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Federal Reserve Bank of Chicago

The Fiscal Theory of the Price Level
in a World of Low Interest Rates
Marco Bassetto and Wei Cui

November 2017
WP 2017-25
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.

*

The Fiscal Theory of the Price Level in a World of Low
Interest Rates∗
Marco Bassetto†and Wei Cui‡
November 16, 2017

Abstract
A central equation for the fiscal theory of the price level (FTPL) is the government
budget constraint (or “government valuation equation”), which equates the real value of
government debt to the present value of fiscal surpluses. In the past decade, the governments
of most developed economies have paid very low interest rates, and there are many other
periods in the past in which this has been the case. In this paper, we revisit the implications
of the FTPL in a world where the rate of return on government debt may be below the
growth rate of the economy, considering different sources for the low returns: dynamic
inefficiency, the liquidity premium of government debt, or its favorable risk profile.
∗

We are indebted to Gadi Barlevy, John Cochrane, Mariacristina De Nardi, Pedro Gomis-Porqueras, François

Gourio, John M. Roberts, Martı́n Uribe, Christopher J. Waller, Stephen D. Williamson, and Randall Wright for
useful comments and to Matthew Easton for valuable research assistance. Financial support from the ADEMU
(H2020, No. 649396) project and from the ESRC Grant #ES/L500343/1 through the Centre for Macroeconomics
is gratefully acknowledged. Wei Cui is also grateful for the financial support from the Hong Kong Institute for
Monetary Research. The views expressed herein are those of the authors and not necessarily those of the Federal
Reserve Bank of Chicago or the Federal Reserve System.
†
Federal Reserve Bank of Chicago, University College London, and IFS.
‡
University College London and Centre for Macroeconomics.

1

1

Introduction

Models of monetary economies are plagued by the presence of multiple equilibria, which weakens
the ability to make tight predictions.1 To select among them, it has become common to appeal
to what Leeper [25] defined as “active monetary policies.” However, these rules imply local determinacy, but not global uniqueness, and are therefore not universally accepted as an equilibrium
selection criterion.2
An alternative approach to price-level determinacy follows what Leeper [25] defined “active
fiscal policies,” in which the requirement that government debt follows a stable trajectory is used
to select an equilibrium. In particular, since Sims [36] and Woodford [39], this approach is known
as the “fiscal theory of the price level” (FTPL from now on). According to the FTPL, price-level
determinacy follows when the present value of government (primary) surpluses does not react to
the price level in a way that ensures government budget balance; rather, government debt is a
promise to deliver “dollars” (either purely a unit of account, or the underlying currency which
by assumption can be freely printed by the government), and the value of a dollar adjusts in
equilibrium so that the present value of surpluses and the real value of debt match.
Whether the FTPL is a valid selection criterion has been controversial.3 Bassetto [2] analyzes
the issue in a game-theoretic environment, where the price level arises explicitly from the actions
of the agents in the economy and their interaction; in this more complete description of the economic environment, the FTPL remains valid when surpluses are always positive, but it requires
adoption of more sophisticated strategies when the desired equilibrium path involves periods of
net borrowing. This distinction is particularly important in the context of our analysis, because
we find that, across a variety of models, low interest rates are compatible with a stable and positive path for debt only when the government runs primary deficits, at least on average, which
is precisely the environment in which the theory is on weakest ground.
1

This issue is addressed in any graduate textbook in macroeconomics; see e.g. Sargent [33], Woodford [40],

or Ljungqvist and Sargent [26]. Woodford [39] contains an exhaustive description of the nature of equilibrium
multiplicity for a cash-in-advance economy under money-supply and interest-rate rules.
2
See Cochrane [13] for a particularly stinging critique of active rules as a device to achieve uniqueness.
3
Examples of criticisms appear in Buiter [7], Kocherlakota and Phelan [23], and Niepelt [31].

2

Percent
8
6
4
2
0
-2
-4
-6

UnitedStates

UnitedKingdom

Canada

-8

France

Italy

Japan

-10

1955

1960

1965

1970

1975

1980

1985

1990

Germany

1995

2000

2005

Figure 1: Real Interest Rates in the G7 countries (11-year centered moving averages of annual
real interest rates). Reproduced from Yi and Zhang [41].
In this paper, we sidestep the controversy and assume that the government can indeed commit
to a sequence of real taxes, independent of the realization of the price level, but we reassess
whether the uniqueness result of the FTPL continues to hold in economies in which the interest
rate on government debt is persistently below the growth rate of the economy. This question
is motivated by the long decline in real interest rates on government debt from the high values
of the 1980s and early 1990s. As an example, Figure 1 (reproduced from Yi and Zhang [41])
plots the experience of the G7 countries and shows that real interest rates on government debt
below the growth rate of the economy might well be the norm rather than the exception for those
countries. When interest rates fall short of the growth rate of the economy, the present value of
primary surpluses may not be well defined, posing a challenge for the FTPL.
The possibility that low interest rates may affect the interaction between fiscal and monetary
policy was discussed by Darby [16] in the context of Sargent and Wallace’s [35] unpleasant
monetarist arithmetic. However, as emphasized by Miller and Sargent [30], the way in which
real interest rates respond to government debt is important to reach valid conclusions: it is
thus important to analyze economies where interest rates are endogenous. We study three main
reasons why interest rates may be low, and we show that the validity of the FTPL is sensitive

3

to the specific reason.
Our first experiment is the most favorable to the FTPL. It studies a stochastic economy and
analyzes what happens when real rates of return are low because of high risk premia, similarly to
Bohn [6]. In this case, an equilibrium still requires the present value of government surpluses to
be (positive and) well defined and the FTPL remains valid, although without the risk adjustment
expected surpluses may well be negative.
Second, we entertain the possibility that the economy might be dynamically inefficient. In this
context, we show that the FTPL is no longer able to select a unique equilibrium, and multiple
price levels are consistent with an equilibrium even when taxes are set in real terms and do not
adjust. Even then, the FTPL can select a range of equilibrium prices even when monetary policy
is run as in Sargent and Wallace [34] and no prediction would be possible otherwise.
The hypothesis of dynamic inefficiency has been revived by Geerolf [20], who rebuts the
negative evidence in Abel et al. [1]. Nonetheless, other authors have emphasized that the rate of
return on capital has not declined in line with that of government debt.4
Our final explanation for the prevalence of low rates of return on government debt is that it
provides liquidity services, so that it has itself some of the characteristics usually associated with
money. This new environment is described by equations that are very similar to the previous
one, since debt played the role of money there too, but with the important difference that
negative holdings are now ruled out. Restricting our attention to deterministic paths, we can
prove that the FTPL holds if primary surpluses are positive (at least asymptotically), but in
this case we can also show that the interest rate will necessarily exceed the growth rate of the
economy (normalized to zero in our case). When instead primary surpluses are zero or negative,
many equilibria where debt has positive value exist, and our results mimic what happens in
a dynamically inefficient economy. This economy is similar to that analyzed by Berentsen and
Waller [5] and by Domı́nguez and Gomis-Porqueras [19]. Berentsen and Waller find results similar
to ours, but choose to focus on cases in which the fiscal theory holds, while Domı́nguez and GomisPorqueras [19] revisit Leeper’s [25] analysis of active vs. passive monetary and fiscal rules and
4

See e.g. Yi and Zhang [41], Marx, Mojon, and Velde [27], and Del Negro et al. [18].

