Can Deficits Finance Themselves?*
George-Marios Angeletos†

Chen Lian‡

Christian K. Wolf§

April 20, 2023

Abstract
We study how fiscal deficits are financed in environments with two key features: (i) nominal rigidity and (ii) a violation of Ricardian equivalence due to finite lives or liquidity constraints. In suchenvironments, deficits contribute to their own financing via two channels: a boom in real economic activity, which expands the tax base, and a surge in inflation, which erodes the real value of
nominal government debt. Our main theoretical result relates the potency of such self-financing
to the timing of fiscal adjustment. Pushing the fiscal adjustment further into the future helps generate a larger and more persistent boom, leading to more self-financing. Full self-financing is possible in the limit as fiscal adjustment is delayed more and more: the government can run a deficit
today, refrain from tax hikes or spending cuts in the future, and nevertheless see its debt converge
back to its initial level. We conclude by arguing that a large degree of self-financing is not only
theoretically possible but also quantitatively relevant.

* For helpful comments and suggestions we thank Andrew Atkeson, Mark Aguiar, David Baqaee, Marco Bassetto, Larry
Christiano, Mikhail Golosov, Veronica Guerrieri, Oleg Itskhoki, Guido Lorenzoni, Nobu Kiyotaki, Alisdair McKay, Emi
Nakamura, Matthew Rognlie, Maurice Obstfeld, Mikkel Plagborg-Møller, Dmitriy Sergeyev, Ludwig Straub, Jón Steinsson, Gianluca Violante, Iván Werning, Sarah Zubairy, and seminar participants at Harvard, Princeton, UC Berkeley, UCLA,
IIES, IU Bloomington, the NBER Monetary Economics Meeting, the NBER Summer Institute (EFCE), the Texas Monetary
Conference, and the Federal Reserve Banks of Boston, San Francisco, and Philadelphia. Chen Lian thanks the Alfred P.
Sloan Foundation for financial support. An earlier version of this project was circulated under the title “FTPL Redux.”†
Northwestern and NBER; angeletos@northwestern.edu.
‡
UC Berkeley and NBER; chen_lian@berkeley.edu.
§
MIT and NBER; ckwolf@mit.edu.

1 Introduction
Suppose the government runs a deficit today in order to stimulate aggregate demand. Suppose further
that there is no “free lunch” of the type considered in the recent “r < g ” literature (Blanchard, 2019;
Reis, 2022); i.e., the government’s net cost of borrowing is positive. What does the government haveto do in order to make sure that public debt eventually returns back to its initial level?
The most conventional answer is fiscal adjustment: sooner or later, the government must adopt a
package of tax hikes and/or spending cuts in order to pay down the accumulated debt. In this paper,
we investigate a different margin—what we refer to as the self-financing of fiscal deficits.
The basic idea is simple. Insofar as a deficit triggers a boom, it can contribute to its own financing
via two complementary channels: by expanding the tax base, which helps generate additional tax
revenue without any adjustment in tax rates; and by triggering inflation, which helps reduce the realvalue of the government’s nominal liabilities (i.e., debt erosion). Our contribution is to shed light on
the theoretical properties and the quantitative relevance of this self-financing mechanism.1
We first verify that some self-financing obtains naturally in environments that combine two key
features: a failure of Ricardian equivalence, so that deficits can stimulate aggregate demand; andnominal rigidity, so that aggregate demand can drive real economic activity and thereby also inflation. We next show that, in such environments, the degree of self-financing increases as the fiscal
adjustment is delayed. Intuitively, the further the adjustment is delayed, the larger the boom inducedby the initial deficit, thus raising the potency of both of our two self-financing channels. Pushing thislogic to its limit, we obtain our headline result: as the fiscal adjustment is delayed more and more, thetax hike needed to bring debt back to trend vanishes. In other words, the initial deficit pays for itself.
Our contribution is completed by evaluating the quantitative relevance of this result.
Environment. Our baseline model is kept purposefully close to the textbook New Keynesian model(Galí, 2008; Woodford, 2003b). The supply block is exactly the same and boils down to the New
Keynesian Phillips curve (NKPC). The only change is in the demand block, which now consists ofoverlapping generations of perpetual-youth consumers (OLG à la Blanchard, 1985). Denote the survival probability by ω ∈ (0, 1]. When ω = 1, our model reduces to the standard permanent-income
representative-agent (PIH-RANK) benchmark. When instead ω < 1, our model shares two key properties with quantitative heterogeneous-agent (HANK) models: (i) consumers discount future disposable
income more heavily than in the PIH benchmark; and (ii) they have a larger short-run propensity toconsume (MPC). As will become clear, our lessons derive not from the OLG structure per se, but rather
from these two more general and empirically relevant properties of consumer demand.
1

The tax base and inflation channels bring to mind, respectively, DeLong and Summers (2012) and the Fiscal Theory of
the Price Level (FTPL). We expand on these relations in due course.

1

Fiscal policy is represented by a rule for how tax revenue (and thereby the primary surplus) responds to changes in aggregate income and in public debt. The dependence on aggregate income
captures the tax base channel—as output goes up, so automatically does tax revenue. The dependance on public debt, on the other hand, captures the speed of fiscal adjustment, or equivalently thehorizon at which the government commits to hike taxes as needed to bring debt back to trend.2 Our
analysis will reveal how this policy parameter is a key determinant of the degree of self-financing.
Finally, monetary policy is represented by a rule for how nominal (and thereby real) rates adjust based on the state of the economy. For the bulk of our analysis, we concentrate on the case inwhich the monetary authority keeps the (expected) real rate constant. This case serves as a convenient benchmark and represents a “neutral” monetary policy in the following sense: it neither offsets
the fiscal stimulus by discouraging private spending nor does it increase fiscal space by allowing thegovernment’s real cost of borrowing to fall.
The self-financing result. Starting from steady state, we shock the economy with a one-off, deficitfinanced lump-sum transfer to households. We consider this experiment as a theoretical proxy for thevarious rounds of “stimulus checks” recently seen in the U.S. We then ask: How much of this deficit
shock can be self-financed in equilibrium via the aforementioned tax base and inflation channels? Or,
conversely, how large are the tax hikes needed, sooner or later, to bring public debt back to its trend?
Our first observation is that some self-financing is possible if and only if ω < 1. When ω = 1 (i.e.,
PIH-RANK), Ricardian equivalence holds, and aggregate demand is invariant to both the aforementioned shock and the timing of the subsequent fiscal adjustment. It follows that both self-financing
channels are idle—the date-0 tax cut is fully financed with later tax hikes.3 When instead ω < 1, Ricardian equivalence fails, and the deficit—which now represents a net transfer from future generations to current generations or, less literally, helps some consumers overcome liquidity constraints—
stimulates aggregate demand. With nominal rigidities, this increase in aggregate demand translates
to an increase in real income and, via the NKPC, to an increase in the nominal price level. The two
self-financing channels are thus operative: the real boom leads to higher tax revenues even withoutchanges in tax rates; and the accompanying inflation lowers the real government debt burden.
The main question of the paper is how large such self-financing can be. We let ν denote the overall
degree of self-financing; i.e., the fraction of the initial deficit that is financed via the tax base and inflation channels. The residual, 1 − ν, then equals the discounted present value of the future tax hikes
needed for each dollar of current deficit. Our headline result is that, if ω < 1, then ν converges mono2

In FTPL jargon, the last statement means that fiscal policy is “passive” or “Ricardian,” which in turn highlights that our
self-financing result is distinct from that found in the FTPL. We return to this point shortly.
3
To be precise, these statements are true as long as we select the conventional or “fundamental” solution of the New
Keynesian model. We elaborate on this below, in the paragraph entitled “Relation to FTPL.”

2

tonically towards 1 as fiscal adjustment is pushed further into the future. In other words, delaying the
fiscal adjustment reduces its magnitude, with full self-financing becoming possible in the limit.
The economics of self-financing. The economic intuition for our self-financing result can be understood through analogy with a simple two-period economy. Periods are indexed by t ∈ {0, 1}; prices
are fully rigid, so only the tax base channel will be operative; and demand at t = 0 is myopic (i.e., independent of outcomes at t = 1) and proportional to concurrent disposable income. Echoing our fulldynamic model, the government pays out transfers of size ε > 0 to households at t = 0, and then hikes
taxes at t = 1 as much as necessary to bring total government debt down to its initial level.
By a simple static Keynesian cross logic, date-0 aggregate output is given as
y=

MPC
× ε,
1 − MPC(1 − τ y )where MPC ∈ (0, 1) is the household’s marginal propensity to consume out of disposable income andτ y ∈ (0, 1) is the rate of proportional income taxation. The boom generates additional tax revenue att = 0 of τ y × y (i.e., the tax base channel), and so the required date-1 tax hike is just ε−τ y × y. It follows
that the overall degree of self-financing in this simple economy is given asν≡τy × yε=τ y × MPC
1 − MPC × (1 − τ y ).

We note two important properties of this expression. First, ν is strictly increasing in the MPC. Thisis because a larger MPC maps to both a larger partial equilibrium effect on consumer spending (numerator) and stronger general equilibrium amplification (denominator). Second, ν = 1 if and only if
MPC = 1. Intuitively, in that case, the partial equilibrium response is 1 and the Keynesian multiplier is111−(1−τ y ) = τ y , so every dollar of initial transfer ends up generating exactly one dollar of additional tax

revenue—i.e., full self-financing obtains.
Our fully-fledged dynamic model closely echoes these intuitions if and only if ω < 1. To see theconnection, translate “t = 0” in the above example to an initial time interval in our model that is bothsufficiently long and sufficiently distant from the eventual tax hike (i.e., the “short run”); “t = 1” tothe time of the delayed tax hike (i.e., the “long run”); and “MPC” as the cumulative MPC over the
initial interval. Two properties of our economy then complete the connection. The first property is
discounting: consumers discount any future disposable income, and hence also any future taxes, at ahigher rate than the interest rate faced by the government. This implies that tax hikes in the far-ahead
future have little effect on consumer demand in the short run, echoing the lack of feedback fromdate-1 taxes to date-0 demand in the two-period economy. The second property is front-loading:consumers spend any additional income receipt quickly. This implies that the cumulative MPC overthe short run can be close to 1, and so the general equilibrium feedback plays out faster than in the3

PIH benchmark. Together, these properties guarantee that the limit of our economy as the tax hike inpushed further into the future is akin to letting MPC → 1 in the simple example.
Tax base vs inflation. The above intuition emphasized the tax base channel by assuming perfectly
rigid prices. More generally, the fiscal stimulus partially translates into inflation rather than fully into
real output, but without otherwise changing the monotonicity and limit properties of ν. The only
material change is, naturally, the relative contribution of the two channels: as prices become more
flexible and so the Phillips curve becomes steeper, more self-financing occurs through debt erosion.
Generality and extensions. We illustrate the versatility of the logic behind our self-financing resultwith a number of extensions.
Our first two extensions generalize the demand block of the economy. The first one accommodates a more flexible specification for consumer demand. This allows us to see more clearly how
our self-financing result depends on measurable properties of the aggregate MPCs out of current income, future income, and wealth. Echoing the intuition provided above, the key conditions for selffinancing to occur are discounting and front-loading: consumers need to respond relatively little toexpectations of future taxes, so that the “stimulus checks” policy can indeed stimulate demand; andthey also need to spend their transfer receipts relatively quickly, so that the resulting Keynesian boom
plays out fast. The second extension accommodates investment. Under standard assumptions aboutfirm behavior, the Keynesian cross logic behind our result continues to apply, and our self-financing
result goes through almost unchanged.
The remaining extensions generalize our assumptions about fiscal and monetary policy. We firstallow for the future fiscal adjustment to be distortionary, rather than lump-sum. If so, future taxes appear as additional cost-push shocks in the model’s supply block, while having no effect on the demandside. Switching from lump-sum to distortionary taxes thus affects the split between tax base and inflation self-financing, but otherwise leaves the core logic of our self-financing results unchanged. By
the same token, our self-financing result also readily extends from “stimulus checks” to government
purchases. Lastly, we ask what happens when the monetary authority moves away from our “neutral” benchmark, either accommodating or leaning against the fiscally-led boom. If nominal rates
rise less than inflation, then real rates fall, so households front-load spending further and our mechanism plays out even faster. Conversely, if real rates are increased, then the deficit-driven boom isdelayed, and so is convergence to the self-financing limit. In particular, if the monetary response is
too aggressive, then full self-financing ceases to be possible.
Relation to FTPL. Our self-financing result shares with the Fiscal Theory of the Price Level (FTPL)
the flavor that deficits are financed not through outright fiscal adjustment, but rather through mar-4

ket movements in prices and/or quantities. The micro-foundations and the economic mechanisms
behind our result, however, are substantially different from those behind the FTPL.4
The prevailing formulation of the FTPL assumes a representative, infinitely-lived, and fully rational consumer, similarly to Barro (1974)’s classic on Ricardian equivalence and thus to the ω = 1 case in
our model. The FTPL breaks Ricardian equivalence—i.e., it allows debt and deficits to drive aggregate
demand and thereby output and inflation—by assuming that the Taylor principle is violated, opening the door to multiple self-fulfilling equilibria, and then selecting a particular such equilibrium. In
our environment, this mechanism is never at play: the failure of Ricardian equivalence is groundedon fundamental reasons (finite horizons or liquidity constraints); the fiscal authority is “passive;” andour self-financing result is robust to an “active” monetary authority that satisfies the Taylor principle.5
Quantitative relevance. To gauge whether a meaningful degree of self-financing is plausible, we discipline the theory with appropriate evidence on its crucial elements: (i) the deviation from the PIH
benchmark; (ii) the horizon or speed of fiscal adjustment; and (iii) the degree of nominal rigidity. For(iii), we draw from ample empirical evidence on the slope of the Phillips curve. For (ii), we turn to recent work that tries to measure the speed of fiscal adjustment in practice (Galí, López-Salido and Vallés, 2007; Bianchi and Melosi, 2017; Auclert and Rognlie, 2018). Finally, for (i), we review the available
microeconomic evidence on the response of household consumption to income shocks (Fagereng,
Holm and Natvik, 2021), including at relatively far-out horizons. We then calibrate an extended version of our baseline model—one that adds a margin of hand-to-mouth consumers—to match thisempirical evidence. This extended, yet still tractable and parsimonious model, is not only capableof matching the relevant empirical evidence, but also serves as a close proxy for aggregate consumer
behavior in fully-fledged HANK environments (Auclert, Rognlie and Straub, 2018; Wolf, 2021a).
We then use our calibrated model to quantify the effects of “stimulus checks” (defined, again, as
deficit-financed lump-sum transfers to households). We find that such policies can be largely selffinanced if fiscal adjustment is as slow as documented in prior empirical work. We further show thatthe tax base channel is the dominant force at work, and that a quantitatively large degree of selffinancing is consistent with relatively small inflationary pressures, unless the Phillips curve is muchsteeper than prevailing estimates tend to suggest.
Literature. Our analysis relates and contributes to several strands of literature. First, we offer a different perspective on fiscal space than the “r < g ” literature (Blanchard, 2019; Mehrotra and Sergeyev,
2021; Reis, 2022). Similar to this literature, our results suggest that deficits can be financed without
4

Throughout we are concerned primarily with the modern, New Keynesian version of the FTPL (e.g., Cochrane, 2018,2023), but similar points apply to its original, neoclassical version (Leeper, 1991; Sims, 1994; Woodford, 1995).
5
By the same token, our self-financing result, unlike that of the FTPL, is also robust the kind of global-game perturbations used in Angeletos and Lian (2023) to remove the equilibrium indeterminacy of the New Keynesian model.

5future tax hikes. However, unlike that literature, we do not require the real interest rate on government debt to be lower than the economy’s real growth rate. Instead, we emphasize how fiscal deficits
can contribute to their own financing by triggering a Keynesian boom. Mian, Straub and Sufi (2022)
touch upon both these issues (“r < g ” and deficit-driven Keynesian booms), but emphasize a selffinancing mechanism different from ours: they focus on how additional debt issuance can reduce thereal interest rate on government debt by triggering inflation along the ZLB.6
Second, we offer a different rationale for why deficits can finance themselves than that found in the
FTPL literature (Leeper, 1991; Sims, 1994; Woodford, 1995; Bassetto, 2002; Cochrane, 2005)—one that
relies on the failure of Ricardian equivalence due to finite horizons or liquidity constraints as opposed
to equilibrium selection. In so doing, we avoid that literature’s most controversial assumption—thethreat to violate the government budget unless a particular equilibrium is selected (Kocherlakota andPhelan, 1999; Atkeson, Chari and Kehoe, 2010),7 and show self-financing can instead be reconciledwith the standard equilibrium selection of the New Keynesian model (i.e., with “passive” fiscal policyand “active” monetary policy). We also shift the focus from the inflation/debt erosion channel—thefocal point of the FTPL literature—to the tax base channel, which in our quantitative explorations
actually turns out to be dominant.
Third, we add to the large literature on the effects of fiscal policy within the New Keynesian framework. Woodford (2011) and Christiano, Eichenbaum and Rebelo (2011) are the classic references forfiscal multipliers in the PIH-RANK benchmark, while Galí, López-Salido and Vallés (2007), Kaplan,
Moll and Violante (2018) and Auclert, Rognlie and Straub (2018) study environments in which Ricardian equivalence fails. It is well-understood within this literature that fiscal deficits can generatedemand-led booms, thus also contributing to their own financing. From this perspective, our paper’s
added value is to characterize the determinants of self-financing and to highlight that self-financing
could even be complete.
Finally, our take-home message echoes that of DeLong and Summers (2012). These authors provide the high-level arithmetic on fiscal multipliers necessary to generate self-financing without amicro-founded model. We instead characterize the economic primitives required for self-financing
to be possible, characterize its determinants, and evaluate its quantitative potential.
Outline. Sections 2 and 3 present the model and characterize its equilibrium. Section 4 then develops the self-financing result. Further extensions and the quantitative analysis follow in Sections 5 and6. Section 7 concludes. Additional results are provided in the Appendix and Supplement.
6This is akin to the form of monetary accommodation that obtains in our setting when we relax the assumption ofneutral monetary policy and, instead, let real rates fall in response to a deficit-driven Keynesian boom.
7
We note that Bassetto (2002) and Cochrane (2005) have disputed this interpretation. This debate, however, is outside
the scope of our paper.

6

2 ModelFor our main analysis we consider a perpetual-youth, overlapping-generations version of the textbook New Keynesian model. Similarly to Del Negro, Giannoni and Patterson (2015), Farhi and Werning (2019), and Angeletos and Huo (2021), mortality risk (finite lives) is a convenient proxy for liquidity
frictions: it breaks Ricardian Equivalence and lets fiscal policy—i.e., debt and deficits—affect aggregate demand. As will become clear, this departure from the PIH benchmark is central to our results.
We will show later how the insights obtained from our baseline model extend to more general aggregate demand structures, including those found in the HANK literature.8
Throughout we study log-linearized dynamics in response to a surprise increase in fiscal deficits.
We use uppercase variables to indicate levels; unless indicated otherwise, lowercase variables denote
log-deviations from the economy’s deterministic steady state. Time is discrete, indexed by t ∈ {0, 1, ...}.

2.1 Households
We index households by i = (i 1 , i 2 ), where i 1 ∈ {0, 1, . . . } denotes their age and i 2 ∈ [0, 1] their name. A
household survives from one period to the next with probability ω ∈ (0, 1], so that 1−ω is the mortality
rate. Whenever a household dies, it is replaced by a new household (with the same name i 2 but age
reset to i 1 = 0). Households do not altruistically value the utility of the future households that replacethem. Taking into account the mortality risk, the expected lifetime utility of any (alive) household i inperiod t ∈ {0, 1, . . . } is therefore given by
"
#
∞ ¡
X
¢k £
¤
Et
βω u(C i ,t +k ) − v(L i ,t +k ) ,

(1)

k=0

where C i ,t +k and L i ,t +k denote household i ’s consumption and labor supply in period t + k (conditional on survival), and preferences take the standard form u(C ) ≡ C

1−1/σ

1− σ1

−1

and v(L) = χ L

1
1+ ϕ

1+ ϕ1

.

