a given b′ depends on parameters, in particular 𝜆, 𝜌, and 𝛼. Note thatin the Ricardian case of 𝛼 = 𝜆 = 0, there is no shift. Prior to 𝑡 ′, the economy is still subject tothe dynamics governed by the original b𝑜 , which are depicted by solid lines and the arrows. The
Phillips curve does not depend on b, and hence the “𝜋¤ = 0” remains stable.
The economy jumps to the trajectory at 𝑡 = 0 and travels along that path until it reaches the
new steady state at exactly 𝑡 = 𝑡 ′. For larger 𝑡 ′, the announcement effect places the economy
closer to the original steady state; for smaller 𝑡 ′, the economy jumps closer to the eventual steadystate. Depending on parameters, the eigenvalues may be complex or real, and the resultant pathmay cycle or not, respectively.
We depict a thick solid portion of the trajectory as an example path. The initial point involves
higher inflation and lower worker consumption. This is combined with positive c¤ and 𝜋.
¤ The
economy’s response can be understood through the logic of the bond market (which is the flipside of the goods market), working backwards in time from 𝑡 ′. The first thing to note is that for𝑡 ≥ 𝑡 ′, there is a steady state at the intersection of the dashed c¤ = 0 line and the Phillips curve.
Due to the Taylor principle, this is an unstable steady state, and hence the economy must be atthis point at 𝑡 ′; otherwise, the explosive dynamics would generate paths that are not consistent
with the equilibrium conditions. At this steady state, inflation and consumption are higher. The
higher inflation is needed to generate a higher real interest due to the monetary policy rule:𝑟 = 𝜄¯ + (𝜃 𝜋 − 1)𝜋, with 𝜃 𝜋 > 1. The higher consumption accompanies the higher inflation due
to the Phillips curve. Both elements work to restore equilibrium in the bond market after theissuance of the additional debt.
23

Figure 2: Response to an Anticipated Bond Issue𝜋𝜋¤ = 0𝑡 = 𝑡′c¤ = 0𝑡 =0c

0c¤ = 0

Dynamics prior to 𝑡 ′ are driven by the anticipation of the bond issuance. In particular, consumers anticipate higher consumption and fiscal transfers at 𝑡 = 𝑡 ′. A higher eventual worker
consumption lowers the demand for the initial (fixed) stock of bonds for standard inter-temporal
substitution reasons. The bond market clears at 𝑡 ∈ [0, 𝑡 ′) with a higher real interest rate, restoring equilibrium at 𝑏𝑜 . Again due to the monetary policy rule, the higher real interest rate mustbe accompanied by higher inflation. Whether the jump in 𝜋 is associated with an increase or
decrease in the 𝑡 = 0 worker consumption depends on the time horizon and the parameters ofthe model.
For the experiment of Figure 2, we considered the case of a monetary rule with 𝜃𝑏 = 0, whichholds the intercept constant at 𝑟★(b𝑜 ). The increased b requires an increase in the real interest
rate, but given the interest rate rule, this must be associated with higher inflation. If the monetary authority increased its intercept at 𝑡 ′, the shift in the c¤ = 0 curve would be dampened. In
particular, the central bank can keep the economy at the zero inflation initial steady state for all𝑡 > 0 by promising to increase the intercept (and hence the nominal interest rate) one-for-one
with the necessary increase in the real interest rate to absorb the new bonds at the initial levelof income. In particular, this outcome is possible by setting 𝜃𝑏 = 𝜃𝑏★ in the rule. The economy
is at the zero-inflation steady state for all 𝑡 > 0, and worker consumption is unchanged. The
only effect of the fiscal expansion is to increase the level of debt and a rotation of the c¤ = 0 attime 𝑡 = 𝑡 ′. We depict this case in Figure 3. At 𝑡 = 𝑡 ′, the monetary policy rule implements areal interest rate increase that rotates the 𝑐¤ = 0 locus through the point (𝑐 ★, 0). Thus, there is no24

Figure 3: Response to an Anticipated Bond Issue: Alternative Monetary Rule𝑡 = 𝑡′𝜋𝜋¤ = 0𝑐 ★, 0

0𝑐𝑐¤ = 0change in consumption at 𝑡 = 𝑡 ′, and hence no anticipatory change in consumption for 𝑡 < 𝑡 ′.
The primary conclusion from this analysis is that in a non-Ricardian environment, the economic consequences of anticipated deficits hinge on the response of the monetary authority’s ruleto the increase in debt. If the central bank follows the policy advice obtained from the Ricardian
benchmark that the long-run target real interest rate is invariant to fiscal policy, the consequences
are higher inflation and avoidable fluctuations in consumption. To an outsider well versed in the
Ricardian literature, the central bank appears to be doing exactly as prescribed; namely, following
a set monetary rule that leans against lax fiscal policy. However, the rigidity of the rule is thecause rather than the cure for the inflation observed in equilibrium.

4.2

A Discount Rate ShockA standard approach to conceptualizing a “demand shock” in the New Keynesian model is atransitory increase in patience.17 We now show an interesting parallel between such a shock andthe analysis above.
Suppose the discount rate of workers, 𝜌, unexpectedly declines at 𝑡 = 0, but is expected (withperfect foresight) to return to its initial level at 𝑡 = 𝑡 ′. Specifically, indexing 𝜌 by time, we have:𝜌(𝑡) =

𝜌

for 𝑡 ∈/ [0, 𝑡 ′)𝜌


for 𝑡 ∈ [0, 𝑡 ′),ˆ of the entrepreneurs remains constant throughout.
with 𝜌 < 𝜌. The discount rate (𝜌)17 For example, see Krugman (1998), Eggertsson and Woodford (2003), and Jung, Teranishi, and Watanabe (2005).

25

Figure 4: Temporary Decline in 𝜌: ELB does not bind𝜋𝜋¤ = 0𝑡 = 𝑡′0cc¤ = 0𝑡 =0c¤ = 0We explore this scenario in Figure 4. The initial (and final) steady state is depicted by theintersection of the solid “c¤ = 0” and “𝜋¤ = 0” lines. The dashed c¤ = 0 locus is drawn for 𝜌 = 𝜌,when we maintain the interest rate rule intercept constant at 𝜄. From (26), a decline in 𝜌 shifts
the locus down in 𝜋 × c space. The Phillips curve does not shift, as we maintain the 𝜌ˆ constantin this scenario.
This figure is the mirror image of Figure 2. Recall in Figure 2, the shift occurs because thefuture flexible price real interest rate (after the debt issuance) is higher than the initial rate. In
the current experiment, the flexible price interest rate declines at 𝑡 = 0. Hence, the curves shiftsimilarly, but in opposite direction.
In both scenarios, the flexible price real interest rate is temporarily low, and hence the dynamics are similar, albeit shifted in 𝜋 × c space. Specifically, the dashed arrows in Figure 4 depict
the dynamics that hold for 𝑡 ∈ [0, 𝑡 ′); that is, relative to the dashed c¤ = 0 locus. Perfect foresight
and worker and firm optimization imply that the equilibrium must be back at the initial steadystate at 𝑡 = 𝑡 ′. The purple line depicts trajectories that depart and return to that steady state. The
precise point on the trajectory that holds at 𝑡 = 0 depends on 𝑡 ′.
As in the case of Figure 2, the potential trajectories correspond to combinations that clear thebond market. In this experiment, b𝑜 is held fixed, but, all else equal, more patient workers desire
to hold more bonds at a given real wage and interest rate. The market clears by a combination¤ both of which reduced demand. The counterpart is a change in theof lower c and higher c,path of 𝜋 that ensures the worker consumption dynamics are consistent with firm optimization,
which corresponds to lower 𝜋 and higher 𝜋.
¤ The inflation path in turn generates a path of the26

real interest rate via the interest rate rule. A temporary decline in the discount rate, therefore,
generates the familiar temporary declines in inflation and consumption.
As before, the monetary authority could keep the equilibrium at the desired zero-inflation
steady state by having 𝜄¯ move one-to-one with 𝜌(𝑡), temporarily declining before returning to its
long run value. From (20), the appropriate 𝜄¯ that clears the bond market at c★ and zero inflationis𝜄¯(𝑡) = 𝜌(𝑡) − 𝛼 + (𝜌(𝑡) + 𝜆)(𝜆 + 𝛼)b𝑜 /c★.
(27)
Interestingly, a temporary fiscal expansion would also stabilize the economy. That is, at 𝑡 = 0
the fiscal authority statiates workers’ increased patience by issuing more bonds, which it willreverse at 𝑡 = 𝑡 ′. From the analysis associated with Figure 2, this would keep the c¤ = 0 locus atits original (and long run) location. This discussion highlights the symmetry between monetaryand fiscal policy, a point recently emphasized by Wolf (2021). However, this symmetry is onlyin terms of aggregate consumption and inflation. As we shall see below, the level of debt affects
the distribution of consumption across cohorts, and hence welfare, in a manner not attainable bymonetary policy alone.
One important caveat is that the decline in 𝜌 was assumed to be small enough that the monetary authority does not run afoul of the ELB. We will take that possibility up in Section 5.2.
However, the preceding paragraph already anticipates one result of that section; namely, that
fiscal policy can substitute for monetary policy at the ELB and deliver the economy away fromthe ELB.