4

find that the link between Leeper’s original classification and determinacy and uniqueness of an
equilibrium breaks down when government debt plays a liquidity role.5
Concerning the conduct of fiscal policy, these results suggest three main conclusions:
• The ability of the FTPL to select a unique equilibrium when interest rates are low is not
robust across specifications; appealing to fiscal policy to achieve a price level target is thus
fraught with even greater difficulties than those that arise in economies where returns are
such that computing present values is always possible.
• It is difficult to blame an excessively conservative fiscal policy for the recent experience
of low inflation, because, over extended periods of time, low interest rates and stable (or
increasing) government debt levels coexist only when fiscal policy entails primary deficits
on average. Moreover, the presence of multiple equilibria makes it problematic to use
comparative statics to study the effects of fiscal expansions.
• While unsuccessful at uniquely pinning down the price level, the FTPL still provides a
lower bound on prices across all the environments that we consider here, which implies
that it is not completely devoid of content.
In Section 2, we start by purely analyzing the government budget constraint and show how
sustained primary deficits are needed to keep debt positive when interest rates fall short of
the growth rate of the economy. The following three sections derive further implications by
considering specific reasons for low interest rates: risk premia, dynamic inefficiency, or liquidity.
Section 6 summarizes the lessons we draw and suggests future directions for analysis.
5

In a model where the liquidity premium is tied to financing constraints, Cui [14] analyzes the local dynamics

around a steady state with positive primary surplus and finds that Leeper’s [25]’s classification still holds; this
matches with our observation that the FTPL holds when the government always delivers surpluses. Canzoneri
and Diba [10] analyze a different transaction technology, which imposes a lower bound on the equilibrium real
value of government-issued liabilities, no matter how high their opportunity cost is. Such a setup is more favorable
to the FTPL.

5

2

Preliminaries

Before we embark on analyzing the validity of the FTPL in specific models that feature low
interest rates, it is useful to probe the implications of low interest rates just from analyzing the
government budget constraint, an equation that holds across all of the models we will consider.
For simplicity, we will concentrate our analysis on one-period debt. Let Bt be promised nominal
debt repayments by the government due at the beginning of period t, Rt be the nominal interest
rate, Pt be the price level, and τt be real taxes. The government budget constraint is
Bt+1
= Bt − Pt τt ,
1 + Rt

(1)

where we abstract from government spending, since its presence does not affect any of our
subsequent results.6
Since the government tax base is related to the size of the economy, it is convenient to rescale
debt and taxes by real output yt . Defining xt := τt /yt , (1) can be rewritten as


Bt+1
(1 + Rt )Pt yt Bt
=
− xt .
Pt+1 yt+1
Pt+1 yt+1
Pt yt

(2)

In this paper, we are interested in environments in which the real return on government debt
is below the growth rate of the economy. Consider first a deterministic economy; then, the
condition becomes simply
(1 + Rt )Pt yt
< α < 1.
Pt+1 yt+1

(3)

Let the (gross) growth rate be 1 + gt+1 = yt+1 /yt and let the (gross) real interest rate be
1 + rt+1 = (1 + Rt )Pt /Pt+1 . Iterating (2) from period 0 forward and assuming that the economy
starts with positive initial debt, this yields
 X


t 
t−1
t
Y
Bt
B0 Y 1 + rs
1 + rv
=
−
xs
Pt y t
P0 y0 s=1 1 + gs
1 + gv
s=0
v=s+1


t−1
t
X
Y
1 + rv
t B0
<α
−
xs
.
P0 y0 s=0 v=s+1 1 + gv
6

(4)

When government spending is present, our results about the signs of taxes should be reinterpreted as applying

to the sign of primary surpluses instead.

6

If taxes converge asymptotically to some value x̄ and government debt is to remain positive (and
bounded away from zero), equation (4) implies that x̄ < 0. This is intuitive: when the interest
rate is below the growth rate of the economy, the debt/GDP ratio shrinks to zero on its own,
and continuing primary deficits are required to prevent debt from vanishing in the limit (or even
becoming negative). If taxes do not converge to a steady state, equation (4) still implies that a
distributed sum must remain negative in order for government debt not to disappear or become
negative.
In stochastic environments, the real rate of return on government debt may or may not exceed
the growth rate of the economy, and we instead study what happens when the expected return
is low. More precisely, the corresponding version of condition (3) that we study is


(1 + Rt )Pt yt
Et
< α < 1.
Pt+1 yt+1

(5)

We then obtain
(


 X
)
t
t 
t−1
Y
B0 Y 1 + rs
1 + rv
xs
−
P0 y0 s=1 1 + gs
1 + gv
v=s+1
s=0
" t−1
#


t
X
Y
1 + rv
t B0
<α
− E0
xs
.
P0 y 0
1
+
g
v
s=0
v=s+1

Bt
= E0
E0
Pt yt

(6)

In order for the expected debt/GDP ratio to remain bounded away from zero in the limit, we
need
lim E0


#
t
Y
1 + rv
< 0.
xs
1 + gv
s=0
v=s+1

" t−1
X

t→∞

(7)

Equation (6) is slightly more involved than (4), since the expectation about the future sum of
taxes also involves covariance terms with realized rates of return; even in this case, under a
suitable change in measure that takes into account the role of risk premia, the sum of future
taxes needs to be negative to sustain a positive debt/GDP ratio.
Without taking a specific stance on the nature of the economy at hand, pure accounting
implies that, in economies with low interest rates, the government will necessarily run recurrent
primary deficits. As discussed in Bassetto [2], this by itself has significant implications for the
FTPL, since the fiscal strategies that support a unique equilibrium are more involved when the
equilibrium features primary deficits.
7

Following Cochrane’s [12] analogy, when the government always runs primary surpluses, government debt looks like a corporate stock paying a stream of positive dividends, and the price
level can be viewed as the inverse of the value of this stock. Just as any mispricing of the stock
would not require an adjustment in the dividends paid by the corporation, any deviation in the
general level of prices would not require an adjustment in the surpluses that the government
raises to reabsorb the money created when nominal debt is repaid. However, when primary
deficits are part of the picture, the proper analogy is with a corporation that may have trouble
raising fresh funds from investors (such as AIG in 2008): in this case, mispricing of the stock
could force the corporation to alter its investment/divestment plans and thus its future dividends
and likewise would be true in the case of the government and its debt.7
In Sections 4 and 5, we highlight even greater challenges that low interest rates may pose for
the FTPL, but first we consider a more benign case.

3

An Economy with High Risk Premia

We study an economy in which the presence of risk implies that households are happy saving
in the form of government debt for precautionary reasons, even though its expected return falls
short of the expected growth of the economy.
We consider a pure-exchange economy with a continuum of identical infinitely lived agents
and a government, similar to that analyzed by Cochrane [12].8
Private agents can save by buying one-period nominal government debt Bt ; they are subject
to a lower bound on debt holdings

Bt
P t yt

≥ −B, which we assume never to be binding (other than

in the limit, through the transversality condition). The government sets a fixed and constant
nominal interest rate R at which debt is issued. In this and the following sections, we abstract
7

The question whether the AIG crisis in 2008 was the result of a coordination failure or whether it reflected the

true weak fundamentals of the company is beyond the scope of our work. The analogy entertains the hypothesis
that it was at least in part the former.
8
As discussed by Cochrane, the presence of cash is not essential for the results, and we continue to abstract
from it.