Households can save and borrow by trading an actuarially fair, risk-free nominal annuity. Conditional on survival, households therefore enjoy a nominal rate of return equal to I t /ω, where I t is
the nominal rate on government bonds. Households furthermore receive labor income and dividend
income, given respectively by Wt L i ,t and Q i ,t (both in real terms). Households also pay taxes. The
real tax payment Ti ,t depends on both the individual’s income and aggregate fiscal conditions: i.e.,
Ti ,t = T (Yi ,t , Z t ), where Yi ,t ≡ Wt L i ,t +Q i ,t is the household’s total real income, Z t captures aggregate
conditions (including outstanding government debt), and T is a function describing tax policy (to be
specified later). Finally, old households are obliged to make contributions to a “social fund” whose8

Also note that, consistent with Woodford (2003b) and Galí (2008), we will consider a “moneyless” economy. There isthus no seignorage revenue and the channel in Sargent and Wallace (1981) will not help finance deficits.

7

proceeds are distributed to the newborn households; the role of this fund is explained momentarily.
All in all, the date-t budget constraint of household i is therefore given as
A i ,t +1 =

¡
¢¢
It ¡
A i ,t + P t · Wt L i ,t +Q i ,t −C i ,t − Ti ,t + S i ,t ,
|
{z
}
ω
|{z}

(2)

Yi ,t

annuitywhere A i ,t denotes i ’s nominal saving at the beginning of date t , P t is the date-t price level, and S i ,t is
the transfer from or contribution to the fund, with S i ,t = S new > 0 for newborns and S i ,t = S old < 0 for
old households (and where (1 − ω)S new + ωS old = 0, ensuring that the fund is balanced).
Compared to Blanchard (1985), the only novelty in our set-up is the social fund. We will set S new =D ss (and therefore S old = − 1−ω
D ss ), where D ss is the real steady-state value of public debt (and so inωequilibrium also the real steady-state value of private wealth). The fund thus ensures that all cohorts,
regardless of age, enjoy the same wealth and hence the same consumption in steady state. This in turnaffords two simplifications. First, it simplifies aggregation when we log-linearize the model around its
steady state, with every cohort equally weighted in aggregate demand. Second, it guarantees that thesteady state of our model is invariant to both ω and the steady-state level of public debt, and hence
is exactly the same as its RANK counterpart. The two models thus differ only in terms of how fiscal
policy influences output gaps, cleanly isolating the mechanism that we are interested in.
It remains to specify how household income is determined. First, we assume that all households
receive identical shares of dividends. Second, we abstract from heterogeneity in labor supply. Specifically, we assume that labor supply is intermediated by labor unions. Those unions bargain on behalfof households, equalizing the (post-tax) real wage and the marginal rate of substitution between consumption and labor supply. Since all households work the same hours, it follows that all households
receive the same labor income. Overall, the resulting household labor supply relation is exactly thesame as in the textbook New Keynesian model.9 Putting the pieces together, we conclude that Yi ,t = Y t
and Ti ,t = T t —in any given period, all households receive the same income and face the same taxes.

2.2 Firms
The production side of the economy is the same as in the textbook New Keynesian model: there is aunit-mass continuum of monopolistically competitive retailers who set prices subject to a standard
Calvo friction, hire labor on a spot market, and pay out all their profits as dividends back to households. Combined with our assumptions on household labor supply, this implies that the supply blockof our economy reduces to the standard New Keynesian Phillips curve (NKPC).
9

To be precise, this statement presumes that the tax distortion is time-invariant, which is indeed the case under our
upcoming specification for taxes. Further details on the labor supply block of our economy are provided in Appendix A.1.

8

2.3 Policy
The government consists of two blocks: a fiscal authority issuing debt and setting taxes, and a monetary authority setting nominal interest rates.
Fiscal policy. We (for now) abstract from government spending and let B t denote the total nominal
public debt outstanding at the beginning of period t . We can then write the nominal flow budget
constraint of the government as follows:1
B t +1 = B t − P t T t ,
It

(3)

R
where T t ≡ Ti ,t d i is the total real tax revenue at t . Letting D t ≡ B t /P t denote the real value of public
debt, Πt +1 ≡ P t +1 /P t the realized inflation between t and t + 1, and R t ≡ I t /Et [Πt +1 ] the (expected)
real rate at t , we can rewrite the government budget in real terms asµ¶Et [Πt +1 ]D t +1 = R t (D t − T t ).
Πt +1
This underscores how an inflation surprise between t and t +1 erodes the real value of the outstanding
nominal debt, thus reducing the tax revenue needed to balance the government budget.
We henceforth log-linearize around a steady state in which inflation is zero, real allocations are
given by their flexible-price counterparts, and the real debt burden is constant at some arbitrary levelD ss . Thanks to the annuities (which offset the mortality risk) and the social fund (which makes surethat all cohorts enjoy identical wealth and consumption in steady state), the steady-state real rateis the same as in the PIH-RANK counterpart: R ss = β1 > 1.10 Steady-state taxes then satisfy T ss =(1 − β)D ss . While we will throughout focus on on the empirically relevant scenario with D ss > 0, we
do wish to accommodate D ss = 0, and so we let d t ≡ (D t − D ss ) /Y ss , b t ≡ (B t − B ss ) /Y ss , and t t ≡T t /Y ss —i.e., we measure fiscal variables in terms of absolute deviations (rather than log-deviations)from steady state, scaled by steady-state output. Re-writing (3) in real terms and linearizing, we obtaind t +1 =1
D ss(d t − t t ) + ss r t
β
Y
|
{z
}

+

expected debt burden tomorrow

D ss(Et [πt +1 ] − πt +1 )
ss
{z
}
|Y

(4)

debt erosion due to inflation surprise

where r t ≡ log (R t /R ss ) , πt +1 ≡ log (Πt +1 /Πss ) , and Y ss is the steady-state level of output. The government must satisfy both the above flow constraint (at each t ) as well as the familiar no-Ponzi condition(in the limit as t → ∞). We can thus go back and forth between the infinite sequence of flow budget
constraints and the corresponding integrated intertemporal budget constraint.11
10

Had we allowed for steady-state growth, this would translate to “r > g ”. Thus, and unlike the literature spurred by
Blanchard (2019), we are in an environment where the real cost of government borrowing is positive.
11
Note that this does not necessarily rule out explosive debt: debt could still explode, provided that it does so at a rate
lower that the steady-state real interest. But such an explosion would be at odds, not only with the spirit of our exercise, butalso with the log-linearization of the economy. Throughout the subsequent analysis, we will therefore restrict attention to9It remains to specify a rule for how taxes adjust over time to balance the intertemporal government
budget. This adjustment occurs in two ways. First, there is a time-invariant distortionary tax τ y ∈[0, 1) on household labor and dividend income (and thus on total income). As a result, tax revenue
fluctuates automatically during booms or recessions, without the fiscal authority needing to adjust
the tax code—i.e., the “tax base” effect that will play a significant role in our subsequent analysis.
Second, there is an additional lump-sum component, which includes both the deficit shock as wellas any subsequent fiscal adjustment. The latter captures any tax hikes used, sooner or latter, to makesure that government debt returns to steady state.12
The picture is completed by parameterizing the speed of fiscal adjustment. We consider two suchparameterizations, each serving a distinct purpose.
1. Baseline fiscal rule. Our baseline rule sets total taxes as follows:¡¢
Ti ,t = τ y Yi ,t + T̄ − E t + τd D t − D ss + E t ,(5)where τ y ∈ [0, 1) is the time-invariant tax rate on income, the “intercept” T̄ is set to guarantee
budget balance at steady state, E t is a mean-zero and i.i.d. deficit shock (e.g., issuance of stimulus checks), and τd ∈ (0, 1) is a scalar that parameterizes the speed of fiscal adjustment. Intuitively, fiscal adjustment is arbitrarily fast for τd → 1− and arbitrarily slow for τd → 0+ . Thanks
to our simplifying assumption that all households receive the same income, we can drop the iindex and rewrite (5), after (log-)linearization, as follows:tt =τy y t| {z }

tax base effectwhere εt ≡ E t /Yss

+ τd · (d t + εt ) −|{z}fiscal adjustment

εt
|{z}

,

(6)

deficit shockis the deficit shock (conveniently rescaled), y t ≡ ln(Y t /Y ss ) is the deviation of

output from steady state, and d t is the corresponding deviation of the real value of public debt.
2. Alternative fiscal rule. Our second rule is a time-dependent variant of (5). This rule sets, after(log-)linearization,
τ y y i ,t − ε0t i ,t = t t = τ y y i ,t




d
t

t =0
t ∈ {1, · · · , H − 1} ,

(7)

t ≥H

Note that this rule shuts down any deficit shocks at dates t ≥ 1, thus allowing us to focus cleanly
policies (and equilibria) that guarantee that, following any given shock, public debt converges eventually back to steadystate. Formally, we require that Et [limk→∞ d t +k ] = 0 for all t and all realizations of uncertainty.
12
While our main analysis treats the required tax hikes as non-distortionary, we emphasize that this is only for the sakeof simplicity in exposition. As we demonstrate in Section 5.4, allowing the tax hikes to be distortionary has minimal impact
on our self-financing results.

10

on impulse responses to a date-0 shock.13 The interpretation of the rule is as follows: following
the date-0 shock, the government abstains from any fiscal adjustment for the first H periods,
before then adjusting taxes from date H onwards to return government debt to steady state. We
can thus identify H as the lag between the initial deficit and the future adjustment.
Intuitively, we can capture a longer delay in fiscal adjustment either through a low τd (first rule) or
through a high H (second rule). This suggests that the two rules are interchangeable for our main
purposes—an intuition that turns out to be correct and that we will make precise in due course.
This high-level interchangeability notwithstanding, we like to study both fiscal rules, as each serves
different auxiliary purposes. On the one hand, the baseline rule (6) facilitates a tractable, recursive
characterization of the equilibrium (which we use for our main theoretical results), as well as a mapping between the theory and some relevant empirical work (which we use for our quantitative exercises). Furthermore, (6) allows a sharp comparison to prior theoretical work.14 On the other hand, thealternative rule (7) captures more transparently the timing of fiscal adjustment and thus allows us todevelop a sharper intuition for our limiting self-financing result. It also makes abundantly clear that
our fiscal policy is “Ricardian” or “passive” in the sense of the classical FTPL literature: (7) embeds
a commitment to raise taxes as needed to make sure that debt is back in steady state at t = H (andthereafter), no matter what path the economy has followed up to that point, and no matter what path
it is expected to follow thereafter. We will return to this point in Section 4.5.
Monetary policy. The monetary authority sets the rate of interest on nominal bonds ashi µ Y ¶φt
Πt +1
ss
I t = R Et Πss
,
Y ss

(8)

for some φ ∈ R. This is equivalent to saying that the monetary authority implements the following
relation between the (expected) real interest rate and real output:r t = φy t .

(9)Monetary policy in our model is thus parameterized by the pro-cyclicality of the real interest rate.
Since deficits will be shown to be expansionary in equilibrium (provided that ω < 1, i.e., that Ricardian equivalence fails), φ also parameterizes the comovement between real rates and deficits. We can
13

See Wolf (2021a) for a discussion of the—conceptually simple but notationally involved—mapping from policy rules
in sequence space to their state-space analogues. Also, the disappearance of the term τ y y i ,t in (7) for t ≥ H may suggestthat the tax distortion on labor supply also disappears at t ≥ H , which would complicate our upcoming NKPC by addinga non-stochastic but time-varying cost-push term. We can abstract from this nuisance by setting individual taxes as t i ,t =d t −τ y y t +τ y y i ,t for t ≥ H . Since y i ,t = y t , this immediately reduces to t t = d t for t ≥ H , in line with specification (7) above,
while also guaranteeing that the marginal tax rate faced by the individual household is time-invariant. It follows that our
upcoming NKPC remains the same as we move from our baseline policy rule (6) to the variant rule (7).
14
In particular, (6) nests the fiscal rule found in Leeper (1991). As in that paper, the term τd · (d t + εt ) captures fiscal
adjustment: the variation in taxes induced by the exogenous deficit shock. The key novelty here is the inclusion of theterm τ y y t , which captures the emphasized tax base channel.

11thus interpret φ < 0 as an “accommodative” monetary authority that, in response to a positive deficit
shock, lets real interest rates fall so as to increase fiscal space. Conversely, we can interpret φ > 0
as a “hawkish” or “fiscally conservative” monetary authority that leans against any boom (and any
inflation) triggered by deficits through rate hikes.
For our main analysis, we let monetary policy be “neutral” in the specific sense that φ = 0; thatis, the real rate is kept fixed throughout. This is the same baseline policy as in Woodford (2011) andallows us to cleanly isolate how the interaction of fiscal policy and private spending shapes the scopefor fiscal deficit self-financing. We will relax this restriction in Section 5.3.15

3 Equilibrium
This section lays the groundwork for our self-financing result by characterizing the economy’s equilibrium. We start by reducing the economy to a system of three equations: one for aggregate demand,
one for aggregate supply, and one for the dynamics of public debt. We then characterize the uniquebounded solution to this system. Throughout this section, we employ our baseline fiscal rule (6).
Derivations for the alternative rule (7) are slightly different and relegated to Appendix A.2, though theeconomic essence is identical.

3.1 Aggregate demand
The consumption-savings problem of a household i is to choose sequences of consumption and asset
holdings to maximize (1) subject to (2). Using the simplifying property that all households receive thesame income (and pay the same taxes), we can express the (log-linearized) consumption function ofhousehold i in period t as follows:
Ã
"
#!
"
#
∞ ¡
∞
X
X
¡
¢
¢k ¡
¢
k
c i ,t = 1 − βω ã i ,t + Et
βω y t +k − t t +k
− γEt(βω) r t +k ,
k=0

(10)

k=0

where ã i ,t denotes the household’s real financial wealth (inclusive of social fund payments) and γ ≡ssσβω − (1 − βω)β P ssÃY ss combines the intertemporal substitution and wealth effects of the real interest rate. When ω = 1, (10) reduces to the consumption function of a standard permanent-income
household: future disposable income is discounted at rate β, and the marginal propensity to consume (MPC) from both financial wealth and permanent income is 1 − β. Relative to this benchmark,ω < 1 maps to both more discounting of future disposable income (and hence of future taxes) and a
15

One may also wonder how our specification of monetary policy relates to more standard Taylor rules, and how φmatters for equilibrium determinacy. As we make clear in due course (Section 5.3), the answers to both of these questions
are of little consequence for the lessons of our paper.

12higher MPC. It is by now well understood how these qualitative properties extend to richer, more realistic, HANK-type models (e.g., see Auclert, Rognlie and Straub, 2018; Farhi and Werning, 2019; Wolf,
2021a); as will become clear in due course, our results are driven by these more general qualitative
properties of consumer demand, and not by the specific micro-foundations behind them.
Under our baseline monetary policy, r t = 0 for all t , so the last term in (10) drops out. Aggregating
across households, and using the fact that aggregate private financial wealth equals total government
debt, we reach the following description of aggregate consumption:
"
#
∞ ¡
X
¡
¢
¡
¢
¢k ¡
¢
c t = 1 − βω d t + 1 − βω Et
βω y t +k − t t +k .

(11)

k=0

Next, using (6) to express future taxes as functions of the current public debt and future output, andreplacing c t with y t (market clearing), we arrive at the following representation of aggregate demand:
"
#
∞ ¡
¡
¢X
¢k
y t = F1 · (d t + εt ) + F2 · Et 1 − βωβω y t +k ,
(12)
k=0

where F1 ≡ ((1−ω)τ y1−βω)(1−ω)(1−τd )and F2 = 1 − 1−ω(1−τ ) . (12) is a key equation of this paper. The first term
1−ω(1−τd )dcaptures the direct or partial equilibrium effect of fiscal deficits on aggregate demand. We see that
F1 > 0 (i.e., deficits enter positively in aggregate demand) if and only if ω < 1 (no Ricardian equivalence) and τd < 1 (no immediate financing). Intuitively, τd < 1 means that deficits today are financed
at least in part with taxes in the future; as long as ω < 1, this means that a deficit today translates to a
positive real transfer from future cohorts to current cohorts, thus increasing aggregate demand. The
second term then captures the general equilibrium feedback between aggregate demand and aggregate income—the “intertemporal Keynesian cross.” Note in particular that F2 measures the “slope”
of this Keynesian cross, in the following particular sense: if we raise expectations of future spendingin all periods by 1, then current spending increases by F2 .16
Finally, consider how equation (12) specializes for ω = 1. In this case, F1 = 0, F2 = 1, and (12) col£¤Pklapses to y t = Et (1 − β) ∞βy, which in turn can be rewritten in recursive form as y t = Et [y t +1 ].
t+k
k=0This makes clear two points. First, that our aggregate demand equation is a natural extension of its
RANK-PIH counterpart, namely the Euler equation of a representative, infinitely-lived, financially unconstrained consumer. And second, that debt and taxes enter this equation because and only because
we have departed from PIH consumer behavior.
16

Also note that F2 = 1 when ω = 1 or τ y = 0, but F2 < 1 as soon as ω < 1 and τ y > 0. The combination of finite lives (orliquidity constrains) and proportional taxes thus attenuates the Keynesian feedback. This helps explain why our economy
features a unique bounded equilibrium, as we discuss below and further in Supplement E.1.

13

3.2 Aggregate supply
By design, the aggregate supply side of our model is exactly the same as its familiar RANK counterpart.
In particular, labor supply is given by1
1ℓt = w t − c t .
ϕ
σ

(13)

Together with market clearing (c t = y t ) and technology (y t = ℓt ), this pins down the real wage as
w t = ξy t , where ξ ≡ ϕ1 + σ1 > 0. Firm optimality, on the other hand, pins down the optimal reset price afunction of current and future real marginal costs. Following standard steps, we can then reduce thesupply-side of the economy to the familiar NKPC:
πt = κy t + βEt [πt +1 ] .

(14)

where κ ≥ 0 depends on ξ (the pro-cyclicality of real marginal costs) and θ (the Calvo reset probability). Since there is a one-to-one mapping between κ and θ, and since this mapping is invariant to bothfiscal and monetary policy, we henceforth treat κ as an exogenous parameter and (re)parameterize thedegree of price flexibility by it.

3.3 Law of motion for public debt
The remaining third equilibrium restriction comes from combining the government’s flow budget
constraint with the fiscal rule (6). This yields the following law of motion for public debt:¢ D ss¡d t +1 = β−1 d t + εt − τd · (d t + εt ) − τ y y t − ss (πt +1 − Et [πt +1 ]) .
{z}|
| {z }|Y{z}
fiscal adjustment

with initial condition17
d0 = −

tax base(15)

debt erosion

D ss
π0 .
Y ss

(16)

Finally, recall that the sequence of government flow budget constraints must be complemented with£¤the usual no-Ponzi restriction limk→∞ Et βk d t +k = 0.

3.4 Equilibrium definition and characterization
A standard equilibrium definition combines (i) individual optimality for consumers, (ii) individual
optimality for firms, (iii) market clearing, and (iv) budget balance for the government (together with
the no-Ponzi constraint). The preceding analysis has log-linearized the model and has reduced thefirst three requirements to equations (12) and (14), and the last requirement to equation (15). These17

ss

Note that d 0 = P1ss b 0 − D
Y ss (π0 − E −1 [π0 ]). Since we start in steady state (i.e., b 0 = E −1 [π0 ] = 0), we recover (16).

14equations, like the log-linearization itself, make sense only insofar the economy remains in a neighborhood of the steady state. Accordingly, our notion of equilibrium is as follows.
Definition 1. An equilibrium is a stochastic path {y t , πt , d t }∞t =0 for output, inflation, and the real valueof public debt that is bounded in the sense of Blanchard and Kahn (1980) and that satisfies aggregate
demand (12), aggregate supply (14), and the law of motion for public debt (15), along with the initial£¤condition (16) and the no-Ponzi game condition Et limk→∞ βk d t +k = 0.
We can now state our first main result.
Proposition 1. Suppose that ω < 1 and τ y > 0. There exists a unique (bounded) equilibrium. Along thisequilibrium, real output and real public debt satisfy
y t = χ (d t + εt )Et [d t +1 ] = ρ d (d t + εt ) ,

and

(17)

for some χ > 0 and ρ d ∈ (0, 1). These coefficients solve the following fixed point problem:
χ=

F1
1−βω
1 − F2 1−βωρ
d

and

¡
¢
ρ d = β−1 1 − τd − τ y χ .

