ADEMU mini-conference on Fiscal Policy in Times of Crisis - Barcelona GSE, 30 March 2017

1. On the Risk of Leaving the Euro∗
Albert Marcet†
Manuel Macera‡
Juan Pablo Nicolini§
March 2017
Abstract
In the aftermath of the recent European sovereign debt crisis, there have been many
proposals to leave the Euro for South-European countries. Presumably, abandoning the
currency union would benefit these countries by allowing them to devalue their currency.
In this paper we argue that the risk of hyperinflation can be significant for these countries
if they leave the Euro with positive and persistent deficits, large value of debt to output
ratios and agents in the economy have beliefs regarding the behavior of the economy that
imply only very small departures from rational expectations.
∗
The views expressed herein are those of the authors and do not necessarily those of the Federal Reserve
Bank of Minneapolis or the Federal Reserve System.
†
ICREA, IAE, MOVE, and Universitat Autonoma de Barcelona.
‡
Colorado State University.
§
Federal Reserve Bank of Minneapolis and Universidad Di Tella.

2. 1. Introduction
Following the great world recession of 2008 and the European sovereign debt crisis of 2012, the
proposal to leave the Euro and reintroduce a national currency has regained support both in
academic and political circles in some south-European countries like Italy, Greece and Portugal.1
The purpose of this paper is to evaluate the potential implications of that policy decision
on the future path of inflation rates for these countries. To do so, we use a simple model of
seigniorage financing with three key assumptions. First, we assume that the initial value of
the deficit is positive, that it is very persistent and that is subject to stochastic shocks. These
features are present in all those south-European countries: we use data for these countries
to discipline the behavior of the deficits in the model. Second, we assume that, following
the departure from the Euro, and given the sovereign debt crisis that unraveled in all these
countries a few years ago, it would be very hard for these countries to finance their deficits in
bond markets. Therefore, they will have to resort in money financing of the deficit, which will
clearly be inflationary. This second assumption is more controversial, and we discuss it further
below. Finally, we allow for small departures from rational expectations, in a sense we make
very precise in the paper. This assumption is still quite controversial, we discuss it at length
in the paper.
The contribution of the paper is twofold. First, we show that the interaction of very per-
sistent deficits and severe difficulties to borrow in bond markets with small deviations from
rational expectations dramatically amplifies the equilibrium inflation rates generated by the
model. Thus, the paper highlights a potential risk of hyperinflations that may follow a de-
parture of the Euro system for countries such as Italy, Greece or Portugal under the current
circumstances. This risk seems to have been overlooked in the recent debate.
Second, we propose a methodology to perform policy analysis in models in which agents are
rational given their beliefs, but their beliefs do not necessarily conform with rational expecta-
tions. We do this in several steps. In the first step we show that in a model with incomplete
markets and heterogenous agents, the amount of information agents need to compute the ra-
tional expectations beliefs is enormous. A possible approach is to follow the Bayesian learning
literature and endow agents with priors regarding all the exogenous variables they do not know
and let them update those priors using data. With incomplete markets and heterogeneity the
problem of updating becomes extremely complex very rapidly. And the theory leaves all the
1
In Italy, following the December 2016 referendum, the political party five stars, who had grown in the
polls by large numbers, has proposed to call for a referendum to leave the Euro. In Greece, there was a heated
debate during the worst months of the debt crisis in 2014 and 2015. In Portugal, a political party campaigned to
leave the Euro, with not much public support. See also https://krugman.blogs.nytimes.com/2010/04/28/how-
reversible-is-the-euro/.
2

3. priors unrestricted, which implies additional assumptions to pin them down.2
Our second step
is to assume that agents in the model have a system of beliefs on the endogenous variable that is
taken as given by them in their optimization process (future price level in our monetary model).
This system of beliefs becomes the rational expectations beliefs as a particular case when they
are centered at the right prior and the precision of the prior is infinite. Thus, by properly
parameterizing the deviation of the initial prior from the true value and the precision of the
prior, we can solve for beliefs that are arbitrarily close to rational expectations. In this way, we
can study how robust are the implications of, in our case, leaving the Euro with positive and
persistent deficits and without the ability to borrow to small deviations from rational expecta-
tions. As we show, for this particular policy decision, the implications of rational expectations
are not robust in this sense. Our third step involves performing a series of tests on the system
of beliefs, and show that for period lengths of between 10 and 15 years, agents in the model
would not reject the hypothesis that their system of beliefs is the correct one. Thus, there
would not be enough evidence, for these sample sizes, to contradict the beliefs agents hold.
The paper proceeds as follows. In the next section we describe a heterogeneous agent model
with incomplete information that delivers the micro foundations of our approach. In Section 3
we introduce seigniorage financing and study Learning Equilibria. In Section 4 we assess the
quantitative performance of our model and show that the presence of learning translates into
recurrent hyper inflationary episodes. In Section 5 we derive testable implications of the belief
system and test whether agents can reject their beliefs based on data generated by the model.
In Section 6 we conclude.
2. A Model with Heterogeneous Agents
Consider a constant cohort size, overlapping generations model in which each agent lives for
two periods. Agents are heterogeneous in their endowments and in their preferences, which are
determined by the moment they are born. The endowments of agent j ∈ [0, 1] born at time t
are 1 when young, and ej
t+1 when old, and her preferences are given by
ln ct + αj
t ln ct+1
The heterogeneity within each cohort is determined as follows:
ej
t+1 = et+1 + ηj
t+1
αj
t = αt + εj
t
2
In discussing this issue, we follow Adam and Marcet (2011) closely.
3