8

from any role for money, so that R measures government promises in an abstract unit of account.
However, introducing a role for money does not change any of our results, as long as money is
elastically supplied at the interest rate R; we illustrate this in Appendix B for the economy of
Section 5.
The representative consumer discounts utility at β ∈ (0, 1), pays (real) lump-sum taxes τt ,
and chooses a sequence {ct , Bt }∞
t=0 to solve
max E0

∞
X

βt

t=0

ct1−γ − 1
1−γ

subject to
Pt ct +

Bt+1
+ Pt τt = Pt yt + Bt ,
(1 + R)

taking as given {yt , τt , Rt , pt }∞
t=0 and the initial bond holdings B0 .
Letting πt+1 = Pt+1 /Pt be gross inflation from t to t + 1 and
zt := β t c−γ
t
be the real stochastic discount factor, the first-order condition for the consumer reduces to


1 + R zt+1
Et
·
= 1,
(8)
πt+1
zt
along with the transversality condition,9

lim Et

s→∞


Bt+s zt+s
= 0.
Pt+s (1 + R)s

The government uses direct lump-sum taxes and new debt to repay its existing obligations every
period, subject to the budget constraint (1). In equilibrium, the transversality condition is
necessary to ensure that consumers are willing to hold debt, which leads to an intertemporal
budget constraint (the core equation of FTPL)

∞ 
X
Bt
zt+s
= Et
τt+s .
Pt
zt
s=0

(9)

This economy is a textbook version of the models used to illustrate the FTPL. The present
value of future primary surpluses must be well defined in a competitive equilibrium, since the
9

For a discussion of the necessity of transversality conditions in stochastic environments, see Kamihigashi [22].

9

transversality condition is necessary for household optimization. If taxes are set exogenously in
real terms, the present value pins down the price level both in period 0 and in all subsequent
periods, as long as the present value itself is positive as of time 0 (otherwise no equilibrium would
exist with B0 > 0).
The only difference with the standard treatment is the observation that equation (9) does not
rule out the possibility that (5) and (7) hold as well.10 In particular, while the present value of
taxes has to be positive, it is quite possible that E(τt+s ) is negative in all periods. We illustrate
this possibility in Appendix A by considering a specific endowment process and fiscal policy for
which government debt is risk free in real terms; equation (5) holds, and expected taxes are
always negative.

4

Dynamic Inefficiency

While the present value of government surpluses was well defined in the previous section, we
now turn to environments where this is no longer the case. In this section, we illustrate the
implications of the FTPL in dynamically inefficient economies. We work with an overlappinggenerations (OLG) economy where people live for two periods, based on Sargent [33], chapter
7. This setup is useful to obtain analytical results that readily generalize to more complex
environments in which other frictions imply dynamic inefficiency.
We consider a pure exchange economy populated by overlapping generations of constant size,
normalized to 1. Each generation lives for two periods. To keep the notation simple, we assume
that the endowment is constant over time, and we abstract from any uncertainty. The income
of each household is wy when young and wo when old. Households have preferences given by
U (cyt , cot+1 ),

(10)

where cyt (cot ) is consumption by the young (old) in period t. The only asset in the economy is
one-period government debt, as introduced in Section 2. In period 0, there is an initial stock of
10

On this point, see Bohn [6] as well.

10

nominal debt B0 . As in the previous section, the government sets a constant nominal interest
rate R. Taxes are raised in a real amount τt from the old;11 this is a transfer if τt < 0.
The budget constraint for the generation born in period t is given by
Pt cyt +

Bt+1
≤ Pt w y ,
1+R

Pt+1 cot+1 ≤ Pt+1 (wo − τt+1 ) + Bt+1 .

(11)
(12)

The old cohort in period 0 simply consumes all of its after-tax endowment and its savings:
P0 co0 = P0 (wo − τ0 ) + B0 .

(13)

With a constant nominal rate R, the real interest rate in the economy between periods t and
t + 1 is
1 + rt+1 := (1 + R)

Pt
,
Pt+1

(14)

and the problem of maximizing (10) subject to (11) and (12) yields a (real) saving function
f (1 + rt+1 ). We assume that U is strictly increasing in both arguments, strictly quasiconcave,
continuously differentiable, that consumption when young and old are gross substitutes, and that
Inada conditions apply. Under these assumptions, f is strictly increasing in the real interest rate,
which allows us to invert the function and obtain the equilibrium real interest rate as a function
of savings by the young: rt+1 = r(st+1 ), where st+1 = cyt − wy . Since the growth rate of this
economy is normalized to zero, we are interested in the case in which f (1) > 0, or, equivalently,
r(0) < 0:12 young households have a sufficient need to save for their old age that they are willing
to do so even at a zero interest rate, which allows for dynamically inefficient equilibria with
positive debt to arise. The government budget constraint in period t is given by (1).
A competitive equilibrium of this economy is given by a sequence {cyt , cot , Pt , rt , Bt+1 }∞
t=0 , such
that households maximize their utility subject to their budget constraints, the government budget
11

Our results do not depend on the way taxes are allocated across generations, as long as the allocation is fixed

over time. Of course, the range of parameters for which dynamic inefficiency arises, with its implications for the
FTPL, does depend on this allocation.
12
If preferences satisfy Inada conditions, this will necessarily happen provided the endowment when young is
sufficiently larger than the endowment when old.

11

constraint holds in each period t, the definition of real rate (14) applies, and markets clear, i.e.,
Pt f (1 + rt+1 ) = Bt+1 /(1 + R).

(15)

We compute competitive equilibria following Sargent [33]. Define πt+1 := Pt+1 /Pt to be the
gross inflation rate between t and t + 1. Combining equations (1), (14), and (15), an equilibrium
must satisfy the following difference equation:
f (1 + rt+1 ) = (1 + rt )f (1 + rt ) − τt ,

t≥1

(16)

with an initial condition
f (1 + r1 ) =

B0
− τ0 .
P0

(17)

It is easier to analyze this equation in terms of the savings by the young, which gives
st+1 = (1 + r(st ))st − τt

t≥1

(18)

and
s1 =

B0
− τ0 .
P0

(19)

We concentrate our attention on the case of constant taxes, τt = τ , t ≥ 0, in which the
difference equation is time invariant and clearer results can be established analytically.
Proposition 1.

• If τ > 0, the difference equation (18) admits exactly two steady states,

one with positive and one with negative savings.
• If τ = 0, the difference equation (18) admits exactly two steady states, one with zero and
one with positive savings.
• If τ < 0, there generically exists an even number of steady states, all of which feature
positive saving. If τ is sufficiently close to zero, the number of steady states is two; and if
it is sufficiently negative, it is zero (no steady states exist).
Proof. A steady state s̄ requires r(s̄)s̄ = τ . The Inada conditions imply lims→wy r(s) = ∞
and lims→−∞ r(s) = −1, which, in turn, means that lims→wy r(s)s = lims→−∞ r(s)s = ∞.
Furthermore,
d[r(s)s]
= sr0 (s) + r(s).
ds
12

Thus, sr(s) is monotonically increasing when both s and r(s) are positive, and monotonically
decreasing when they are both negative. sr(s) = 0 when either s = 0 or r(s) = 0. Since r(0) < 0
and r is increasing, r(s) = 0 occurs exactly at one point, at a value ŝ > 0, which proves the
statement for the case in which τ = 0. For τ > 0, the derivative and limits above imply that there
exists exactly one steady state in (−∞, 0) and one in (ŝ, ∞), which again proves the relevant
statement. Finally, for τ < 0, we know sr(s) > τ for all values of s ∈ (−∞, 0] ∪ [ŝ, wy ). By
continuity, the number of steady states in the interval (0, ŝ) must be generically even.13 We can
also remark that, again by continuity, the number of steady states for τ < 0 but sufficiently
close to zero must be exactly two (as in the case of τ = 0), and no steady state will exist if τ is
sufficiently negative, since sr(s) attains an interior minimum, which completes the proof.
To complete the characterization of the equilibria, the following proposition analyzes the
dynamics of the system away from the steady state.
Proposition 2.