(18)κFinally, inflation satisfies πt = 1−βρyt .
dCondition (17) contains two relations. The first relation expresses the equilibrium level of output
as a proportion χ of the private sector’s real financial wealth (which itself equals d t ) and of the fiscal
transfer (the deficit shock εt ). Note that χ > 0—i.e., deficits trigger booms. As emphasized previously,
this is so because of the two key features of our model environment: the failure of Ricardian equivalence, which lets deficits stimulate aggregate demand; and the nominal rigidity, which lets aggregate
demand drive output. The second relation gives the (expected) evolution of the real value of public
debt, with ρ d measuring the (expected) persistence of debt. But since y t is proportional to d t , we seethat ρ d here also measures the expected persistence of the Keynesian boom triggered by deficits.
Condition (18) summarizes the fixed-point relation between χ and ρ d —i.e., the two-way feedback
between aggregate demand and fiscal conditions. On the one hand, as long as τ y > 0, higher aggregate
demand contributes to higher output, higher tax revenue and thereby to lower public debt tomorrow.
This feedback is reflected in the second part of condition (18), which pins down ρ d as a function ofχ and of the two fiscal policy parameters (τd , τ y ). On the other hand, as long as ω < 1, more delay infiscal adjustment, or more persistence in public debt, will translate to a larger effective transfer fromgenerations in the far future to generations in the present and the near future, thus stimulating aggregate demand both directly (the partial equilibrium effect) and indirectly (the general equilibrium
Keynesian cross). This is reflected in the first part of condition (18), which pins down χ as a function ofρ d and of the relevant aggregate-demand parameters (F1 , F2 ,βω). We emphasize that the feedback
15

from deficits to aggregate demand is present only when ω < 1, while the feedback from aggregate
demand to tax revenue and thereby to public debt dynamics is present only when τ y > 0. The textbook model assumes away both feedbacks. We return to this point at the end of Section 4 and also in
Supplement E.1, where we explain the conceptual differences between our analysis and the classical
FTPL. For now, we wish to emphasize that the aforementioned two-way feedback is responsible bothfor the uniqueness of the equilibrium and for our upcoming self-financing results.18
We close this section with an important remark about the role of inflation. While Proposition 1 applies regardless of κ, there is a subtle difference as we move from κ = 0 to κ > 0. When prices are rigid,
d t is predetermined in the beginning of period t , and so the second part of (17) holds state-by-state,
and not just in expectation. When instead prices can move, then d t is no longer predetermined—it
depends on the concurrent innovation in the price level, which itself depends on the concurrent innovation in εt . This is the inflation surprise term in equation (15). Proposition 1 can then be read assaying that, in general, date-0 inflation rescales the initial real debt burden d 0 , but otherwise has nofurther effect on the dynamic shape of the output, tax, and debt impulse responses. In fact, this separation between “scale and dynamics” extends to an arbitrary Phillips curve: the latter’s specificationinfluences the equilibrium mapping of the innovations in εt to innovations in d t + εt , but it does notchange at all the dynamic two-way feedback between real private wealth (also, real government debt)
and real economic activity characterized in Proposition 1. This observation not only explains why our
upcoming self-financing result is robust to the value of κ or the specification of the NKPC, but will also
be important for some of the further extensions that we consider in Section 5. All in all, Proposition 1
and the two-way feedback encapsulated in it hold the key to the rest of our analysis.

4 Self-financing of fiscal deficits
This section presents our headline result on the possibility of self-financing deficits. We first use theintertemporal government budget constraint to provide a quantitative measure of the degree of selffinancing. We then show that complete self-financing is possible in the limit as fiscal adjustment isdelayed further and further (in the sense of τd → 0 or H → ∞), and we explain the economics behind
this result. Finally, we zero in on the roles played by our environment’s three other key parameters (κ,τ y , and ω) and we contrast the economics behind our result to those behind the FTPL.
18

Under the alternative rule (7), instead, uniquenessrequiresone of the following minor modifications: (i) a strength£¤
ening of the notion of boundedness to limk→∞ Et y t +k = 0, which amounts to saying that expectations “at infinity” areanchored to the steady state; or (ii) the reinterpretation of φ = 0 as the limit of φ → 0 from above, which is basically a(limit) Taylor principle. The sole role of either one of these modifications is to remove a class of sunspot equilibria that are
inherited from the standard New Keynesian model. See Appendix A.2 for details.

164.1 Sources of fiscal financing£¤
Iterating the debt relations (15) and (16) forward and using limt →∞ E0 βt d t = 0 (since ρ d ∈ (0, 1)), we
obtain the following present-value restriction on fiscal policy:
Ã
!
∞
∞
X
X
£ ¤
ε 0 = τd ε 0 +τ y β k E0 y k
βk E0 [d k ] +
|{z}
k=0
k=0
deficit
{z
}
|
{z
}
|
fiscal adjustment

D ss(π0 − E−1 [π0 ])
ss
|Y
{z
}

+

self-financing via tax base

.

(19)

self-financing via debt erosion

The left-hand side of (19) is the exogenous shock in the initial deficit, while the right-hand side contains the three ways in which this shock will be financed over time: the first term captures the adjustment in current and future taxes triggered by the shock and any resulting accumulation of public
debt; the second term collects the present value of the extra tax revenue generated by the deficitdriven boom in real economic activity; and the third term gives the erosion in the real debt burdencaused by the associated innovation in inflation. Put differently, the first term captures the conventional notion of fiscal adjustment—the government actively adjusts its primary surplus to stabilize
its debt—while the second and third terms reflect our two sources of self-financing. Finally, we notethat a fourth source of financing—monetary accommodation—emerges if the monetary authority depresses real rates in response to deficits. In our main analysis, the assumption that φ = 0 means that
this channel is not operative.
We can now define the (overall) degree of self-financing as follows:
Definition 2. The degree of self-financing is the fraction of the initial deficit that is financed by an
expansion in the tax base and/or an erosion in the real debt burden:£ ¤ D ssP∞kτβEy k + Y ss π0y0ν ≡ k=0ε0(20)This definition applies regardless of whether fiscal policy obeys the baseline rule (6) or the variant rule(7). Note next that the overall degree of self-financing can be decomposed into its two components:ν ≡ ν y + νp
where
νy ≡

∞
X
£ ¤
1
τy
βk E0 y kε0 k=0andνp ≡1 D ssπ0ε0 Y ss(21)

measure, respectively, the tax-base and debt-erosion components of self-financing. Finally, because£ ¤P
the NKPC implies that π0 = κ ∞βk E0 y k , it is immediate that
k=0
ss

νp =

κD
Y ss
τy

νy .

(22)

This relation suggests that once we understand the two-way feedback between real economic activity
17and fiscal deficits in the rigid-price benchmark (κ = 0), it will be straightforward to extend the analysis
to the general case (κ > 0). In the rest of this section we therefore proceed as follows: we first state ourheadline result for the general case; we next expand on the economics behind it under the special
case of κ = 0 (rigid prices); and we finally discuss the role of letting κ > 0.
Before proceeding to our main results we close with a brief remark on our two fiscal policy specifications (6) and (7). As anticipated in Section 2.3, these two specifications are indeed equivalent waysof thinking about the effects of delays in fiscal adjustment.
Lemma 1. Suppose that ω < 1 and τ y > 0. There exists a strictly decreasing mapping T : N → (0, 1],with limH →∞ T (H ) = 0, such that the degree of self-financing generated by policy (7) is the same as thatgenerated by policy (6) if and only if τd = T (H ).
Importantly, this result establishes that “infinite financing delay” in the sense of H → ∞ under therule (7) maps to τd → 0 under the rule (6), and vice versa.

4.2 The self-financing result
We can now state our main theoretical result on the possibility of self-financing.
Theorem 1. Suppose that ω < 1 and τ y > 0, and let fiscal policy follow either our baseline rule (6) or thevariant rule (7). The equilibrium degree of self-financing, ν, has the following properties:
1. It is increasing in the delay of fiscal adjustment; i.e., ν is decreasing in τd for the fiscal rule (6) andincreasing in H for the fiscal rule (7)).
2. It converges to 1 as fiscal adjustment is delayed further and further; i.e., ν → 1 (from below)
as τd → 0 (from above) or as H → ∞. Furthermore, these two limits induce the same equilib©
ª∞rium paths y t , πt , d t t =0 , and in this common limit self-financing is sufficiently strong to return real government debt to steady state (i.e., limk→∞ Et [d t +k ] → 0 for the baseline rule (6) andlimH →∞ E0 [d H ] → 0 for the variant rule (7)).
Theorem 1 is our core result. Its main implication is that the failure of Ricardian equivalence—here encapsulated in ω < 1—opens the door for fiscal deficits to finance themselves. First, as fiscal
adjustment is delayed, the initial fiscal deficit induces both a larger and more persistent Keynesianboom, thus increasing the share of self-financing through higher tax revenue and a larger date-0 pricelevel jump. Second, the limit as τd → 0 or H → ∞ is one of complete self-financing: the deficit-driven
boom is large and fast enough to cover the cost of the initial fiscal outlay ε0 and to make sure thatpublic debt automatically returns back to steady state, without any fiscal adjustment.
18

B ASELINE FISCAL POLICYA LTERNATIVE FISCAL POLICY

Figure 1: Top panel: impulse responses of output y t , government debt d t , and the self-financing shareν to the deficit shock ε0 as a function of τd . Bottom panel: same as above, but as a function of H .

19

A visual illustration. We provide a visual illustration of Theorem 1 in Figure 1.19 The figure shows theeffects of a deficit shock ε0 under different assumptions about fiscal adjustment. The left and middle
panels in the top half of the figure begin by showing impulse responses of output y t and government
debt d t as a function of the fiscal adjustment parameter τd in our baseline fiscal rule (6). Consistently
with Theorem 1, we see that smaller fiscal response coefficients correspond to larger impact output
booms (i.e., larger χ) and more persistent deviations of output and government debt from steady state(i.e., larger ρ d ). This boom then contributes to financing of the initial deficit ε0 through our two selffinancing channels: a tax base expansion and a jump in the date-0 price level. The top right panelof the figure then reports this degree of self-financing ν—as well as the split into ν y and νp —as afunction of the fiscal adjustment parameter τd . We see that ν is decreasing in the strength of fiscal
adjustment τd , i.e., increasing in the delay of fiscal adjustment. In particular, as τd declines towards
zero, the degree of self-financing converges to one.
The bottom half of Figure 1 provides a different perspective on the same logic, using instead thealternative fiscal rule (7), in which taxes adjust after some (finite) horizon H to perfectly balance thebudget. We see again that ν is increasing in the delay of fiscal adjustment, and in particular again
converges to one as fiscal adjustment is delayed further and further (H → +∞). This not only offers acomplementary interpretation of what “delay” means, but also proves the following important point:
our self-financing result is consistent with fiscal policy being “Ricardian” or “passive” in the strong
sense that it commits on bringing debt back to steady state at t = H + 1 regardless of the path that theeconomy has taken up to that point.

4.3 The economics behind the self-financing result
To understand the economics behind Theorem 1 as transparently as possible, we in this subsection
restrict attention to the special case of fully rigid prices (i.e., we set κ = 0). Based on the discussion at
the end of Section 3, shifting from this rigid-price case to the general sticky-price case will be straightforward, only requiring us to re-scale all impulse responses without changing their shape. We also
focus on the fiscal “H -rule” (7). As anticipated before, this fiscal rule is pedagogically useful because
it makes clear both what we mean by delay in fiscal adjustment and why our results are consistent
with fiscal policy being “passive” or “Ricardian” in the sense of FTPL literature.
The policy experiment studied in the remainder of this section is thus as follows: the fiscal authority pays out a lump-sum transfer to households at date 0 and promises to hike taxes at date H in order19

We only emphasize qualitative features of our results here, so we do not in detail discuss the model parameterization;
it suffices to note that we set ω = 0.75—a meaningful departure from Ricardian equivalence—and κ = 0.1—a rather steep
NKPC, allowing a clean visual illustration of our two sources of self-financing. A more serious quantitative investigation isrelegated to Section 6.

20

to return debt to its steady-state value at date H + 1. The questions of interest are how this policy affects equilibrium outcomes, how large the required tax hike at H turns out to be, and what happens as
we increase H . We address these questions in two steps. First, to build intuition, we analyze a simple
two-period example. Second, we show how the intuition from this simple, essentially static example
sheds light on the workings of our full dynamic economy.
A simple static example. We consider a two-period economy in which the government pays out atransfer ε to households at t = 0, generates automatic tax revenue τ y for every dollar of output, andtaxes households to return debt to trend at t = 1 (as necessary). Prices are fully rigid, so output at t = 0
is fully demand-determined. We assume that consumer demand in period 0 is given as
c = MPC · y disp
where MPC ∈ (0, 1) is the marginal propensity to consume andy disp = (1 − τ y )y + ε
is disposable income. We note that this set-up embeds a myopia assumption: date-0 consumption isinvariant to date-1 outcomes, thus allowing us to characterize the date-0 equilibrium without reference to what happens later. By imposing market clearing (y = c), we immediately see that the date-0
equilibrium level of income is given by
y=

MPC
×ε
1 − (1 − τ y )MPC

(23)

This equation is just the solution of the familiar, static Keynesian cross: MPC is the partial equilibrium
effect of a unit transfer; (1 − τ y )MPC is the slope of the Keynesian cross; and 1−(1−τ1y )MPC is the generalequilibrium multiplier.
Consider now the government’s budget constraint. Since the government hands out the transfer
ε and collects taxes τ y y, the net deficit at the end of date 0 is ε − τ y y. The amount of public debt
inherited at date 1 is thus given bydebt tomorrow = R(ε − τ y y),(24)

where R is the real interest rate between the two periods. Plugging (23) into (24), we conclude thatdebt tomorrow = R(1 − ν)ε
where
ν≡

τy y
ε

=

τ y MPC
1 − (1 − τ y )MPC

(25)

is the degree of self-financing.
Equation (25) reveals two important insights. First, we see that a higher MPC maps both to a larger
21

partial equilibrium effect (numerator) and to a higher general equilibrium multiplier (denominator),and therefore overall to a larger degree of self-financing ν. Second, in the limit as MPC → 1, the partial
equilibrium effect converges to 1, the multiplier converges to τ1y , and ν converges to 1—i.e., there iscomplete self-financing.
Back to the full model. To what extent is the simple, effectively static, example offered above informative about what is actually going on in our full dynamic economy? Note that complete selffinancing (ν → 1) in the static model relies on two key properties: first, that the expected date-1 tax
hike does not affect date-0 spending behavior; and second, that the date-0 transfer as well as all the
additional income it generates are fully spent at date 0 (MPC → 1), thus generating enough tax revenue to stabilize debt before the promised date-1 tax hike. The core intuition is that, as the financing
delay H increases, our dynamic economy starts to emulate those two features of the static example.
To see why this is so, we begin by highlighting two important properties of our economy’s aggregate consumption function. Formal details are provided in Lemma E.1 in the Supplement.
1. Discounting. Consider first how consumption demand at date t ≥ 0 responds to an anticipatedfuture change in disposable income at some future date t + ℓ, with ℓ ≥ 0. We write this response
as Mt ,t +ℓ —the (t , t + ℓ) element of the matrix of intertemporal MPCs studied by Auclert, Rognlieand Straub (2018). It is straightforward to see that β−ℓ Mt ,t +ℓ = (1 − β) in the PIH benchmark(ω = 1). With finite horizons (ω < 1), on the other hand, it can be shown that β−ℓ Mt ,t +ℓ isstrictly decreasing in ℓ, and in particular thatlim β−ℓ Mt ,t +ℓ = 0.

ℓ→∞

In words, as long as ω < 1, an income change of a fixed present value that occurs further and further into the future has a diminishing and eventually vanishing effect on current consumption.
We refer to this property of aggregate demand as “discounting.”
2. Front-loading. Consider next how changes in disposable income at date t ≥ 0 affect consumption demand at some future date-t +ℓ, with ℓ ≥ 0. We write this response as Mt +ℓ,t —the (t + ℓ, t )element of the intertemporal MPC matrix. In the PIH benchmark, we have that Mt +ℓ,t is invariant to ℓ, and in particular that Mt +ℓ,t β−t = (1 − β), reflecting perfect consumption smoothing.
With finite horizons, we instead have that Mt +ℓ,t is decreasing in ℓ, withlim Mt +ℓ,t = 0.

ℓ→∞

In words, consumers tend to spend any income receipt faster than in the PIH benchmark, andany income shock today has a vanishing effect on consumption far in enough in the future. We
refer to this property as “front-loading”.
22Figure 2: Direct, partial equilibrium demand effect (blue) and full general equilibrium output impulse
response (grey) to a date-0 deficit shock with fiscal financing at date H = 70.
These two properties of the aggregate consumption function allow us to connect our full dynamic
economy with the simple two-period example; a visual illustration to accompany the following discussion is furthermore provided in Figure 2.
The first object shown in the figure (in blue) is what we refer to as the “partial equilibrium” effect of the policy. It captures the following thought experiment: Suppose the government pays out
a transfer of size 1 at date 0, but then counterfactually assume that aggregate output and inflation
were to not move in response to that policy. Then, debt would grow at rate R ss = β−1 all the way up
until date H , when the government hikes taxes to stabilize government debt. The blue line gives the
response of consumption demand to this combination of date-0 transfer and date-H tax hike.20 The
discounting and front-loading properties allow us to understand the shape of this partial equilibrium
effect. By “discounting”, the date-0 cohort is essentially unaffected by the future announced tax hike;by “front-loading”, it spends its lump-sum transfer receipt relatively quickly, with the cumulative MPC
approaching 1 faster than in the PIH benchmark. The cohorts born shortly after date-0 are also essentially unaffected by the future tax hike, so overall spending demand is back to trend after around20 periods. It is only around t = 60 that expectations of the future tax hike at H = 70 start to depress
demand. Our dynamic economy thus echoes the static example: the future tax hike does not affectMathematically, the blue line plots M · t PE , where M is the full matrix of intertemporal MPCs and t PE is the “partial
equilibrium” tax-and-transfer vector, which equals −1 at date 0, β−H at date H , and 0 otherwise.
20

23

“short-run” spending behavior, and the “short-run” cumulative MPC approaches 1. The key is to interpret the “short run” as an interval of time that is long enough but also sufficiently distant from thetax hike; in the figure, this corresponds to, roughly, the first 20 quarters.
The second object shown in the figure (in grey) is the full general equilibrium effect of our policy
experiment—the object we formally characterized in Theorem 1. In general equilibrium, since prices
are rigid, the initial increase in consumer demand generates additional income. Importantly, again
by the front-loading property of consumer demand, this income is spent quickly. Since the cumulative short-run MPC is 1, this delivers a front-loaded Keynesian multiplier of size τ1y . Crucially, thisKeynesian boom increases tax revenue (in date-0 present value terms) by τ y × τ1y = 1, thus returning
government debt to trend far before date H (or “the long-run”). As a result, the promised subsequent
tax hike at H —together with its negative effect on spending—vanishes, yet again echoing what happens as MPC → 1 in the simple example above.
While the above discussion focused on the limit as H → ∞, the underlying intuition also readily
connects with the monotonicity of ν in H . As H is increased, the effect of the anticipated tax hikeon short-run demand decreases, thereby strengthening the initial partial equilibrium demand boom.
Similarly, the larger H , the longer the general equilibrium feedback—i.e., the Keynesian cross—canplay out before being moderated by the future tax hike. The size of the short-run boom thus is increasing in H , and by extension so is the endogenously raised tax revenue. Finally, we note that theexact same logic also explains why the equilibrium boom becomes larger and more persistent forsmaller τd under our baseline fiscal rule (6). Either way, as fiscal adjustment is delayed further andfurther, the short-run Keynesian boom on its own becomes big enough to stabilize debt.
A comment on generality. We emphasize that the intuitions offered above transcend the particular
OLG economy that underlies Theorem 1 and our graphical illustration in Figure 1. We will later makethis claim precise in Section 5.1, where we generalize our self-financing result to much richer aggregate demand structures. This generalization will identify sufficient conditions that—in line with theintuition offered above—ensure (i) deficits inducing in partial equilibrium a positive and front-loaded
response in aggregate demand, and (ii) this translating in general equilibrium to a boom in real economic activity that is both large enough and fast enough to raise the required tax revenue prior to thepromised future tax hike.

