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 ReserveBank 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.

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

1In Italy, following the December 2016 referendum, the political party five stars, who had grown in thepolls by large numbers, has proposed to call for a referendum to leave the Euro. In Greece, there was a heateddebate during the worst months of the debt crisis in 2014 and 2015. In Portugal, a political party campaigned toleave the Euro, with not much public support. See also https://krugman.blogs.nytimes.com/2010/04/28/how-reversible-is-the-euro/.

2

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 ejt+1 when old, and her preferences are given by

ln ct + αjt ln ct+1

The heterogeneity within each cohort is determined as follows:

ejt+1 = et+1 + ηjt+1

αjt = αt + εjt

2In discussing this issue, we follow Adam and Marcet (2011) closely.

3

where et and αt are aggregate shocks and ηjt and εjt 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 (ejt+1 < 1 for all t and j). Second, we assume that

agents have a relative preference for consumption when old (αjt > 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:

Ptcjt +M j

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+1xjt+1 ≤M j

t + ejt+1Pt+1 (2)

for all Pt+1. Hence, the problem of agent of type j born in period t consists in maximizing:

Ejt [ln ct + αjt 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 −M jt

= Ejt

[αjt

M jt + ejt+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

3As agents cannot issue money, the constraint M jt ≥ 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 bindingas long as the inflation rate is not too high, thus we ignore this constraint in our theoretical analysis. In thenumerical section, we impose this constraint on the equilibrium.

4

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:

M jt

Pt= φjt

(1− γjtE

jt

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

φjt(Pt − γjtE

jtPt+1

)dj (5)

Observe that if we impose homogeneous expectations we could write EjtPt+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

ηjt+1 = εjt = 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

4The linearization and the expressions for φjt and γjt are detailed in the Appendix.

5

distribution of ejt+1 and αjt 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

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− γπet+1

)(7)

where πet+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:

M st = M s

t−1 + dtPt (8)

where M st is the money supply and dt denotes seigniorage, which evolves according to:

dt = (1− ρ)δ + ρdt−1 + εt (9)

where εt 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:

πet+1 = EPt[Pt+1

Pt

](10)

5In the quantitative section, we allow for policy regimes in which the government is able to deplete interna-tional reserves to finance its deficit.

7

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:

πet+1 = Et[Pt+1

Pt

](11)

In equilibrium we must have Mdt = M s

t = Mt, which allows us to combine the money demand

equation (7) and the government budget constraint (8) to obtain:

πt =φ− φγπet

φ− φγπet+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 πet = 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

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 − 2γ12 ))

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 6= 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

0 20 40 60 80 100t

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

πt

d0 < δ

π1(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 statestarts 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:

πt =δ

φ− φγπ1(δ)− δdt (17)

dt = ρdt−1 + εt (18)

where we are using the notation xt = lnxt − ln x, with bold letters indicating steady state

values. We can express inflation recursively as:

πt = ρπt−1 + νt (19)

6See Appendix C for details.

10

where νt ≡ δεt/(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) π?t + ρππt−1 + ut (20)

π?t = π?t−1 + ηt

where ut ∼ N(0, σ2u) and ηt ∼ N(0, σ2

η) are i.i.d. and independent of the shock to the seigniorage

εt. We allow for ρπ 6= 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−17. We assume agents observe the realizations of inflation but

not those of π?t and ut separately. Thus, the learning problem consists of filtering out long run

inflation π?t .

We will use βt to denote the prior belief regarding π?t entering period t. Notice that if we

make σ2η = 0 and assign probability 1 to β0 = π1(δ), we obtain:

πet = (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

7In principle, we could allow for a general formulation such as:

πt = (1− ρπ)π?t + ρππt−1 + ψεt + ut

π?t = π?t−1 + ηt

where agents would use information about the innovations to the deficit to adjust their expectations regardingfuture inflation. In equilibrium, we must require that ψ is consistent ex-post with the generated equilibriumprocess for inflation.

11

0 20 40 60 80 100time

-6

-4

-2

0

2

4

% d

evia

tio

n f

rom

ste

ad

y s

tate

×10-4

ρ = 0.75

inflation seignoriage

0 20 40 60 80 100time

-6

-4

-2

0

2

4

% d

evia

tio

n f

rom

ste

ad

y s

tate

×10-4

ρ = 0.25

inflation seignoriage

Figure 2: Sample paths for inflation and seignoriage around the low inflation steadystate in the linearized rational expectations equilibrium.