4.3

Monetary Policy and Cost-Push Shocks

The fact that both monetary and fiscal policies can shift and rotate the c¤ = 0 has implications
for how government debt plays a role in the response to monetary policy shocks and so-called“cost-push” shocks. In this subsection, we explore how the level of debt influences the economy’s
response to a monetary policy shock and a cost-push shock.
As standard, we consider monetary policy shocks as temporary shifts to the rule (16). Specifically, consider a contractionary monetary policy shocks such that the intercept of the rule 𝜄¯(𝑡)unexpectedly increases at 𝑡 = 0 but is expected to return to its initial level at 𝑡 = 𝑡 ′𝜄¯(𝑡) =𝜄¯ + ∆
for 𝑡 ∈/ [0, 𝑡 ′)


𝜄¯


for 𝑡 ∈ [0, 𝑡 ′),with ∆ > 0. The effects of this contractionary monetary policy announcement are qualitatively

27

similar to those depicted in Figure 4. The c¤ = 0 locus shifts down upon the announcement whilethe 𝜋¤ = 0 locus is unaffected. Note that given the aggregate Euler equation, (24), the magnitudeof the downward shift of the c¤ = 0 is equal to ∆ measured in the y-axis. The impact effects ofthis announcement are a fall in consumption and inflation. Over time consumption and inflation
increase such that at 𝑡 = 𝑡 ′ the economy is back at the initial steady state.
Although qualitatively the effects of contractionary monetary policy are unaffected by government debt, quantitatively higher debt lowers the economic response to monetary policy shocks.
To see this, consider the case of zero debt (which maps also into the representative agent case).
The c¤ = 0 in this case is horizontal and the downward shift of the c¤ = 0 locus crosses at a level ofinflation that is equal to −∆. For a given 𝑡 = 𝑡 ′, the shift in inflation and consumption will be ofhigher magnitude and the trajectory will be faster between [0, 𝑡 ′). With high government debt,
in contrast, the c¤ = 0 schedule is steep and the same monetary contraction which generates thesame vertical shift, results in smaller impact effects on consumption and inflation. We conclude
therefore that the effectiveness of monetary policy shocks diminishes as government debt rises.
During the monetary contraction, we have held constant fiscal policy. As in the previous
section, the fiscal authority could keep the equilibrium at the desired zero-inflation steady statewith an appropriate path of debt that offsets the monetary policy contraction.
We can study a cost-push shock also in the standard manner; namely, an exogenous shift in
the Phillips curve up in 𝜋 × c space. From the phase diagram, we can see that to what extent
the shift generates inflation versus a decline in consumption depends on the slope of the c¤ = 0
locus. As already derived in the above examples, the slope of this line depends both on the level ofgovernment debt as well as the responsiveness of monetary policy. In particular, a highly indebted
economy will have a steep c¤ = 0 evaluated at c★, assuming monetary policy follows the nonRicardian Taylor Rule. Hence, for a given monetary rule, high debt implies that inflation bears
the brunt of the cost push shock, rather than consumption, and vice versa for low indebtedness.
This is illustrated in Figure 5. The panels show the response of the economy to a temporary costpush shock. It is assumed that the economy is originally at 𝜋 = 0 and c = c★ in both panels. In
both panels, the dashed 𝜋¤ = 0 locus is the corresponding original Phillips Curve locus. The solid𝜋¤ = 0 locus represents a temporary shift in the Phillips curve that lasts from 𝑡 = 0 to some 𝑡 ′.
The economy must be back to the original intersection at 𝑡 = 𝑡 ′, that is, once the cost-push shock
disappears. The main difference between the panels is in the underlying level of debt, which leads
to a rotation of the c¤ = 0 locus. Panel (a) represents a low debt case; while panel (b) represents ahigh debt case.

28

Figure 5: Cost-Push Shock𝜋𝜋c¤ = 0𝜋¤ = 00c¤ = 0𝜋¤ = 0c0(a) Low Debt

4.4𝜋¤ = 0𝜋¤ = 0c(b) High Debt

Welfare Implications of Government Debt

We now turn to the question of whether and how increasing the stock of government debt affects
the distribution of consumption and welfare in our New Keynesian environment with heterogeneous agents.
We begin by noticing that the only aggregate store of value for workers in this economy isgovernment bonds. To understand how the OLG structure breaks Ricardian equivalence, consider
a balanced growth path on which 𝐵 > 0 and 𝑟 > 𝑔 + 𝑛. In this case, a bond is a claim on futuretaxes, some of which is paid by future generations, and hence perceived as net wealth by thosecurrently alive. That is, the newborn cohort pays taxes but holds no bonds, which represents
a net transfer to older cohorts. This is the intuition that is perhaps the most familiar from theliterature on Ricardian non-equivalanece.
A more useful intuition for our purposes is to consider the fact that newborn agents whowish to save must purchase a bond from older cohorts. The value of this perspective is it makes
clear that the same logic holds even if 𝑟 < 𝑔 + 𝑛. In this case, the government earns money fromoutstanding debt, which it lump-sum transfers to those currently alive. Nevertheless, if an ownerof a bond perceives that they will be able to sell the bond at a positive price to future generations not yet alive, it will be considered net wealth. This future bond demand from generations
not yet born represents the “social contrivance” of Samuelson (1958) and encapsulates the main
inefficiency of the OLG environment.

29

4.4.1

Pareto Improvements: Revisiting Samuelson

Suppose we start from an initial BGP with b(0) = b𝑜 ≥ 0 and a corresponding real rate 𝑟 𝑜 . We willconsider a simple policy change: at 𝑡 = 0, the government issues new debt, moving the total stockof debt to b′ from b𝑜 , and adjusts taxes on (or transfers to) workers to satisfy its budget constraint.
Based on the analysis above, we will consider the case of the monetary rule in (16), with 𝜃𝑏 = 𝜃𝑏★
and appropriate intercept, such that the nominal interest rate tracks 𝑟★(b) with zero inflation. We
will assume the economy is away from the ELB, but consider the alternative in Section 5.2.
Along a BGP, we can compute the welfare of the newborn cohort in closed form, which we
present in Appendix B. Intuitively, welfare is increasing in the real wage, which represents theshare of output going to workers. However, the key implication for our purposes is that welfare
depends non-monotonically on 𝑟 . A greater 𝑟 implies lower discounted lifetime wealth, lowering
initial consumption, but a higher return to saving and a faster growth rate of consumption going
forward. As shown in Appendix B, for 𝑟 ∈ [𝜌 − 𝛼 + 𝑔, 𝜌 + 𝜆 + 𝑔 + 𝑛), the range of rates consistent
with a BGP according to (21), 𝑈 𝑡 is a strictly concave function in 𝑟 with a maximum at 𝑟 = 𝑔 + 𝑛:
Lemma 4. For a given wage, the welfare of a newborn worker in a BGP is maximized when𝑟 = 𝑔 + 𝑛.
In an equilibrium, individual workers may see a net return to bonds different than 𝑔 + 𝑛. For
example, the autarkic interest rate (when b = 0) is 𝜌 + 𝑔 − 𝛼 which may be less than the value of𝑔 + 𝑛 for 𝑛 or 𝛼 large enough. From their Euler equation, an 𝑟 ̸= 𝑔 + 𝑛 distorts the inter-temporalpath of consumption relative to the social optimum, which in turn distorts the cross-sectional
allocation of the exogenous amount of resources. This allows for an increase in the supply ofbonds to improve upon the equilibrium allocation by raising the interest rate and facilitatinginter-generational trades.
Newborn welfare is maximized at 𝑟 = 𝑔 + 𝑛, but a Pareto improvement must also accountfor existing cohorts at the time of the policy change. If b𝑜 = 0, then existing cohorts have zero
wealth and are thus identical to newborns regarding the welfare consequences of a change in 𝑟 ,but in addition get the initial distribution b′ − b𝑜 . However, if b𝑜 > 0, then existing cohorts havepositive wealth and are thus sensitive to changes in 𝑟 beyond the effects on newborn cohorts.
The positive savings give them an extra benefit from higher real interest rates. For a given levelof 𝑎(𝑠, 𝑡) ≥ 0, utility is strictly increasing in the interest rate for 𝑟 < 𝑔 + 𝑛. Contrary to newborns,
if 𝑎 > 0, utility peaks at 𝑟 strictly greater than 𝑔 + 𝑛, due to the presence of positive assets. This
restricts attention to policies that increase 𝑟 starting from 𝑟 𝑜 < 𝑔 + 𝑛.
Turning to the entrepreneurs, they are indifferent to equilibria with the same real wage andrate of inflation. In particular, profits are declining in 𝑤 (the mirror of worker welfare), but

30

non-monotonic in inflation, as inflation changes the markup as well as represents a deadweightloss. Thus, if 𝑟 𝑜 < 𝑔 + 𝑛, the government can implement a Pareto improvement by issuing debt
and moving 𝑟 closer to 𝑔 + 𝑛, while ensuring entrepreneurs’ profits do not decrease. For thisoutcome, it is imperative that the monetary authority accommodates the expansion of debt witha higher nominal interest rate target, which is achieved with the rule that responds to debt withcoefficient 𝜃𝑏★. Without such a response, the economy experiences inflation, reducing the welfare
of entrepreneurs.
The following lemma summarizes this result.
Lemma 5. Suppose the economy is in an initial BGP with 𝑟 0 < 𝑔 + 𝑛. Then there exists a new
BGP with a combination of fiscal and monetary policies that is a Pareto improvement.
We want to highlight that the key inefficiency in the model behind Lemma 5 is the presenceof a never-ending flow of new generations of workers (or the “infinite hotel” of Shell, 1971). Thisis the seminal insight of Samuelson (1958), and there is a simple way to see it in our current environment. Set 𝛼 = 0, and consider a situation without new cohorts being born. This corresponds
to the case where 𝑛 = −𝜆, that is, population shrinks at the death rate 𝜆. In that case, 𝐴(𝑡) drops
out from the aggregate Euler equation (10).18 There is then a unique real rate consistent with a
BGP, 𝑟 = 𝜌 +𝑔 > 𝑔 +𝑛, and thus, there is no role for government bonds in improving upon (or evenaffecting) the market equilibrium. Moreover, in this case the monetary rule should not respondto debt.
4.4.2

Robust Pareto Improvements

To implement the above Pareto improvement the government must alter taxes and transfers, 𝑇 .
Given the knowledge of the utility function, it knows how much agents are willing to trade offhigher taxes (or lower transfers) in exchange for a higher return to savings. This assumes a levelof knowledge that is not realistic. For example, if there were a negligible fraction of consumers
that are “hand-to-mouth” consumers, the change in taxes may not be worth the higher interest
rate. Such concerns motivated our use of a “robust” criteria in our earlier work (Aguiar, Amador,and Arellano, 2022). In this subsection, we revisit “Robust Pareto Improvements” (RPI) in a New
Keynesian setting.
Aguiar, Amador, and Arellano (2022) defines an RPI to be a policy that induces a change inprices and taxes such that the budget set of any agent is guaranteed to be weakly expanded atany state and time. In our current environment:
18 Buiter (1988) referred to this as the debt neutrality case. Note that our equations do not work in the limit of𝑛 = −𝜆 when 𝛼 > 0, as there is no effective aggregate amount of labor left in the long-run in this case.