• If no steady state of the difference equation (18) exists, then st+1 is mono-

tonically increasing and it would eventually exceed wy , where the difference equation ceases
to be defined: hence, no competitive equilibrium exists.
• Let there be 2N steady states, ordered (s̄1 , . . . , s̄2N ). If the initial saving rate is s1 < s̄2 ,
then st converges monotonically to s̄1 . If N > 1 and s1 ∈ (s̄2k , s̄2k+2 ),

k = 1, ...N − 1, st

converges monotonically to s̄2k+1 . Finally, if s1 > s̄2N , then st is monotonically increasing
and eventually exceeds wy , which implies that no equilibrium exists for such an initial
condition.
Proof. First, lims→wy r(s)s = ∞ (or lims→−∞ r(s)s = ∞), along with continuity, implies that
st+1 > st always if no steady state exists. In this case, equilibrium would require a monotonically
increasing sequence {st }∞
t=1 , which cannot converge, because any convergence point would have
y
to be a steady state. Since {st }∞
t=1 is bounded by the endowment w , a contradiction ensues,

and no equilibrium can exist.
13

For a measure-zero set of values of τ , sr(s) will have a tangency point to τ , in which case an odd number of

steady states can occur. Proposition 2 applies to this case if one interprets the tangency point as two coincident
steady states.

13

When 2N steady states exist, the same limits imply st+1 > st when st < s̄1 , st > s̄2N , or
st ∈ (s̄2k , s̄2k+1 ), k = 1, . . . , N − 1. Conversely, st+1 < st when st ∈ (s̄2k−1 , s̄2k ), k = 1, . . . , N .
Furthermore, given any steady state s̄i , we have
st+1 = st + r(st )st − τ = st + r(st )st − r(s̄i )s̄i .
For st > s̄i ,
st+1 = st + r(st )st − r(s̄i )s̄i > st + r(s̄i )(st − s̄i ) > s̄i ,
with the converse being true for st < s̄i . Hence given any initial condition s1 , the sequence of
saving will never “jump over” a steady state, but rather it will monotonically converge to an
odd-numbered steady state. The exception is the case in which s1 > s̄2N , where the sequence
increases monotonically up to the point at which no solution exists.
Propositions 1 and 2 describe the behavior of the economy given s1 ; more specifically, provided
τ is not too negative, they show that a continuum of values of s1 ∈ (−∞, s̄MAX ] is consistent
with a competitive equilibrium, where s̄MAX is the largest steady state of the difference equation.
While in the representative-agent economy it is impossible for debt to have positive value when
τ ≤ 0 for sure, this is not the case in the OLG economy, where dynamic inefficiency implies that
money (or unbacked debt) can have value: hence, provided τ is not too negative, the economy
can still admit positive values of debt.
Figures 2–4 illustrate the evolution of the economy for the simplest case in which st+1 is a
convex function of st , so that there are exactly two steady states. When τ > 0 (Figure 2), one
steady state features positive saving, whereas the other one has negative saving. The steady state
with positive saving (s̄2 ) corresponds to the traditional solution that would apply in a standard
treatment of the FTPL: with constant endowment, a representative agent, and lump-sum taxes,
the real interest rate would be independent of government debt, and the solid line would become
straight and steeper than the 45-degree line. In this steady state, the interest rate is positive.
When household saving is at s̄2 , taxes are exactly sufficient to pay interest on government debt.
For higher household saving (and hence higher government debt), taxes are insufficient to repay
interest and debt explodes. Vice versa, for lower saving, taxes are more than enough and debt
14

st+1

s̄2

(0,0)

st

s̄1

Figure 2: Evolution of st when τ > 0.
decreases. While this is true both in the representative-agent and in the OLG economy, in
the latter the interest rate drops as st becomes lower, flattening the relationship between st+1
and st . Eventually the solid line is flatter than the 45-degree line and a second, stable steady
state emerges. In this steady state, young households are borrowing from the government at
a negative interest rate, so that their repayment in old age is smaller than what they receive
from the government when young. Taxes are used by the government to replenish its assets and
be able to lend the same amount to the next generation. Any value of s1 ∈ (−∞, s̄2 ] leads to
a stable evolution going forward: the economy always converges to s̄1 , except if it starts (and
stays) at s̄2 .
When taxes are zero, the solid line shifts to the position indicated by Figure 3. Here, the
interest rate in the higher steady state s̄2 is exactly zero, but government debt still has value

15

st+1

s̄2

s̄1
(0,0)

st

Figure 3: Evolution of st when τ = 0.
as fiat money: it is passed from one generation to the next at a constant price. Any initial
value above s̄2 would imply a positive interest rate, so that debt would explode in the absence of
taxes, which is inconsistent with an equilibrium, since household saving is bounded above by wy .
Conversely, any lower value implies a negative interest rate; in the absence of taxes or transfers,
household saving shrinks to zero over time, where the lower steady state s̄1 is located.
Finally, if τ < 0, but not too large, Figure 4 emerges: in this case, both steady states feature
positive saving and a negative interest rate, so that the government repays to the households
less than it borrowed, but the additional transfer payments make up for the negative interest. If
τ is too negative, then the solid line is always above the 45-degree line, and no equilibrium can
exist, since debt would always explode.
In order to fully characterize the set of competitive equilibria of the economy, one last step is

16

st+1

s̄2

s̄1
(0,0)

st

Figure 4: Evolution of st when τ < 0.
required: we need to relate the starting level of saving s1 to the initial price level, P0 . We obtain
B0
= τ + s1 .
P0
Assuming that the economy starts with positive values of government debt and that s̄MAX +τ > 0
(s̄2 + τ > 0 in the figures), the requirement that s1 ∈ (−∞, s̄MAX ] implies that there exists a
continuum of equilibria indexed by the initial price level, with14
P0 ∈ [(s̄MAX + τ )/B0 , ∞).
14

(20)

The condition s̄MAX + τ > 0 is an implicit characterization, since s̄MAX is itself a function of τ . However,

from proposition 1, we know that s̄MAX > 0, and it is straightforward to prove that it is a decreasing function of
τ . Hence, s̄MAX + τ > 0 will always hold for τ ≥ 0, and also for τ < 0, provided it is not too negative. Finally,
note that there is a range of negative values of τ for which steady states exist but s̄MAX + τ ≤ 0: for these values,
competitive equilibria would exist for negative initial amounts of debt, but none exists if B0 > 0.