12

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:

πet+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 πet+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:

πet+1 =(1− ρ2π

)βt + ρ2ππt−1 (24)

Using this equation, we rewrite (12) as follows:

πt =φ− φγ((1− ρ2π) βt−1 + ρ2ππt−2)

φ− φγ((1− ρ2π) βt + ρ2ππt−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)− φγφ− φγπi(δ)− δ

< 1 (26)

As long as the denominator in (26) is positive10, we can verify that this is indeed the case if

8In 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.

9Note that under this assumption, there is no need to include the innovations to the seigniorage in (20), asproposed in Footnote (7), since they would be independent of past observations. Put differently, εt would beindependent of information used at t to forecast inflation.

10Whenever there is a positive price that clears the money market, the denominator will be positive. We

13

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. β 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 σε 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 theexistence of such a price, in the spirit of Marcet and Nicolini (2003).

11The extended model that considers asset accumulation is detailed in the appendix.12The 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

Table 1: Parameters for Baseline Economy

Parameter Symbol ValuePersistence of deficit ρ .9334SD of shocks to deficit σε .0146Persistence of inflation ρ?π 9334Money Demand Parameter φ .36Money Demand Parameter γ .39ERR 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

Table 2: Probabilities of Hyperinflations for βU = 4.00 and T = 100

Probability of n hyperinflationsDeficit mean δ = 0.0%, Initial Deficit d0 = 1.0%1/α 0 1 2 ≥ 30.05 41.54 26.68 15.96 15.820.04 42.06 26.38 15.80 15.760.03 43.38 26.12 15.34 15.160.02 43.98 25.72 14.54 15.760.01 66.12 20.64 7.56 5.680.00 100.00 0.00 0.00 0.00Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%1/α 0 1 2 ≥ 30.05 31.70 28.12 17.84 22.340.04 32.58 28.08 18.56 20.780.03 33.58 27.96 17.80 20.660.02 35.10 27.82 17.38 19.700.01 59.52 22.76 10.06 7.660.00 100.00 0.00 0.00 0.00Deficit mean δ = 4.0%, Initial Deficit d0 = 1.0%1/α 0 1 2 ≥ 30.05 6.16 13.86 18.02 61.960.04 6.48 13.90 19.00 60.620.03 7.92 15.44 18.60 58.040.02 18.26 21.06 18.92 41.760.01 70.86 14.18 6.74 8.220.00 100.00 0.00 0.00 0.00Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%1/α 0 1 2 ≥ 30.05 4.22 10.90 16.40 68.480.04 4.38 11.02 17.92 66.680.03 5.28 12.84 17.94 63.940.02 13.04 17.90 18.98 50.080.01 63.58 15.98 9.24 11.200.00 100.00 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

0 20 40 60 80 1000

1

2

3

4

5

6

7 8 9 10

quart

erly inflation r

ate

(lo

g s

cale

)

1/α = 0.051/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 equilibriumwith a positive constant gain and the solid red line corresponds to an equilibrium with fixedexpectations 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

Table 3: Rejection Frequencies at the 5% level for βU = 4This table reports rejection frequencies obtained from testing Restriction 1 of Proposition 2using simulated data, including up to three lags of xt as an instrument. All restrictions includea constant as an instrument.

T40 60 100 200

Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01Restriction 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 = 0.05Restriction 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 = 0.01Restriction 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 = 0.05Restriction 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:

QT = T

(1

T

T∑t=1

yt

)′S−1

(1

T

T∑t=1

yt

)→ χ2

n

where S is the sample counterpart of 13:

S = 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

13Here we exploit the MA(1) property of et and use the fact that beyond the the first lead and lag, theseauto-covariances matrices must be equal to zero.

18

Table 4: Rejection Frequencies at the 5% level for βU = 4This table reports rejection frequencies obtained from testing the Restrictions 2-4 of Proposition2 using simulated data.

T40 60 100 200

Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01Restriction 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 = 0.05Restriction 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 = 0.01Restriction 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 = 0.05Restriction 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 QT . However, we also want to consider testing restrictions by assuming

that agents use the small sample properties of QT , for different values of T .

In the case of Restrictions 1, 2 and 4, asymptotic theory implies that QT → χ2n 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

14Notice that by including a constant, Restriction 4 is embedded in the joint hypothesis testing performedfor Restriction 1.