31

Definition 1 (RPI). We say that the fiscal policy of increasing debt from b𝑜 to b′ > b𝑜 given amonetary rule generates an Robust Pareto Improvement (RPI) if (i) 𝑇 ′ ≤ 𝑇 𝑜 , (ii) for any (𝑠, 𝑡),𝑟 ′𝑎𝑜 (𝑠, 𝑡) ≥ 𝑟 𝑜 𝑎𝑜 (𝑠, 𝑡), where 𝑎𝑜 (𝑠, 𝑡) is the original equilibrium choice of assets for cohort 𝑠 attime 𝑡, (iii) 𝑤 ′ ≥ 𝑤 𝑜 , and (iv) Π′ − 𝑓 (𝜋 ′) ≥ Π𝑜 − 𝑓 (𝜋 𝑜 ).
The first condition is that taxes weakly decrease; the second states that financial income
weakly increases; the third condition is that real wages weakly increase; and the final condition requires that the profits do not fall. Condition (iv) ensures the entrepreneurs are no worse
off in the new equilibrium.
From the workers’ perspective, the first three conditions ensure that their budget set weakly
increases. Given that the initial transfer to workers is strictly positive (b′ > b𝑜 ), generations alive
at 𝑡 = 0 have a strict increase in their budget sets. In aggregate, worker consumption may notincrease, but nevertheless, individual workers prefer the new allocation due to the better intergenerational distribution. An important element here is that it is not necessary to detail how
aggregate consumption is reallocated across workers.19
In the appendix (see the proof of Lemma 6), we show that in the BGP, 𝑎(𝑠, 𝑡) ≥ 0 for all 𝑠 and𝑡. That is, no cohort has a negative asset position in the initial equilibrium, and thus condition(ii) requires 𝑟 ′ ≥ 𝑟 𝑜 .20
One advantage of the RPI criteria is that it ensures a Pareto improvement regardless of how
agents trade-off consumption intertemporally (or across uncertain states in a richer environment).
In an RPI, with weakly greater flow income at all dates, any agent’s initial equilibrium consumption path remains affordable. Thus their welfare cannot fall and must increase if the budget set
expands strictly. A second advantage of the RPI criteria is that to check for the existence of an
RPI, only knowledge about how aggregate private savings respond to real interest rate changes
is needed. The potentially rich household heterogeneity at the micro level only affects the conditions for the existence of an RPI through the shape and interest rate elasticity of this aggregate
savings response.
In the present model, that elasticity is endogenous and partially controlled by the monetary
policy response to fiscal policy. This raises the question of whether the appropriate monetary
policy for generating an RPI delivers 𝑟★(b) and zero inflation or whether a different monetary
19 This was our main motivation when introducing the RPI in the context of a Bewley-Aiyagari model. It is notnecessary to specify how agents trade off consumption across states and time. As will be true here as well, only theaggregate savings supply is needed to determine the feasibility of an RPI.
20 Every cohort has a consumption profile that changes at a constant rate, 𝑟 𝑜 − 𝜌, and a labor earnings profile(net of taxes) that changes at a constant but weakly lower rate, 𝑔 − 𝛼 ≤ 𝑟 𝑜 − 𝜌. The latter is a requirement for the
economy to sustain non-negative levels of government debt in the stationary equilibrium, see equation (20). Given
that newborns have zero assets, it follows that the asset holdings of every cohort must remain non-negative overtheir lifecycle.

32

configuration introduces an additional tool to achieve an RPI. In this section, we show that theanswer is no.
We say that the monetary authority implements the flexible price equilibrium if the monetaryrule features 𝜃𝑏 = 𝜃𝑏★ and it delivers 𝑟★(b) and zero inflation after any debt issuance.21 In particular,
suppose we begin on a balanced growth path with 𝜋 𝑜 = 0 and worker wages at 𝑤 𝑜 = 𝑤 ★. From
the Phillips curve, aggregate worker consumption is 𝑐 ★. Let initial debt be b(0) = b𝑜 ≥ 0, with acorresponding real rate 𝑟 𝑜 = 𝑟★(b𝑜 ). At 𝑡 = 0, the government issues new debt, moving the total
stock of debt (in efficiency units) to b′ from b𝑜 , and adjusts taxes on workers to satisfy its budget
constraint.22
After 𝑡 = 0, there are no additional policy changes, debt remains constant at b′, and taxes
are adjusted to satisfy the government budget constraint. From equation (17), it follows that 𝑟 (𝑡)must also be constant, at a value we denote by 𝑟 ′. To implement the flexible price equilibrium
response, the monetary authority simultaneously increases the nominal interest rate from 𝑖 = 𝑟 𝑜
to 𝑖 = 𝑟 ′. From equation (18) it follows then that transfers remain constant at a value 𝑇 ′. Taken
together, this means that after the policy change, the economy jumps to a new BGP, and there
are no transitional dynamics.23
We can combine the government budget constraint at the new level of debt, b′, and the aggregate Euler equation, using the definition of b★ in (23), to obtain the following relation between
taxes, debt, and the real rate
𝑇 ′ = (𝑟 ′ − (𝑔 + 𝑛))b★(𝑟 ′).

(28)

where b★(𝑟 ′) = b′. Note that b★(𝑟 ′) is the aggregate savings supply schedule of the economy in
the new BGP, and uniquely determines 𝑟 ′, given b′.
Now, a local increase in b from b𝑜 must be matched by an increase in 𝑟 from 𝑟 𝑜 . For this togenerate a decrease in taxes, 𝑇 , it suffices that−b★′ (𝑟 𝑜 )𝑟 𝑜 − (𝑔 + 𝑛)
> 1.
b𝑜

(29)

21 We assume that the ELB is not binding and characterize the new BGP that arises as the result of the policy. For

the ELB not to bind at the initial BGP, it must be that 𝑟★(b𝑜 ) ≥ 0. Given that we are assuming that the ELB is notbinding after the policy change, the economy must transfer to a new BGP, given the unstable dynamics.
22 As noted above, the case of a bond issuance is equivalent to the government distributing bonds directly to theexisting worker cohorts, increasing aggregate savings by b′ − b𝑜 .
23 The lack of transitional dynamics is in our view a main difference with respect to Bewley-Aiyagari type models,
where the distribution of wealth needs time to converge to a new ergodic state after a policy change. In our model,
the distribution of wealth does not need to adjust after the policy change, a feature driven by the linearity of thepolicy functions. A potential extension of the present model, that will generate transitory dynamics, is to introduce
additional heterogeneity on the workers’ side as in Gertler (1999).

33

Thus, if the aggregate savings elasticity with respect to the real interest rate (net of 𝑛 + 𝑔) is higher
than unity, then an RPI exists. Given that b★′ (𝑟 𝑜 ) > 0, it follows from (29) that 𝑟 𝑜 < 𝑔 + 𝑛. This is
not sufficient however, b★′ (𝑟 𝑜 ) must also be large enough.
The elasticity condition (29) is related to the debt Laffer curve: it guarantees that the economy
starts from a region below the peak of the curve, and thus it is possible to increase government
revenue by increasing debt.24 Figure 6 shows the debt Laffer curve and its relationship with theexistence of an RPI. In our current model, we can go a bit farther, as we can characterize the peakof the Laffer curve in closed form:
Lemma 6 (Existence of an RPI). Suppose the economy is in an initial BGP. There exists a debt
issuance at 𝑡 = 0 that, combined with a monetary policy that implements the flexible priceequilibrium, generates an RPI if and only if 𝑟 𝑜 − 𝑔 − 𝑛 < (𝜌 − 𝛼 − 𝑛)/2 < 0.
In Figure 6 we denote the point where 𝑟 = 𝑔 + 𝑛 as the “Samuelson’s point”. At this point, theeconomy is at a Pareto optimum with respect to the allocation of resources among workers. For
values of debt below this point, the equilibrium allocation is Pareto dominated by Samuelson’s
point. However, if the economy started from a lower level of debt and policy moved it towards thispoint, although interest rates would increase towards 𝑔 + 𝑛, transfers do not uniformly increase.
As the figure shows, transfers decrease with debt for levels of debt to the right of the peak of thetransfer curve. Such a decline in transfers is the reason increasing debt up to the Samuelson’s
point does not constitute an RPI.
The debt Laffer curve in Figure 6 assumes that monetary policy implements the flexible priceequilibrium. We now turn to the question of how the curve changes under alternative configurations for the monetary policy rule, and whether such changes expand the scope for RPIs. The
monetary policy rule is given by
𝑖 ′ = 𝜄¯ + 𝜃 𝜋 𝜋 ′ + 𝜃𝑏 (b′ − b𝑜 ).
The parameter 𝜃𝑏 controls the response of monetary policy to government debt. Using the above,
we can solve for the level of taxes at the new BGP as a function of the aggregate savings schedule
as follows
𝑇 ′ = (𝑟 ′ − (𝑔 + 𝑛))b𝑚 (𝑟 ′),(30)

24 As mentioned in the introduction, several other papers have studied the debt Laffer curve in the context of 𝑟 < 𝑔.

See for example Bassetto and Sargent (2020) and Mian, Straub, and Sufi (2022). The first is very related. In particular,
their discussion in their Section 4.1 represents an example of an RPI within an OLG model, although focused on
steady states.

34

Figure 6: The Debt Laffer Curve and RPIs in the Flex Price Equilibrium−𝑇 ′Initial debt
levelswhere
RPI exists

Samuelson’s
point

𝐵

−𝑇 𝑜

0

𝐶

𝐴

𝒃𝑜

𝒃′
𝒃 ′ generates
RPI given 𝒃 𝑜Notes: The horizontal axis represents the new levels of debt, b′, and the vertical axis represents
the negative of the corresponding tax levels, 𝑇 ′. For any initial value of b𝑜 that lies from the origin
to point B, there exists an RPI. Given the b𝑜 plotted, the ranges of b′ from point A to C represent
a corresponding RPI.

where b𝑚 (𝑟 ′), the aggregate savings supply schedule in the economy for arbitrary parameters in
the monetary policy rule, is now given by:






𝜃𝑏★ − 𝜃𝑏 (𝑟 ′ − 𝑟 𝑜 )b𝑚 (𝑟 ′) = b★(𝑟 ′) 1 +𝜃𝑏 (𝑟 ′ − 𝑟 𝑜 ) + 𝜃𝑏★ (𝛾 + 𝜃𝑏 b𝑜 ) 




−1)𝜅
with 𝜃𝑏★ ≡ c★ and 𝛾 ≡ (𝜃 𝜋𝜌c
ˆ ★ .
A comparison of 𝑇 ′ in (30) with 𝑇 ′ under the flex-price equilibrium, (28), reveals an importantdistinction: the shape of the aggregate savings supply schedule is now a function of the monetary
policy rule coefficients, as well as the Phillips curve parameters. When 𝜃𝑏 = 𝜃𝑏★, the monetary rule
implements the flexible price equilibrium, but alternative rules trace out different Laffer curves.
′ (𝑟 𝑜 ) > b′ (𝑟 𝑜 ). In this case,
For any value of 𝜃𝑏 ∈ [0, 𝜃𝑏★), we have that b𝑚 (𝑟 ′) > b★(𝑟 ′) and b𝑚★
the monetary model features a total savings schedule that is more elastic to interest rates than
the flex price model. Thus, the range of debt levels for which debt issuances reduce taxes has
expanded in comparison.
In Figure 7, we compare two distinct debt Laffer curves under two distinct values of 𝜃𝑏 . The
solid line is the same Laffer curve as in Figure 6, which corresponds to 𝜃𝑏 = 𝜃𝑏★. The dashed
𝜇35