4. where et and αt are aggregate shocks and ηj
t and εj
t are agent-specific. All shocks are assumed to
be mutually independent and i.i.d. both over time and across agents. We make two additional
assumptions on these stochastic processes. First, we restrict the endowment when old to be
smaller than the endowment when young (ej
t+1 < 1 for all t and j). Second, we assume that
agents have a relative preference for consumption when old (αj
t > 1 for all t and j). These
assumptions are made to ensure that as long as the return on savings is not too low, young
agents would save in equilibrium.
Markets are incomplete in the sense that the only asset agents can hold is fiat money. Thus,
at any point in time, there is only one spot market in which agents can exchange goods for
money, at a price Pt. When young, agents chose how many units of money to hold for next
period, given the price level that prevails at time t. The budget constraint when young is given
by3
:
Ptcj
t + Mj
t ≤ Pt (1)
In the following period, they consume their endowment plus whatever they can buy with the
money previously held, so their budget constraint when old is:
Pt+1xj
t+1 ≤ Mj
t + ej
t+1Pt+1 (2)
for all Pt+1. Hence, the problem of agent of type j born in period t consists in maximizing:
Ej
t [ln ct + αj
t ln ct+1] (3)
by choosing consumption and money holdings, subject to the budget constraints (1) and (2).
The expectation is taken with respect to the price level Pt+1, which due to the presence of ag-
gregate uncertainty, it is a random variable. Notice that by indexing the expectations operator,
we are suggesting that agents could hold heterogeneous expectations.
Since the budget constraints will hold with equality, once we substitute them in (3), an
interior solution requires:
1
Pt − Mj
t
= Ej
t
αj
t
Mj
t + ej
t+1Pt+1
(4)
which defines implicitly the individual money demand equation. Importantly, money demand
must be measurable with respect to the information set available when young. Since the only
source of uncertainty, namely Pt+1, appears in the denominator of the right hand side, we
3
As agents cannot issue money, the constraint Mj
t ≥ 0 must be imposed. However, the assumption that the
endowment in the second period is smaller than in the first period implies this constraint will not be binding
as long as the inflation rate is not too high, thus we ignore this constraint in our theoretical analysis. In the
numerical section, we impose this constraint on the equilibrium.
4

5. cannot solve for the money demand equation in closed form. In order to make progress, we
study the linearized version of it, which can be written as4
:
Mj
t
Pt
= φj
t 1 − γj
t Ej
t
Pt+1
Pt
which corresponds to the money demand by each agent of generation t. Thus, the aggregate
money demand is given by:
Mt =
1
0
φj
t Pt − γj
t Ej
t Pt+1 dj (5)
Observe that if we impose homogeneous expectations we could write Ej
t Pt+1 = EtPt+1 and
then:
Mt = φt (Pt − γtEtPt+1) (6)
If agents have full information regarding the relevant aggregate states, homogeneous expecta-
tions can be attained in two different ways. First, we could shut down heterogeneity by making
ηj
t+1 = εj
t = 0 for all j and t. In this case, the model permits a representative agent formulation,
which trivially rules out heterogeneous expectations. Alternatively, we could assume that in
addition to agents observing all the relevant aggregate states, they use the exact same mapping
from aggregate states to prices. Thus, homogeneous expectations also follows trivially. Notice
however that the informational burden in the latter case is much greater than in the former
since, as it is usual in models with heterogeneous agents, the set of relevant aggregate states
comprises not only the shocks et+1 and αt, but also the distribution of the individual states
across agents.
In this paper we want to consider situations in which agents do not directly observe the
aggregate state. More precisely, we assume that each agent observes her own type, but is unable
to separate the individual component from the aggregate one. Since agents with different types
will now face a different inference problem, the computation of the right hand side of (5)
becomes a much more complicated task. In particular, it requires each agent knowing the
inference problem solved by all other agents in the economy. Even if we endow each agent with
knowledge of the distribution of types of all other agents in the economy, it is apparent that
individual maximization becomes a much more challenging problem.
To overcome this feature of the model, we want to consider modeling choices that bring
us back to the homogeneous expectations case. One possible solution - the one proposed
by Bayesian learning models - is to assume that agents holds a common prior regarding the
4
The linearization and the expressions for φj
t and γj
t are detailed in the Appendix.
5

6. distribution of ej
t+1 and αj
t for all j ∈ [0, 1], which they update relying solely on price level data.
This implies entertaining a continuum of priors and updating them using a single observation
on the aggregate price level. This approach delivers homogeneous expectations because it
effectively washes out individual differences, although it doesn’t make individual problems any
simpler.
We take an alternative route: given that agents only care about one single variable - the
price level next period - we model their expectations with a single prior, that is updated using
the single variable they care about. Thus, we attach to each agent, a common prior on the
inflation rate, which given the current price level is equivalent to a prior on next period price
level, and an updating rule of that prior using the observations that the model generates. At
first sight, this may seem innocuous: instead of a continuum of updating rules with a single
observation, we just update a single variable - the one of interest which is the price level -
with that single observation. But the procedure involves one degree of freedom: when doing
Bayesian learning, the updating scheme is given by Bayes’ rule, which is the rational way of
updating. Once we decide to update directly on the variable of interest, which rule should one
use? We discuss this in the next section.
An advantage of our approach is that if we assume that agents have no uncertainty regarding
their prior, we obtain the rational expectations solution - the reason being that we start agents
with priors that are centered at the right value. Thus, our approach can be viewed as a
robustness check: how does the equilibrium outcome changes, given a policy, if we allow for
small departures from fully rational expectations? In the next Section, we discuss in detail our
approach.
3. Introducing Seigniorage Financing
The model in the previous section highlighted the fact that representative agent models hide
valuable insight regarding how expectations must be formed. In this section we consider a
version of the economy just described in which we shut down heterogeneity by taking the
variance of all shocks to zero. We also introduce seigniorage financing and switch the focus of
the analysis to the way expectations are formed rather on the presence of heterogeneity. We
presume that heterogeneity on itself is not important for the evolution of inflation.
In order to consider deviations from rational expectations that are small, we proceed in the
following way. First, we compute the stochastic properties that inflation follows in the rational
expectations equilibrium. We then endow agents with a system of beliefs regarding the process
of inflation - that agents rightly perceive as exogenous to their decisions - that is consistent
with the behavior of inflation in the rational expectations equilibrium. But we assume that
6

7. agents are not completely sure regarding a parameter in that process. In the background, we
can think of this uncertainty as stemming from not knowing the distributions governing the
shocks affecting all the agents in the economy.
The system of beliefs for the process of inflation that we endow the agents with is centered at
the right value, but we allow some uncertainty regarding this value. Given this system of beliefs,
agent rationally use the data generated by the model to update their prior. In particular, given
the system of beliefs, agents rationally use the Kalman filter to obtain a more precise estimate
of the parameters they are uncertain about.
3.1. Equilibrium Conditions
We carry out the analysis by focusing on three equations: the money demand equation, the
government budget constraint, the law of motion for the level of seigniorage. The demand for
real balances is given by:
Md
t
Pt
= φ 1 − γπe
t+1 (7)
where πe
t+1 denotes the expected gross inflation rate. Notice that the money demand parameters
are no longer time varying since we purposefully shut down the volatility of all shocks introduced
in the previous section.
The only potential source of uncertainty in this model comes from the level of seigniorage.
In particular, the government budget constraint is given by:
Ms
t = Ms
t−1 + dtPt (8)
where Ms
t is the money supply and dt denotes seigniorage, which evolves according to:
dt = (1 − ρ)δ + ρdt−1 + t (9)
where t denotes an i.i.d. perturbation term. As mentioned in the introduction, this formulation
is supposed to capture the feature that upon abandoning a currency union, a country is unable
to issue debt and must finance government deficit through money printing5
.
To close the model, we need to specify how expectations regarding future inflation are
formed. In full generality, we postulate:
πe
t+1 = EP
t
Pt+1
Pt
(10)
5
In the quantitative section, we allow for policy regimes in which the government is able to deplete interna-
tional reserves to finance its deficit.
7