4.4 The roles of κ, τ y , and ω
So far we have focused on how ν depends on the delay in fiscal adjustment (τd or H ). We now study
how ν changes with our model’s three other key parameters: the degree of price flexibility (κ), thefeedback from aggregate income to tax revenue (τ y ), and the proximity to PIH behavior (ω).
24

The degree of price flexibility. The intuition offered in the previous subsection considered the special case of κ = 0 (rigid prices), but extends with little change to κ > 0. In this general case, the Keynesian boom brings with it inflation, reducing the real value of public debt. Relative to the rigid-pricecase, this only increases the potency of self-financing. Furthermore, since the innovation in inflation
is—via the NKPC—proportional to the innovation in the present discounted value of aggregate income, we can readily compute the relative importance of the two sources of self-financing, as shownin equation (22). Putting everything together, we have the following result.
ss

Proposition 2. Let ω < 1, τ y > 0, κ > 0 and D> 0, and consider either of our two fiscal policies.
Y ss
1. The overall degree of self-financing, ν, increases with the degree of price flexibility κ, and thesssteady-state debt-to-GDP ratio, D
.
Y ss
ss

2. The contribution of the inflation/debt erosion channel, νp /ν, is increasing in κ and D
Y ss .
The more flexible are prices, the larger the share of self-financing that comes via prices rather thanquantities. In particular, as κ → ∞, all self-financing comes through the date-0 jump in prices.
The strength of the tax base channel. Let us now zero in on the role of τ y , starting with the case of
rigid prices (κ = 0) and focusing on the tax base channel. In principle, τ y has two offsetting effects on
ν: first, mechanically, a larger τ y means more tax revenue for any given boom; second, by weakening
the intertemporal Keynesian cross (see (12) and (18)), a larger τ y decreases the size and persistence ofthe boom. The next result verifies that the first effect dominates.
Proposition 3. Let ω < 1, τ y > 0, and κ = 0, and consider either of our two fiscal policies. The overall
degree of self-financing, ν, increases with τ y .
When κ > 0, a third effect enters the picture: the aforementioned dampening of the real boomssalso dampens the inflation response, thus weakening the debt erosion channel. When κ and/or D
Y ss

are large enough, this effect may dominate. Quantitatively, however, this is never the case in the neighborhood of the model parameterization that we will consider in Section 6.
Distance from PIH. Finally, consider the importance of the deviation from permanent-income consumption behavior (i.e., ω < 1). For this purpose, we find it convenient to focus on our main fiscal
policy specification (6)—a specification that allows us to summarize the persistence of government
debt in the single coefficient ρ d .
Proposition 4. Let ω < 1 and τ y > 0 and consider the fiscal policy (6).
1. For any τd > 0, a lower ω raises ν and decreases ρ d ; that is, a larger departure from the permanentincome benchmark yields both larger and faster self-financing.
25

full

2. Let ρ dbe the limit of ρ d , the persistence of government debt and output, as τd → 0.
full
full
ρ d < 1 for any ω < 1 but ρ d → 1 as ω → 1.

Then,

The first part of Proposition 4 is straightforward: for any given finite delay in fiscal adjustment (i.e.,
for any given τd > 0), a larger departure from the PIH benchmark (i.e., a smaller ω) implies a larger
and quicker Keynesian boom, thus increasing ν and decreasing ρ d . The second part then zeroes inon how the magnitude of this departure matters for the limit of an infinite delay in fiscal adjustment(i.e., τd → 0). Provided that ω < 1, we can always achieve complete self-financing by pushing the fiscal
adjustment sufficiently far in the future. However, the closer ω is to 1, the smaller and less front-loaded
the resulting Keynesian boom is, and hence the longer it takes for public debt to return to steady state.
Figure E.1 in the Supplement provides a visual illustration. Consistent with Theorem 1, the selffinancing limit still exists: ν → 1 as τd → 0. However, convergence to this limit is exceedingly slow: νremains very close to 0 for all values of τd other than those in a small neighborhood of 0. This in turnsuggests that the practical relevance of Theorem 1 critically depends on how far real-world consumer
behavior is from the PIH. We tackle this question in Section 6.

4.5 Relation to the FTPL
We now zero in on the difference between our results and the Fiscal Theory of the Price Level (FTPL),
which itself is built upon the PIH benchmark (i.e., ω = 1). In this benchmark, and therefore also in the
FTPL, aggregate demand reduces to"∞X

c t = Et (1 − β)#βk y t +k .

k=0

Ricardian equivalence thus holds in the following partial equilibrium sense: holding constant the
expected path of aggregate income, optimal individual spending is invariant to fiscal policy. However,Ricardian equivalence could still fail in general equilibrium, due to a “self-fulfilling prophesy.” To see
what we mean by this, combine the above demand relation with output market-clearing to obtain:"#∞Xy t = Et (1 − β)βk y t +k ,k=0or equivalently y t = Et [y t +1 ]. It is immediate that this equation admits a continuum of bounded solutions. One solution of special interest is y t = 0 for all t and all histories. This solution, which guarantees that Ricardian equivalence extends from partial equilibrium to general equilibrium, is known
as the fundamental or minimum state variable (MSV) solution; it is the one selected by the Taylor
principle; and it is also the limit of our economy’s unique equilibrium as ω → 1 from below. But when
ω is exactly 1, and when in addition the Taylor principle is violated, uniqueness is lost: consumers
can coordinate on multiple self-fulfilling levels of aggregate spending, and fiscal policy can be used as26

a coordinating device. This is the crux of how the FTPL allows deficits to drive output and inflation,
despite the fact that they do not enter into either aggregate demand or aggregate supply.
The plausibility of this mechanism has long been questioned. Inter alia, Kocherlakota and Phelan (1999) and Buiter (2002) have argued that the FTPL boils down to a threat by the fiscal authority
to “blow up” its budget unless consumers coordinate on a very specific self-fulfilling solution. Morerecently, Angeletos and Lian (2023) have shown that all such solutions can be ruled out by small perturbations of social memory. These points notwithstanding, the controversies surrounding the FTPL
are outside the scope of our paper. Instead, the purpose of the above discussion is to clarify the difference between our paper and the FTPL: our self-financing mechanism derives exclusively from a partial
equilibrium failure of Ricardian equivalence, and not from any kind of self-fulfilling prophesy.21

5 Extensions
This section discusses several extensions of our headline self-financing result, the first two generalizing the model and the last three considering alternative specifications of fiscal and monetary policy. First, we consider a reduced-form aggregate demand relation that nests—but also goes materially beyond—our baseline OLG environment. Second, we discuss an extension featuring investment.
Third, we relax our assumption of a “neutral” monetary policy. Fourth, in Section 5.4 we allow for fiscal adjustment to be distortionary. Finally, we show that our results extend with little change to fiscal
stimulus in the form of government purchases rather than lump-sum transfers.

5.1 A more general aggregate demand structure
Extending our self-financing result to a more general aggregate demand relation allows us to: shedfurther light on the economics of our result; understand better the kind of environments that allow(or do not allow) for meaningful self-financing; and build a bridge to state-of-the-art HANK models.
Generalizing the demand block. Recall that, in our baseline OLG environment, aggregate consumer
demand was given by the following function of current household wealth as well as current and future
income net of taxes:Ã
c t = 1 − βω d t + Et
¡

¢

"

∞ ¡
X

#!
¢k ¡
¢
βω y t +k − t t +k
.

(26)

k=0

This form of the aggregate demand relation embeds economically meaningful restrictions on consumer behavior: the MPC out of current income and wealth is the same (equal to 1 − βω), and the
MPCs out of the discounted presented value of future disposable income decline at a constant rate21

We further elaborate on these ideas and on the characterization of the ω = 1 benchmark in Supplement E.1.

27

ω. We consider a generalized aggregate demand relationship that relaxes these constraints while preserving tractability:c t = Md · d t + M y ·Ã
¡"y t − t t + δ · Et
¢

∞ ¡
X

#!
¢k ¡
¢
βω y t +k − t t +k

.

(27)

k=1

Equation (27) generalizes (26) in three meaningful ways. First, it allows for different MPCs out ofcurrent income and wealth, denoted respectively by M y ∈ (0, 1) and M d ∈ (0, 1), and where we willassume that M d ≤ M y . Second, it disentangles the geometric discounting rate ω from the contemporaneous MPCs (now M y and M d instead of 1 − βω). And third, it additionally allows for constantdiscounting at rate δ ∈ [0, 1]. We will elaborate on the economics of these relaxations and give examples of several familiar models that can be nested in this structure.
A general sufficient condition for self-financing. We now revisit our self-financing result with thegeneralized demand relation (27) replacing our simpler baseline OLG demand block. It turns outthat self-financing obtains under two key restrictions on (27), echoing the “discounting” and “frontloading” properties highlighted previously.
Assumption 1. The aggregate consumption function features positive geometric discounting: ω < 1.
In words, MPCs out of future disposable income relative to MPCs out of current income and wealthdecline strictly faster than the rate of interest β. Consistent with our discussion in Section 4.3, this issufficient to ensure that far-ahead future tax hikes—i.e., tomorrow’s tax hike in the analogy in Section
4.3—have vanishingly small effects on current aggregate demand, similar to the baseline OLG model.
The fiscal deficit shock will thus lead to a demand boom around date 0.
Assumption 2. Intertemporal MPCs are sufficiently front-loaded in the particular sense that
Ã
!
∞ ¡
X
¢
¢k
1−β ¡
1−β
Md +
1 − τy M y 1 + δ
βω
>
.
τy
τy
k=1
For (28) to hold for all τ y ∈ (0, 1], the sufficient and necessary condition is
µ
¶
βω
≥ 1.
M d > 1 − β and M y 1 + δ
1 − βω

(28)(29)(28) is the condition required to ensure that the persistence of government debt ρ d in the limiting
full self-financing equilibrium is strictly less than 1—i.e., that government debt will return to trend
even as the future tax hike becomes vanishingly small. Intuitively, this requires the general equilibrium Keynesian boom to be sufficiently front-loaded, which in turn requires households to spend any
income gains sufficiently quickly. If MPCs out of income and wealth are large enough—in the precise
sense of (29) or (28)—then household spending is indeed sufficiently fast.22
22Specifically, the first condition in (29) (i.e., that M d > 1 − β) corresponds to the second property of the consumption

28

Together, Assumptions 1 and 2 suffice to generalize our self-financing result.
Theorem 2. Suppose that aggregate demand takes the generalized form (27) and that Assumptions 1
and 2 hold. As τd → 0 (from above) or as H → ∞, there exists a unique bounded equilibrium, and it issuch thatν → 1 (from below).
That is, complete self-financing obtains as fiscal adjustment is indefinitely delayed. Moreover, selffinancing is sufficiently strong to return real government debt to steady state (i.e., limk→∞ Et [d t +k ] → 0
for the baseline rule (6) and limH →∞ E0 [d H ] → 0 for the variant rule (7)).
Theorem 2 together with its underlying assumptions exactly echoes our intuitive discussion offered in Section 4.3. First, Assumption 1 guarantees that the future tax hike is discounted, so deficits
will lead to a short-run boom. Second, Assumption 2 ensures that any additional income—both
the initial transfer as well as all higher-order general equilibrium income gains—is spent sufficiently
quickly to deliver a front-loaded boom that raises the required tax revenue before the promised future tax hike becomes necessary. In the remainder of this section we will discuss examples of specificmodels of household demand that fit into the general form (27) and either satisfy or violate our two
key conditions in Assumptions 1 and 2.
What environments are consistent with self-financing? Our generalized aggregate demand block(27) is consistent with many familiar models of household consumption-savings decisions.
1. Permanent-income consumers. The canonical PIH model readily fits into our generalized aggregate demand structure with M d = M y = 1 − β and δ = ω = 1. It is immediate that Assumptions
1 and 2 are violated. First, a deficit today together with (finitely) delayed financing does not
induce a demand boom, simply because future tax hikes are not discounted further. Second,even if there was a general equilibrium boom, it would never be “quick”, as permanent-income
households postpone part of their spending into the infinite future, violating (28).
2. Liquidity constraints. Our baseline OLG model can be interpreted as a reduced-form representation of occasionally binding liquidity constraints (see Farhi and Werning, 2019). This interpretation extends to the present generalization, with the following additional flexibility: since(27) disentangles the MPC out of wealth M d and income M y , the present structure is also consistent with “hybrid” models that add a margin of hand-to-mouth spenders. Such models havereceived attention in recent work because they tend to provide a relatively accurate approximation of aggregate demand in richer HANK models (Auclert, Rognlie and Straub, 2018; Wolf,
function (“front-loaded MPCs”) discussed in Section 4.3—that is, it ensures that limℓ→∞ Mt +ℓ,t = 0. The second conditionin (29) guarantees that the general equilibrium Keynesian cross feedback is in fact front-loaded enough to stabilize debt.

292021a). They are nested herein by letting ω equal the survival rate of the OLG households and by(1−µ)(1−βω)setting M y = µ + (1 − µ)(1 − βω), M d = (1 − βω), and δ = µ+(1−µ)(1−βω) , where µ ∈ (0, 1) is the fraction of hand-to-mouth spenders. It is then straightforward to verify that, as long as ω < 1,bothAssumptions 1 and 2 are satisfied, and hence our self-financing result holds. Our quantitative
analysis in Section 6 will rely on a particular, calibrated version of this hybrid model.
It is instructive to also consider an alternative environment with a mixture of hand-to-mouthand permanent-income consumers—i.e., the “spender-saver model” of Campbell and Mankiw(1989) and Galí, López-Salido and Vallés (2007). This corresponds to the knife-edge case of theprevious hybrid example in which ω = 1. Both Assumptions 1 and 2 then fail. The presence ofa margin of permanent-income savers means that (i) the effect of the future tax hike on date-0
consumption never vanishes, and (ii) the boom is never fast enough to return public debt backto trend. As a result, the date-0 boom is always and invariably offset by a subsequent bust,with the self-financing share ν equal to zero.23 This stark result is a feature rather than a bugof our theory: classical permanent-income savers correspond to an idealistic infinite-horizon,
infinite-liquidity, and infinite-rationality scenario—a limit that can be refuted empirically on
multiple grounds. These include: (i) an infinite long-run elasticity of household asset demandwith respect to interest rates (e.g., Moll, Rachel and Restrepo, 2022); (ii) counterfactually low andbackloaded MPCs (see Section 6.1); and (iii) unrealistically strong responses to news about the
future (e.g., as emphasized in the forward guidance puzzle literature). Finite horizons, wealthin the utility function of savers (Michaillat and Saez, 2021), liquidity constraints (as in HANK),and certain kinds of behavioral frictions (as discussed further below) all break this unrealistic
benchmark and return us to our self-financing result.
3. Informational or behavioral frictions. Our general demand block (27) can also capture the essenceof how aggregate demand is influenced by incomplete information (Angeletos and Lian, 2018),
limited rationality (Farhi and Werning, 2019; Vimercati, Eichenbaum and Guerreiro, 2021; Angeletos and Sastry, 2021), and cognitive discounting (Gabaix, 2020). The common implicationof these frictions is to dampen forward-looking behavior, which translates to a lower ω in (27).
This in turn has two conflicting effects. First, the initial partial equilibrium demand boom is amplified: the future tax hike is discounted even further, and so the initial boom is larger. Second,
the intertemporal Keynesian cross is weakened, as future gains in income do not feed back asstrongly to today’s consumer spending. We can see this in Assumption 2: a smaller ω decreases
the left-hand side of (28), reflecting a Keynesian boom that is less front-loaded. We provide aquantitative illustration of these effects in Section 6.2 and Supplement E.8.2.
23

A visual illustration is provided in Supplement E.5.

30

5.2 Investment
Our second extension illustrates the robustness of our insights to the presence of firm investment. We
sketch the main ideas here and delegate the details to Appendix B.1.
Relative to our baseline model, this extension features a meaningfully generalized supply block,with firms employing both labor and capital, and with capital owned and accumulated by the firms
themselves. Firms continue to distribute all their earnings, now net of investment costs, as dividends
to the households. It follows that, similar to our baseline model, aggregate firm saving (now net ofphysical investment) is zero and all public debt is still owned and accumulated by the households.
For our purposes, the main insight is that equation (12) continues to hold, just now in terms of
consumption c t rather than output y t . Put differently, the Keynesian cross applies to output net ofinvestment. Furthermore, since investment is not taxed, the law of motion for public debt in equation(15) also continues to hold if we replace y t by c t .24 It follows that the equilibrium continues to be
characterized as in Proposition 1, with y t re-interpreted as c t , and so nothing of essence changes.
In particular, the only change has to do with how the dynamics of investment and capital influence
real marginal costs and thereby date-0 inflation. This complication is absent when prices are rigid; inthis case, Theorem 1 remains the same. More generally, the introduction of investment modifies the
inflation/debt erosion channel. However, as emphasized at the end of Section 3, this channel has only
a “scale” effect: it moderates the effective innovation in real public debt (and real household wealth),
without affecting its dynamic propagation, or the nature and shape of the deficit-led Keynesian boom.
Our limiting self-financing result thus again applies without change.25

5.3 More general monetary policy
We now return to our baseline environment, but relax the assumption of a “neutral” monetary policy.
Specifically, we now consider the more general monetary rule (9), restated here for convenience:r t = φy t

(30)

The φ > 0 scenario can be interpreted as a hawkish monetary authority that leans against the deficitled Keynesian boom. The φ < 0 scenario, on the other hand, can help capture two kinds of monetaryaccommodation: the case of a monetary authority that purposefully lets the government’s cost of
24

Note that the households continue to face a proportional tax on the sum of labor and dividend income at rate τ y ; but
since dividend income is now net of investment, this tax is effectively a proportional tax on their consumption.
25
Additional subtleties would arise if we were to relax the assumption that firms do not save or borrow. If they did, thenthis could influence how much of the increase in public debt translates to an increase in household wealth. This mechanism is beyond the scope of our paper. On the other hand, it is straightforward to see that nothing in our arguments hinges
on the abstraction from adjustment costs or variable capacity utilization, so the result presented here is fully consistent
with the production structures considered in quantitative business-cycle models (e.g., Smets and Wouters, 2007).

31

borrowing fall when fiscal space is tight; and that of a monetary authority constrained by the ZLB.26
We continue to identify self-financing with the combination of the tax base and debt erosion effects,27 and we ask how φ ̸= 0 affects their potency. For this we will need to restrict φ ∈ [−1/σ, φ̂), where
φ̂ ≡

τy
ss

βD
Y ss

. These bounds have a simple interpretation: when φ = −1/σ, the monetary accommodationand the deficit-driven boom turn out to be so large that there is immediate self-financing; and whenφ ≥ φ̂, the increase in the interest-rate costs of public debt is so large that the tax-base channel is fully
counteracted. We now have the following generalized self-financing result.
Theorem 3. Consider our OLG-NK environment with ω < 1µ and τ y¶ > 0, and let fiscal and monetary
policy obey, respectively, (6) and (30). Then there exists a φ̄ ∈ 0,

τy

ss

βD
Y ss

such that:

1. If φ < φ̄, complete self-financing is attained as the fiscal adjustment is infinitely delayed, i.e.,ν → 1 and limk→∞ Et [d t +k ] → 0 as τd → 0 (from above).
2. If φ > φ̄, there exists a ν̄ ∈ (0, 1), such that any bounded equilibrium has degree of self-financing νwith ν < ν̄.
The intuition underlying Theorem 3 is straightforward. If φ < 0 (i.e., monetary accommodation),
then, in response to the fiscally-led boom, real rates fall, which not only helps the government budget
but also leads to households front-loading their spending even more. The boom is thus even quicker
and debt is even less persistent—i.e., we obtain a smaller ρ d than Theorem 1. It follows that, with
monetary accommodation, even less of a delay in the promised fiscal adjustment is needed to ensurematerial self-financing. Conversely, if φ > 0 (i.e., the monetary authority leans against the boom),then real rates rise, leading households to postpone their spending, thus delaying the boom and giving larger ρ d . The cutoff φ̄ is exactly the point where the monetary policy-induced delay prevents
self-financing from returning real government debt to steady state, delivering ρ full(φ̄) = 1. For any
dstrictly more aggressive monetary policy (φ > φ̄), complete self-financing is not possible and the requisite fiscal adjustment is bounded away from zero: for any φ > φ̄, there is a threshold τd such that a¡ ¢(bounded) equilibrium exists, and government debt returns to steady state, if and only if τd > τd φ .
In particular, in this case, self-financing is necessarily partial, with ν bounded above by ν̄ < 1.