17

Equation (20) proves that the FTPL breaks down in our OLG economy: even with a fixed nominal
interest rate and a fixed amount of real taxes, a continuum of possible price levels emerges. In
Woodford [39], the FTPL explicitly appears as a way of selecting among equilibria in monetary
economies, to remedy the fact that the monetary side alone typically does not achieve global
uniqueness of an equilibrium. However, in OLG economies in which dynamic inefficiency is
possible, government debt itself is akin to money, and hence the original multiplicity reemerges.
With a continuum of possible equilibrium price levels, we cannot rely on comparative statics
for tight predictions on the inflationary consequences of lowering τ , providing a cautionary tale
for using fiscal policy as a substitute for monetary policy to achieve a desired price target. While
uniqueness fails in our environment, equation (20) still imposes a lower bound on prices, so that
the FTPL is not completely devoid of content even in a dynamically inefficient economy.
Remark 1. A possible criticism of the analysis above is that the general specification of the
FTPL does not require that a sequence of real taxes should be independent of the initial price
level, but rather that the present value of the sequence should be. With the possibility of dynamic
inefficiency, computing present values becomes problematic, because they may not be well defined.
However, we can easily construct an example in which even this version of the FTPL fails.
Specifically, consider a sequence in which τ0 > 0 and τt = 0, t > 0. For this sequence, the present
value of taxes is always τ0 , independent of the initial price level, and yet there exists a continuum
of equilibria indexed by
P0 ∈ [(s̄MAX
τ =0 + τ0 )/B0 , ∞),
where the subscript to s̄MAX makes it explicit that it refers to the (maximal) steady state of the
economy with no taxes (since taxes will indeed be zero from period 1 onwards).
Remark 2. The two-period OLG economy and our assumptions on f imply relatively that all
deterministic equilibria feature monotone dynamics. By relaxing these assumptions, it would be
possible to obtain cycles or even chaotic dynamics. An early survey of these possibilities appears
in Woodford [38]. Furthermore, in a stochastic environment, sunspot equilibria would emerge.
Our conclusion is robust to these more exotic environments: specifying an exogenous path for
real taxes is insufficient to pin down the initial price level, and the FTPL breaks down.
18

5

Debt as a Source of Liquidity

In the previous section, government debt bears a low rate of return because households have a
large desire to save and there are no other assets that would allow them to earn a better rate of
return.15 Here, we study an economy which is dynamically efficient and in which private assets
pay a rate of return that is higher than the growth rate of the economy (normalized to zero);
government debt pays a lower rate of return instead because it plays a special liquidity role,
allowing some transactions that cannot be completed by exchanging private assets.
In common with the previous section, government debt has itself the characteristics of money,
and the only difference is the role money plays. This leads to many similarities, but also to
some important differences. When government debt is an asset like any other, as in the OLG
context, equilibria in which households borrow from the government are possible. However, when
government debt provides a special liquidity service, this service is predicated on the government
being the debtor and the households being creditors. We prove below that this restores the
validity of the FTPL for τ > 0, albeit it also implies that interest rates below the growth rate of
the economy are impossible in this case.
We develop our analysis in the context of the model developed by Lagos and Wright [24]; the
analysis would be similar if we considered a cash-in-advance model, or alternative models where
debt facilitates transactions and/or relaxes liquidity constraints.16
15

As is well known, similar results apply even if physical capital was present, as long as it is subject to a

sufficiently decreasing rate of return.
16
One important assumption that is implied by the Lagos-Wright model is that the demand for governmentissued means of transacting vanishes as their opportunity cost is sufficiently high. The importance of this assumption is well understood from the analysis of monetary models. In models in which the value of liquidity services
may become unbounded, the indeterminacy we describe here may not arise. In the context of monetary models,
this is what happens when preferences for cash goods are unbounded from below (see e.g. Woodford [39]). To
mention an extreme example, if money velocity is unresponsive to the interest rate and output is exogenous, then
real balances are fixed and the nominal quantity of money automatically pins down the price level. An example in
which this channel is at work is Canzoneri and Diba [10], where velocity is variable but bounded above. Assuming
that the utility from liquidity services is unbounded from below (or, equivalently, that velocity of the liquidity
instruments is bounded above, no matter the cost) strikes us as implausible: if households faced unboundedly

19

5.1

The Basic Environment

We consider an economy populated by a continuum of identical infinitely lived households and
a government. Each period is divided into two subperiods.
In the first subperiod (the “morning”), households disperse in bilateral anonymous markets,
where they have an opportunity to buy a good that they like with probability χ ∈ (0, 1), and they
have an opportunity to produce the good that the other party likes with the same probability.
Double-coincidence meetings are ruled out. In these meetings, private credit and privately issued
assets cannot be recorded and/or recognized: only government debt can be used in exchange for
the desired good. We assume that buyers make a take-it-or-leave-it offer to the sellers, which,
given the preferences below, is equivalent to competitive pricing.17
In the second subperiod (the “evening”) a centralized market opens, where a good is traded
that all households value and can produce. In this market, a record-keeping technology is present,
and households can trade the evening good, privately issued claims, and government debt. The
government levies taxes according to an exogenous real sequence {τt }∞
t=0 , repays maturing debt,
and supplies new nominally risk-free debt at a set interest rate R. In Appendix B, we add money
paying no interest, so that R is the opportunity cost of holding money vs. government debt.
Preferences of each household are given by
E0

∞
X

β t [u(qt ) − nt + ct − yt ] ,

t=0

where qt is the consumption of the morning good, u is strictly increasing and strictly concave
with u(0) = 0, ct is the consumption of the evening good, and nt and yt are the production of
the morning good and evening good, respectively. Production enters negatively in the utility
function because it requires labor effort. We assume that there exists q ∗ ∈ (0, +∞) such that
u0 (q ∗ ) = 1. We will concentrate on equilibria where aggregates are deterministic, hence the
large costs from acquiring government-issued means of exchange, they would presumably find alternatives to
money or government bonds, such as other currencies, gold, cigarettes, etc. In the context of the Lagos-Wright
model, such an alternative is implicitly represented by having finite utility, even without consuming any of the
goods supplied in the market for which money or government bonds are essential.
17
Similar results would apply to different bargaining protocols.

20

expected value is taken with respect to the idiosyncratic history of matches encountered by a
household.

5.2

Characterizing the Economy

Let Wt (b, a) be the value for an agent who enters the centralized market with b units of bonds
and a units of private claims, and let Vt (b, a) be the value of the same position upon entering
the decentralized market. We have
Wt (b, a) = max
{c − y + βEt Vt+1 (b0 , a0 )} s.t.
0 0

(21)

a0
b0
+
≤ Pt (y − τt ) + a + b,
1 + Rtp 1 + R

(22)

c,y,b ,a

Pt c +

a0 ≥ −aPt ,

(23)

b0 ≥ 0.

(24)

and

Pt is the price of goods in term of nominal claims in the centralized market, and Rtp is the
nominal interest rate on private claims between periods t and t + 1. Equation (23) imposes a
borrowing limit on households so that they cannot engage in Ponzi schemes; equation (24) bans
naked short selling of government debt. While privately-issued claims and government bonds are
perfect substitutes when entering into the centralized market, the two assets are different in the
decentralized market, and equation (24) formalizes the constraint that only the government can
issue the latter.
The solution to maximizing (21) subject to (22), (23), and (24) yields
Wt (b, a) = Ŵt +

a+b
,
Pt

(25)

where Ŵt depends on subsequent choices, but is independent of the current state. When buyers
and sellers meet in the decentralized market, in any meeting in which buyers exchange b̃ units
of government bonds for q̃ units of goods, equation (25) implies that the sellers’ participation
constraint is
− q̃ + b̃/Pt ≥ 0.
21

(26)

Since buyers make a take-it-or-leave-it offer, equation (26) is equivalent to a “bonds-in-advance”
constraint in which buyers can purchase the good at the linear price Pt (the same price that will
prevail in the centralized market), up to their endowment of government bonds.
Using equation (26), and exploiting the fact that all the surplus from decentralized trade is
appropriated by the buyers, the value function at the beginning of the period, before it is known
who will be a seller, who will be a buyer, and who will not trade, is given by
Vt (b, a) = Wt (b, a) + χ max[u(q) − q] = Ŵt +
q

a+b
+ χ max[u(q) − q],
q
Pt

(27)

subject to
Pt q ≤ b.