8. which indicates that the expectation is taken using the subjective probability measure P. This
probability measure specifies the joint distribution of {Pt, dt}∞
t=0 at all dates and it is fixed at
the outset. Under rational expectations, market prices are assumed to carry only redundant
information because agents know the exact mapping from the history of seigniorage levels
to prices, Pt(dt
). For notational simplicity, in that case we will drop the superscript in the
expectation operator and write:
πe
t+1 = Et
Pt+1
Pt
(11)
In equilibrium we must have Md
t = Ms
t = Mt, which allows us to combine the money demand
equation (7) and the government budget constraint (8) to obtain:
πt =
φ − φγπe
t
φ − φγπe
t+1 − dt
(12)
where πt ≡ Pt/Pt−1 denotes the realized gross inflation rate. This equation governs the evolution
of inflation in any equilibrium, regardless of how expectations are formed, and we will use it
repeatedly. We start by studying the rational expectations benchmark.
3.2. The Rational Expectations Benchmark
We now study equilibria under rational expectations, restricting attention first to a deterministic
environment. We focus on the case with persistence in the level of seigniorage, which embeds
the case studied in Marcet and Nicolini (2003).
In the absence of uncertainty, imposing rational expectations amounts to require πe
t = πt
for all t. Plugging this condition into the main equation (12) and rearranging delivers:
πt+1 = (1 − ρ)
φ + φγ − δ
φγ
−
1
γπt
+ ρ
φ + φγ − dt−1
φγ
−
1
γπt
(13)
This equation will govern the dynamics of inflation in equilibrium.
The initial position of the economy is given by d0. Notice that if d0 = δ, then dt = δ for all
t and the equilibrium will be stationary. In such a case, (13) admits two stationary equilibria,
which are obtained as the solutions to the following quadratic equation:
φγπ2
− (φ + φγ − δ)π + φ = 0 (14)
One could use this equation to trace out a stationary Laffer Curve, depicting the inflation rates
that allow the government to finance the level of seigniorage δ. We use {π1(δ), π2(δ)} to denote
the two roots of (14), where the small root π1(δ) corresponds to the good side of the Laffer
8

9. Curve.
In the case in which d0 differs from δ, we define xt ≡ (πt, dt) and write the dynamic system
composed of (9) and (13) as follows:
xt = G (xt−1) ≡



(1 − ρ)F(πt−1, δ) + ρF(πt−1, dt−1)
(1 − ρ)δ + ρdt−1


 (15)
where
F(π, d) =
φ + φγ − d
φγ
−
1
γπ
(16)
In a deterministic environment, dt will always revert to its long run mean δ. Hence, to charac-
terize equilibria, it suffices to understand the behavior of πt, conditional on the initial position
d0. To this end, it will prove convenient to ensure that stationary inflation rates are always
positive and well-defined, for which we assume the following:
Assumption 1 δ ∈ D ≡ [0, φ(1 + γ − 2γ
1
2 ))
One can easily check that under this assumption, stationary inflation rates are always within
the interval [1, γ−1
]. Moreover, the upper bound of D can be interpreted as the maximum
level of seigniorage that the government can finance, given the primitives of the economy. The
following proposition summarizes the behavior of inflation under Rational Expectations:
Proposition 1 Under Assumption 1, for any d0 ∈ D there exists π(d0) such that:
1. If π0 < π(d0), then limt→∞ πt = −∞
2. If π0 = π(d0), then limt→∞ πt = π1(δ)
3. If π0 > π(d0), then limt→∞ πt = π2(δ)
The proof is relegated to Appendix B. Notice that in the special case d0 = δ, one can show
that π(d0) = π1(δ) and the equilibrium is equivalent to that corresponding to the case with no
persistence.
The equilibria characterized in this proposition for the case d0 < δ is depicted in Figure 1.
The line with circles that starts at π(d0) represents the stable path that converges to the low
inflation steady state under Rational Expectations. For π0 = π(d0), the lines with crosses show
the inflation paths that either converge to the high inflation steady state or diverge to infinity.
In the remainder of this section, we linearize (15) and introduce a small amount of uncer-
tainty in the segnioriage to learn about the properties of the inflation process around the low
inflation steady state.
9

10. 0 20 40 60 80 100
t
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
πt
d0
< δ
π
1
(δ) π
1
(d
0
) π
2
(δ) π
2
(d
0
)
Figure 1: Inflation paths under Rational Expectations that evolve according to (13).
The horizontal lines correspond to the solutions to the quadratic equation when d = δ (solid)
and when d = d0 (dashed). The circled line that converges to the low inflation steady state
starts at π0 = π(d0) which is characterized in Proposition 1.
3.3. Inflation Persistence under Rational Expectations
To learn more about the stochastic properties of inflation in equilibrium, we linearize the main
equation (12) around the low inflation steady state and introduce a small amount of uncertainty
in the level of seigniorage. The linearization boils down to6
:
ˆπt =
δ
φ − φγπ1(δ) − δ
ˆdt (17)
ˆdt = ρ ˆdt−1 + t (18)
where we are using the notation ˆxt = ln xt − ln x, with bold letters indicating steady state
values. We can express inflation recursively as:
ˆπt = ρˆπt−1 + νt (19)
6
See Appendix C for details.
10