5.4 Distortionary tax hikes
Our main analysis has treated the fiscal adjustment margin as non-distortionary. We here investigate
what happens if adjustment is instead distortionary. In particular, we now replace our baseline fiscal
26

To understand the latter case, note that, as long as κ > 0, any deficit-led boom comes together with inflation; and aslong as the ZLB binds, such inflation translates one to one to negative real rates, or equivalently to φ < 0.
27
While this makes sure that our notion of self-financing remains the same as we move from φ = 0 to φ ̸= 0, it requires
an adjustment in the formal definition of ν. See Appendix B.2 for details.

32rule (5) with the following alternative:
Ti ,t = τ y,t Yi ,t − E t(31)

where the time-varying distortionary tax rate τ y,t is now given byτ y,t = τ y + τd

D t − D ss
.
Y ss

(32)

After log-linearization, this extended model variant maps into exactly the same aggregate demand relation and law of motion for public debt as before; intuitively, in those equations, what matters is only
how much tax revenue is extracted from the private sector, not whether it is done in a distortionary or
lump-sum way. The only model equation that is affected is the NKPC, where the time-varying distortion in labor supply now manifests itself as a cost-push shock:πt = κ(y t + ζd t ) + βEt [πt +1 ](33)

where ζ is a function of model primitives such that ζ > 0 if and only if τd > 0 (see Appendix B.3).
Intuitively, higher debt maps into higher distortionary taxes, and thereby to a higher labor wedge andhigher real marginal cost for a given level of output.
This modification of the model again has very limited effect on our headline results. First, if κ = 0,then nothing changes, and in particular the entirety of Theorem 1 continues to hold. Second, even if
κ > 0, then nothing of essence changes: as we have already emphasized, our characterization of theequilibrium dynamics of y t and d t is robust to the specification of the NKPC, and thus in particular
to the presence of additional cost-push shocks. Furthermore, in the limit of interest (i.e., as τd → 0),
those cost-push terms in fact vanish, and so the second part of Theorem 1 remains intact even if κ > 0.
The economics of this limit case are particularly transparent: because our original self-financing limit
result guarantees that fiscal adjustment is never needed, it makes no difference whether the fiscal
adjustment was hypothesized to be distortionary or lump sum.

5.5 Government spending
While we have so far considered a policy that hands out lump-sum transfers to households, it isstraightforward to see that our results extend with almost no change to deficit-financed government
purchases. We sketch the main points here and relegate details to Appendix B.4.
The only change relative to our baseline economy is that the government now consumes someamount G t of the final good. We assume that G t is a stochastic, mean-zero spending shock. The
linearized government budget constraint becomes
·
¸
1
D ss
d t +1 =d t − t t + g t + β ss (i t − πt +1 ) ,
β
Y
33

(34)

where g t = YGsst . We next specify taxes as follows:t t = τd · (d t + (1 − τ y )g t ) + τ y y t ,(35)

where the presence of (1 − τ y ) in front of g t ensures that τd = 1 again corresponds to a period-byperiod balanced budget.28 Finally, the aggregate output market-clearing condition is replaced by
yt = ct + g t .

(36)

Relative to our baseline environment, switching to government purchases has two effects. First,
the deficit shock immediately and one-to-one maps into demand, rather than leading to a gradual,
moderately persistent increase in spending (e.g., recall Figure 2). Second, g t translates to a cost-pushshock in the NKPC, with the formula given in Appendix B.4. We have already discussed why the second effect will always be immaterial. The first effect, on the other hand, implies that even immediately
tax-financed government purchases have real effects—the well-known unit fiscal multiplier result ofWoodford (2011), Auclert, Rognlie and Straub (2018) and Hagedorn, Manovskii and Mitman (2019).
Our added insight here is that the fiscal multiplier increases monotonically with the delay in tax financing, and that government spending can pay for itself if tax hikes are pushed far enough.

6 Quantitative analysis
We now ascertain the quantitative relevance of our insights. Specifically, we ask: how important is
the self-financing margin for realistic departures from PIH, delays in fiscal adjustment, and nominal
rigidities? We address this question in two steps. First, we present and calibrate a quantitatively relevant extension of our baseline model. Second, we use the model to compute the self-financing shareν under different, empirically plausible, assumptions on delays of fiscal financing.

6.1 Model and calibration strategy
We consider a variant of our model in Section 2 with the baseline fiscal rule (6), but with one twist:
we add a margin µ ∈ (0, 1) of hand-to-mouth spenders, along the lines discussed in Section 5.1. These
agents receive the same income and pay the same taxes as the perpetual-youth consumers, but only
the latter save and hold government debt. Further details are presented in Appendix C.1.
This hybrid spender-OLG model is the ideal environment for our quantitative analysis, for two
reasons. First, it remains simple enough to fit into the generalized demand block of Section 5.1. As
a result, we can readily invoke Theorem 2 to verify whether complete self-financing obtains in the28

To see this, one can guess and verify that, under the above specification, τd = 1 translates in equilibrium to y t = g t ,t t = g t , and d t = 0 for all t (i.e., the fiscal multiplier is one, the primary surplus is zero, and debt stays in steady state).

34

Parameter

Description

Value

TargetValueµ

Share of HtM

0.111

Average MPC

0.22

ω

OLG survival rate

0.88

Short-run MPC slope

0.88

Tax feedback{0.085, 0.026, 0.004}

Literature, see text

σ

EIS

1

Standard

βDiscount factor

0.998

Annual real rate

0.01

Tax rate

0.3

Average Labor Tax

0.3

Gov’t debt

1.04

Liq. wealth holdings

1.04

NKPC slope

0.01

Standard

Consumer spending

Fiscal adjustment
τdRest of the model

τy
ss

D /Y
κ

ss

Table 1: Hybrid OLG-spender model, calibration.
limit as τd → ∞. Second, and more importantly, it is rich enough to agree with quantitative HANK
models on their implied intertemporal MPCs (Auclert, Rognlie and Straub, 2018; Wolf, 2021a) andto match the relevant empirical evidence. Since the micro-foundations of aggregate demand affectthe degree of self-financing only through those intertemporal MPCs (see Supplement E.2 for the formal discussion), it follows that our calibrated model is a suitable laboratory for evaluating whether
a quantitatively meaningful degree of self-financing can obtain for realistic parameterizations of theintensity of fiscal adjustment, as herein parameterized by τd .29
Calibration. We discuss the model calibration in three steps: evidence on household consumption
behavior to pin down the spender share µ and the OLG coefficient ω; evidence on fiscal adjustment
to pin down τd ; and all other parts of the model.
Consumer spending behavior. Empirical evidence on the household-level consumption response
to income gains suggests two salient features of consumer spending behavior (Fagereng, Holm andNatvik, 2021). First, the average MPC out of income gains is elevated, with a standard quarterly valueof around 0.22. Second, the income gain is spent gradually. Our baseline OLG environment of Section
2 provides a tight joint restriction on the level of the MPC and its dynamics: the level MPC (i.e., entry
M0,0 of the matrix of MPCs) is 1 − βω, while the slope of the spending profile (i.e., the ratio of Mℓ,0 toMℓ−1,0 ) is ω. This model-implied connection between level and slope is, unfortunately, inconsistent
with the data; in particular, relative to the impact MPC, income in the data is subsequently spentmuch more quickly than predicted by the theoretical OLG-implied relationship. The spender-OLG
29

We note that the same is not true for our baseline OLG model; see the first point of the discussion on calibration below.

35hybrid model instead allows us to disentangle the level and the intertemporal pattern of the spending
response to income gains in a way that is consistent with empirical evidence; in particular, we choose
the spender share µ and the OLG coefficient ω to jointly match (i) impact and (ii) short-run (i.e., up
to two years out) spending responses to lump-sum transfer receipt, as estimated by Fagereng, Holmand Natvik. A visual illustration of the implied intertemporal MPCs is provided in the top left panel of
Figure 4, and a discussion of several alternative calibration strategies will follow in Section 6.2.
Current and future MPCs out of today’s income gains—i.e., the estimand of Fagereng, Holm andNatvik—is of course not all that matters for our theory; for our general equilibrium Keynesian cross,
it is similarly important how anticipated income changes in the (far) future affect spending today(i.e., entries Mt ,t +ℓ for large ℓ in the matrix of MPCs). Given the lack of empirical evidence on such
responses to (far-away) income news shocks, our baseline exercise simply extrapolates these spending responses through the structure of the model. Reassuringly, as discussed in Wolf (2021a), ourspender-OLG model extrapolates in a way very similar to quantitative HANK models. Nevertheless,we later also consider how results change with additional discounting due to limited knowledge orlimited rationality, consistent with suggestive evidence from Ganong and Noel (2019).
Delay in fiscal adjustment. We henceforth focus on fiscal rule (6), in line with a large literature in
the footsteps of Leeper (1991). Because prior work has estimated this type of rule, we can draw from it
to discipline the calibration of τd . In particular, we consider a range of values taken from Galí, LópezSalido and Vallés (2007), Bianchi and Melosi (2017), and Auclert and Rognlie (2018), displayed in the
middle part of Table 1. A detailed literature review is provided in Auclert, Rognlie and Straub (2020).30
Rest of the model. The remaining model parameters are set to standard values. First, we set σ = 1,
giving log preferences. Second, we set the discount factor to hit a steady-state real rate of interestof one per cent. Third, we set τ y = 0.3, in line with the discussion in DeLong and Summers (2012):
for every dollar of additional output, we assume that the primary surplus automatically rises by 30
cents. Fourth, we set the slope of the NKPC to 0.01, a value consistent with the related empiricaland quantitative literatures (e.g., see Galí and Gertler, 1999; Christiano, Eichenbaum and Evans, 2005;
Hazell et al., 2022; Barnichon and Mesters, 2020). In one of our robustness exercises, we will also
consider a slope of 0.1—an ad-hoc, materially larger value that could be relevant in the inflationary
post-covid environment.
30

Specifically, our three values for τd correspond to their lower bound, preferred estimate, and upper bound for the
fiscal adjustment coefficient (ψ in their notation). We note that all three values correspond to “passive” fiscal rules in theterminology of Leeper (1991). See Supplement E.7 for further details.

36

Figure 3: Impulse responses of output y t , inflation πt , and the total self-financing share ν to the deficit
shock ε0 as a function of τd . The left and middle panels show the impulse responses for the threevalues of τd discussed in Section 6.1. In the right panel these three points are marked with circles.

6.2 Are stimulus checks plausibly self-financed?
It is straightforward to verify that our calibrated spender-OLG model satisfies the sufficient conditions
identified in Theorem 2 required for complete self-financing to be possible in the limit as τd → ∞.
What remains is to ascertain how close we are to this limit for the range of τd suggested by the relevant
empirical literature. Results are displayed in Figure 3.
The figure shows output (left panel) and inflation (middle panel) impulse responses to a date-0
deficit shock for our three fiscal adjustment coefficients τd (shades of grey); the right panel furthermore shows the degree of self-financing ν as a function of τd over the entire unit interval. As expected,
the share of self-financing ν is decreasing in τd and approaches one as τd → 0, consistent with Theorem 2. For the purposes of the quantitative analysis here, the key takeaway is that we are already quite
close to this limit for the values of τd estimated in the aforementioned works, with ν up to 0.95. Furthermore, given our assumed—empirically disciplined—flat NKPC, almost all of this self-financing is
coming through the tax base channel rather than through inflation.
The remainder of this section discusses several robustness exercises and extensions.
Assumptions on consumer behavior. As remarked in Section 6.1, our calibration strategy relied on
two standard but potentially material assumptions: first, we disciplined the slope of the spending profile (i.e., entries Mℓ,0 ) from evidence on relatively short-run spending behavior (small ℓ); and second,
we relied on model structure to extrapolate from responses to contemporaneous income changes tospending behavior following income news shocks (i.e., entries Mt ,t +ℓ ). We now investigate how our37

conclusions change with alternative calibration strategies and modeling assumptions.
Spending responses in the tails. As we emphasized throughout Sections 4 and 5.1, sufficiently
front-loaded intertemporal MPCs are important for our self-financing results—they are what ensuresa front-loaded Keynesian boom. We thus now consider alternative calibration strategies that try tomore directly leverage information on spending responses to lump-sum transfer receipt in the tails(i.e., Mℓ,0 for large ℓ). These tail responses are crucial to determine how fast cumulative MPCs converge to 1 and thus how front-loaded the Keynesian boom turns out to be.31
Results are reported in Figure 4. Here the three panels correspond to three different models: thecalibrated spender-OLG model we have worked with so far (top panel) and two alternatives (middle
and bottom panel). Our empirical targets, reported as the grey areas (corresponding to 95 per centconfidence intervals) in the three “cumulative MPC” panels, are again taken from Fagereng, Holmand Natvik (2021) (see their Figure 2).
Consider first the top panel. By construction, this model matches MPCs—and thus also cumulative MPCs—over the first couple of years after income receipt. However, as revealed by the middle panel, the implied cumulative MPC appears to converge to 1 somewhat too quickly, reflecting
intertemporal MPCs that converge to 0 relatively fast, at rate ω. The middle panel—labeled “lower
bound calibration”—instead disciplines the model parameters (µ, ω) by matching (i) the same impact
MPC as before and then (ii) targeting the lower bound of the estimated confidence interval five years
after the income receipt. This calibration strategy unsurprisingly delivers a larger ω (equal to 0.94),
mapping into materially flatter intertemporal MPC profiles (left part), inconsistent with evidence on
short-run spending responses (middle part). As expected, convergence to our self-financing limit isslowed, though even in this calibration the degree of self-financing remains rather substantial for our
various fiscal adjustment rules. Finally, in the bottom panel, we consider an extended model—one
featuring spenders together with two types of OLG blocks, with heterogeneous ω1 and ω2 —that isrich enough to provide a tight fit to the entire dynamic profile of cumulative MPCs, up to five years
out. The left part of the bottom panel reveals that this model looks rather similar to our baseline calibration in the periods around income receipt, but then has somewhat flatter tail MPCs. Thus, as in
the middle panel, the speed of convergence to the self-financing limit is slowed down, though thepredicted degree of self-financing again remains meaningful.
Spending response to news about future income. All models discussed above are calibrated to beconsistent with evidence on consumer spending behavior in response to income shocks today; thesimilarly important consumption response to “news shocks” (i.e., Mt ,t +ℓ ), on the other hand, is ex31

Yet another calibration strategy is presented in Supplement E.6. There we discipline the discounting coefficient ω
through empirical evidence on long-run elasticities of household asset demand. Interestingly, this very different (andmuch less direct) approach suggests values of ω reasonably close to our baseline calibration strategy.

38

B ASELINE CALIBRATIONL OWER BOUND CALIBRATIONE XTENDED THREE - TYPE MODEL

Figure 4: Top panel: iMPCs, cumulative MPCs, and self-financing share ν (as a function of τd ) in thebaseline model. Middle and bottom panels: same as above, but for models calibrated to (also) match
far-ahead tail MPCs.
39

trapolated through the model structure. We here briefly discuss results from two additional models
that extrapolate somewhat differently, with details in Supplements E.8.1 and E.8.2. First, in Supplement E.8.1, we consider a fully-fledged quantitative HANK model. Consistent with the results in Auclert, Rognlie and Straub (2018) and Wolf (2021a), we find that this model extrapolates MPCs acrosshorizons in almost the same way as our reduced-form spender-OLG hybrid model, so our conclusions
on self-financing are largely unchanged. Second, in Supplement E.8.2, we consider a variant of our
spender-OLG model with cognitive discounting. As discussed in Section 5.1, we expect this behavioral
friction to introduce two offsetting effects on ν—the future tax hike is discounted by more, but higher
future income feeds back less strongly to the present. Our model simulations confirm this intuition:
we find that ν is higher than in our baseline model for intermediate values of τd , but converges to thefull self-financing limit somewhat more gradually.32
Steeper Phillips curve. In all quantitative experiments displayed above, self-financing occurred almost exclusively through the tax base channel, reflecting the fact that our assumed NKPC was—consistent with empirical evidence and much quantitative modeling—very flat. Larger values of theslope coefficient κ have two effects, as discussed in Section 4.4: first, for any given τd , the total amountof self-financing ν increases; and second, the split of self-financing moves from the tax base effect toinflation. In Appendix C.2, we repeat our analysis with somewhat more flexible prices, setting κ = 0.1.
This value lies at the upper end of recent empirical estimates pertaining to the inflationary post-covid
episode. We find that, with this alternative calibration, around 20 per cent of self-financing comesthrough date-0 inflation.
Monetary policy reaction. Our baseline exercises assumed a monetary authority that neither accommodates the fiscal deficit nor leans against the resulting Keynesian boom. Conceptually, it allows
us to cleanly isolate the quantitative potential of the interaction between consumer spending behavior and fiscal financing delays at the heart of our paper. Practically, it at least broadly consistent withthe recent post-covid U.S. policy experience. In Appendix C.3, we nevertheless consider what happens when the monetary authority leans more aggressively against the fiscal boom, consistent with astandard Taylor-type rule. The results from this exercise are consistent with the insights in Theorem
3. First, for moderately active monetary reaction, full self-financing is still possible, though convergence as τd → 0+ is now slowed down. Second, if the monetary response is very aggressive, then for abounded equilibrium to exist fiscal adjustment needs to be sufficiently strong, and so complete selffinancing is not possible anymore. That being said, for empirically relevant calibrations of the Taylor
rule and of the slope of the NKPC, the degree of self-financing can still be substantial.
32

We expect similar results in behavioral model variants that replace cognitive discounting à la Gabaix, 2020 with incomplete information à la Angeletos and Lian (2018) and/or bounded rationality à la Farhi and Werning (2019), Vimercati,
Eichenbaum and Guerreiro (2021), and Angeletos and Sastry, 2021.

40

7 Conclusion
The central contribution of this paper is to clarify the conditions under which fiscal deficits can finance themselves. Our analysis applies to environments with two empirically relevant features: (i)nominal rigidities and (ii) a violation of Ricardian equivalence due to finite lives or liquidity constraints. The headline result is that, as the fiscal authority delays the tax adjustment, the demand-led
boom gets larger and more persistent, and this contributes to a larger share of self-financing. In thelimit, deficits can even fully pay for themselves.
Our results have important conceptual as well as practical implications. Conceptually, they suggest that one specific idea from the classical Fiscal Theory of the Price Level—that future tax hikes
are not necessary to finance today’s deficits—is more robust than commonly believed. Our accommodation of this idea does not rely on the threat of a government violating its budget constraint, nor
is it vulnerable to debates regarding equilibrium selection. Rather, it is grounded in realistic departures from permanent-income consumer behavior, together with sufficient delay in fiscal adjustment.
Practically, our results suggest that a meaningful degree of self-financing is empirically plausible, especially in the absence of aggressive monetary policy responses. These insights seem relevant for recent U.S. fiscal stimulus experiments and for the related debate on “excess savings” (Auclert, Rognlieand Straub, 2023): fiscal deficits without a quick tax offset are predicted to lead to a long-lived boom
and to largely pay for themselves.
We leave several important avenues for future work. First, to assess the practical relevance ofour results, we disciplined the theory with the best available evidence on consumer behavior andthe degree of nominal rigidity. A more direct empirical test would compare the dynamic causal effects of equally large but differentially financed changes in government spending, using the standard
macroeconometric toolkit (see Ramey, 2016). Second, a meaningful limitation of our theory was theassumption of a closed economy. In an open economy, some of the deficit-led demand boom will leakabroad, dampening the general equilibrium Keynesian amplification. We therefore expect our mechanism to be more relevant for large closed economies (like the U.S.) than for small open economies.
Third, note that our analysis here has been entirely positive, not normative. The implications of ourresults for optimal fiscal policy seem like another important direction for future work.