(28)

∂Vt (b,a)
(b,a)
= P1t for any b > Pt q ∗ and ∂Vt∂b
= χu0 (b/Pt ) + (1 −
∂b
(b,a)
that ∂Vt∂a
= P1t always. Taking this into account, the necessary

It is straightforward to verify that
χ)(1/Pt ) for any b < Pt q ∗ , and

and sufficient conditions for optimality in maximizing (21) subject to (22), (23), and (24) are
given by

β(1 + Rtp )Pt
=1
Pt+1

(29)


 
 

Bt+1
Pt+1
0
− 1 = χ max u
,1 − 1 .
β(1 + R)Pt
Pt+1

(30)

and

Unless households are satiated with liquidity (Bt+1 /Pt+1 ≥ q ∗ ), the rate of return on government
bonds will be lower than the one on private assets, and it could even be negative in real terms.

5.3

Does the FTPL Hold when Debt Provides Liquidity?

We can invert equation (30) to obtain the demand for real bonds as a correspondence of the real
interest rate on government debt, which yields the same equation as (15) for the OLG economy,
with f defined as

f (1 + rt+1 ) :




=

1+R
(u0 )−1
1+rt+1



1
−(1−χ)
β(1+rt+1 )

χ



∈ [β(1 + R)q ∗ , ∞)

22


for rt+1 < 1/β − 1
(31)
for rt+1 = 1/β − 1.

Since we are interested in equilibria where the real rate on government debt is below the growth
rate of the economy (zero, in our case), a necessary condition is that f (1) > 0, as in the OLG
economy, and we assume this to be the case: the demand for liquidity must be enough for
households to be willing to hold government debt even at a zero real interest rate. As in the
OLG economy, we assume f to be increasing in r.18 Furthermore, no equilibrium is possible if
rt+1 > 1/β − 1, since households would have an incentive to accumulate indefinite savings at
that interest rate.
With this definition of f , an equilibrium is characterized by the same difference equation as
in the previous section, which is (16) and (17) in terms of the interest rate and (18) and (19) in
terms of real purchases of government bonds.
While the equilibria of the two economies are described by the same difference equation, so
that several results will be similar, nonetheless some important differences must be noted:
• The domain of st is different: in an OLG economy, st can take values in (−∞, wy ), while
in the liquidity economy its values must be contained in [0, ∞). While the difference at the
top has no effect on the results, the inability to borrow from the government has important
implications for equilibrium selection. This is not surprising, since we encountered in the
previous section an instance in which the stable steady state of the difference equation is
negative.
• In the OLG economy, r(st ) approaches infinity as st → wy ; in contrast, here r(st ) is
constant at 1/β − 1 when st ≥ β(1 + R)q ∗ .
• In the OLG economy, r(st ) is finite at st = 0. In contrast, r(st ) may approach −1 at st = 0
if u satisfies Inada conditions.
To characterize the set of equilibria, we now study the steady states and convergence properties
of the difference equations, as we did in Propositions 1 and 2 for an OLG economy.
18

As an example, if u exhibits constant relative risk aversion γ, a sufficient condition is that γ < 1, which is

already necessary to obtain u(0) = 0. See the discussion in footnote 16.

23

Proposition 3.

• If τ > 0, the difference equation (18) admits exactly one steady state, with

positive savings.
• If τ = 0, the difference equation (18) admits exactly two steady states, one with zero and
one with positive savings.
• If τ < 0, there generically exists an even number of steady states, all of which feature
positive saving. If τ is sufficiently close to zero, the number of steady states is two; and if
it is sufficiently negative, it is zero (no steady states exist).
Proof. As in Proposition 1, a steady state s̄ requires r(s̄)s̄ = τ , and sr(s) is monotonically
increasing when both s and r(s) are positive. We also know r := lims→0 r(s) ≥ −1, with equality
if limq→0 u0 (q) = ∞, and r(s) = 1/β − 1 for s ≥ βq ∗ (1 + R), so that lims→∞ sr(s) = ∞. sr(s) = 0
when either s = 0 or r(s) = 0. From f (1) > 0 we know r(0) < 0, so r(s) = 0 occurs exactly
at one point, at a value ŝ > 0, which proves the statement for the case in which τ = 0. For
τ > 0, the monotonicity properties of sr(s) and its limits imply that there exists exactly one
steady state, in (ŝ, ∞), which again proves the relevant statement. Finally, for τ < 0, we know
sr(s) > τ for all values of s ∈ [ŝ, ∞), and at s = 0. By continuity, the number of steady states in
the interval (0, ŝ) must be generically even.19 Continuity also implies the remaining properties
of the last bullet.
Away from a steady state, the dynamics are described in the following proposition:
Proposition 4.

• If no steady state of the difference equation (18) exists, then st+1 is mono-

tonically increasing and government debt eventually explodes exponentially, violating a
household’s transversality condition, which cannot happen in an equilibrium.
• If τ > 0, such that a unique steady state s̄ exists, then for s1 > s̄, {st+1 }∞
t=0 is monotonically
increasing and government debt eventually explodes exponentially, violating a household’s
transversality condition. For s1 < s̄, {st+1 }Tt=0 is monotonically decreasing until sT +1 [1 +
r(sT +1 )] − τ < 0, at which point the difference equation no longer has a solution.
19

Footnote 13 applies here as well.

24

• Let there be 2N steady states, ordered (s̄1 , . . . , s̄2N ). If the initial saving rate is s1 < s̄2 ,
then st converges monotonically to s̄1 . If N > 1 and s1 ∈ (s̄2k , s̄2k+2 ),

k = 1, ...N − 1,

st converges monotonically to s̄2k+1 . Finally, if s1 > s̄2N , then government debt eventually
explodes exponentially, violating a household’s transversality condition.
Proof. First, lims→∞ r(s)s = ∞, along with continuity, implies that st+1 > st always if no steady
state exists. In this case, equilibrium would require a monotonically increasing sequence {st }∞
t=0 ,
which cannot converge, since any convergence point would have to be a steady state. Once
st ≥ βq ∗ (1+R), the difference equation becomes st+1 = st /β −τt+1 , so that limt→∞ st+1 /st = 1/β.
The household transversality condition requires



Bt
Bt ∗
t 0
lim β u max
,q
= 0.
t→∞
Pt
Pt
For st ≥ βq ∗ (1 + R),

Bt+1
Pt+1

≥ q ∗ , hence exponential growth in st at a rate β would not be optimal

from the households’ perspective. The same reasoning applies if τ > 0 and s1 > s̄, or if τ ≤ 0
and s1 ≥ s̄2N .
When 2N steady states exist, continuity and the boundary properties of sr(s) at s = 0
and s = ∞ imply st+1 > st when st < s̄1 , st > s̄2N , or st ∈ (s̄2k , s̄2k+1 ), k = 1, . . . , N − 1.
Conversely, st+1 < st when st ∈ (s̄2k−1 , s̄2k ), k = 1, . . . , N . The rest of the proof is identical to
Proposition 2.
While in the OLG economy a continuum of initial values of s1 is consistent with an equilibrium
(provided τ was such that an equilibrium exists), for the economy with liquidity this is only true
if τ ≤ 0. When τ ≤ 0, Figures 3 and 4 are still valid, provided we only concentrate on the
upper-right quadrant. In this case, the FTPL breaks down in the same way it did in the OLG
economy. A continuum of values of s1 ∈ [0, s̄MAX ] is consistent with a competitive equilibrium.
Unless the economy starts at the highest steady state s̄MAX , it converges to a lower steady state,
which involves positive debt if τ < 0 and no debt if τ = 0. Correspondingly, if the economy starts
with positive values of government debt and s̄MAX + τ > 0, a continuum of initial levels of prices
is consistent with an equilibrium, as described by equation (20); the same considerations about
comparative statics and the presence of a lower bound for prices that we discussed in Section 4
25

apply here as well. All of these results are reminiscent of the properties of equilibria with moneysupply rules in cash-in-advance economies, as in Matsuyama [28, 29] or Woodford [39]. This is
not surprising, since debt plays the same role as fiat money for this environment.20

st+1

s̄

(0,0)

st

Figure 5: Evolution of st when τ > 0.
When τ > 0, the dynamics of the OLG economy and the economy in which debt provides
liquidity services differ. The restriction to st > 0 bites in this case, as shown by Figure 5:
now all trajectories that do not start at the positive steady state s̄ end up violating either
the nonnegativity constraint or the transversality condition, hence s̄ is the unique solution to
the difference equation when debt provides liquidity services. Hence, when τ > 0, equilibrium
20

Remarks 1 and 2 apply to this section as well. In addition to the papers by Matsuyama and Woodford, the

potential for complicated dynamics is analyzed by Rocheteau and Wright [32] in an environment closer to ours.