11. where νt ≡ δ t/(φ−φγπ1(δ)−δ). Hence, around the low inflation steady state, inflation behaves
as an AR(1) process that inherits the persistence of seigniorage. Figure 2 displays sample paths
of both inflation and seigniorage according to this linearized system.
3.4. Learning Equilibria
We are now ready to analyze equilibria with learning. Consider agents that hold the following
beliefs regarding the inflation process:
πt = (1 − ρπ) πt + ρππt−1 + ut (20)
πt = πt−1 + ηt
where ut ∼ N(0, σ2
u) and ηt ∼ N(0, σ2
η) are i.i.d. and independent of the shock to the seigniorage
t. We allow for ρπ = ρ and assume that agents know the value of both persistence parameters.
The intuition behind the proposed belief system (20) is that at period t, all the information
of the deficit is embedded in πt−1
7
. We assume agents observe the realizations of inflation but
not those of πt and ut separately. Thus, the learning problem consists of filtering out long run
inflation πt .
We will use βt to denote the prior belief regarding πt entering period t. Notice that if we
make σ2
η = 0 and assign probability 1 to β0 = π1(δ), we obtain:
πe
t = (1 − ρπ) π1(δ) + ρππt−1 (21)
which, as long as ρπ = ρ, is equivalent to (19) for small deviations around the low inflation
steady state. In that sense, this setup encompasses rational expectations equilibria as a special
case.
In what follows, it will prove convenient to assume that the initial prior β0 is normally
distributed and centered at the low inflation steady state, with a precision equal to the inverse
of the Kalman filter uncertainty about βt. Optimal learning then implies that the posterior
will also be normal, with the same precision, but centered at (πt − ρππt−1)/(1 − ρπ). Optimal
7
In principle, we could allow for a general formulation such as:
πt = (1 − ρπ) πt + ρππt−1 + ψ t + ut
πt = πt−1 + ηt
where agents would use information about the innovations to the deficit to adjust their expectations regarding
future inflation. In equilibrium, we must require that ψ is consistent ex-post with the generated equilibrium
process for inflation.
11

12. 0 20 40 60 80 100
time
-6
-4
-2
0
2
4
%deviationfromsteadystate
×10-4
ρ = 0.75
inflation seignoriage
0 20 40 60 80 100
time
-6
-4
-2
0
2
4
%deviationfromsteadystate
×10-4
ρ = 0.25
inflation seignoriage
Figure 2: Sample paths for inflation and seignoriage around the low inflation steady
state in the linearized rational expectations equilibrium.
12

13. updating implies that βt will evolve recursively according to:
βt = βt +
1
α
πt − ρππt−1
1 − ρπ
− βt−1 (22)
where α denotes the optimal Kalman gain8
.
The belief system implies that:
πe
t+1 = (1 − ρπ)βt + ρππt (23)
Notice that if we plug in this equation into (12), multiplicity regarding the inflation rate would
arise. To sidestep this problem, we assume that when expectations regarding πe
t+1 are formed
at period t, agents still do not know the realization of πt. Therefore, in order to form their
expectations regarding future inflation, they forecast inflation two periods ahead using πt−1 as
follows9
:
πe
t+1 = 1 − ρ2
π βt + ρ2
ππt−1 (24)
Using this equation, we rewrite (12) as follows:
πt =
φ − φγ((1 − ρ2
π) βt−1 + ρ2
ππt−2)
φ − φγ((1 − ρ2
π) βt + ρ2
ππt−1) − dt
(25)
To provide intuition about the behavior of inflation in this case, let us write (25) as
H(βt, βt−1, πt, πt−1, πt−2, dt) = 0
and define h(β, π, d) ≡ H(β, β, π, π, π, d). The function h is useful to provide an approximation
of the behavior of inflation in a situation in which dt = δ, βt ≈ βt−1 and πt ≈ πt−1 ≈ πt−2 ≈
βt−1. In such a case, (25) boils down to the quadratic equation (14), which implies that the
rational expectations stationary inflation rates are also stationary inflation rates under learning.
However, notice that the stationary inflation rate πi(δ) is stable under learning only if:
∂π
∂β β=πi(δ)
= −
∂h/∂β
∂h/∂π β=πi(δ)
=
φγπi(δ) − φγ
φ − φγπi(δ) − δ
< 1 (26)
As long as the denominator in (26) is positive10
, we can verify that this is indeed the case if
8
In the quantitative section, we will modify (22) to incorporate information regarding (πt −ρππt−1)/(1−ρπ)
with a lag, in order to avoid the simultaneity between πt and βt.
9
Note that under this assumption, there is no need to include the innovations to the seigniorage in (20), as
proposed in Footnote (7), since they would be independent of past observations. Put differently, t would be
independent of information used at t to forecast inflation.
10
Whenever there is a positive price that clears the money market, the denominator will be positive. We
13

14. and only if:
πi(δ) <
φ + φγ − δ
2φγ
(27)
Using (14), it is easy to show that this condition is satisfied by the smallest root π1(δ), but not
by the largest π2(δ). Therefore, as pointed out in Marcet and Nicolini (2003), it is the small
Rational Expectations stationary inflation rate the one which is stable under learning.
4. Quantitative Performance
In this section we assess the quantitative performance of the model under learning through
simulation. We introduce two monetary policy rules in the model described in the previous
section. The first one corresponds to regime switching in the spirit of Marcet and Nicolini
(2003). The idea is that as long as the government has enough reserves to deplete, it can
always switch to an Exchange Rate Regime (ERR hereafter) that allows it to target any level
of inflation. The second monetary policy rule prevents deflationary episodes by allowing the
government to use money to accumulate assets when the demand for real balances is too high
relative to the level of seigniorage11
. We introduce the following additional notation:
1. βL
denotes the lower bound on realized inflation that triggers asset accumulation.
2. βU
denotes the upper bound on expected inflation that triggers ERR
3. β denotes the inflation target under ERR
4. ˆβ denotes the upper bound on the stable set that deactivates ERR.
We start by setting βL
= 1 which means that negative inflation rates are never observed. ERR
is triggered whenever expected inflation exceeds βU
or to restore equilibrium12
. The baseline
parameterization is summarized in Table 1. Observe that we set ρπ = ρ, which is a reasonable
assumption as we show in the appendix. Money demand parameters target the inflation rate
that maximizes the stationary Laffer curve and the maximum seigniorage as a percentage of
GDP for the case of Argentina, and they are taken from Marcet and Nicolini (2003). The values
of ρ and σ correspond to the estimation of an AR(1) process for Greece’s primary balance as
a percentage of GDP.
can always extend the model to include the case in which there are reserves that can be depleted to ensure the
existence of such a price, in the spirit of Marcet and Nicolini (2003).
11
The extended model that considers asset accumulation is detailed in the appendix.
12
The two cases in which ERR is required to restore equilibrium are if the money demand becomes negative,
or if given the realization of seigniorage, the money demand is too low for an equilibrium to exist.
14