41

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44

Appendix for:Can Deficits Finance Themselves?

This appendix contains supplemental material for the article “Can Deficits Finance Themselves?”. We
provide: (i) supplementary discussion of our baseline model; (ii) details for the extensions consideredin Section 5; (iii) additional analysis and alternative results for our quantitative investigation in Section 6. The end of this appendix contains all proofs. Further supplementary materials are provided in
the Supplement Angeletos, Lian and Wolf (2023).
Any references to equations, figures, tables, assumptions, propositions, lemmas, or sections that are
not preceded by “A.”—“D.” refer to the main article.

A Further model details
We here provide some additional discussion of our baseline model. Subsection A.1 characterizes labor
supply and explains how our model’s supply block reduces to the standard NKPC. Subsection A.2
explains how we characterize equilibria under the alternative fiscal policy rule (7).

A.1 The supply block
Unions equalize the post-tax real wage and the average marginal rate of substitution between labor
supply and consumption. Since we abstract from heterogeneity in labor supply L i ,t = L t , the optimallabor supply relation is1
ϕ(1 − τ y )Wt = R 1

χL t

−1/σ
0 C i ,t d i

,

(A.1)

With Log-linearizing, we obtain (13).
Optimal firm pricing decisions as usual give inflation as a function of real marginal costs. With astandard constant-returns-to-scale, labor-only production function this gives (e.g., see the textbook
derivations in Woodford, 2003a; Galí, 2008)
πt = κ̃w t + βEt [πt +1 ] ,

45

(A.2)

where κ̃ is a function of firm-side primitives, including in particular the stickiness of prices. Combining (A.2) with (13) and imposing that c t = ℓt = y t we obtainµ¶
1 1πt = κ̃+y t + βEt [πt +1 ] ,ϕ σ| {z }(A.3)≡κ

as required.

A.2 Equilibrium characterization with the alternative fiscal rule
We characterize the equilibrium in our OLG-NK environment with ω < 1, τ y > 0, and the alternative
fiscal rule (7) here. The aggregate demand relation (11) together with monetary policy (8) and market
clearing y t = c t lead to the following recursive aggregate demand curve33
yt =

¡
¢
1 − βω (1 − ω)βω£¤(d t − t t ) + Et y t +1 ,(A.4)

where we use that Et [d t +1 ] = β1 (d t − t t ) from (4).
We characterize the bounded equilibrium path through backward induction. Given the alternative
fiscal rule (7), we know that, for t ≥ H ,£¤d t − t t = 0 =⇒ y t = Et y t +1 .
We focus on the equilibrium with y t = 0 for t ≥ H . As discussed in Footnote 18, this equilibrium
selection can be justified in two ways: strengthening the boundedness requirement in the equilibrium
definition or considering limits to φ = 0 from above. The sole role of any of these modifications is toremove a class of sunspot equilibria that are inherited from the standard New Keynesian model. Given©
ªH −1
this selection we use (A.4) to find the equilibrium path of y t , d t t =0 through backward induction
starting fromy H = χ0 d H with χ0 = 0.

(A.5)

For t ≤ H − 1, substitute the alternative fiscal rule (7) into (A.4), giving(1−βω)(1−ω)
yt =
33

βω

1+ (

1−βω)(1−ω)τyβω(d t + εt ) +£¤1
Et y t +1 .
1−βω)(1−ω)
1+ (
τ
βω

y

Note that (12) does not apply here, since its derivation uses the baseline fiscal policy (6).

46

As a result, for t ≤ H − 1,(1−βω)(1−ω)
y t = χH −t (d t + εt ) with χH −t =βω1−βω)(1−ω)
1+ (
τ
βω

+
y

1
β

¡
¢
1 − τ y χH −t1−βω)(1−ω)
1+ (
τ
βω

χH −t −1 ,

(A.6)

y

where, according to (7), εt = 0 for all t ̸= 0. Rearranging terms, we find the following recursive formulafor the χs:χH −t =

(1−βω)(1−ω) + χH −t −1
βω

1+ (
³

β

1−βω)(1−ω)χ+ H β−t −1
βω

´

τy

¡
¢
= g χH −t −1 ,

(A.7)

where

¡ ¢
g χ =

(1−βω)(1−ω) + χ
βω

¡ ¢ 1
and g ′ χ = ³ββ1−βω)(1−ω)
χ
1+ (
+ τ

³

´

βω

β

y

1
1−βω)(1−ω)χ
1 + τ y ( βω+β³´´2 ≥ 0 ∀χ ≥ 0.

(A.8)

We thus know thatχk ∈ (0,1) ∀k ≥ 1 and χk increases in k.
τy

(A.9)

From (4), r t = 0, (7), and (A.6), we also know thatE0 [d t ] =¢1 t −1 ¡Π1−τχ(d 0 + ε0 )yH−j
βt j =0(A.10)

To further characterize the equilibrium we begin by considering an alternative economy with rigid
prices (κ = 0) but otherwise identical to the baseline economy. Let ν′ denote the self financing sharein this alternative economy, i.e.,ν′ · ε0 = ν′y · ε0 ≡

∞
X

£ ¤
τ y β k E0 y k .

k=0

In this economy, there is no t = 0 price level jump and so the real value of public outstanding at t = 0,
d 0 = b 0 = 0 is pre-determined. From (A.6) and (A.10), we have that
ν′ =

HX
−1
t =0

¡
¢
¡
¢
H −1
Πtj −1
1
−
τ
χ
τ
χ
=
1
−
Π
1
−
τ
χ
.
y
H
−
j
y
H
−t
y
H
−
j
j =0
=0

(A.11)

We can now return to the general case with κ ≥ 0. From the NKPC (14) as well as the definitions in (20)– (21), we have that

ss

νp =

κD
Y ss
τy

ss

νy =

47

κD
Y ss

ss

τy + κ D
Y ssν

(A.12)

Finally, from the formula of d 0 (16), we know
d 0 = −νp ε0¡¢ ′and ν y = 1 − νp ν .

Together, we have
ss

τy

ν=

ν′

νy =

ss

τy
ss

τ y +κ D
Y ss

+

κD
Y ss

ss

τ y +κ D
Y ss

ν′

ss
τ y +κ D
Y ss

κD
Y ss

ν′
,

ss

τy
ss

τ y +κ D
Y ss

+

κD
Y ss

ss

τ y +κ DY ssν′and νp =ssτ y +κ D
Y ss

ν′
.

ss

τy
ss

τ y +κ D
Y ss

+

κD
Y ss

ss

τ y +κ D
Y ss(A.13)ν′This completes our characterization of the equilibrium.

B Details on model extensions
We elaborate on some of the model extensions considered in Section 5: a model with investment inAppendix B.1; equilibria under more general monetary rules in Appendix B.2; fiscal financing throughdistortionary taxes in Appendix B.3; and government purchases in Appendix B.4.

B.1 Investment
The household block changes very little. Households still receive labor income and dividends; wenow denote this total household income by e t (which in equilibrium will be equal to total household
consumption rather than total aggregate income). The linearized household demand relation is now
Ã
"
#!
"
#
∞ ¡
∞
X
X
¡
¢
¢k
c t = 1 − βω d t + Et
βω (e t +k − t t +k ) − γEt(βω)k r t +k ,
k=0

(B.1)

k=0

while labor supply still satisfies
1
1ℓt = w t − c t
ϕ
σ

(B.2)

The firm block on the other hand changes materially relative to our baseline model. Since thisproduction side is entirely standard our discussion here will be brief and only present linearized optimality conditions; a detailed discussion of an almost identical model is offered in Wolf (2021b). The
production sector consists of three parts: perfectly competitive intermediate goods producers who
accumulate capital and hire labor on spot markets; monopolistically competitive retailers who purchase the intermediate good and costlessly differentiate it, subject to nominal rigidities; and a competitive final goods aggregator. Profits of the corporate sector as a whole are returned to households,
subject to the time-invariant tax τ y . The relevant equilibrium relations follow from the behavior of48

the intermediate goods producers and the retailers.
1. Intermediate goods producers. The production function takes a standard Cobb-Douglas form
with capital share α, and capital depreciates at rate δ. We let p tI denote the real relative price of
the intermediate good. Optimal labor demand gives the static relationw t = p tI + αk t −1 − αℓt(B.3)

while optimal capital accumulation gives34
¶
µ
£
¤
1
1
− 1 + δ × Et p tI +1 + (α − 1)k t + (1 − α)ℓt +1(i t − Et [πt +1 ]) =
β
β

(B.4)

By our assumptions on the production function total output is given as
y t = αk t −1 + (1 − α)ℓt

(B.5)

1
(k t − (1 − δ)k t −1 )
δ

(B.6)

and finally investment x t satisfies
xt =2. Retailers. Optimal price-setting as usual relates real marginal costs—here the relative price of
the intermediate good, p tI —to aggregate inflation:πt = κp tI + βEt [πt +1 ](B.7)

Aggregating dividend payments from intermediate goods producers and retailers, we obtain
Q t = Y t − Wt L t − X t(B.8)

which implies that total household income E t (in levels) is given as
E t = Wt L t +Q t = Y t − X t(B.9)

Aggregate output market-clearing dictates that
C ss
X ss
y t = ss c t + ss x t
YY
34(B.10)

Adjustment costs on the capital stock or investment flows would complicate this relation but not affect any of our
subsequent arguments.

49

Finally we return to the government. The monetary rule (8) and the government budget constraint(4) are unchanged. The fiscal policy rules (6) or (7) are also unchanged up to the tax base revenue term:since the government taxes labor and dividend income, this term now equals τ y × E t .
Equilibrium characterization. Our key building block result is that we can reduce the equilibriumof this extended model to a system of equations almost as simple as that of our baseline model in Section 2. First, combining market-clearing and the policy rules with private-sector demand we obtain"c t = F1 · (d t + εt ) + F2 · Et

¡

#
∞ ¡
¢X
¢k
1 − βω
βω c t +k .

(B.11)

k=0

Relative to our baseline model, the only change is that this equilibrium demand relationship is in aggregate consumption c t rather than aggregate output y t . We emphasize that this is possible precisely
because the government taxes dividend and labor income, which as discussed above in equilibrium
is equal to total consumption. Second, the law of motion for aggregate debt is now



ss


 D
d t +1 = β−1 d t + εt − τd · (d t + εt ) − τ y c t  − ss (πt +1 − Et [πt +1 ]),
|
{z
}
|{z}
|Y
{z
}
fiscal adjustment

tax base(B.12)

price levelwith real debt at date 0 given as
d0 = b0 −

D ss
D ss
−
E
=
−
π0 .
(π
[π
])
0
−1
0
Y ss
Y ss

(B.13)

Again, relative to the baseline model, the only change is that now it is aggregate consumption rather
than aggregate output appearing in (B.12).
We note that (B.11) - (B.12) is a system in {c t , d t }∞t =0 that depends on the rest of the economy—andso in particular the investment block—only through the presence of π0 . π0 on the other hand can beobtained as a function of the consumption path {c t }∞t =0 by solving the system (B.2), (B.3), (B.4), (B.5),(B.6), (B.7) and (B.10) given consumption, and with the monetary policy rule (8) imposed. We write
this function as¡¢
π0 = Π0 {c t }∞t =0

(B.14)The equilibrium described by equations (B.11) - (B.14) is straightforward to characterize given
our earlier analysis of the model without investment in Sections 2 and 4. We begin with the case of
perfectly rigid prices (κ = 0), and for simplicity restrict attention to the limiting self-financing case(τd → 0 or H → ∞). In that case π0 = 0, so we can focus on the bivariate system (B.11) - (B.12) in{c t , d t }∞t =0 . Crucially, this system is exactly the same as that covered in Theorem 1, so the equilibrium
50

characterization underlying that result applies unchanged, with c t replacing y t .
We now turn to the case of general κ. To this end let c t ,0 denote the solution of the rigid-pricesystem, and furthermore let p tI ,0 denote the corresponding equilibrium intermediate goods price obtained by solving the system (B.2), (B.3), (B.4), (B.5), (B.6), and (B.10) for p I given {c t ,0 }∞t =0 . Proceeding
analogously to the proof of Theorem 1, we will now construct the equilibrium for general κ by simply
scaling the κ = 0 equilibrium. To this end conjecture that equilibrium consumption satisfies a × c t ,0 ,for some scalar a. It is then immediate that then we would also have p tI = a × p tI ,0 . But then, from
(B.7), we have thatπ0 = a × κ ×

∞
X
t =0

βt p tI ,0(B.15)

Finally it follows from the government budget constraint that—again in our limiting self-financing
equilibrium—we must have∞XD ss
ε0 = a × τ y ×
β c t ,0 + a × ss × κ ×
βt p tI ,0
Y
t =0
t =0
∞
X

t

Solving this equation for a we obtain consumption and thus inflation as well as government debt in
the general sticky-price equilibrium. In particular we see that self-financing yet again obtains exactly
as in our baseline economy. We summarize these observations in the following corollary.
Corollary B.1. Consider the extended OLG-NK environment with investment. Fiscal adjustment isnever necessary in equilibrium—that is, ν → 1—if the tax response is infinitely delayed, i.e., τd → 0
or H → ∞. These two limits induce the same equilibrium paths {c t , πt , d t }∞t =0 , and in this common
limit self-financing is sufficient to return government debt to steady state (i.e., ρ d ∈ (0, 1)).

B.2 More general monetary policy
With variable real rates, the government’s intertemporal budget constraint in (19) has to be re-written
as follows:
Ã
!
+∞
+∞
X k+1
X k
X
£ ¤ D ss
D ss +∞
ε0 + ssβ E 0 [r k ] = τd ε0 +β E 0 [d k ] +τ y βk E 0 y k + ss (π0 − E−1 [π0 ]) .
Y k=0
Y
k=0
k=0
ss

The right hand side is the same as before, while the new term on the left-hand side, D
Y ss

P+∞

k=0

βk+1 E 0 [r k ],captures how the time-varying real interest rate in (9) changes the costs of servicing the outstanding
public debt. We interpret this new term as analogous to the deficit shock itself and accordingly define
the share of self-financing asss
τ βk E0 y k + Dπ
Y ss 0
k=0 y
ν≡
P
ss
+∞ k+1ε0 + Dβ E 0 [r k ]
Y ss
k=0

P∞

£

51

¤

(B.16)P ERMANENT- INCOME CONSUMERS (ω = 1)

F INITE - HORIZON CONSUMERS (ω < 1)

Figure B.1: Determinacy regions for ω = 1 (left panel) and ω < 1 (right panel) as a function of the fiscal
and monetary policy rule coefficients τd (y-axis) and φ (x-axis).
This reduces to the original definition when φ = 0.
The two panels in Figure B.1 provide a visual illustration of equilibrium determinacy—they showwhether a bounded equilibrium (i.e., the standard solution concept of Blanchard and Kahn (1980))
exists and is unique under different assumptions on monetary policy (φ) and fiscal policy (τd in (6)),
for a standard permanent-income model (left panel) and our OLG economy (right panel).
The figure reveals that the determinacy properties of the two economies are materially different.
Results for the permanent-income model are well-known and require little explanation: equilibrium
uniqueness requires that fiscal policy is passive (τd > 1 − β) and monetary policy is active (φ > 0), or
vice-versa; if both rules are active then no bounded equilibrium exists, and if both are passive thenthere are multiple bounded equilibria. With discounting on the household side (i.e., ω < 1), the regions of equilibrium determinacy look rather different. Perhaps most importantly, the benchmark
monetary rule of φ = 0 now induces unique bounded equilibrium for any τd , consistent with Proposition 1. Intuitively, with ω < 1, determinacy comes from the fact that public debt directly enters theaggregate demand relation. Moreover, existence comes from the fact, with τ y > 0, output also directly
affects the government budget. Self-financing is now strong enough to pull debt as well as spending
towards zero, even if interest rates do not provide any further Euler equation tilting. This automaticstabilization of government debt also shrinks the equilibrium non-existence region in the bottom
right corner of the figure. See the discussion in Supplement E.1 for further details.

52

B.3 Distortionary tax hikes
With time-varying distortionary taxes the optimal labor supply relation becomesR 1 ϕ1
χ 0 L i ,t d i(1 − τ y,t )Wt = R 1−1/σ
0 C i ,t d i(B.17)

1
1
1
ct −τby,t = ℓtσ
1 − τyϕ(B.18)

Log-linearizing, we find thatwt −where τbt ≡ τ y,t − τ y . Next note that the firm optimal pricing relationship is still
πt = κ̃w t + βEt [πt +1 ](B.19)

Combining (B.18), (B.19), and the modified baseline fiscal (32), we obtain










1

´¡
πt = κ  y t + ³
τd d t ¢ + βEt [πt +1 ] ,
1
1


+
1
−
τ
y
ϕ
σ


{z
}
|

(B.20)≡ζ³´
where κ = κ̃ ϕ1 + σ1 . All other equilibrium relations of the model are unaffected.
Note that, as τd → 0, we obtain that ζ → 0 for all t . It thus follows that the equilibrium characterization of Theorem 1 for the self-financing limit applies without any change to the alternative economy
in which fiscal adjustment is distortionary. Intuitively, since the adjustment is not necessary in equilibrium, it is immaterial whether the adjustment would have been distortionary or lump-sum.

B.4 Government spending
The government spending policy experiment was already described in Section 5.5. The only missing
ingredient for equilibrium characterization is the adjusted NKPC. By standard arguments (e.g., see
Galí, 2008), it is now given as
πt = κy t + βEt [πt +1 ] − κ 1

ϕ

1
σ

+ σ1

gt

Intuitively, the last term reflects the fact that, if higher output comes from higher government purchases (rather than higher consumption), then household labor supply is larger for standard wealtheffects reasons—i.e., a negative cost-push shock. Since the overall analytics of the self-financing result

53with government purchases are exactly analogous to our baseline “stimulus checks” case, we do notrepeat those derivations here. See Supplement E.8.3 for a visual illustration.

C Quantitative analysis
This section supplements our quantitative analysis in Section 6. We: provide some missing details
on the specification of our spender-OLG hybrid model in Appendix C.1; and consider model variants
with more flexible prices and more aggressive monetary reactions in Appendices C.2 and C.3.

C.1 Further details on the hybrid model
The only change relative to our baseline model of Section 2 is that we generalize the household block
to also feature a margin µ of spenders—that is, households who do not hold any assets and immediately spend any income they receive. The remaining fraction 1 − µ of households are exactly as
described in Section 2.1. Both groups of households receive labor income as well as dividends andpay taxes, but only the OLG block holds government bonds.
We will make assumptions ensuring that both groups of households receive the same labor anddividend income, pay the same taxes (up to a between-group steady-state transfer), and have identicalsteady-state consumption. First, we assume that unions assign identical hours worked to both groups,
and that dividends also accrue equally to both. Second, we assume that the government in lump-sum
fashion redistributes between the two groups to ensure identical steady-state consumption; giventhat government bonds are held by the OLG block, this will generally require lump-sum transfers tospenders. Under those assumptions, it is first of all immediate that the supply block of the economy—notably (14)—is unchanged. Next, the demand block of the economy generalizes (26) as follows:
Ã
"
#!
∞
X
£
¤
¡
¢
c t = (1 − βω) · d t + µ + (1 − µ)(1 − βω) · (y t − t t ) + (1 − µ)Et
(ωβ)k y t +k − t t +k
.

(C.1)

k=1

Replacing (26) by (C.1) is the only difference between our baseline OLG economy and the generalized
hybrid model. Relative to (26), the most important change in (C.1) is that we allow the MPC out of
income to be larger than that out of wealth. As we discuss next, this minimal departure from ourbaseline OLG model is all that is needed to ensure (approximate) consistency with consumptionsavings behavior even in quantitative HANK-type models.

54

Figure C.1: Impulse responses of output y t , inflation πt , and the total self-financing share ν to thedeficit shock ε0 as a function of τd , with more flexible prices. The left and middle panels show theimpulse responses for the three particular values of τd discussed in Section 6.1. In the right panel
these three points are marked with circles.