19

Table 5: Rejection Frequencies at the 5% level for βU = 4This table reports rejection frequencies obtained from testing Restriction 1 of Proposition 2using simulated data, including up to three lags of xt as an instrument. All restrictions includea constant as an instrument.

T40 60 100 200

Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01Restriction 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 = 0.05Restriction 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 = 0.01Restriction 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 = 0.05Restriction 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

Table 6: Rejection Frequencies at the 5% level for βU = 4This table reports rejection frequencies obtained from testing the restrictions of Proposition 2using simulated data.

T40 60 100 200

Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01Restriction 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 = 0.05Restriction 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 = 0.01Restriction 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 = 0.05Restriction 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

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

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 mjt ≡ M j

t /Pt and πt+1 ≡ Pt+1/Pt. The

linearization boils down to:(1

1− mj

)2

(mjt − mj) = −

(αjt

mjt + ejt+1πt+1

)21

αjt

{Ejt [m

jt − mj] + ejt+1E

jt [π

jt − πj]

}where the tilde variables represent steady state variables. In steady state we must have that

1

1− mj=

αjt

mj + ejt+1π

which also implies that1 + αjt

αjtmj +

ejt+1

αjtπ = 1

Using these two expressions above and rearranging we obtain:

mjt =

αjt

1 + αjt

{1−

ejt+1

αjtEjt [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 6= 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 6= 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

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

C Linearization of Equation 12

To linearize Equation 12, we treat πt, πet and πet+1 as different variables. Thus we obtain:

πt =δ

φ− φγπ1(δ)− δdt −

φγπ1(δ)

φ− φγπ1(δ)πet +

φγπ1(δ)

φ− φγπ1(δ)− δπet+1

Rational Expectations implies that, around the low inflation steady state, πet = 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 + εt (28)

πt =φ− φγ(ρ2ππt−2 + (1− ρ2π) βt−1)

φ− φγ(ρ2ππt−1 + (1− ρ2π) 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 εt is small enough so that the system remains stable.

We generate N samples of length T for each variable defined in (28)-(30). Let ρπ(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 6= ρδ and restrict attention to positive autocorrelation

coefficients. Figure 4 displays ρπ(N, T ) as a function of ρπ. The figure shows that for any ρ,

there is a unique ρπ such that ρπ = ρπ(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 6= 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 ρ?π = ρπ = ρπ(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

0 0.2 0.4 0.6 0.8 1ρπ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Boots

trap e

stim

ate

of ρπ

ρ = 0.05 ρ = 0.48 ρ = 0.91

Figure 4: Inflation persistence: Bootstrap estimate vs True Parameter: The points inwhich the solid lines cross the dashed line represent the fixed points ρπ = ρπ(N, T ). For thissimulation 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 thefixed points portrayed in Figure 4 for different values of ρ. For this simulation we set N = 2500and T = 2500.

26

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 + ψεt + ut −(π∗t−1 + ψεt−1 + ut−1

)= ηt + ψεt − ψεt−1 + ut − ut−1

whereas

dt − ρdt−1 = (1− ρ)δ + εt

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 + εt)(ηt + ψεt − ψεt−1 + ut − ut−1)] = ψσ2ε

while

E [(dt−1 − ρdt−2)et] = E [((1− ρ)δ + εt−1)(ηt + ψεt − ψεt−1 + ut − ut−1)] = −ψσ2ε

so that Restriction 2 also holds.

Restriction 3. Note that:

E [etet−1] = E [(ηt + ψεt − ψεt−1 + ut − ut−1)(ηt−1 + ψεt−1 − ψεt−2 + ut−1 − ut−2)]

= −ψ2σ2ε − 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

where

Cov [et, dt − ρdt−1] = Cov [ηt + ψεt − ψεt−1 + ut − ut−1, εt] = ψσ2ε

Var [dt − ρdt−1] = σ2ε

so that the projection is given by ψεt, and the variance of the projection is ψ2σ2ε . Plugging

these results into the right hand side of Restriction 3 delivers:

E [et, et−1] + b′Σb = −ψ2σ2ε − σ2

u + ψ2σ2ε = −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 (dt, bt) 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

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:

bt = 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(P et+1 − P e

t )

So, in order to avoid deflation we must have that

Mt −Mt−1 ≥ φγ(P et − P e

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(P et − P e

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

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