15. Table 1: Parameters for Baseline Economy
Parameter Symbol Value
Persistence of deficit ρ .9334
SD of shocks to deficit σ .0146
Persistence of inflation ρπ 9334
Money Demand Parameter φ .36
Money Demand Parameter γ .39
ERR trigger βU
4
4.1. Probability of Hyperinflations
Table 2 reports the implications of the model on the probability of hyperinflations, given dif-
ferent values of δ and d0. The values for 1/α constitute the weight that agents place on past
inflation to update their beliefs. Notice that the probability of experiencing a hyperinflation
vanishes as the learning equilibrium approaches the rational expectations one (e.g. 1/α → 0).
This feature is also present in the case without persistence studied in Marcet and Nicolini
(2003), since 1/α = 0 merely keeps expectations constant and ensure the economy stays around
the low inflation steady state. In terms of comparative statics, both the long run deficit and the
initial deficit increase the probability of experiencing a hyperinflation, with the largest effect
coming from the former.
Figure 3 shows sample paths for inflation in a learning equilibrium. The solid blue line
corresponds to the learning equilibrium with a positive constant gain and the solid red line
corresponds to an equilibrium with fixed expectations that considers the same realizations of
the shocks to seigniorage.
5. Testable Restrictions
In this section we study the conditions under which agents would question their belief system
in a learning equilibrium. In order to do this, we consider the implications of the equilibrium
conditions and the belief system for the vector xt = (et, dt − ρdt−1), where et ≡ (πt − ρππt−1) −
(πt−1 − ρππt−2), and we evaluate these implications using simulated data. Given the dataset
{xt}, the following proposition lists a set of necessary conditions to conclude that the data was
indeed generated by the model.
Proposition 2 If {xt} satisfies the equilibrium conditions and the belief system, then the fol-
lowing restrictions must hold:
1. E[xt−iet] = 0 for all i ≥ 2.
15

16. Table 2: Probabilities of Hyperinflations for βU
= 4.00 and T = 100
Probability of n hyperinflations
Deficit mean δ = 0.0%, Initial Deficit d0 = 1.0%
1/α 0 1 2 ≥ 3
0.05 41.54 26.68 15.96 15.82
0.04 42.06 26.38 15.80 15.76
0.03 43.38 26.12 15.34 15.16
0.02 43.98 25.72 14.54 15.76
0.01 66.12 20.64 7.56 5.68
0.00 100.00 0.00 0.00 0.00
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%
1/α 0 1 2 ≥ 3
0.05 31.70 28.12 17.84 22.34
0.04 32.58 28.08 18.56 20.78
0.03 33.58 27.96 17.80 20.66
0.02 35.10 27.82 17.38 19.70
0.01 59.52 22.76 10.06 7.66
0.00 100.00 0.00 0.00 0.00
Deficit mean δ = 4.0%, Initial Deficit d0 = 1.0%
1/α 0 1 2 ≥ 3
0.05 6.16 13.86 18.02 61.96
0.04 6.48 13.90 19.00 60.62
0.03 7.92 15.44 18.60 58.04
0.02 18.26 21.06 18.92 41.76
0.01 70.86 14.18 6.74 8.22
0.00 100.00 0.00 0.00 0.00
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%
1/α 0 1 2 ≥ 3
0.05 4.22 10.90 16.40 68.48
0.04 4.38 11.02 17.92 66.68
0.03 5.28 12.84 17.94 63.94
0.02 13.04 17.90 18.98 50.08
0.01 63.58 15.98 9.24 11.20
0.00 100.00 0.00 0.00 0.00
2. E[((dt − ρdt−1) + (dt−1 − ρdt−2))et] = 0.
3. b Σb + E[etet−1] < 0.
4. E[et] = 0.
where Σ is the variance of (dt − ρdt−1) and b corresponds to the coefficient of a regression of et
on (dt − ρdt−1).
16

17. 0 20 40 60 80 100
0
1
2
3
4
5
6
7
8
9
10
quarterlyinflationrate(logscale)
1/α = 0.05
1/α = 0.00
Figure 3: Sample Path for Inflation in Learning Model. The parameterization is dis-
cussed in the Quantitative Section. The solid blue line corresponds to the learning equilibrium
with a positive constant gain and the solid red line corresponds to an equilibrium with fixed
expectations that considers the same realizations of the shocks to seigniorage.
The proof is presented in the appendix. We test these moment restrictions using the parame-
terization displayed in Table 1.
5.1. Statistics
Restrictions 1, 2 and 4 represent first moment restrictions of the form E[yt] = 0 for yt = etqt,
where qt ∈ Rn
. In order to test these restrictions, we estimate E[yt] through its corresponding
sample mean and define the statistic:
QT =
1
T
T
t=1
yt V −1 1
T
T
t=1
yt
where V is the variance of the sample mean. In order to calculate QT , we need an estimator
of V . We use the Newey-West estimator, which requires also to estimate the auto-covariance
17

18. Table 3: Rejection Frequencies at the 5% level for βU
= 4
This table reports rejection frequencies obtained from testing Restriction 1 of Proposition 2
using simulated data, including up to three lags of xt as an instrument. All restrictions include
a constant as an instrument.
T
40 60 100 200
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 1a 6.3 % 6.6 % 8.3 % 8.2 %
Restriction 1b 7.9 % 7.9 % 9.1 % 9.0 %
Restriction 1c 8.2 % 7.7 % 8.5 % 7.7 %
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 1a 4.5 % 5.1 % 5.8 % 5.8 %
Restriction 1b 5.6 % 6.0 % 6.7 % 6.5 %
Restriction 1c 6.7 % 6.4 % 6.7 % 5.9 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 1a 6.6 % 6.7 % 8.4 % 18.2 %
Restriction 1b 8.9 % 8.3 % 9.9 % 20.1 %
Restriction 1c 9.3 % 8.3 % 8.7 % 17.5 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 1a 5.2 % 4.6 % 5.2 % 10.5 %
Restriction 1b 7.3 % 6.4 % 6.6 % 11.1 %
Restriction 1c 8.5 % 6.6 % 7.0 % 11.1 %
matrices of yt. Hence:
ˆQT = T
1
T
T
t=1
yt
ˆS−1 1
T
T
t=1
yt → χ2
n
where ˆS is the sample counterpart of 13
:
ˆS = T · E [(¯yt − y)(¯yt − y)) ] =
∞
ν=−∞
Γν = Γ−1 + Γ0 + Γ1
In order to test Restriction 3, we use a one-sided test of the form H0 : α < 0, where α is set to
satisfy
E [((dt − ρdt−1)b + et−1)et − α] = 0
and b satisfies:
E [(dt − ρdt−1) ((dt − ρdt−1)b − et)] = 0
13
Here we exploit the MA(1) property of et and use the fact that beyond the the first lead and lag, these
auto-covariances matrices must be equal to zero.
18