Figure 7: The Debt Laffer Curve and RPIs in the Monetary Model−𝑇 ′𝜃𝑏 = 0−𝑇 𝑜𝜃𝑏 = 𝜃𝑏★
0

𝜋↑

𝒃′

𝒃𝑜

Notes: The horizontal axis represents the new levels of debt, b′, and the vertical axis represents
the negative of the corresponding tax levels, 𝑇 ′. The curves represent the levels of transfers in the
new BGP for two different monetary policy rules. The solid curve is the case where 𝜃𝑏 = 𝜃𝑏★ which
replicates the flex-price economy. The dashed curve is the case with 𝜃𝑏 = 0, where the monetary
policy rule facilitates larger transfers, but at the cost of higher inflation and lower profits.

line corresponds to a curve with a monetary policy rule with 𝜃𝑏 = 0. Such a rule increases theelasticity of the aggregate savings schedule at b𝑜 to the interest rate, and thus, the peak of the
Laffer curve shifts to the right.
However, to achieve this, the monetary policy necessarily boosts inflation after the fiscal
expansion, and as a result, the share of output that is allocated to labor increases: 𝑤 ′ > 𝑤 𝑜 and𝜋 ′ > 0. Both of these reduce entrepreneurial profits. Thus, the fiscal policy does not generate an
RPI unless the monetary policy sets 𝜃𝑏 = 𝜃𝑏★.
To summarize, in our model, monetary policy does not create a free lunch, since it inevitably
redistributes resources between workers and entrepreneurs. We did not let the government compensate entrepreneurs, however, through, for example, subsidies for potential profit losses due toa fiscal expansion. And a more accommodative monetary policy rule could generate additional
fiscal revenue to fund such subsidies, and thereby facilitate the existence of an RPI. Allowing
transfers from workers to entrepreneurs makes the model less tractable, and, arguably, less realistic. For this reason, we leave this question for future work.
This subsection extends the insight of Samuelson to a New Keynesian environment.25 The
25 Balasko and Shell (1980) extended Samuelson (1958) to provide precise conditions on the existence of Pareto

improvements in OLG models. Balasko and Shell show that a stationary equilibrium is Pareto efficient if and only if𝑟 ≥ 𝑔 + 𝑛. This subsection extends that result to the standard New Keynesian framework, including the necessary36

crucial lesson learned here is that expanding the stock of safe assets is not sufficient to improvewelfare. In the New Keynesian model, this will have potentially unappealing inflationary anddistributional consequences. However, the pairing of debt issuance with an appropriate monetary rule that responds to the fiscal expansion rescues the traditional insight that increasing thequantity of safe assets may improve the distribution of a fixed amount of income. As noted above,
this insight may have implications for the broader HANK literature.

5

Additional Monetary and Fiscal Policy Interactions

In this section, we use the phase diagram to explore equilibrium dynamics for various additional
scenarios that feature monetary and fiscal interactions. In doing so, we use the phase diagram toclarify and/or reconcile several results in the literature. In all of these scenarios, we set 𝑔 = 𝑛 = 0;
that is, no technological or population growth. We do this for simplicity.

5.1

The Forward Guidance Puzzle

The phase diagram is also a transparent analytical tool to understand the “forward guidance puzzle” of Del Negro, Giannoni, and Patterson (2023) and McKay, Nakamura, and Steinsson (2016).
In a standard representative agent New Keynesian model, Del Negro, Giannoni, and Patterson(2023) showed that an announcement to temporarily reduce real interest rates at some point in
the future had a large effect on consumption and inflation in the announcement period, and all
periods leading up to the interest rate cut. Moreover, pushing the reduction arbitrarily far into
the future does not dampen the economy’s initial response to the announcement.
Using alternative quantitative models, Del Negro, Giannoni, and Patterson (2023), McKay,
Nakamura, and Steinsson (2016), and Kaplan, Moll, and Violante (2018) showed that breaking
Ricardian equivalence mitigates this puzzle. Del Negro, Giannoni, and Patterson (2023) builds on
Blanchard-Yaari, and hence is closest to our framework. The other two papers build on Aiyagari(1994), but draw similar lessons.
As a counterpoint, Farhi and Werning (2019) uses a Blanchard-Yaari model and argues that
incomplete markets (or the presence of new generations in our model) are not sufficient to resolve
the forward guidance puzzle. We use our tractable framework to understand these approaches
and why they differ in their conclusions. In particular, we show how the different conclusions
can be mapped into different assumptions on the underlying fiscal policy.
We follow McKay, Nakamura, and Steinsson (2016) and consider a situation where the monadjustment of the monetary policy rule. Moreover, these improvements can be made with simple policies that do
not rely on complex changes to the tax schedule.

37

Figure 8: Forward Guidance Puzzle𝜋𝜋𝜋¤ = 0c¤ = 0𝜋¤ = 0

𝑝0
𝑝0

𝑝′
0

c

𝑝

0

𝑝′
c

𝑝
c¤ = 0(a) No Debt Scenario(b) Debt Scenario

etary authority promises to temporarily lower the real rate in the future. Let 𝑟 𝑜 be the stationary
real interest rate 𝑟★(b𝑜 ) that clears the bond market at zero inflation (equation 22). At 𝑡 = 0, themonetary authority announces that at 𝑡 0 > 0 it will reduce the real interest rate below 𝑟 𝑜 for theinterval [𝑡 0, 𝑡 1 ), before resuming its original policy. This is equivalent in our setting to a monetaryrule with 𝜃 𝜋 = 1, 𝜃𝑏 = 0, and an intercept that shifts down for 𝑡 ∈ [𝑡 0, 𝑡 1 ). Specifically, for some
∆ > 0, monetary policy follows:𝑟 𝑜 + 𝜋(𝑡)𝑖(𝑡) =
𝑟 𝑜 − ∆ + 𝜋(𝑡)

for 𝑡 ∈/ [𝑡 0, 𝑡 1 )(31)

for 𝑡 ∈ [𝑡 0, 𝑡 1 ).

After 𝑡 1 , we assume that the economy is back at the zero inflation steady state, which the monetary
authority implements by following the corresponding monetary rule.
We can replicate the “forward guidance puzzle” by considering the case with zero government
bonds. This scenario also corresponds to the representative agent case. From (10), we recover anaggregate Euler Equation that is identical to that of a representative agent economy:c¤ = (𝑟 (𝑡) − 𝜌 + 𝛼)c(𝑡).

(32)

The stationary real interest rate is 𝜌 −𝛼, which for the current experiment will be 𝑟 𝑜 . Panel (a)of Figure 8 depicts the associated phase diagram. The 𝜋¤ = 0 curve remains unchanged from theprevious scenarios. However, for the current experiment, any level of c is stationary for 𝑡 < 𝑡 0 ,38and hence we drop the c¤ = 0 locus. When 𝑡 ∈ [𝑡 0, 𝑡 1 ) and 𝑟 (𝑡) = 𝑟 𝑜 − ∆ < 𝜌 − 𝛼, we have c¤ < 0, forall 𝑐. The consumption dynamics for 𝑡 ∈ [𝑡 0, 𝑡 1 ) are depicted with a dashed arrow.
The equilibrium can be solved backwards from 𝑡 1 . At 𝑡 = 𝑡 1 , the real interest rate returns
to 𝑟 𝑜 = 𝜌 − 𝛼, and the economy must be back at its zero inflation steady state, labeled 𝑝. For𝑡 ∈ [𝑡 0, 𝑡 1 ), we can solve (32) with 𝑟 (𝑡) = 𝑟 𝑜 − ∆ backwards in time from the boundary conditionc(𝑡 1 ) = c★. Associated with this path for c(𝑡) there is a path for 𝜋(𝑡) that satisfies (14) with theboundary condition 𝜋(𝑡 1 ) = 0. As c¤ < 0 over this time interval for any c, we know the saddle
path lies to the right of 𝑝. Moreover, given the dynamics associated with the Phillips curve, thehigher c is associated with a higher 𝜋. Hence, the saddle path lies to the north-east of 𝑝 andbelow the 𝑝¤ = 0 locus. This trajectory is depicted as the path leading from the point labeled 𝑝 ′ tothe zero-inflation steady state 𝑝.
For 𝑡 ∈ [0, 𝑡 0 ), the monetary authority sets 𝑟 (𝑡) = 𝑟 𝑜 = 𝜌 − 𝛼, and c¤ = 0. The economy at𝑡 = 0 thus jumps to a point directly above 𝑝 ′ and follows a vertical trajectory that reaches 𝑝 ′ at𝑡 = 𝑡 0 . This is depicted as point 𝑝 0 . Note that at announcement, the economy jumps to a pointwith higher inflation and higher c. Moreover, the further in the future is 𝑡 0 , keeping the intervalof low rates 𝑡 1 − 𝑡 0 constant, the longer time is spent on the vertical trajectory. As 𝑡 0 → ∞, theeconomy starts closer and closer to the 𝜋¤ = 0 line and spends longer in the high inflation-highconsumption situation. The puzzle has two parts: first, the economy immediately responds to thepromise of a future decline in real rates, no matter how arbitrarily far in the distant future thisdecline occurs; and second, the further in the future the reduction is, the larger the initial effecton inflation and the longer the economy experiences high inflation and high consumption.
We now show why government debt in our non-Ricardian model can resolve/remove thispuzzling feature of the standard model. To do so, we assume the economy starts with a positive
level of government bonds b𝑜 > 0. The monetary policy continues to be given by (31), but now
𝑟 𝑜 = 𝑟★(b𝑜 ), the stationary real interest rate given by (22).
The phase diagram is depicted in Panel (b) of Figure 8. The main difference between the twopanels concern the dynamics of consumption. When b > 0, for each 𝑟 there is a unique c thatkeeps c¤ = 0. The solid vertical line depicts the c¤ = 0 line associated with 𝑟 𝑜 and is relevant for𝑡 ∈ [0, 𝑡 0 ). The dashed vertical line is the case for 𝑟 𝑜 − ∆. From (20), this lies to the right of thesolid line.26 The solid and dashed arrows correspond to the dynamics over the two intervals,
respectively.
To explore the equilibrium response to forward guidance, we again work backwards throughtime. The trajectory for 𝑡 ∈ [𝑡 0, 𝑡 1 ) behaves in a qualitatively similar fashion as in the representative agent case. As c¤ < 0 over the low-interest rate interval in both, we have a similar trajectory
26 Here, we assume that 0 < ∆ < 𝑟 𝑜 −𝜌+𝛼; otherwise there is not a well-defined stationary locus for the low-interest

environment.