26

requires s1 = s̄ and we can recover the price level uniquely from the condition
B0
= τ + s1 .
P0

(32)

However, when τ > 0, the steady state is such that s̄r(s̄) > 0, which implies r(s̄) > 0: the
real interest rate on government debt is necessarily positive (above the zero growth rate of the
economy). The FTPL holds in this case, but it negates the premise of our paper.
Our analysis has important implications when evaluating the effectiveness of fiscal policy to
fight deflation. According to the standard FTPL, lowering τ will increase prices.21 Hence, a
commitment to smaller fiscal revenues will lead to an immediate jump to higher prices. This
policy prediction ceases to be true in an economy in which government debt offers liquidity
services and the real interest rate is negative.22 As discussed in Section 2, observing positive
debt and a persistently negative interest rate in this economy is by itself evidence that households
already expect primary deficits, at least in the long run, and that government debt retains positive
value only because it also provides liquidity services.23

6

Conclusion

The FTPL is not a robust equilibrium selection criterion when the interest rate is persistently
below the growth rate of the economy: whether the theory does or does not hold depends on
the specific economic forces that lead to low rates. In this paper, we have shown three broad
classes of models in which government bonds feature low returns (or low expected returns); the
situation is further complicated by the possibility that these reasons interact with each other.
21

In our perfect foresight economy, lowering τ corresponds to a surprise revaluation of government debt, and is

subject to Niepelt’s [31] criticism. However, the same result applies in an economy in which τ is ex ante stochastic
and we are comparing across different realizations. See Daniel [15].
22
More precisely, what is relevant is whether the interest rate is negative asymptotically, since our proofs rely
on limiting dynamics of the difference equation.
23
A knife-edge situation arises for time-varying paths of taxes in which τt > 0 (at least asymptotically) but
taxes decay exponentially to zero. Adapting Proposition 1 in Tirole [37], one can then prove that there exists
a unique equilibrium, even though the interest rate is asymptotically negative. The FTPL would hold in this
knife-edge case. We are indebted to Gadi Barlevy for pointing this out.

27

As an example, government debt might have a specific liquidity role because of its favorable risk
profile; see Caballero and Farhi [8]. In turn, the greater liquidity role of debt in recessions might
limit the need for procyclical fiscal policy to support the debt’s favorable risk profile.24
In this paper, we have concentrated on stationary environments. As Figure 1 showed, the real
return on bonds has varied a great deal in the past decades, and it is possible that interest rates
will exceed the growth rate again in the future. In such richer environments, the validity of the
FTPL would depend on the frequency and duration of low-rate episodes, in ways that could be
analyzed using a regime-switching model such as Chung, Davig, and Leeper [11] and Davig and
Leeper [17]. However, to the extent that policymakers are not confident about their ability to
estimate the true stochastic process of interest rates at a secular frequency, our analysis suggests
caution in relying on fiscal surprises to manage inflation.

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32

Appendix A

A Positive Net Present Value with Negative
Expected Taxes

To discuss a specific example of Section 3, we posit that the (log) endowment grows stochastically
over time according to:
ln yt − ln yt−1 = ln ∆ + εt ,

(33)

where εt is independent across time, with an exponential distribution with coefficient λ.
By using (33) and imposing market clearing (yt = ct ), we rewrite the first-order condition (8)
at time t as
1 =β(1 + rt+1 )∆−γ Et [exp{−γεt+1 }] .
It then follows that
1 + rt+1 =

∆γ (γ + λ)
.
βλ

(34)

In this economy, the level of the real interest rate is independent of government policy, as long
as an equilibrium exists. Given the real interest rate computed above, equation (5) is satisfied if
and only if
∆γ−1 (γ + λ)
< 1.
β(λ + 1)

(35)

In order for the household problem to be well defined, parameters must be such that the utility
of consuming the endowment must be finite, which requires
λβ∆1−γ < (γ + λ − 1).

(36)

Equations (35) and (36) are mutually compatible only if γ > 1, i.e., when agents are sufficiently
risk averse; in this case, there is a range of values for ∆ and λ such that the downside risk (the
inverse of ∆) and volatility (λ) are sufficiently elevated that (35) hold, but not as large as would
yield infinitely negative utility for the household.
Consider taxes next. We study a tax rule in which xt (= τt /yt ) is a function x(εt ) of the
contemporaneous shock εt only and government debt is risk free in real as well as in nominal

33

terms. In order for this to be the case, the price level Pt+1 must be time-t measurable. From
equation (9),
Pt+1 =

Bt+1
P
x(εt+1 )yt+1 + Et+1 ∞
s=2

=


yt+1

zt+s
τ
zt+1 t+s

Bt+1
.
P∞ s−1  yt+s 1−γ
x(εt+1 ) + Et+1 s=2 β
x(εt+s )
yt+1

(37)

In equation (37), the assumption of i.i.d. growth and that the
surplus/GDP
ratio is

primary

1−γ
P
yt+s
s−1
x(εt+s ) is a constant,
only a function of the current shock implies that Et+1 ∞
s=2 β
yt+1
which we define as ρ.25 It is straightforward to prove that ρ must be positive for government
debt to also be positive in the future: intuitively, the present value of future taxes must remain
positive. From equation (1), Bt+1 is predetermined (that is, time-t measurable). Hence, Pt+1 is
time-t measurable if and only if
x(εt+1 ) = x̄e−εt+1 − ρ

(38)

for some constant x̄ > 0. Iterating on the definition of the present value of taxes ρ, we can
recover how it is related to x̄:
"

∞
X

#
1−γ
y
t+s
ρ =Et+1 βx(εt+2 )
β s−1
+
x(εt+s )
yt+1
s=3
"
1−γ 
1−γ #



y
x̄β∆1−γ λ
y
t+2
t+2
.
=βEt+1 x̄e−εt+2 − ρ
+
ρ =
yt+1
yt+1
λ+γ


yt+2
yt+1

1−γ



(39)

Combining (38) and (39) and taking expected values, we obtain


x̄λ
β∆1−γ (λ + 1)
Et (xt+1 ) =
1−
< 0,
λ+1
λ+γ
where the last inequality follows from assuming that the interest rate is low, more precisely, that
(5) and hence (35) hold. Hence, in this i.i.d. economy, when parameters are such that (5) holds
and fiscal policy stabilizes the debt/GDP ratio, expected taxes one period ahead are always
negative.
25

Note however that ρ depends on the function x(.), so we need to solve for both of them jointly in what follows.