19. Table 4: Rejection Frequencies at the 5% level for βU
= 4
This table reports rejection frequencies obtained from testing the Restrictions 2-4 of Proposition
2 using simulated data.
T
40 60 100 200
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 2 11.9 % 13.0 % 15.1 % 18.4 %
Restriction 3 8.1 % 4.7 % 2.8 % 1.3 %
Restriction 4 0.0 % 0.0 % 0.0 % 0.0 %
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 2 8.7 % 9.4 % 10.7 % 10.3 %
Restriction 3 6.1 % 3.8 % 2.0 % 0.9 %
Restriction 4 0.0 % 0.0 % 0.0 % 0.0 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 2 12.0 % 12.3 % 17.3 % 34.4 %
Restriction 3 4.5 % 1.8 % 0.3 % 0.0 %
Restriction 4 0.0 % 0.0 % 0.0 % 0.0 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 2 8.8 % 8.3 % 9.9 % 18.4 %
Restriction 3 3.7 % 1.5 % 0.2 % 0.0 %
Restriction 4 0.0 % 0.0 % 0.0 % 0.0 %
GMM sets the estimate of b to the OLS coefficient of a regression of et on (dt − ρdt−1) and the
estimate of α precisely to b Σb + et−1et.
5.2. Rejection Frequencies
Observe that for all restrictions, the rejection of any of the null hypotheses implies that the data
{xt} was not generated by the belief system. Therefore, the belief system can be evaluated by
checking whether the rejection frequencies exceed a predetermined significance level. If agents
are using the wrong model of inflation, they should expect that as the sample size increases,
rejection frequencies also increase for at least some of the restrictions being tested.
We calculate rejection frequencies in two different ways. We first use the theoretical asymp-
totic distribution of ˆQT . However, we also want to consider testing restrictions by assuming
that agents use the small sample properties of ˆQT , for different values of T.
In the case of Restrictions 1, 2 and 4, asymptotic theory implies that ˆQT → χ2
n as the
sample size increases. In testing Restriction 1, we use as many as three lags of the instrument
(dt−ρdt−1) and we always include a constant term 14
. In the case of Restriction 3, the asymptotic
14
Notice that by including a constant, Restriction 4 is embedded in the joint hypothesis testing performed
for Restriction 1.
19

20. Table 5: Rejection Frequencies at the 5% level for βU
= 4
This table reports rejection frequencies obtained from testing Restriction 1 of Proposition 2
using simulated data, including up to three lags of xt as an instrument. All restrictions include
a constant as an instrument.
T
40 60 100 200
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 1a 11.4 % 14.6 % 16.9 % 20.3 %
Restriction 1b 10.0 % 12.2 % 15.2 % 19.6 %
Restriction 1c 8.3 % 10.2 % 13.3 % 15.4 %
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 1a 8.9 % 11.5 % 13.1 % 14.8 %
Restriction 1b 8.0 % 10.4 % 11.9 % 15.2 %
Restriction 1c 7.2 % 9.2 % 11.6 % 11.8 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 1a 12.4 % 15.1 % 19.8 % 34.7 %
Restriction 1b 12.5 % 14.5 % 19.7 % 36.4 %
Restriction 1c 9.2 % 12.3 % 16.0 % 31.1 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 1a 10.4 % 11.7 % 12.9 % 23.5 %
Restriction 1b 10.4 % 11.4 % 13.3 % 23.8 %
Restriction 1c 8.7 % 9.7 % 11.9 % 21.8 %
properties of the GMM estimator of α imply that under the null hypothesis it will be normally
distributed and centered at 0. To evaluate this restriction using the small sample distribution,
we simply estimate b and α using OLS and the appropriate sample counterparts.
Rejection Frequencies using the Asymptotic Distribution. Tables 3 and 4 display the
results of testing the restrictions of Proposition 1 using the asymptotic theoretical distribu-
tion of the statistics described in the previous paragraphs. Since Restriction 1 required some
discretionary choice regarding the set of instruments, we single out its results.
The results indicate that agents will find difficult to reject their beliefs based on the obser-
vation of realized inflation. This is especially true when agents have only observed a short path
of realizations and when the signal to noise ratio of their beliefs is higher, since this implies a
higher stationary 1/α. As for the comparative statics with respect to the deficit, the higher
the long run deficit, the higher the rejection frequencies. This is indeed related to the fact
that when hyperinflations are more frequent, the path of inflation is explosive, rather than a
regression as the AR(1) process of beliefs suggests.
The rejection frequencies for the remaining restrictions also look good. As agents gather
more information regarding inflation they should find more difficult to validate their beliefs.
20

21. Table 6: Rejection Frequencies at the 5% level for βU
= 4
This table reports rejection frequencies obtained from testing the restrictions of Proposition 2
using simulated data.
T
40 60 100 200
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 2 17.6 % 20.2 % 23.0 % 30.7 %
Restriction 3 0.2 % 0.1 % 0.1 % 0.1 %
Restriction 4 6.0 % 5.7 % 6.0 % 4.8 %
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 2 13.8 % 15.9 % 17.5 % 19.6 %
Restriction 3 0.1 % 0.1 % 0.1 % 0.0 %
Restriction 4 5.8 % 5.6 % 5.9 % 4.5 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 2 18.4 % 21.7 % 30.7 % 50.7 %
Restriction 3 0.1 % 0.0 % 0.0 % 0.0 %
Restriction 4 9.9 % 9.7 % 10.5 % 10.0 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 2 14.7 % 15.2 % 19.8 % 33.2 %
Restriction 3 0.1 % 0.0 % 0.0 % 0.0 %
Restriction 4 10.1 % 8.9 % 9.1 % 8.7 %
This remains true for all restrictions, with the exception of Restriction 3. One can interpret
this restriction as stating there is no better predictor of current inflation than past inflation.
Notice however that when ERR is not in place, the deficit must be a good predictor of inflation.
Therefore, we shall expect to reject this restriction less frequently if ERR is constantly triggered.
As a matter of fact, as T increases the economy will spend more time in ERR, and the link
between inflation and seigniorage weakens, as shown in Table 4.
Rejection Frequencies using the Small Sample Distribution. Tables 5 and 6 show the
rejection frequencies calculated using the small sample distribution of the statistics presented
in the previous section. As T increases, now not only the distribution of the model-generated
statistic changes, but the benchmark small sample distribution also does. In principle, there is
no way to know for sure how the critical values will change and how rejection frequencies will
respond.
Qualitatively, the rejection frequencies respond in the same manner as they do when agent
use the asymptotic distribution, and they remain within reasonable values. We conclude that
even when testing the model using small sample properties, agent will tend to validate their
beliefs, especially if they observe a short path of inflation realizations.
21