39

from high inflation/high consumption back towards the zero-inflation steady state. Thus, during
the low interest rate period, the two economies operate similarly.
However, the anticipatory behavior prior to 𝑡 0 is quite different. In panel (b), consumption
cannot remain constant for 𝑡 ∈ [0, 𝑡 0 ) without a discontinuity in the anticipated path of consumption (which is inconsistent with equilibrium). To avoid a discontinuity, consumption must follow
the “solid line” dynamics, which implies a path that leaves from 𝑝 and heads northeast toward 𝑝 ′.
At the announcement, the economy jumps to a point 𝑝 0 on this trajectory that ensures arrival at
𝑝 ′ at 𝑡 = 𝑡 0 . The further in the future is 𝑡 0 , keeping 𝑡 1 − 𝑡 0 constant, the longer the span spent on
this trajectory, and the closer to the original (and final) steady state 𝑝 the economy starts at 𝑡 = 0.
Hence, there is no longer a “forward guidance puzzle”: The further in the future is the plannedinterest rate cut, the less the initial economy reacts.
The crucial difference between panels (a) and (b) is that in the no debt case, which replicatesthe representative agent scenario, the economy can spend an indefinite amount of time at any
level of consumption, as long as 𝑟 (𝑡) = 𝜌 − 𝛼. Conversely, with debt, the stationary interest ratedepends on b/c, and hence there is a unique c = c★ that is stationary, given b𝑜 . This is a property
shared by the HANK models of McKay, Nakamura, and Steinsson (2016) and Kaplan, Moll, andViolante (2018). Moreover, given the unstable dynamics around the zero-inflation steady state,
the dynamics pick up speed as we move away from 𝑝, and hence long trajectories must start close
to that steady state.
Note that the only difference between panels (a) and (b) is the level of government debt. Del
Negro, Giannoni, and Patterson (2023) appeal to a different intuition for the elimination of the
puzzle. Specifically, they emphasize the finite lives of the agents in the economy, which generates
an additional “discounting term” in their log linearization of the Euler equation around a steadystate with positive aggregate wealth. This extra discounting term disappears with zero aggregate
wealth. Our analysis does not rely on a log-linear approximation to the Euler equation, nor does
it restrict attention to scenarios with strictly positive debt. When debt is zero, even though agents
have finite lives, the model replicates the result of the representative agent. This makes clear that
it is not the finite lives, but the non-Ricardian aspect of the model with positive debt that leads
to the elimination of the forward guidance puzzle.
5.1.1

Alternative Fiscal Policies and Forward Guidance

As noted above, Farhi and Werning (2019) uses a Blanchard-Yaari model to argue that incomplete
markets (or the presence of new generations in our model) are not sufficient to resolve the forward
guidance puzzle. In this subsection, we present a formulation inspired by their analysis. We use
this to show the role that fiscal policy plays in determining the efficacy of forward guidance. We
will illustrate that if fiscal policy is set to increase debt one-for-one with consumption in response40

to the forward guidance announcement, then the puzzle can be restored.
In the analysis associated with Figure 8, we kept the stock of debt fixed. As debt is the only
store of value for workers, this implied aggregate wealth was constant, and the ratio of wealthto consumption varied with c. In the model of Farhi and Werning (2019), the equilibrium ratio
of wealth to output is constant.27 In our environment, this is akin to the ratio b(t)/c(t) being
constant. More precisely, suppose fiscal policy sets b(𝑡)/c(𝑡) = 𝛿 for all 𝑡, where 𝛿 > 0 is thetarget ratio of debt to worker consumption. We now explore how does this alternative fiscal
policy affects the forward guidance puzzle.
The policy associated with 𝛿 can be implemented by ensuring aggregate tax revenues areproportional to c; for example 𝑇 (𝑡) = 𝜏c(𝑡).28 The government budget constraint becomes¤ = (𝑟 (𝑡) − 𝑔 − 𝑛)b(𝑡) − 𝜏c(𝑡).
b(𝑡)

(33)

¤ = c/c¤ we see that for a given target ratio
Together with the Euler equation (17), and setting b/b
of b/c there is a unique tax policy 𝜏 that implements it (regardless of the path of 𝑟 (𝑡)). Specifically,𝜏 = 𝜇𝛿 2 + (𝜌 − 𝑛 − 𝛼)𝛿, where 𝜇 is defined in equation (17).
We can now revisit the forward guidance puzzle. To use the phase diagram, we follow Farhi
and Werning (2019) and assume prices cannot adjust at all; that is, 𝜋(𝑡) = 0 regardless of monetary
policy and the Phillips curve is completely flat. This eliminates 𝜋(𝑡) as a state variable, and we
can rewrite the dynamic system in terms of b(𝑡) and c(𝑡) and analyze the equilibrium using a2-dimensional phase diagram.
The dynamics, and hence the phase diagram, depends on the choice of 𝑟 (𝑡). We first solve fora balanced growth path along which b = 𝛿c. Setting c¤ = 0 in equation (17) and b/c = 𝛿, we seethat c is stationary as long as𝑟 (𝑡) = 𝑟 ≡ 𝜌 + 𝑔 − 𝛼 + 𝜇𝛿.
Similarly, from (33), and substituing for 𝜏 given 𝛿, we see that b is also stationary as long as𝑟 (𝑡) = 𝑟 . We depict this case in Panel (a) of Figure 9. The stationary points lie on a ray from theorigin with slope 𝛿, the targeted value of b/c. For a given c, moving above this locus puts thegovernment in the situation of 𝜏c < 𝑟 b, and hence b¤ > 0. The reverse is true below the stationaryline. From the Euler equation, an decrease in c relative to the c¤ = 0 line decreases c¤ < 0. This
gives the dynamics depicted by the arrows to the northwest of the c¤ = b¤ = 0. The dynamics tothe southeast are reversed.
The key element induced by this fiscal policy is that any level of consumption is consistent
27 In Farhi and Werning (2019), the constant wealth to income ratio is obtained in an equilibrium in which savers

hold an asset whose dividends are proportional to output.
28 We maintain our assumption that taxes are lump-sum. This avoids introducing a distortion in labor supply. But
we could also assume that the government sets a proportional tax on labor income.

41

Figure 9: Forward Guidance Puzzle: An Alternative Fiscal Policyb¤ = 0bbc¤ = 0b¤ = 0𝑝′b′c¤ = 0b𝑜𝑝c★

c(a) 𝑟 = 𝑟c′c(b) Dynamics with Forward Guidancewith a steady state when 𝑟 (𝑡) = 𝑟 . That is, fiscal policy adjusts to ensure that any c can besustained indefinitely. This indeterminacy of steady-state consumption is exactly the key featureof the no-debt economy in Figure 8 Panel (a) that generated the forward guidance puzzle. It will
be no surprise, therefore, that we can replicate the puzzle in the current environment under thisalternative fiscal policy, which we do next.
Suppose the economy starts at a BGP on which c = c★ and 𝑟 = 𝑟 . As before, the monetary
policy is announced at 𝑡 = 0: the real interest rate will be reduced at 𝑡 0 to 𝑟 − ∆ for 𝑡 ∈ [𝑡 0, 𝑡 1 ),after which it returns to 𝑟 . As before, we assume the economy returns c★ after 𝑡 1 .29
Panel (b) of Figure 9 depicts the phase diagram for 𝑟 (𝑡) < 𝑟 . From (33), the b¤ = 0 locus becomes:
𝜏c¤b = 𝑟𝜏c−∆ > 𝑟 = 𝛿, where the last equality is by definition of 𝑟 . Thus, the b = 0 locus rotates up
relative to 𝛿. The Euler equation evaluated with 𝑟 < 𝑟 implies that the c¤ = 0 rotates down relative
to the ray with slope 𝛿. Panel plots these stationary lines, along with a dashed ray from the origin
with slope 𝛿 for reference. Note that the dashed ray also contains the equilibrium trajectory forthe economy, due to the fiscal rule keeping b(𝑡) = 𝛿c(𝑡) for all 𝑡.
We can use Panel (b) of Figure 9 to describe the dynamics of the economy. Working backwards
through time, at 𝑡 = 𝑡 1 , the economy must be at its stationary point, depicted as (c★, b𝑜 = 𝛿c★).
For 𝑡 ∈ [𝑡 0, 𝑡 1 ), the economy moves along the dashed line in a southwesterly direction. Thus, at
𝑡 = 𝑡 0 , the economy must be at a point with c(𝑡 0 ) > c★, which we label 𝑝 ′. Government debt isaccordingly b′ = 𝛿c′ .
Prior to 𝑡 0 , all points along the dashed ray are stationary points. Thus, to avoid a discontinuity
in consumption at 𝑡 0 , the economy is at 𝑝 ′ for 𝑡 ∈ (0, 𝑡 0 ), as well. This also implies that b(𝑡) jumps to29 This is harder to justify in an environment with a flat Phillips curve, but keeps the analysis simple and keeps us

close to the assumption made in the literature.

42

b′ on announcement.30 As in the representative agent case, there are no consumption dynamics
leading up to 𝑡 = 𝑡 0 . The entire response happens on announcement, regardless of how far in the
future the interest rate decline is.
The dynamics shown in panel (b) are similar to the dynamics that would arise in a representative agent model. That is, c jumps on announcement, remains constant while the interest rate
remains unchanged, then slowly decreases back to 𝑐 ★ starting at 𝑡 = 𝑡 0 (following the Euler equation with a lower real rate). At time 𝑡 = 𝑡 1 the economy is back to its starting point. The horizonof the change in interest rates does not affect the change in consumption at announcement (asin the analysis in Panel (a) of Figure 8). Hence, despite the non-Ricardian agents, the forward
guidance puzzle remains as strong as it was in the corresponding representative agent model.31
An interesting lesson arises: the strength of forward guidance depends on the associated fiscal
policy. In particular, an expansionary fiscal policy makes forward guidance more effective. The
lesson should not be a surprise as, for example, the effects of monetary policy depend on theunderlying assumptions about fiscal policy in HANK models (Kaplan, Moll, and Violante, 2018;
Kaplan, Moll, and Violante, 2016).32
The use of forward guidance as a policy tool arose out of the need to enhance the toolkit
of central banks when facing the zero lower bound constraint on nominal interest rates. The
exercise in this last section highlights that fiscal policy can help make forward guidance more
effective. But also hints that fiscal policy on its own may be sufficient to boost the economy outof a liquidity trap. We proceed to study this case in the next section.