C.2 Self-financing with more flexible prices
Figure C.1 repeats our analysis of Section 6.2 in a variant of the baseline model with more flexible
prices, setting κ = 0.1—a value at the large end of recent empirical evidence and arguably more relevant for the inflationary post-covid environment. With more flexible prices, a non-trivial share—herearound 20 per cent—of self-financing comes from adjustments in prices rather than quantities.

C.3 Active monetary reaction
For our final extension we consider a variant of the baseline quantitative model in which the monetary
authority instead follows a standard Taylor-type rule, setting
i t = ϕπt
We consider ϕ ∈ {1.25, 1.5, 2}—a range of values, all corresponding to active monetary policy responses
in the classical terminology of Leeper (1991). The three panels in Figure C.2 show the self-financing
share ν as a function for τd for the three displayed values of ϕ. In the middle and right panels the redregions indicate equilibrium non-existence for the underlying policy mix.
The figure illustrates the robustness of our conclusions to active monetary policy reactions. First,
for a moderate monetary response (ϕ = 1.25), complete self-financing is still possible, consistent with55

Figure C.2: Total self-financing share ν in response to the deficit shock ε0 as a function of τd , forthe standard Taylor-type monetary policy rule i t = ϕπt , with ϕ ∈ {1.25, 1.5, 2}. The red areas indicate(bounded) equilibrium non-existence.
Theorem 3. Compared to our baseline exercises in Section 6.2, the more aggressive monetary reaction
delays the fiscal boom, so ν is strictly smaller for any τd > 0, though still converging to 1 as τd → 0+ .
Second, for even stronger monetary reactions (ϕ = 1.5 or ϕ = 2), existence of a bounded equilibrium
requires that τd is sufficiently far above 0, as indicated by the shaded red areas. We neverthelesssee that, for small values of τd outside of these non-existence regions, there can still be meaningfuldegrees of self-financing, even for a central bank that quite aggressively leans against the fiscal boom.

D Proofs
D.1 Proof of Proposition 1
Note that we restrict that ω ∈ (0, 1), τ y ∈ (0, 1), and τd ∈ [0, 1). We first write (12) recursively:£¤
y t − F1 · (d t + εt ) = (1 − βω)F2 · y t + βωEt y t +1 − F1 · (d t +1 + εt +1 )·¸¤
1£
= (1 − βω)F2 · y t + βωEt y t +1 − F1 ·(1 − τd ) (d t + εt ) − τ y y t .
β56

After rearranging terms and using the formula of F1 and F2 (as stated after (12)), we have£¤βω
(1 − ω (1 − τd )) F1
Et y t +1(d t + εt ) +1 − ωτ y F1 − (1 − βω)F2
1 − ωτ y F1 − (1 − βω)F2(1−βω)(1−ω) − τ
(1
d)
£
¤
1
βω
=
Et y t +1 .
(d t + εt ) +1−βω)(1−ω)1−βω)(1−ω)
1+ (
τ
1+ (
τ

yt =

βω

y

y

βω

Applying period-t expectations Et [·] to (15), we have 1−τd
Et [d t +1 ]β£¤  =  (1−βω)(1−ω)(1−τd )Et y t +1
−
βω


τy

−β
1−βω)(1−ω)
(
1+
τ
βω



d t + εt





yt



y

(D.1)

The two eigenvalues of the system are given by the solutions of¶
11
1 − βωλ −λ
τ y (1 − ω) + (1 − τd ) = 0,
(1 − τd ) + 1 +
β
βω
β
2

µ

with
³

λ1 =
³

>

1−βω
1
β (1 − τd ) + 1 + βω τ y (1 − ω)

´ r³
´2
1−βω
1−βω
1 − β1 (1 − τd ) − βω τ y (1 − ω) + 4 βω τ y (1 − ω)
+

2
¯
´ ¯
¯
¯
1−βω
1−βω
1
1
+
1
−
τ
−
ω)
τ
−
ω)
−
τ
−
τ
+
1
+
−
(1
(1
(1
(1
)
)
¯
¯
y
y
d
d
β
βω
β
βω
2

(D.2)

≥1

and
³

λ2 =
³

<

1−βω
1
β (1 − τd ) + 1 + βω τ y (1 − ω)

´ r³
´2
1−βω
1−βω
1 − β1 (1 − τd ) − βω τ y (1 − ω) + 4 βω τ y (1 − ω)
−

2
¯
´ ¯
¯1
¯
1−βω
1−βω
1
−
τ
−
ω)
τ
−
ω)
−
1
−
τ
−
τ
+
1
+
+
(1
(1
(1
(1
)
)
¯
¯
y
y
d
d
β
βω
β
βω
2

(D.3)

≤ 1,

¡
¢′
with λ2 > 0 too since λ1 λ2 = β1 (1 − τd ) > 0. Let 1, χ2 denote the eigenvector associated with λ2 , where
¢
1¡
λ2 =
1 − τ d − τ y χ2β(1−βω)(1−ω)
and χ2 =βω(1 − τd )1+ (1−βω)(1−ω)τ y − λ2
βω

> 0.

(D.4)

©
ª+∞
This means that any bounded equilibrium path d t , y t t =0 of (D.1)35 takes the form of

y t = χ (d t + εt )and Et [d t +1 ] = ρ d (d t + εt ) ,35£¤
Boundedness means that limk→+∞ Et [d t +k ] and limk→+∞ Et y t +k are bounded for any t , d t + εt , and y t , similar toBlanchard and Kahn (1980).

57

where χ and ρ d are uniquely given byχ = χ2 > 0 and ρ d = λ2 ∈ (0, 1) .

(D.5)

In other words, the equilibrium takes the form of (17) and satisfies (18).36
To prove equilibrium uniqueness, note that the total amount of nominal public debt outstanding at thestart of t = 0, B 0 = B ss , is given. From (14) and (16), we know d 0 is uniquely pinned down by
ss

χ

Y

1−βρ dκD
X k £ ¤
D ss +∞D ss
χD ss
Y ss 1−βρ d
β E0 y k = −κ ss
d 0 = − ss π0 = −κ ss(d 0 + ε0 ) = −
ss
χ
Y
Y k=0
Y 1 − βρ d
1 + κ D ss

ε0 .

(D.6)

Similarly, for t ≥ 1,
d t − Et −1 [d t ] = −X k¡ £¤£¤¢D ss
D ss +∞−E=−κ
β Et y t +k − Et −1 y t +k−1
(π
[π
])
t
t
−1
t
ss
ss
Y
Y k=0
ss

χ

Y

1−βρ dκDD ss
χ
Y ss 1−βρ d= −κ ss(d t − Et −1 [d t ] + εt ) = −ssχ
Y 1 − βρ d
1 + κ D ss

εt .

©
ª+∞
This pins down the unique bounded equilibrium path of πt , d t , y t t =0 with (14) and (17).

D.2 Proof of Lemma 1
The result follows directly from two facts in the proof of Proposition 1 and Theorem 1. First, under fiscal rule(6), ν decreases in τd , limτd →0+ ν → 1, and limτd →1− ν → 0. Second, under fiscal rule (7), ν increases in H , ν = 0
with H = 0, and limH →∞ ν → 1.

D.3 Proof of Theorem 1
We start with the case based on the baseline fiscal policy (6). We begin by considering an alternative economy
with rigid prices (κ = 0) but otherwise identical to the baseline economy. Let ν′ denote the self financing sharein this alternative economy, which similarly to (21) is given as
ν′ · ε0 = ν′y · ε0 ≡

∞
X

£ ¤
τ y β k E0 y k .

k=0Note that in this alternative economy all self-financing comes from tax base changes. In particular, there is not = 0 price level jump and so the real value of public outstanding at t = 0, d 0 = b 0 = 0 is pre-determined. From(17) and (21) we know thatν′ = τ y36

χ.
1 − βρ d

To see the first part of (18), combine (12) with (17).

58

(D.7)

Now, consider the general case with κ ≥ 0. From NKPC (14) and the definitions in (20) – (21), we have
ss

ss

νp =

κD
Y ss
τy

νy =

κD
Y ss

ss

τy + κ D
Y ss

ν

(D.8)

From the formula of d 0 (16), we know¡¢
and ν y = 1 − νp ν′ .

d 0 = −νp ε0
Together, we have

ss

τy

ν

′

ν=

νy =

ss

τy
ss

τ y +κ D
Y ss

+

κD
Y ss

ss

τ y +κ D
Y ss

ν′

ss
τ y +κ D
Y ss

ss

+

κD
Y ss

ss

τ y +κ D
Y ss

ν′

ss

τ y +κ D
Y ss

and νp =,ssτyτ y +κ D
Y ss

From (18), we know that

κD
Y ss

ν′

ν′
.

ss

τy
ss

τ y +κ D
Y ss

+

κD
Y ss

ss

τ y +κ D
Y ss

(D.9)

ν′

χ
χ
.
=
1 − βρ d τd + τ y χ(D.10)

From (D.3) and (D.5), we know
ρ d = λ2 = f (a, b) ≡where a = β1 (1 − τd ) > 0 and b =a +b +1−p(a + b − 1)2 + 4b2

(D.11)∂f1−βω1p (a+b−1)βω τ y (1 − ω) > 0. Since ∂a = 2 − 2 (a+b−1)2 +4b > 0,we know that ρ d decreases(1−βω)(1−ω)

with τd . From (D.4) and (D.5), we then know χ =(1−τd )βω1−βω)(1−ω)
(
1+
τ −ρ
βω

y

also decreases in τd . From (D.10), we knowdχ1−βρ d decreases in τd . Finally, from (D.7) and (D.9), we know ν decreases in τd . This finishes the proof of Part 1.

For Part 2, from (D.3) and (18), we know that ρ d and χ are continuous in τd ∈ [0, 1), and³

ρ fulld ≡ lim + ρ d =1−βω
1
β + 1 + βω τ y (1 − ω)

´ r³
´2
1−βω
1−βω
−
1 − β1 − βω τ y (1 − ω) + 4 βω τ y (1 − ω)

τd →0

<

2
¯
¯
¯
¯
1−βω
1−βω
1
1
+
1
+
τ
−
ω)
−
+
τ
−
ω)
−
1
(1
(1
¯
¯
y
y
β
βω
β
βω
2

χfull ≡ lim + χ =τd →0

1 − βρ full
d
τy

>0

(D.12)≤1(D.13)

χ

From (D.10), we know limτd →0+ 1−βρ d = τ1y . From (D.7) and (D.9), we know limτd →0+ ν = 1. Finally, limk→∞ Et [d t +k ] →0 follows directly from ρ full
< 1.
d
Now we turn to the alternative fiscal policy rule in (7), for which we use the equilibrium characterization in
Appendix A.2. For the case with rigid prices (κ = 0), one can see from (A.11) that ν′ increases in H , which proves© ª∞Part I. For Part II and to find limH →∞ ν′ , first note that, from (A.9), χk k=0 is a bounded, increasing sequence.

59

´
¡
¢ ³
As a result, there exists χfull,NM such that limk→∞ χk = χfull,NM and χfull,NM = g χfull,NM ∈ 0, τ1y . From (A.11),

we know that limH →∞ ν′ = 1. From (A.10), we know that limH →∞ E0 [d H ] = 0. From (D.4) and (D.5), we also know¡¢that g χfull = χfull where χfull defined in (D.13) parametrizes the output response in the complete self-financing
limit (τd → 0) with the baseline fiscal rule (6). From the definition of g (·) in (A.8), we know that there is a unique¡ ¢χ > 0 such that g χ = χ when ω < 1 and τd ∈ (0, 1) . As a result, χfull,NM = χfull < τ1y and limk→+∞ χk = χfull . Thatis, these two limits (τd → 0 and H → ∞) share the same equilibrium path. Finally, for the general case withκ ≥ 0, the desired result follows directly from the rigid price case together with (A.13) and (D.9).

D.4 Proof of Proposition 2
For part 1 of the Proposition, we start with the case of the baseline fiscal policy (6). From (D.11), we know thatss

ρ d is independent of the degree of price flexibility κ and the steady-state debt-to-GDP ratio D
Y ss . From (18) and(D.7), we know that ν′ is independent of the degree of price flexibility κ and the steady-state debt-to-GDP ratio
D ss
Y ss . From (D.9), we know that ν is increasing in the degree of price flexibility κ and steady-state debt-to-GDP
ssratio DY ss .

Now we turn to part 1 with the alternative fiscal policy in (7), for which we use the equilibrium characterization in Appendix A.2. From (A.7) and (A.11), we know that ν′ is independent of the degree of price flexibilityssκ and the steady-state debt-to-GDP ratio D
Y ss . From (A.13), we know that ν increases in the degree of price flexiss

bility κ and the steady-state debt-to-GDP ratio D
Y ss .
Part 2 of the Proposition follows directly from (22), which holds for both fiscal policies and gives
νy =

ss

τy
ss

τy + κ D
Y ss

× ν and νp =

κD
Y ss

ss

τy + κ D
Y ss

× ν.

D.5 Proof of Proposition 3
We start with the case of the baseline fiscal policy (6). From (D.11), we knowρd =a +b +1−p

2a(a + b + 1)2 − 4a=,p2
a + b + 1 + (a + b + 1)2 − 4a1−βωβω τ y (1 − ω) . As a result, ρ d decreases in τ y . From the second part of 18, we know thatτy χ
τ y χ increases with τ y . From (D.7) and D.10, we know that, when κ = 0, ν = τd +τy χ increases with τ y .

which decreases in b =Now we turn to part 1 with the alternative fiscal policy in (7), for which we use the equilibrium characterization in Appendix A.2. From (A.7), we know thatτ y χH −t =(1−βω)(1−ω) τ + χH −t −1 τ
βω

y

β

y

1−βω)(1−ω)χ
1+ (τ + H −t −1 τ
βω

y

β

60

¡
¢
= G τ y , χH −t −1 τ y ,ywhereG x, y =¡(1−βω)(1−ω) x + 1 y
βω

¢

β

1−βω)(1−ω)
x+ 1y
1+ (
βω

β

increases in x ∈ (0, 1) and y > 0. Together with (A.5), we know that τ y χk increases with τ y for each k ≥ 1. Together
with (A.11), when κ = 0, ν increases with τ y .

D.6 Proof of Proposition 4
Consider the baseline fiscal policy (6). From (D.11), we knowρd =a +b +1−

where a = β1 (1 − τd ) ∈ (0, 1) and b =decreases in b =p

2a(a + b + 1)2 − 4a=,p2
a + b + 1 + (a + b + 1)2 − 4a1−βωβω τ y (1 − ω) > 0. For the second part of the equation, we know that ρ d1−βωβω τ y (1 − ω) .

From the second half of (18), we know χ decreases in ω. From, (D.7), (D.9) and (D.10), we know that ν
decreases in ω. This proves Part 1.
For Part 2, from Theorem 1, we know that ρ full
< 1 for any ω < 1. From (D.12), we know that limω→1 ρ full
=
d
d
³

´ r³

1
+1 −
β

1− β1

´2

2

= 1.

D.7 Proof of Theorem 2
We start with the baseline fiscal policy (6). We focus on a bounded equilibrium similar to (17), taking the formofy t = χd d t + χ ε ε t ,E t [d t +1 ] = ρ d d t + ρ ε εtwith χd , χε > 0, ρ d ∈ (0, 1).

(D.14)For (D.14) to be an equilibrium, it needs to satisfy (15) and (27). For (D.14) to satisfy to the government budget(15), we need
ρd =

¢
1¡
1 − τ d − τ y χdβand ρ ε =¢1¡
1 − τ d − τ y χε .
β

(D.15)For (D.14) to satisfy aggregate demand (27) (together with market clearing c t = y t ), we need
³
´
!
Ã
δβωρ
+∞
M d − τd M y 1 + 1−βωρdd
X¡
£¡
¢
¤
¢k
´.
χ d = M d + M y 1 − τ y χd − τ d 1 + δ
βωρ d
=
¡
¢³
δβωρ
k=1
1 − M y 1 − τ y 1 + 1−βωρddandÃχε = M y

Ã

X
¢
¡¡
¢
¢ +∞
¡
1 + 1 − τ y χε − τ d + δ 1 − τ y χ d − τ d(βω)k ρ k−1
d ρε
k=1

61

(D.16)!!

.

(D.17)

(D.15) and (D.16) together mean that ρ d needs to be the root of the following equation:³´δβωρ d
M
−
τ
M
1
+
y
d
d
¡
¢ 1 − τd − βρ d1−βωρ d³
´ = 0,−h ρ d ; τd =
¡
¢
δβωρ
τy
1 − M y 1 − τ y 1 + 1−βωρddwith limτd →0+ h (0; τd ) = τ1y − 1−M Md1−τ ≥ 0 because τ y > 0, M y ∈ (0, 1) and M y ≥ M d .
y(
y)
¢
¡
; τd = 0.
∈ [0, 1) such that limτd →0+ h ρ full
When Assumption 2 holds, we first show that there exists ρ full
d
d
´
¡
¢³
δβω
There are two cases. First, M y 1 − τ y 1 + 1−βω > 1. In this case, there exists ρ̄ ∈ (0, 1) such that
µ
¶
¡
¢
δβωρ̄
M y 1 − τy 1 += 1,
1 − βωρ̄¢¡¡¢and limτd →0+ ,ρ d →(ρ̄ )− h ρ d ; τd = −∞. As a result, there exists a unique ρ full; τd =∈ [0, 1) such that we have limτd →0+ h ρ full
d
d

0.37
´
¡
¢³
δβω
Second, M y 1 − τ y 1 + 1−βω < 1. In this case,lim + h (1; τd ) =

τd →0

Md
1−β
´ <0
−
¡
¢³
δβω
τy
1−M 1−τ 1+
y

y

1−βω

¡
¢
from Assumption 2. As a result, there exists a unique ρ full
∈ [0, 1) such that limτd →0+ h ρ full; τd = 0.
dd¡¢Since h ρ d ; τd is continuous, we know that, for each τd in a right neighborhood of 0, there exists a unique¡¢ρ d (τd ) ∈ [0, 1) such that h ρ d (τd ) ; τd = 0 and limτd →0+ ρ d (τd ) = ρ full
∈ [0, 1). For each τd in this neighborhood,dgiven ρ d (τd ) , one can find ρ ε (τd ) and χd (τd ) , χε (τd ) > 0 from (D.15) – (D.17) to constitutethe unique bounded¢¡Pkτd ε0 + ∞k=0 β E0 [d k ]= 0 andε0
Finally, limk→∞ Et [d t +k ] → 0 follows directly from ρ full
∈ [0, 1). Thisdequilibrium in the form of (D.14). Next note that from boundedness we have limτd →0+limτd →0+ ν = 1, using also (19) and (20).

finishes the proof with the baseline fiscal policy (6).
We now turn to the case with (7). Together with market clearing c t = y t , we first write the aggregate demandin (27) recursively
·
¸
£
¤
My
My
My
Md
Md
dt −
t t + δβωEt y t +1 − t t +1 + βωEt y t +1 −d t +1 +t t +1
1 − My1 − My1 − My1 − My1 − Myµ¶£¤M y − ωM d
1 − (1 − δ) M yMy
M d (1 − ω)=dt −t t + βωEt y t +1 + βω
(D.18)(1 − δ) Et [t t +1 ] .
1 − My1 − My1 − My1 − Myyt =From (7), we know that t t = d t for all t ≥ H . As a result, d t +1 = 0 for all t ≥ H . Similar to the argument in Appendix
A.2, we can then focus on the case that y t = d t = 0 for t ≥ H + 1. At t = H , from (D.18), we have
yH =

37

¡
¢
− M y − Md

1 − My¡¢Note that, for ρ d ∈ ρ̄, 1 , limτd →0+

d H = χ0 d H with χ0 =¡¢− M y − Md

1 − My.