34

Appendix B

A Model with Government Debt and Money

We reconsider the economy of Section 5, but we now explicitly introduce money, so that money
and government bonds circulate at the same time. The environment is the same as in Section 5,
except that government debt is no longer accepted with probability one in bilateral meetings,
but only with probability ζ ∈ (0, 1). In contrast, the central bank issues “money,” which is
perfectly durable, divisible, and intrinsically useless, and yields a zero nominal return. Money
is always recognized in bilateral meetings and can therefore be used with probability one. We
again assume that buyers make a take-it-or-leave-it offer to the sellers, and they make their offer
knowing whether the seller is able to accept only money or both money and bonds in exchange
for goods.
In the centralized market, government debt is now a commitment by the government to
deliver the face value in money at maturity, justifying the assumption that it is nominal debt.
The central bank policy of setting an interest rate R implies that there is an infinitely elastic
supply of new one-period bonds vs. money, at a relative price 1/(1 + R).26
The government budget constraint is now modified to
Bt+1
+ Mt+1 = Bt + Mt − Pt τt .
1+R

B.1

(40)

Characterizing the Economy

As before, denote with W and V the value functions when entering the centralized and decentralized market, respectively. They now depend on money holdings m, in addition to government
debt holdings and private claims. We have
Wt (m, b, a) = max
{c − y + βEt Vt+1 (m0 , b0 , a0 )} s.t.
0 0
c,y,b ,a

Pt c +
26

a0
b0
+ m0 ≤ Pt (y − τt ) + a + b + m,
p +
1 + Rt
1+R

(41)
(42)

See Bassetto and Phelan [4] for a discussion of the consequences of imposing limits on the central bank’s

ability to convert new bonds into money and vice versa.

35

(23), (24), and
m0 ≥ 0.

(43)

As government debt, money cannot be sold short.
The solution to the maximization problem above yields
Wt (m, b, a) = Ŵt +

a+b+m
,
Pt

(44)

where Ŵt is independent of the current state. When entering the centralized market, private
claims and government bonds are both nominal, so they are a commitment to deliver money
in that market: hence, they are perfect substitutes for money at that stage. This implies that,
when able to accept government bonds, sellers are indifferent between receiving bonds or money.
When buyers and sellers meet in the decentralized market, let ˜l be the amount of nominal
claims that sellers receive in exchange for q̃ units of goods. Depending on the meeting, this can
take the form of money only, if the seller cannot accept government bonds or both bonds and
money. Equation (44) then implies that the sellers’ participation constraint is
− q̃ + ˜l/Pt ≥ 0.

(45)

Buyers face again a linear price Pt when making their take-it-or-leave-it offer.
Using equation (26), and exploiting the fact that all the surplus from decentralized trade is
appropriated by the buyers, the value function at the beginning of the period, before it is known
who will be a seller, who will be a buyer, and who will not trade, is given by
Vt (m, b, a) =Wt (m, b, a) + χζ max[u(q) − q] + χ(1 − ζ) max[u(q̂) − q̂]
q

q̂

a+b+m
=Ŵt +
+ χζ max[u(q) − q] + χ(1 − ζ) max[u(q̂) − q̂],
q
q̂
Pt

(46)

subject to
Pt q ≤ m + b

(47)

Pt q̂ ≤ m.

(48)

and

36

We thus get

∂Vt (m,b,a)
∂a

=

1
,
Pt



1

∂Vt (m, b, a)
Pt
=



∂b
χζu0 m+b + (1 − χζ)(1/Pt )
Pt

if m + b > Pt q ∗
otherwise,

and


1



Pt

 
∂Vt (m, b, a) 
= χ(1 − ζ)u0 m + (1 − χ(1 − ζ))(1/Pt )
Pt

∂m


 




χζu0 m+b + χ(1 − ζ)u0 m + (1 − χ)(1/Pt )
Pt
Pt

if m > Pt q ∗
if m ∈ (Pt q ∗ − b, Pt q ∗ )
otherwise.

Taking this into account, the necessary and sufficient conditions for optimality in maximizing
(41) subject to (42), (23), (24), and (43) are given by (29),

 
 

Pt+1
Mt+1 + Bt+1
0
− 1 = χζ max u
,1 − 1 ,
β(1 + R)Pt
Pt+1

(49)

and

 
 


 
 

Mt+1 + Bt+1
Pt+1
Mt+1
0
0
−1 = χζ max u
, 1 − 1 +χ(1−ζ) max u
, 1 − 1 . (50)
βPt
Pt+1
Pt+1
We will assume that R > 0, for otherwise money and government bonds would have the same
opportunity cost and households would only hold bonds if they were satiated with liquidity.
In this case, the rate of return on money is always lower than the rate of government bonds;
equations (49) and (50) then imply that households will never find it optimal to accumulate
enough money so as to buy q ∗ with money only. Government bonds will have a lower rate of
return than private assets when liquidity is scarce even in meetings where money and bonds are
traded ((Mt+1 + Bt+1 )/Pt+1 ≤ q ∗ ).
We obtain an excess demand for money relative to government bonds
 


Pt+1
Mt+1
Pt+1
0
−
= −χ(1 − ζ) u
−1 ,
β(1 + R)Pt
βPt
Pt+1

(51)

which means that money carries a liquidity premium over government bonds.
We invert equation (49) to obtain the demand for total liquidity (money plus bonds), which
is given by
Bt+1 + Mt+1
1
= f (1 + rt+1 ) :=
(u0 )−1
Pt (1 + R)
1 + rt+1
37

"

1
β(1+rt+1 )

− (1 − χζ)
χζ

#
.

(52)

Using this equation and (40), we obtain a difference equation
f (1 + rt+1 ) = (1 + rt )f (1 + rt ) −

R Mt+1
− τt
1 + R Pt

(53)

with an initial condition
f (1 + r1 ) =

B0 + M0 − RM1 /(1 + R)
− τ0 .
P0

Equation (53) differs from equation (16) due to the presence of seigniorage revenues. If taxes are
set so as to offset these revenues, i.e., if
τt = τ −

R Mt+1
,
1 + R Pt

(54)

the difference equation coincides with the cashless economy and the same results apply.27 In this
case, given an equilibrium path for the real interest rate {rt+1 }∞
t=0 that satisfies the difference
equation
f (1 + rt+1 ) = (1 + rt )f (1 + rt ) − τ,

t≥1

along with the initial condition
f (1 + r1 ) =

B0 + M0
− τ,
P0

the additional equilibrium condition, which is equation (51), can be used to determine real (and
nominal) money balances.28
27

Equation (54) assumes that the fiscal authority observes and can react to Mt+1 and Pt . These variables

are observed in the centralized (evening) market of period t, where taxes are also levied. The issue of joint
determination of macroeconomic aggregates and policy variables is at the heart of complications discussed in
Bassetto [2, 3]. This is a side issue from the perspective of the current paper. However, for completeness, (54)
is a well-specified government strategy under the following description of the centralized market. First, the
government repays maturing bonds in money. Second, households participate in the goods market according to
some Walrasian mechanism (e.g., an auction), where money is used as numeraire. Households then purchase
new government bonds with money at the set nominal rate R. At this stage, households in the aggregate are
left with Bt + Mt − Bt+1 /(1 + R) units of money. The government sets and collects nominal taxes equal to
Pt τt = Pt τ (1 + R) − R[Bt + Mt − Bt+1 /(1 + R)], to be settled in money. Households are left in the aggregate with
Mt+1 = Bt + Mt − Bt+1 /(1 + R) − Pt τt , and simple algebra shows that (54) holds. Finally, an auction opens for
private borrowing and lending.
28
Note that knowing rt+1 is equivalent to knowing πt+1 , given the nominal interest rate R and the definition
of real rate (14).

38

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