5.2

The Effective Lower BoundIn the previous sections, we have abstracted from the possibility of the zero lower bound on
nominal interest being binding. In this section, we complete the model description by stating
the dynamics of the economy including the ELB, and discussing the monetary and fiscal policy
interactions at the bound.
The phase diagram in Figure 1 depicted equilibrium trajectories when the monetary authority
is unencumbered by the ELB. Here the monetary authority sets 𝑖(𝑡) = 𝜄¯+𝜃 𝜋 𝜋(𝑡), and this will be thecase for 𝜋(𝑡) > −¯𝜄 /𝜃 𝜋 . For 𝜋(𝑡) < −¯𝜄 /𝜃 𝜋 , the monetary rule cannot be followed without runningafoul of the ELB.
30 In Farhi and Werning (2019), the increase in b is effected by a capital gain on financial assets.
31 The stationary interest rate 𝑟 , however, will differ in our economy relative to the representative agent. A higher

target ratio of debt would tend to increase the stationary interest rate in our model, which in turn would dampen
the rotations of the b¤ = 0 and c¤ = 0 at announcement.
32 Farhi and Werning (2019) discuss how having liquidity (in our model, 𝐴(𝑡)) that co-moves with income (in ourmodel, workers labor income) is important in their result. We reinterpret this from the point of view of our modelby noting that the amount of liquidity is a policy choice, driven by fiscal policy.

43

When the ELB binds, the nominal interest rate is zero, and the Fisher relation implies 𝑟 (𝑡) =−𝜋(𝑡). Substituting this into (17), we have
¤ = (−𝜋(𝑡) − 𝜌 − 𝑔 + 𝛼) c(𝑡) − 𝜇b𝑜 .
c(𝑡)

(34)

Hence, for 𝜋(𝑡) < −¯𝜄 /𝜃 𝜋 , the stationary points for c are given by:
𝜋 = −(𝜌 + 𝑔 − 𝛼) − 𝜇

b𝑜
.
c

(35)

When the ELB binds, we have a positive relationship between 𝜋 and c that keeps c(𝑡) constant.
A lower real interest rate is obtained by higher inflation under the ELB, and hence bond-market
clearing requires a higher c for a higher 𝜋 when c¤ = 0. Note that equation (34) does not dependon any aspects of monetary policy but is sensitive to the amount of government debt. We returnto this below when we discuss how to use fiscal policy to escape the ELB.
In Figure 10 we add this additional locus to that of Figure 1. The horizontal line labeled 𝐸𝐿𝐵
demarcates the threshold 𝜋 = −¯𝜄 /𝜃 𝜋 . Note that if b𝑜 = 0, both schedules are independent ofc, which would be the case in a Ricardian environment in which the stationary real interest is
independent of the level of debt and income.
From (34), we see that as 𝜋 increases for a given c relative to the c¤ = 0 locus, c¤ < 0. Thus, thedirection of change for c and 𝜋 are the same between the two c¤ = 0 loci, regardless of whether
we are above or below the ELB threshold. Hence, there is no discontinuity in trajectories at the
ELB threshold.
Figure 10 depicts a situation where there are two stationary points. The first steady state isthat depicted already in Figure 1, at which the ELB does not bind. At the other steady state, the
ELB is binding, and the dynamics around it are saddle path stable. The existence of this steadystate, and the dynamics around it, follows the logic spelled out in Benhabib, Schmitt-Grohé, andUribe (2001).33
5.2.1

Discount Factor Shocks and Avoiding the ELB

We now revisit the transitory decline in workers’ discounting analyzed in Section 4.2. However,
we now assume that the decline in 𝜌 is so severe that the monetary authority is constrained by
the ELB. This will be the case if 𝜌 is sufficiently low such that implementing the flex price real
33 Note that it is possible to have further steady states in the ELB region. If the lower part of the c¤ = 0 curve

had intersected the 𝜋¤ = 0 curve twice, we would have a third steady state, which would be unstable, and if threetimes (which is possible due to the vertical section of the Phillips curve), then there is an additional stable point, aswell. The fact that there could be multiple possible steady states in the ELB region is a consequence of the failure ofRicardian equivalence in the model.

44

Figure 10: Phase Diagram𝜋𝜋¤ = 0c¤ = 0

𝜋
c★, 0
0



c

𝜋𝑎

𝐸𝐿𝐵

𝐸𝐿𝐵 𝑏𝑖𝑛𝑑𝑠𝜋𝑏𝜋c¤ = 0Notes: ELB represents the value 𝜋 = − 𝜃𝜄¯𝜋 . For inflation levels below this value, the ELB constraint
is binding; while it is not above. The values of 𝜋𝑎 and 𝜋𝑏 represent the asymptotes of the respective𝜄¯−(𝜌+𝑔−𝛼)c¤ = 0 lines. Their values are 𝜋𝑎 = 𝜃 𝜋 −1 and 𝜋𝑏 = −(𝜌 + 𝑔 − 𝛼). The graph is drawn for the casewhere 𝜋𝑎 > − 𝜃𝜄¯𝜋 > 𝜋𝑏 .

45

Figure 11: Temporary Decline in 𝜌: Binding ELB and Fiscal Policyc¤ = 0𝜋𝜋c¤ = 0𝜋¤ = 0𝜋¤ = 0↑b

𝐸𝐿𝐵𝑡 = 𝑡′
0

0
c 𝐸𝐿𝐵

c

𝑡 =0

(a) ELB Binds(b) Increase in binterest rate at zero inflation requires a negative nominal interest rate. In particular, that 𝜄¯ inequation (27) evaluated at 𝜌(𝑡) = 𝜌 is negative.
This case is depicted in Figure 11 panel (a). The downward and upward sloping c¤ = 0 lines
represent equations (26) and (35), respectively, with 𝜌 set to 𝜌. The “ELB”’ line is the inflation
rate at which the monetary rule with intercept (27) hits the ELB; that is, 𝜋 = −¯𝜄 /𝜃 𝜋 .
The decline in 𝜌 makes the zero-inflation steady state not achievable for 𝑡 ∈ [0, 𝑡 ′), as 𝜋 = 0lies below the ELB point. The dynamics for 𝑡 ∈ [0, 𝑡 ′) are governed by the upward sloping “c¤ = 0”
locus, and the intersection of this curve with the 𝜋¤ = 0 Phillips curve is an unstable steady state(see footnote 33). The trajectory that leads away from this intersection and reaches the zeroinflation steady state at 𝑡 = 𝑡 ′ is the equilibrium.34 This trajectory follows the logic set out in
Werning (2011) in his “no-commitment” policy, and echoes that of Krugman (1998), Eggertsson
and Woodford (2003), and Jung, Teranishi, and Watanabe (2005). There is initially a large decline
in worker consumption and inflation, and the economy slowly recovers anticipating a return to“normal” at 𝑡 = 𝑡 ′.
Eggertsson and Woodford (2003), Jung, Teranishi, and Watanabe (2005), and Werning (2011)
argue that commitment to future low interest rates (after the ELB no longer binds) can improveupon the no-commitment policy. Commitment to this post-𝑡 ′ inflationary policy “pulls up” the34 Here, we ignore the possibility that there are additional deflationary steady states, which may be possible given

the (omitted) vertical portion of the Phillips curve.

46

trajectory: at time 𝑡 ′, the equilibrium has higher inflation and income than the zero-inflation
steady state, and via anticipation, the entire trajectory adjusts. This logic underlies the nowstandard “forward guidance” policy prescription when at the ELB.
As anticipated in Section 4.2, fiscal policy can provide an alternative solution that does not
require the same type of commitment as forward guidance. To see this, suppose that the monetary
policy rule features 𝜃𝑏 = 𝜃𝑏★, and the fiscal policy expands, b′ > b𝑜 . This shifts the upward-sloping
portion of the “c¤ = 0” locus down. For modest increases in b′, the intersection anchoring the
trajectory in Figure 11 moves closer to the zero-inflation steady state. For large enough increases,
the curve shifts enough that the ELB no longer binds, which occurs when the associated real rate
at zero inflation is no longer negative. This case is shown in Figure 11 panel (b): fiscal policy is
sufficiently large to render the ELB irrelevant, and the monetary authority can implement thetarget of c = c★ and 𝜋 = 0. Again in this case, there is a need for monetary and fiscal policy
coordination as after the fiscal expansion, the monetary authority must adjust its interest rate
rule as the underlying permanent real rate at zero inflation has increased.35
Eggertsson, Mehrotra, and Robbins (2019) study an OLG economy with nominal rigidities
where the ELB permanently binds (the economy is in “secular stagnation”). They show that there
are cases where a permanently binding ELB is the only equilibrium outcome, and features consumption permanently depressed with accompanying deflation. They also show that the equilibrium is locally determinate, and that fiscal policy can help move the economy away from thedeflationary state.36 We can replicate their case using our model. For this we need that 𝑟★(b𝑜 ) < 0.
If the monetary authority targeted the zero-inflation steady state by setting 𝜄 = 𝑟★(b𝑜 ), in Figure
11 the economy would be at the intersection of the 𝜋¤ = 0 line and the lower c¤ = 0, and would havepermanent deflation and c < c★. Note that such an outcome is also locally determinate in ourmodel, and will be unique if this happens to be the only intersection. The monetary authority has
the option to avoid the ELB by raising the intercept of its interest rate rule, setting 𝜄¯ > 0 > 𝑟★(b𝑜 ).
This would shift the upper c¤ = 0 locus out, and the ELB threshold down, creating the opportunity
for a positive inflation steady state with c > c★. At this steady state, 𝑖 > 0 and the ELB would
not bind, but inflation and worker consumption are higher than the original target of c = c★ and𝜋 = 0. Fiscal policy provides an alternative path to the zero-inflation steady state. An increase ingovernment debt can increase the long-run real interest rate to the point that the zero-inflation
35 Note that the analysis relies on the non-Ricardian features of our environment. Mertens and Ravn (2014) study

a binding ELB in a representative agent model with nominal rigidities. They analyze how changes in government
spending and in distortionay taxes affect the economy and can help when the ELB binds. Our Figures 10 and 11(a)are our model-equivalent of their Figures 1 and 2, respectively. However, differently from our case, their model is
Ricardian, given the presence of lump sum taxes, and government debt is not a relevant policy variable.
36 Correia et al. (2013), Mian, Straub, and Sufi (2022), and Wolf (2021) also discuss dealing with the ELB using fiscal
policy tools. The last two papers explicitly study the role of debt in a non-Ricardian environment with a binding
ELB.