22. 6. Conclusions
TO BE WRITTEN
References
Adam, K. and A. Marcet (2011, May). Internal rationality, imperfect market knowledge and
asset prices. Journal of Economic Theory 146(3), 1224–1252.
Adam, K., A. Marcet, and J. P. Nicolini (2016, 02). Stock Market Volatility and Learning.
Journal of Finance 71(1), 33–82.
Marcet, A. and J. P. Nicolini (2003). Recurrent hyperinflations and learning. The American
Economic Review 93(5), 1476–1498.
22

23. A Linearization of the Money Demand Equation
We linearize the individual money demand around a fictitious steady state with no aggregate
uncertainty. In this appendix, we use the notation mj
t ≡ Mj
t /Pt and πt+1 ≡ Pt+1/Pt. The
linearization boils down to:
1
1 − ˜mj
2
(mj
t − ˜mj
) = −
αj
t
mj
t + ej
t+1˜πt+1
2
1
αj
t
Ej
t [mj
t − ˜mj
] + ej
t+1Ej
t [πj
t − ˜πj
]
where the tilde variables represent steady state variables. In steady state we must have that
1
1 − ˜mj
=
αj
t
˜mj + ej
t+1˜π
which also implies that
1 + αj
t
αj
t
˜mj
+
ej
t+1
αj
t
˜π = 1
Using these two expressions above and rearranging we obtain:
mj
t =
αj
t
1 + αj
t
1 −
ej
t+1
αj
t
Ej
t [πt+1]
which corresponds to the expression in the main text.
B Proof of Proposition 1
If d0 = δ, it is straightforward that π(d0) = π1(δ). Hence, the goal is to show that π(d0) exists
when d0 = δ. The logic consists in showing that, given d0, one can find a value for π0 so as to
be exactly at π1(dt) in period t, where π1(dt) denotes the solution to the quadratic equation
(14) given the level of seignoriage dt. Notice that such value is well defined for all dt as long as
d0 ∈ D.
Lemma 1 If d0 = δ, there exists a monotone and bounded sequence of initial conditions {βt}
such that for all t, π0 = βt implies πt = π1(dt).
This result indicates that limt→∞ βt is well defined. We denote this limit by π(d0), making
explicit its dependence on the initial value of seigniorage d0. Furthermore, it also indicates
that if π0 = π(d0), then limt→∞ πt = limt→∞ π1(dt) = π1(δ). Hence, the second statement of
Proposition 1 can be viewed as a corollary of Lemma 1. Although the two cases d0 < δ and
23

24. d0 > δ need to be considered separately, the proof is analogous so we present only the one
corresponding to d0 < δ.
Proof of Lemma 1. The proof is by induction. For the initial step, observe that
π0 = π1(δ) =⇒ π1 = (1 − ρ)π1(δ) + ρF(π1(δ), d0) > π1(δ) > π1(d1)
where the first inequality follows from the following property about F:
F(π, d) > π ⇐⇒ π ∈ (π1(d), π2(d))
and the second follows from the fact that d < δ implies π1(d) < π1(δ). On the other hand, we
also have that:
π0 = π1(d0) =⇒ π1 = (1 − ρ)F(π1(d0), δ) + ρπ1(d0) < π1(d0) < π1(d1)
Since F is continuous and monotone, it follows that there exists a unique β1 ∈ (π1(d0), π1(δ))
such that if π0 = β1 then π1 = π1(d1).
For the inductive step, suppose there exists βt such that π0 = βt implies πt = π1(dt). Then
it follows that
πt = π1(dt) =⇒ πt+1 = (1 − ρ)F(π1(dt), dt) + ρπ1(dt) < π1(dt) < π1(dt+1)
and we have again that
πt = π1(δ) =⇒ πt+1 = (1 − ρ)π1(δ) + ρF(π1(δ), dt) > π1(δ) > π1(dt+1)
Hence there exists βt+1 ∈ (βt, π1(δ)) such that π0 = βt+1 implies πt+1 = π1(dt+1). This also
implies that βt+1 > βt for all t, so the sequence is monotone and bounded.
To complete the proof of the proposition, take an arbitrary sequence {πt} that evolves
according to (13) and suppose π0 > π(d0). Two cases need to be considered. First, if π0 ≥
max{π1(δ), π1(d0)}, then (13) and the properties of F imply that statement 3 is satisfied.
Second, if π0 ∈ (π(d0), max{π1(δ), π1(d0)}), then the proof of Lemma 1 indicates there exists a
period t in which πt > max{π1(δ), π1(dt)} and therefore (13) and the properties of F again tell
us that statement 3 holds. The proof of the first statement, when π0 < π(d0), follows exactly
the same steps.
24

25. C Linearization of Equation 12
To linearize Equation 12, we treat πt, πe
t and πe
t+1 as different variables. Thus we obtain:
ˆπt =
δ
φ − φγπ1(δ) − δ
ˆdt −
φγπ1(δ)
φ − φγπ1(δ)
ˆπe
t +
φγπ1(δ)
φ − φγπ1(δ) − δ
ˆπe
t+1
Rational Expectations implies that, around the low inflation steady state, ˆπe
t = 0 for all t.
Therefore, the last two terms in the right hand side cancel out and we obtain the equation in
the main text.
D Estimating Inflation Persistence
To calculate inflation paths we use the following system
dt = (1 − ρ)δ + ρdt−1 + t (28)
πt =
φ − φγ(ρ2
ππt−2 + (1 − ρ2
π) βt−1)
φ − φγ(ρ2
ππt−1 + (1 − ρ2
π) βt) − dt
(29)
βt = βt−1 +
1
α
πt−1 − ρππt−2
1 − ρπ
− βt−1 (30)
with the initial condition {π−1, d0, α0, β0} = {π1, δ, α, π1}, where π1 is the low inflation steady
state in the RE equilibrium, and α is just a positive integer. We assume that the variance of
the i.i.d shock t is small enough so that the system remains stable.
We generate N samples of length T for each variable defined in (28)-(30). Let ˆρπ(N, T) be
the bootstrap estimate of the persistence of inflation, which is a function of all primitives of the
model, including ρπ and ρ. We allow ρπ = ρδ and restrict attention to positive autocorrelation
coefficients. Figure 4 displays ˆρπ(N, T) as a function of ρπ. The figure shows that for any ρ,
there is a unique ρπ such that ρπ = ˆρπ(N, T). We are interested in those fixed points because
they correspond to the case in which agents are able to estimate ρπ using past data. Since
we proved that in a RE equilibrium, ρ = ρπ, considering ρπ = ρδ implies that we are allowing
the persistence of inflation to be biased in a learning equilibrium. The nature of that bias is
portrayed in Figure 5, which plots the fixed point ρπ = ρπ = ˆρπ(N, T) as a function of ρ. The
figure indicates that the bias tends to be negative for low persistence of seigniorage and positive
for high persistence of seigniorage.
25