³´δβωρM d −τd M y 1+ 1−βωρd¡¢d³´ < 0 and so limτ →0+ h ρ d ; τd > 0.
δβωρ dd1−M y (1−τ y ) 1+ 1−βωρd62(D.19)

©
ªH −1
Similar to the main analysis in Appendix A.2, we will now use (D.18) to find the equilibrium path of y t , d t t =0

through backward induction. At t = H − 1, from (7) and (D.18)

y H −1 =³M d (1−ω)1−M yd H −1 + βω

1−(1−δ)M y
1−M y

´

M y (1−δ)
1−M y

χ0 +

dH
M y −ωM d
1 + τ y 1−M y³´
1−(1−δ)M y
M y (1−δ)M d (1−ω)χ0 + 1−M y ¡
¢
1−M y
1−M y
=
d
+
ω
d
−
τ
y
H
−1
H
−1
y
H
−1
M y −ωM d
M y −ωM d
1 + τ y 1−M y
1 + τ y 1−M yh³´
i
1−(1−δ)M y
M y (1−δ)M d (1−ω)
+
ω
χ
+
0
1−M y
1−M y
1−M y
h³
´
i,
= χ1 d H −1 with χ1 =M y −ωM d
1−(1−δ)M y
M y (1−δ)
1 + τ y 1−M y + ωτ y
χ0 + 1−M y
1−M y
1 + τyM y −ωM d
1−M ywhere χ1 can be re-rewritten as
ωM y
M d (1−ω)
1−M y + 1−M y

χ1 =
1+

³

Md
1−M d
M y − δ 1−M y

M y −ωM d
ωM y
1−M y τ y + 1−M y

³

´

Md
1−M d
M y − δ 1−M y

´

τy

.

(D.20)For 1 ≤ t ≤ H − 2, from (7) and (D.18),

yt =

=

M d (1−ω)
1−M y d t
M y −ωM d
1 + 1−M y τ y
M d (1−ω)
1−M y d t
M y −ωM d
1 + 1−M y τ y

+ βω

+ω

1−(1−τ y )(1−δ)M y
1−M y

£
¤
Et y t +1

M y −ωM d
1 + 1−M y τ y1−(1−τ y )(1−δ)M y¡
1−M y
M y −ωM d
1 + 1−M y τ y¢d t − τ y y t χH −t −1

1−(1−τ y )(1−δ)M y
M d (1−ω)
1−M y + ω
1−M y³

= χH −t d t with χH −t =´χH −t −1´
1−(1−τ y )(1−δ)M y
M y −ωM d
1 + 1−M y τ y + ωτ y χH −t −1
1−M y³(D.21)

Finally, for t = 0, from (7) and (D.18), we know

y0 =

=

M y −ωM d
M d (1−ω)
1−M y d 0 + 1−M y ε0
M y −ωM d
1 + 1−M y τ y
M y −ωM d
M d (1−ω)
1−M y d 0 + 1−M y ε0
M y −ωM d
1 + 1−M y τ y

+ βω

+ω

1−(1−τ y )(1−δ)M y
1−M y

£ ¤
E0 y 1M y −ωM d
1 + 1−M y τ y1−(1−τ y )(1−δ)M y¡
1−M y
M y −ωM d
1 + 1−M y τ y¢
d 0 + ε0 − τ y y 0 χH −1

1−(1−τ y )(1−δ)M y
M d (1−ω)
1−M y + ω
1−M y³

= χH d t + χεH ε0 with χH =

µ

(

´

χH −1
,
´
M y −ωM d
1−(1−τ y )(1−δ)M y
1 + 1−M y τ y + ωτχ
y H −1
1−M y
³

)

¶
χH −1
µ
¶
and χεH = M −ωM. Define
1−(1−τ y )(1−δ)M yy
1+ 1−M y d τ y +ω
τ y χH −11−M y
1− 1−τ y (1−δ)M y
M y −ωM d
+ω
1−M y
1−M y

63

(D.22)

1−(1−τ y )(1−δ)M y
M d (1−ω)
1−M y + ω
1−M y

³

¡ ¢
g χ ≡

1+

´

χ
´

1−(1−τ y )(1−δ)M y
M y −ωM d
1−M y τ y + ω
1−M y

³

M y −M d1
1−M y + τ y
1
−
=
.
³
´
1−(1−τ y )(1−δ)M y
M y −ωM dτ χ τy 1 +
τ +ω
τ χ
y

y

1−M y

(D.23)

y

1−M y

³ ′´
From (D.20), we know that χ1 = g χ0 with
µ
¶
My
Md
1 − Md
¡
¢
χ0 =
−δ
.
1 − My1 − 1 − τ y (1 − δ) M y M y′(D.24)¡¢¡ ¢From (D.21) and (D.22) we have χk = g χk−1 for all k ∈ {2, · · · , H } . We first find the fixed point of f χ :1−(1−τ y )(1−δ)M y
M d (1−ω)
1−M y + ω
1−M y³

χM SV =

´

χM SV
,
´
1−(1−τ y )(1−δ)M y
M y −ωM dτχ
1 + 1−M y τ y + ω
y
M
SV
1−M y

(D.25)³

which is equivalent to
Ã

ω

!
¡
¢
1 − 1 − τ y (1 − δ) M y

1 − MyÃτ y χ2M SV + χM SV

1+

M y − ωM d
1 − MyÃτy − ω

!!
¡¢1 − 1 − τ y (1 − δ) M y

1 − My−M d (1 − ω)
= 0.
1 − My(D.26)

Let χM SV,1 denote the smaller root and χM SV,2 denote the larger root:
³
³
´´ r³
³
´´
³
´
M y −ωM
1−(1−τ y )(1−δ)M y
1−(1−τ y )(1−δ)M y 2
1−(1−τ y )(1−δ)M y
M y −ωM
M d (1−ω)− 1 + 1−M y d τ y − ω−
1 + 1−M y d τ y − ω+ 4 1−M
ω
τy
1−M y
1−M y
1−M y
y
χM SV,1 =³´
1−(1−τ y )(1−δ)M y
2ω
τy
1−M y
r
³
³
´´
³
´´
³
´
³
M y −ωM
1−(1−τ y )(1−δ)M y
1−(1−τ y )(1−δ)M y 2
1−(1−τ y )(1−δ)M y
M y −ωM
M d (1−ω)
1 + 1−M y d τ y − ω++ 4 1−Mωτy− 1 + 1−M y d τ y − ω
1−M y
1−M y
1−M y
y
.
χM SV,2 =³´
1−(1−τ y )(1−δ)M y
2ω
τ
y
1−M y
(D.27)

If Assumption 1 holds (ω < 1), we know that χM SV,1 χM SV,2 < 0 so χM SV,1 < 0 and χM SV,2 > 0. Note that 1 +
³
´
¡ ¢
¡
¢
¡ ¢
M y −ωM d
1−(1−τ y )(1−δ)M yτ+ωτ y χM SV,1 > 0,38 we have g χ > χ if χ ∈ χM SV,1 , χM SV,2 and g χ < χ if χ ∈
y
1−M y
1−M y
¡ ¢
£
¤
(χM SV,2 , +∞). From (D.23), we also know that g χ increases if χ ∈ χM SV,1 , +∞ .
′Moreover, from (D.24), we know that χ0 ≥ χM SV,1 . To see this, define the left-hand side of (D.26) as
!
Ã
Ã
!!
Ã
¡
¢
¡
¢
¡ ¢
1 − 1 − τ y (1 − δ) M y
M y − ωM d
1 − 1 − τ y (1 − δ) M y
M d (1 − ω)2τy − ω.
h χ =ωτy χ + 1 +χ−
1 − My1 − My1 − My1 − My³´³´M y −ωM d
1−(1−τ y )(1−δ)M y
1−(1−τ y )(1−δ)M y
M d (1−ω)If 1 + 1−M y τ y + ω
τ
χ
<
0,
+
ω
χM SV,1 < 0, and χM SV,1 > 0
y M SV,1
1−M y
1−M y
1−M y
from (D.25), a contradiction.
3864

We have
!!
Ã
Ã
¡
¢
µ
¶
³ ′´
M y − ωM d
τy M y Md
1 − 1 − τ y (1 − δ) M y′
M d (1 − ω)
1 − Md+1+
χ0 −h χ0 = ω−δτy − ω
1 − My My
1 − My1 − My1 − My1 − My
µ
µ
¶¶
µ
¶
My
τy M y′1 − MdMd
M d (1 − ω)
1 − Md= 1+
1 − ωδ
χ0 − ω
−−δ
1 − My1 − My1 − My My
1 − My1 − My³³´´τy M y1−M¶·¸M y 1 + 1−M y 1 − ωδ 1−Mdy µ M
M y 1 − Md
Md
1 − Mdd¡¢−1 − ωδ=−δ
My
1 − My1 − MyMd 1 − M y
1 − 1 − τ y (1 − δ) M y
¡
¢2
¡
¢
µ
¶
·
¸
1 − M y + τ y M y 1 − M y − ωδ (1 − M d ) M d
M y 1 − Md
M y 1 − Md
Md
¡
¢
=
1−δ
−
1 − ωδ
1 − MyMd 1 − M y
1 − MyMd 1 − M y
1 − 1 − τ y (1 − δ) M y³ ′´′
M y 1−M d
M y 1−M d(1−M y )2 +τy M y −τy M y (M y +ωδ(1−Md ))
<
1
−
ωδ
,
we
know
that
h
χ0 < 0 so χ0 ≥
<
1
and
1
−
δ
M d 1−M y
M d 1−M y1−(1−τ y )(1−δ)M y
¡ ¢
© ª∞
′
χM SV,1 . The fact that χ0 ≥ χM SV,1 together with the aforementioned property of g χ means that χk k=0 is a

Sincebounded, monotonic sequence converging to limk→+∞ χk = χM SV,2 > 0.
1−β

If Assumption 2 holds (ω < 1), χM SV,2 > τy . To see this, we have
Ã
!¡
Ã
!!
¡
¢
¡
¢
¢2 Ã
¶
1 − 1 − τ y (1 − δ) M y 1 − β
1 − 1 − τ y (1 − δ) M y 1 − β M d (1 − ω)M y − ωM d
1−β
h
=ω
+ 1+
τy − ω
−τy1 − Myτy1 − My1 − Myτy1 − My
Ã
!
¡
¢
¡
¢
µ
¶
1 − 1 − τ y (1 − δ) M y 1 − β
My
1 − βω M d
1−β
= −βω
+ 1+
τy
−
1 − Myτy1 − Myτy1 − My
´i 
³

¡
¢h
¡
¢
Ã
!
¡
¢
βω
1 − βω 1 − 1 − τ y M y 1 + δ 1−βω
1 − 1 − τ y (1 − δ) M y 1 − β
My
 1−β
< −βω
+ 1 +τy −1 − Myτy1 − My1 − My
τy
!
Ã
!
Ã
¡
¡
¢
¢
1 − 1 − τ y M y (1 − δ) 1 − β
1 − 1 − τ y (1 − δ) M y 1 − β
+ βω
= 0.
= −βω
1 − Myτy1 − MyτyµSimilar to (A.10),
E0 [d t ] =¢1 t −1 ¡Π j =0 1 − τ y χH − j (d 0 + ε0 ) .
t
β

1−β

Since limk→+∞ χk = χM SV,2 > τy , we know that limH →∞ E0 [d H ] → 0. This finishes the proof with the alternative
fiscal policy (7).

D.8 Proof of Theorem 3
As mentioned in the main text, we restrict φ ∈ [−1/σ,

τy
ss

βD
Y ss

). Aggregating individual demand relation (10), to-gether with monetary policy (30), goods and asset market clearing, and the government budget (4) lead to thefollowing aggregate demand relation:
Ã
"
#! µ
"
#
¶
∞ ¡
∞
X
X
¡
¢
¢k ¡
¢
D ss
k
y t = 1 − βω d t + Et(βω) y t +kβω y t +k − t t +k
− σω − (1 − βω) ss βφEt
Y
k=0
k=0

(1−βω)(1−ω)=βω

1−βω)(1−ω) D ss
1 + σφ − (φ Y ssω(d t − t t ) +11−βω)(1−ω) D ss
1 + σφ − (φ Y ssω65£¤
Et y t +1

Together with the baseline fiscal policy (6) we arrive at the following equation:(1−βω)(1−ω)(1 − τd )£¤1
Et y t +1 .
(d t + εt ) +
¢
¡
¢
ss
ss
1−βω)(1−ω) ¡
1−βω
(1−ω)
(
(
)
1 + σφ +1 + σφ +τ y − βφ D
τ y − βφ D
βω
Y ss
βω
Y ss
βω

yt =

Applying the period-t expectation operator Et [·] to (4), we have




ss



1−τd
β

E [d ]
 t £ t +1¤  = 1−βω)(1−ω)(1−τd )Et y t +1−(βω

1 + σφ +τ y −βφ D
Y ss
−
β
³
´
ss
(1−βω)(1−ω) τy −βφ DY ss





βω

d t + εtyt

(D.28)The two eigenvalues are given by the solutions of
¡
¢
µ
¶¶
¡
¢ 1 − τd1 − βω (1 − ω)
1 − τdD ssλ −λτ y − βφ ss + 1 + σφ
+ 1 + σφ +
= 0.
β
βω
Y
β
µ

2

From φ ∈ [−1/σ,

τy
ss

βD
Y ss

(D.29)] and τd ∈ [0, 1], we know that λ1 + λ2 ≥ 0 and λ1 λ2 ≥ 0, so λ1 ≥ 0 and λ2 ≥ 0.

We first prove Part 1 of Theorem 3. That is, if

φ < φ̄ ≡(1−βω)(1−ω) τ

τy
y
ω
¡
¢ (1−βω)(1−ω) D ss < D ss ,β Y ssσ 1−β +
β Y ssω(D.30)

complete self-financing is achieved as the fiscal adjustment is infinitely delayed. That is, there exists a boundedequilibrium of the form (17) with limτd →0+ ρ d ∈ (0, 1) and limτd →0+ ν = 1.
Since the eigenvalue of (D.29) is continuous in τd at 0, we have³
lim λ1 =τd →0+³=ss(1−βω)(1−ω)
1+ 1 + σφ +
τ y − βφ D
β
βω
Y ss

³

´´ r³
´´
³
ss 2
1+σφ
(1−βω)(1−ω)
1++ 1 + σφ +
τ y − βφ D
−4 β
β
βω
Y ss

2
r³
´´
³
´´
´
¡
¢ (1−βω)(1−ω) ³
ss
ss 2
ss
(1−βω)(1−ω)(1−βω)(1−ω)
11
1 + σφ − β+ 1 + σφ +
τ y − βφ D+−
τ y − βφ D
+ 4 1 + σφ
τ y − βφ D
β
βω
βω
βω
Y ss
Y ss
Y ss
³

2

and

³
lim λ2 =τd →0+³=(1−βω)(1−ω)
1
β + 1 + σφ +
βω

³
´´ r³
´
(1−βω)(1−ω) ¡τ − βφ D ss ¢ 2 − 4 1+σφ
1D ssτ y − βφ Y ss −
+
1
+
σφ
+
ss
y
β
βω
Y
β

2
r³
³
´´
´
(1−βω)(1−ω) τ − βφ D ss −(1−βω)(1−ω) ¡τ − βφ D ss ¢ 2 + 4 ¡1 + σφ¢ (1−βω)(1−ω) ¡τ − βφ D ss ¢
1
1
+
1
+
σφ
+
1
+
σφ
−
−
ss
ss
y
y
y
β
βω
Y
β
βω
Y
βω
Y ss
2
(D.31)

When φ ∈ [−1/σ, 0), from (D.31),³
lim λ2 ≤τd →0+(1−βω)(1−ω)
1
β + 1 + σφ +
βω

³
´´ ¯
³
´¯
ss
¯
(1−βω)(1−ω) τ − βφ D ss ¯1
τ y − βφ D
−
+
σφ
−
−
¯1
¯
ss
ss
y
Y
β
βω
Y
2

66

≤ 1 + σφ < 1.

When φ ∈ [0, φ̄), from (D.30), we have
¡
¢
µ
¶
µ
¶
1 − βω (1 − ω)D ss1
τ y − βφ ss > σφ
−1 .
βω
Y
β

Hence
1+σφ

2 β
lim λ2 = ³
r³
´
´
+
τd →0
(1−βω)(1−ω) ¡τ − βφ D ss ¢ +(1−βω)(1−ω) ¡τ − βφ D ss ¢ 2 − 4 1+σφ
1
1
+
1
+
σφ
+
+
1
+
σφ
+
ss
ss
y
y
β
βω
Y
β
βω
Y
β
1+σφ

<

(D.32)

1+σφ

2 β
2 β
¯
¯ = 1.
=r³´2¯
1+σφ ¯ 1+σφ
σφ
1+σφ
1+σφ
1 + β + ¯ β − 1¯
1
+
1
+
+
1
+
−
4
β
β
β
β

Thus, as long as (D.30) holds, a bounded equilibrium in the form of (17) with ρ d = λ2 and χ =1−τd −βρ d>0τywill be a solution of (D.28), with limτd →0+ ρ d < 1. Given (17), one can then back out {πt , d t − Et −1 [d t ]} from+
(14) – (16).
¶ τd →0 ν → 1 follows directly from the definition of ν in (B.16) and the fact that
µ The fact that lim

limτd →0+ τdPkε0 + +∞
k=0 β E 0 [d k ]ε0= 0 given limτd →0+ ρ d ∈ (0, 1).

For Part 2 of Theorem 3, note that, when φ > φ̄, we from (D.30) have
¡
¢
µ
¶
¶
µ
1 − βω (1 − ω)D ss1
τ y − βφ ss < σφ
−1 .
βω
Y
β

Hence, from (D.32),
1+σφβlim λ2 >τd →0+

σφ1
β +1+ β +

2
r³

1+σφ
1+σφ 2
−4 β
1+ β

= 1.

´

As a result, there exists no bounded equilibrium if the fiscal adjustment is infinitely delayed (i.e., if τd → 0 fromabove). For τd > 0, we have39

λ2 = f (a, b) ≡a + b + 1 + σφ −

q¡

¡
¢¢2
¡
¢
a + b − 1 + σφ + 4 1 + σφ b2´ssa+b−(1+σφ)∂f1qτ y − βφ D>
Y ss > 0. Since ∂a = 2 −2
2 (a+b−(1+σφ)) +4(1+σφ)b¡ ¢0, we know that ρ d decreases with τd and limτd →1− λ2 =0. As a result, for φ > φ̄, there exists an τd φ ∈ (0, 1)¡ ¢such that λ2 < 1 if and only if τd > τd φ . As a result, for φ > φ̄, any bounded equilibrium exists if and only if¡ ¢τd > τd φ .

where f (a, b) , a = β1 (1 − τd ) > 0, and b = (391−βω)(1−ω)βω³

The formula for the function f is slightly adjusted compared to (D.11) in the baseline analysis, to accommodate φ ̸= 0.

67

From (14) and (16) ,
ss

χ

Y

1−βρ dκD
X k £ ¤
D ss +∞
χD ss
D ss
Y ss 1−βρ d
β E0 y k = −κ ss
d 0 = − ss π0 = −κ ss(d 0 + ε0 ) = −
ss
χ
Y
Y k=0
Y 1 − βρ d
1 + κ D ss

ε0 .

From the definition (B.16), we know³´ssχ
τ y +κ D
Y ss 1−βρ
χ

ss

ν=

d
where ρ d = 1−τ
β −

β

d
χ

ss

1+
ss
τ y −βφ D
Y ss

d

1+κ D
Y ss 1−βρ

βD
φ 1−βρ
Y ss
ss

χ

d

1+κ D
Y ss 1−βρ

³ ss
´
´
ss
χ
κD
κD
Y ss + τ y 1−βρ d
Y ss + τ y χ=¢ ,¡¢ ss χ =¡ ss
1 + κ + βφ D
τd + κ D
Y ss 1−βρ d
Y ss + τ y χ³dµχ from (D.28). For any φ ∈ φ̄,

τy
ss

βD
Y ss

¡ ¢
ν̄ φ < 1. This proves Part 2 of Theorem 3.

68

¶

³ ss
´
1−τ̄d (φ)
κD
+τ y
ss
Y ss
τ y −βφ D ss
1−τ̄d (φ)
1−τd −βρ d
³ ss
´ Y
,χ=
D ss ≤
D ss and ν ≤
1−
τ̄
d (φ)
τ y −βφ Y ss
τ y −βφ Y ssτ̄d (φ)+ κ D
+τ y
Y ss
D ss
τ y −βφY ss≡