47

steady state is attainable with 𝑖 > 0. Escaping the ELB with fiscal policy can increase the welfare of workers because the policy increases wages and real rates from below zero. Moreover, it
can also lead to increased entrepreneur welfare if the elimination of the costs from disinflation is
sufficiently large.

6

Conclusion

This paper investigated the interaction of fiscal and monetary policy in a non-Ricardian New Keynesian model. We demonstrated how increasing government bond supply can generate Pareto
improvements, and characterized the mix of fiscal policy and monetary rule shifts needed to engineer them. We developed a phase diagram to characterize the global equilibrium dynamics.
We used the diagram to analyze anticipated deficit effects and revisited how the global dynamics of the “forward guidance puzzle” are sensitive to the knife-edge Ricardian equivalence case.
We also explored using fiscal policy to escape a liquidity trap. We think that the insights fromthis tractable non-Ricardian framework provide lessons for more general heterogeneous agent
models, including quantitative HANK models.

48

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52

A

Proofs

Proof of Lemma 1
Proof. Given the log-log preferences, the inequality constraints will not bind at an optimum at almost all 𝑡, andhence we will ignore them in what follows.
Letting 𝜇 denote the (current value) co-state on assets, the Hamiltonian for the worker’s problem is:
H (𝑠, 𝑡, 𝑐, 𝑛, 𝜇) =ln 𝑐(𝑠, 𝑡) + 𝜓 ln(1 − 𝑙(𝑠, 𝑡)) + 𝜇(𝑠, 𝑡) (𝑤(𝑡)𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡) + (𝑟 (𝑡) + 𝜆)𝑎(𝑠, 𝑡) − 𝑐(𝑠, 𝑡) − 𝑇 (𝑠, 𝑡)) .
The first-order conditions for 𝑐 and 𝑙 are:1
= 𝜇(𝑠, 𝑡)𝑐(𝑠, 𝑡)𝜓= 𝑤(𝑡)𝑧(𝑠, 𝑡)𝜇(𝑠, 𝑡).
1 − 𝑙(𝑠, 𝑡)

(36)
(37)

Eliminating 𝜇 by combining (36) and (37) generates (7).
The evolution of the co-state is given by (suppressing 𝑠 and 𝑡)𝜇¤ = (𝜌 + 𝜆)𝜇 −𝜕H= (𝜌 − 𝑟 )𝜇.
𝜕𝑎

From this and (36), we obtain the familiar Euler Equation (6).
We can integrate the Euler Equation forward to obtain:
Z∞

Z∞
𝑅(𝑡, 𝜏)𝑐(𝑠, 𝜏)𝑑𝜏 = 𝑐(𝑠, 𝑡)Z𝑡∞𝑡= 𝑐(𝑠, 𝑡)R𝜏

𝑅(𝑡, 𝜏)𝑒 𝑡 (𝑟 (𝑚)−𝜌)𝑑𝑚𝑑𝜏
𝑒 −(𝜌+𝜆)(𝜏 −𝑡 )𝑑𝜏 =𝑡𝑐(𝑠, 𝑡)
.
𝜌 +𝜆

(38)

Substituting this into the budget set (4), we obtain the “consumption function” that relates consumption at time
𝑡 to financial assets and “human wealth,” net of taxes, (8).
□Proof of Lemma 2
Proof. The static labor-consumption condition (7) can be integrated across cohorts to obtain (11). Aggregating
the consumption function (8) gives (12).
Taking the time derivative of the aggregate consumption definition we haveZ𝑡

Z𝑡¤ 𝑡)𝑑𝑠𝑐(𝑠,¤ 𝑡)𝜙(𝑠, 𝑡)𝑑𝑠 +𝑐(𝑠, 𝑡)𝜙(𝑠,−∞Z𝑡Z𝑡= 𝑐(𝑡, 𝑡)(𝜆 + 𝑛)𝑒 𝑛𝑡 +(𝑟 (𝑡) − 𝜌)𝑐(𝑠, 𝑡)𝜙(𝑠, 𝑡)𝑑𝑠 − 𝜆𝑐(𝑠, 𝑡)𝜙(𝑠, 𝑡)𝑑𝑠

𝐶¤ 𝑤 (𝑡) = 𝑐(𝑡, 𝑡)𝜙(𝑡, 𝑡) +

−∞

−∞

−∞

= (𝑟 (𝑡) − 𝜌 − 𝜆)𝐶 𝑤 (𝑡) + (𝜆 + 𝑛)𝑐(𝑡, 𝑡)𝑒 𝑛𝑡
where the second line uses the first order condition for household consumption and the evolution of 𝜙.

53

Note that using 𝑎(𝑡, 𝑡) = 0, we have



𝜌 +𝜆
𝑐(𝑡, 𝑡) =(ℎ(𝑡, 𝑡) − T (𝑡, 𝑡)).
1 +𝜓Note as well that using the process for 𝑧, we haveℎ(𝑠, 𝑡) = 𝑒 −𝛼(𝑡 −𝑠)ℎ(𝑡, 𝑡)
and thusZ𝑡𝑒 −𝛼(𝑡 −𝑠)𝜙(𝑠, 𝑡)𝑑𝑠 =𝐻 (𝑡) = ℎ(𝑡, 𝑡)−∞(𝜆 + 𝑛)𝑒 𝑛𝑡ℎ(𝑡, 𝑡)𝛼 +𝜆 +𝑛

Similarly we obtain
T (𝑡) =(𝜆 + 𝑛)𝑒 𝑛𝑡
T (𝑡, 𝑡).
𝛼 +𝜆 +𝑛

Thus,

𝑛𝑡

(𝜆 + 𝑛)𝑒 𝑐(𝑡, 𝑡) = (𝜆 + 𝑛)𝑒

𝑛𝑡


𝜌 +𝜆
(ℎ(𝑡, 𝑡) − T (𝑡, 𝑡))
1 +𝜓

𝜌 +𝜆
(𝛼 + 𝜆 + 𝑛) (𝐻 (𝑡) − T (𝑡))
1 +𝜓
𝜌 +𝜆
(𝛼 + 𝜆 + 𝑛) 𝐴(𝑡).
= (𝛼 + 𝜆 + 𝑛) 𝐶 𝑤 (𝑡) −1 +𝜓=Plugging this back into the aggregate Euler equation, we have:𝐶¤ 𝑤 (𝑡) = (𝑟 (𝑡) − 𝜌 − 𝜆)𝐶 𝑤 (𝑡) + (𝜆 + 𝑛)𝑐(𝑡, 𝑡)𝑒 𝑛𝑡= (𝑟 (𝑡) − 𝜌 + 𝛼 + 𝑛)𝐶 𝑤 (𝑡) −𝜌 +𝜆(𝛼 + 𝜆 + 𝑛) 𝐴(𝑡).
1 +𝜓which delivers equation (10).
Using the definition of aggregate assets, and the first order condition for household labor, we get¤ = 𝑎(𝑡, 𝑡)𝜙(𝑡, 𝑡) +
𝐴(𝑡)

Z𝑡

¤ 𝑡))𝑑𝑠(𝑎(𝑠,¤ 𝑡)𝜙(𝑠, 𝑡) + 𝑎(𝑠, 𝑡)𝜙(𝑠,−∞= 𝑟 (𝑡)𝐴(𝑡) + 𝑤(𝑡)𝑍 (𝑡) − (1 + 𝜓 )𝐶 𝑤 (𝑡) − 𝑇 (𝑡)= (𝑟 (𝑡) − 𝜌 − 𝜆)𝐴𝑡 + 𝑤(𝑡)𝑍 (𝑡) − 𝑇 (𝑡) − (𝜌 + 𝜆)(𝐻 (𝑡) − T (𝑡)),and using (11) delivers (13), and completes the proof.
□

Proof of Lemma 3
Proof. Given the CES aggregator across intermediates, we have demand for a good with price 𝑝 𝑗 = 𝑝 given by𝑦 𝐷 (𝑝, 𝑡) =

 𝑝  −𝜂𝑌,
𝑃

where 𝑃 = 𝑃(𝑡) is the aggregate price index and 𝑌 = 𝑌 (𝑡) is final demand. Facing a real wage of 𝑤(𝑡), this implies54

that real profits at time 𝑡 are
𝑝 𝐷
𝑦 (𝑝, 𝑡) − 𝑤(𝑡)𝑦 𝐷 (𝑝, 𝑡)
𝑃(𝑡)



𝑝 −𝜂
𝑝
− 𝑤(𝑡)
𝑌 (𝑡).
=𝑃(𝑡)𝑃(𝑡)Π(𝑝, 𝑡) =The firm’s Hamiltonian is given by:
H (𝑡, 𝑝, 𝑥, 𝜇) = Π(𝑝, 𝑡) − 𝑓 (𝑥)𝑌 (𝑡) + 𝜇(𝑡)𝑥(𝑡)𝑝(𝑡),¤where we repurpose 𝜇(𝑡) to be the co-state on the price adjustment equation 𝑝(𝑡)= 𝑥(𝑡)𝑝(𝑡). The first-ordercondition with respect to 𝑥(𝑡) is:
H𝑥 = 0 ⇒ 𝑓 ′ (𝑥)𝑌 (𝑡) = 𝜇(𝑡)𝑝(𝑡)¤Imposing symmetry across all firms, we have that 𝑝(𝑡) = 𝑃(𝑡) and 𝑥(𝑡) = 𝑃(𝑡)/𝑃(𝑡)≡ 𝜋(𝑡). Then,𝑓 ′ (𝜋)𝑌 (𝑡) = 𝜇(𝑡)𝑃(𝑡).
The first-order condition with respect to the state 𝑝(𝑡) is:ˆ𝜇(𝑡)¤ = 𝜌𝜇(𝑡)− H𝑝 (𝑡, 𝑝(𝑡), 𝑥(𝑡), 𝜇(𝑡))





𝑝 −𝜂
1
1 𝑝
ˆ
= 𝜌𝜇(𝑡)−𝑌 (𝑡)−𝜂− 𝑤(𝑡) − 𝜇(𝑡)𝑥(𝑡)
𝑃(𝑡)
𝑃(𝑡)
𝑝 𝑃(𝑡)𝑌 (𝑡)ˆ(1 − 𝜂 + 𝜂𝑤(𝑡)) − 𝜇(𝑡)𝜋(𝑡),= 𝜌𝜇(𝑡)−𝑃(𝑡)where the last line imposes symmetry.
Differentiating the condition H𝑥 = 0 with respect to time and substituting we get¤𝜇(𝑡)¤𝑌¤ (𝑡) 𝑃(𝑡)
𝜋(𝑡)
¤
+
=
+
.
𝜋(𝑡) 𝑌 (𝑡) 𝑃(𝑡) 𝜇(𝑡)Solving for 𝜋¤ and using the previous equation for 𝜇,¤ we get:𝜋¤ = (𝜌ˆ − 𝑔𝑌 )𝜋 + 𝜅 𝑤 ★ − 𝑤

if 𝜋 ∈ [𝜋, 𝜋]

For 𝜋 ∈/ [𝜋, 𝜋], we have:

if 𝜋 < 𝜋★if 𝜋 > 𝜋 .