26. 0 0.2 0.4 0.6 0.8 1
ρπ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bootstrapestimateofρπ
ρ = 0.05 ρ = 0.48 ρ = 0.91
Figure 4: Inflation persistence: Bootstrap estimate vs True Parameter: The points in
which the solid lines cross the dashed line represent the fixed points ρπ = ˆρπ(N, T). For this
simulation we set N = 2500 and T = 2500.
0 0.2 0.4 0.6 0.8 1
ρδ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρπ
Figure 5: Inflation persistence vs. seigniorage Persistence: The solid line represent the
fixed points portrayed in Figure 4 for different values of ρ. For this simulation we set N = 2500
and T = 2500.
26

27. E Proof of Proposition 2
The proof considers the general formulation in which agents use information about the innova-
tions to the deficit to adjust their expectations regarding future inflation.
Restriction 1. Note that, according to the belief system:
et = π∗
t + ψ t + ut − π∗
t−1 + ψ t−1 + ut−1
= ηt + ψ t − ψ t−1 + ut − ut−1
whereas
dt − ρdt−1 = (1 − ρ)δ + t
so that Restriction 1 holds for i ≥ 2.
Restriction 2. To prove Restriction 2, observe that:
E [(dt − ρdt−1)et] = E [((1 − ρ)δ + t)(ηt + ψ t − ψ t−1 + ut − ut−1)] = ψσ2
while
E [(dt−1 − ρdt−2)et] = E [((1 − ρ)δ + t−1)(ηt + ψ t − ψ t−1 + ut − ut−1)] = −ψσ2
so that Restriction 2 also holds.
Restriction 3. Note that:
E [etet−1] = E [(ηt + ψ t − ψ t−1 + ut − ut−1)(ηt−1 + ψ t−1 − ψ t−2 + ut−1 − ut−2)]
= −ψ2
σ2
− σ2
u
Now consider the projection of dt − ρdt−1 on et. The projection is given by
Cov [et, dt − ρdt−1]
Var [dt − ρdt−1]
(dt − ρdt−1)
27

28. where
Cov [et, dt − ρdt−1] = Cov [ηt + ψ t − ψ t−1 + ut − ut−1, t] = ψσ2
Var [dt − ρdt−1] = σ2
so that the projection is given by ψ t, and the variance of the projection is ψ2
σ2
. Plugging
these results into the right hand side of Restriction 3 delivers:
E [et, et−1] + b Σb = −ψ2
σ2
− σ2
u + ψ2
σ2
= −σ2
u < 0
Restriction 4. Since all the perturbation terms have zero expectation, Restriction 4 follows
immediately.
F Model with Assets
The exogenous process is given by sequences ( ˆdt,ˆbt) such that b0 = 0, d0 = δ and
dt = (1 − ρ)dt−1 + ρδ + ξt
Thus, the true deficit evolves as before, and there is an initial value of assets set to zero.
Consider then the begining of period t, given bt−1, after the draw of dt. Define the new variable:
dt = max{0, dt − (1 + r)bt−1}
which is interpreted as the part of the deficit that can’t be financed with existing assets. Define
also:
bt−1 = max{(1 + r)bt−1 − dt, 0}
which corresponds to the stock of assets left after the deficit is financed. Notice that these
definitions imply that:
dt − bt−1 = max{dt − (1 + r)bt−1, 0} − max{(1 + r)bt−1 − dt, 0}
= max{dt − (1 + r)bt−1, 0} − min{dt − (1 + r)bt−1, 0}
= dt − (1 + r)bt−1
Now, we define a monetary policy rule that avoids deflation. The idea behind this policy rule is
that all money printing is used to finance what is left of the deficit once assets are depleted. In
that case, there are no additional resources created. However, money printing can exceed what
28

29. is needed to finance the deficit and in that case we assume the extra resources are accumulated
as assets. This amounts to set the law of motion for assets as follows:
ˆbt = max{bt−1, bt−1 + (mt − mt−1 − dt)} (31)
where mt denotes real money balances. In principle, total assets next period should be equal
to end of period assets bt−1 which corresponds to the first term in the max operator. However,
if at zero inflation, real balances were more than enough to cover the deficit after assets were
depleted, then those extra resources are indeed printed and used to purchase additional assets.
To understand the logic of this policy rule, notice that the money demand equation dictates
that:
φ(Pt − Pt−1) = (Mt − Mt−1) + φγ(Pe
t+1 − Pe
t )
So, in order to avoid deflation we must have that
Mt − Mt−1 ≥ φγ(Pe
t − Pe
t+1) (32)
Deflation becomes an issue only if prices are expected to move down. In that case, the gov-
ernment must print money to avoid deflation. If the money printed exceeds the amount of the
deficit to be financed, then it is used to purchase assets. In other words, if expectations are
such that:
φγ(Pe
t − Pe
t+1) > (dt − bt−1)Pt
then according to the government budget constraint, no asset accumulation implies:
Mt − Mt−1 = (dt − bt−1)Pt
so that (32) is violated and deflation ensues. In such a case, assets need to be accumulated to
ensure that φ(Pt − Pt−1) = 0. Since that implies
Mt − Mt−1 > (dt − bt−1)Pt
The excess money printing is used to purchase assets. In other words, the modified government
budget constraint is given by
Mt − Mt−1 + (bt−1 − dt)Pt = btPt
29

30. which, expressed in real terms, it can be written as:
mt −
mt−1
πt
+ bt−1 − dt = bt
Evaluated at zero realized inflation, the left hand side of the previous equation corresponds to
the second argument in the max operator of (31). Since it is increasing in πt, if at zero inflation
it is still positive, the excess resources printed are used to purchase extra assets.
30