(𝜌ˆ − 𝑔𝑌 ) 𝜋 = 𝜅 𝑤 − 𝑤 ★
(𝜌ˆ − 𝑔𝑌 ) 𝜋 = 𝜅 𝑤 − 𝑤□

55

Proof of Lemma 6
Proof. The only missing element in the derivation of Lemma 6 is to establish that no cohort has a negative
asset position in the original BGP. Given the premise of a stationary equilibrium, we drop time arguments whenpossible. The budget constraint for cohort 𝑠 at time 𝑡 is given by equation (3):𝑎(𝑠,¤ 𝑡) = (𝑟 + 𝜆)𝑎(𝑠, 𝑡) + 𝑤𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡) − 𝑐(𝑠, 𝑡) − 𝑇 (𝑠, 𝑡).
We have from (7),𝑤𝑧(𝑠, 𝑡)𝑙(𝑠, 𝑡) = 𝑤𝑧(𝑠, 𝑡) − 𝜓𝑐(𝑠, 𝑡),
and from (2) 𝑇 (𝑠, 𝑡) = 𝑧(𝑠, 𝑡)𝑇 (𝑡) where 𝑇 (𝑡) = 𝑇 , given that we are in a BGP. Finally, from (8), we have
𝜌 +𝜆(𝑎(𝑠, 𝑡) + ℎ(𝑠, 𝑡) − T (𝑠, 𝑡))𝑐(𝑠, 𝑡) =
1 +𝜓



Z∞
𝜌 +𝜆
=
𝑅(𝑡, 𝜏)𝑧(𝑠, 𝜏)𝑑𝜏 ,𝑎(𝑠, 𝑡) + (𝑤 − 𝑇 )
1 +𝜓
𝑡


where the second line uses the definition of ℎ and T . Also,Z∞
𝑅(𝑡, 𝜏)𝑧(𝑠, 𝜏)𝑑𝜏 =𝑡𝑧(𝑠, 𝑡)𝑟 +𝛼 +𝜆 −𝑔which requires 𝑟 + 𝛼 + 𝜆 − 𝑔 > 0 for bounded human wealth. Substituting the consumption into the 𝑎¤ equation
above, and rearranging, we have:𝑟 − 𝜌 −𝑔 +𝛼𝑎(𝑠,¤ 𝑡) = (𝑟 − 𝜌)𝑎(𝑠, 𝑡) + 𝑧(𝑠, 𝑡)(𝑤 − 𝑇 )𝑟 +𝛼 +𝜆 −𝑔In a BGP with b𝑜 ≥ 0, (20) implies 𝑟 ≥ 𝜌 + 𝑔 − 𝛼. Non-negative consumption and zero assets at birth imply
𝑤 − 𝑇 ≥ 0. Thus, the last term is positive. Starting from zero assets at birth, this indicates assets can never go
below zero, as 𝑎¤ ≥ 0 when 𝑎 = 0, which completes the proof.
□

B

Worker Welfare on the BGP

In this appendix, we derive in closed form the value function of a worker along a BGP.
Lemma 7. On a BGP with real interest rate 𝑟 ∈ [𝜌 − 𝛼, 𝜌 + 𝜆 + 𝑔 + 𝑛) and real wage 𝑤 > 0, a
newborn worker at time 𝑡 has utility
1 +𝜓𝑟 − 𝜌 log 𝑤
𝑔𝑡
𝑈 (𝑟, 𝑤) =log(𝜌 + 𝜆 + 𝑔 + 𝑛 − 𝑟 ) ++++ 𝑢0 ,𝜌 +𝜆
𝜌 + 𝜆 1 +𝜓 1 +𝜓𝑡where 𝑢 0 is a combination of parameters.
Before proving this lemma, note that the proof of Lemma 4 is immediately obtained by setting
𝜕𝑈 𝑡 /𝜕𝑟 = 0 and verifying that 𝜕 2𝑈 𝑡 /𝜕𝑟 2 < 0.

56

Proof. Utility of cohort 𝑠 at time 𝑡 is given by:Z∞
𝑈 (𝑠, 𝑡) =𝑒 −(𝜌+𝜆)(𝜏 −𝑡 ) [log 𝑐(𝑠, 𝜏) + 𝜓 log(1 − 𝑙(𝑠, 𝜏))] 𝑑𝜏 .

𝑡

In a BGP, the Euler Equation implies 𝑐(𝑠, 𝜏) = 𝑐(𝑠, 𝑡)𝑒 (𝑟 −𝜌)(𝜏 −𝑡 ) , and the static first-order condition
1 − 𝑙(𝑠, 𝜏) =𝜓𝑐(𝑠, 𝜏) 𝜓𝑐(𝑠, 𝑡)𝑒 (𝑟 −𝜌)(𝜏 −𝑡 )=.
𝑤𝑧(𝑠, 𝜏)𝑤𝑧(𝑠, 𝜏)Substituting in, we have
𝜌 +𝜆𝑟 − 𝜌 𝜓 log 𝑤− 𝜁 0 (𝑠, 𝑡)
𝑈 (𝑠, 𝑡) = log 𝑐(𝑠, 𝑡) +
−
1 +𝜓
𝜌 +𝜆
1 +𝜓
where
Z∞
𝜁 0 (𝑠, 𝑡) ≡ 𝜓𝑒 −(𝜌+𝜆)(𝜏 −𝑡 ) log 𝑧(𝑠, 𝜏)𝑑𝜏𝑡

𝜓 (𝑔 − 𝛼)𝜓𝛼𝜓 (𝑔 − 𝛼) 𝜓 log 𝑧 0
=
𝑡+
𝑠+
+
.
𝜌 +𝜆
𝜌 +𝜆
(𝜌 + 𝜆)2
𝜌 +𝜆In a BGP,ℎ(𝑠, 𝑡) =𝑧 0𝑤𝑒 𝑔𝑡 −𝛼(𝑡 −𝑠)𝑟 +𝜆 +𝛼 −𝑔

T (𝑠, 𝑡) =𝑧 0𝑇 𝑒 −𝛼(𝑡 −𝑠),𝑟 +𝜆 +𝛼 −𝑔This implies
𝑧 0 (𝑤 − 𝑇 )𝑒 𝑔𝑡 −𝛼(𝑡 −𝑠)(1 + 𝜓 )𝑐(𝑠, 𝑡) = (𝜌 + 𝜆) 𝑎(𝑠, 𝑡) +.
𝑟 +𝜆 +𝛼 −𝑔
Substituting into the expression for 𝑈 (𝑠, 𝑡) we obtain







𝑟 − 𝜌 𝜓 log 𝑤
𝜌 +𝜆
𝜌 +𝜆
𝜌 +𝜆
𝑧 0 (𝑤 − 𝑇 )𝑒 𝑔𝑡 −𝛼(𝑡 −𝑠)
𝑈 (𝑠, 𝑡) = log 𝑎(𝑠, 𝑡) +
+
−
+ log
−
𝜁 0 (𝑠, 𝑡),
1 +𝜓
𝑟 +𝜆 +𝛼 −𝑔
𝜌 +𝜆
1 +𝜓
1 +𝜓
1 +𝜓

Using that 𝑤 = (1 + 𝜓 )c and that 𝑇 = (𝑟 − 𝑔 − 𝑛)b together with (20) we have







𝜌 +𝜆
𝑟 − 𝜌 log 𝑤
𝜌 +𝜆
𝜌 +𝜆
𝑎(𝑠, 𝑡) 𝜌 + 𝜆 + 𝑔 + 𝑛 − 𝑟
𝑈 (𝑠, 𝑡) = log+
𝑧 0𝑒 𝑔𝑡 −𝛼(𝑡 −𝑠) +
+
+ log
−
𝜁 0 (𝑠, 𝑡).
1 +𝜓𝑤(𝛼 + 𝜆 + 𝑛)(𝜌 + 𝜆)
𝜌 + 𝜆 1 +𝜓
1 +𝜓
1 +𝜓

Using the value of 𝜁 0 (𝑠, 𝑡), we obtain after some manipulation:





𝜌 +𝜆
𝑎(𝑠, 𝑡)(𝜌 + 𝜆)(𝛼 + 𝜆 + 𝑛)−𝛼(𝑡 −𝑠)𝑈 (𝑠, 𝑡) = log+ (𝜌 + 𝜆 + 𝑔 + 𝑛 − 𝑟 )𝑧 0𝑒+
1 +𝜓
𝑤𝑧 0𝑒 𝑔𝑡𝑟 − 𝜌 log 𝑤
𝜓
1
+
+
+
𝑔𝑡 +
𝛼(𝑡 − 𝑠) + 𝑢 0
𝜌 + 𝜆 1 +𝜓 1 +𝜓
1 +𝜓

where
𝑢 0 ≡ − log((𝛼 + 𝜆 + 𝑛)(1 + 𝜓 )) −

57

𝜓 𝑔−𝛼
1
+
log 𝑧 01 +𝜓 𝜌 + 𝜆 1 +𝜓

Evaluating at 𝑠 = 𝑡, we obtain:
log 𝑤 𝑟 − 𝜌𝜌 +𝜆1
𝑈 (𝑡, 𝑡) = log (𝜌 + 𝜆 + 𝑔 + 𝑛 − 𝑟 ) +++𝑔𝑡 + 𝑢 0
1 +𝜓
1 +𝜓 𝜌 + 𝜆 1 +𝜓□58