BANCO DE PORTUGAL
Economic Research Department
UNIQUE EQUILIBRIUM WITH SINGLE MONETARY
WP 12-05November 2005
The analyses, opinions and findings of these papers represent the views of the
authors, they are not necessarily those of the Banco de Portugal.
Please address correspondence to Banco de Portugal, Economic Research Department,
Av. Almirante Reis, no. 71 1150–012 Lisboa, Portugal;
Bernardino Adão, tel: # 351-21-3128409, email: email@example.com;
Isabel Correia, tel: # 351-21-3128385, email: firstname.lastname@example.org;
Pedro Teles, tel: # 351-21-3130035, email: email@example.com.
Unique Equilibrium with Single Monetary
Banco de Portugal
Banco de Portugal, Universidade Catolica Portuguesa and CEPR
Banco de Portugal, Universidade Catolica Portuguesa,
Federal Reserve Bank of Chicago, CEPR.
We consider standard cash-in-advance monetary models and show that
there are interest rate or money supply rules such that equilibria are unique.
The existence of these single instrument rules depends on whether the econ-
omy has an in…nite horizon or an arbitrarily large but …nite horizon.
Key words: Monetary policy; interest rate rules; unique equilibrium.
JEL classi…cation: E31; E40; E52; E58; E62; E63.
In this paper we revisit the issue of multiplicity of equilibria when monetary policy
is conducted with either the interest rate or the money supply as the instrument of
?This paper had an earlier version with the title "Conducting Monetary Policy with a Single
Instrument Feedback Rule". We thank Andy Neumeyer, Stephanie Schmitt–Grohe and Martin
Uribe for comments. We gratefully acknowledge …nancial support of FCT. The opinions are
solely those of the authors and do not necessarily represent those of the Banco de Portugal,
Federal Reserve Bank of Chicago or the Federal Reserve System.
policy. There has been an extensive literature on this topic starting with Sargent
and Wallace (1975), including a recent literature on local and global determinacy
in models with nominal rigidities.We show that it is possible to implement
a unique equilibrium with an appropriately chosen interest rate feedback rule,
and similarly with a money supply feedback rule of the same type. This is a
surprising result because while it is well known that interest rate feedback rules
can deliver a locally unique equilibrium, it is no less known that they generate
multiple equilibria globally.
We show that the reason for the results is the model assumption of an in…nite
horizon. In …nite horizon economies, the number of degrees of freedom in con-
ducting policy does not depend on the way policy is conducted. The number is
the same independently of whether interest rates are set as constant functions of
the state, or as backward, current or forward functions of endogenous variables.
In analogous …nite horizon economies, the number of degrees of freedom in
conducting policy can be counted exactly. The equilibrium is described by a
system of equations where the unknowns are the quantities, prices and policy
variables. There are more unknowns than variables, and the di¤erence is the
number of degrees of freedom in conducting policy. It is a necessary condition for
there to be a unique equilibrium that the same number of exogenous restrictions
on the policy variables be added to the system of equations. Single instrument
policies are not su¢cient restrictions. They always generate multiple equilibria.
This is no longer the case in the in…nite horizon economy, as we show in this
Whether the appropriate description of the world is an in…nite horizon economy
or the limit of …nite horizon economies, thus, makes a big di¤erence for this
particular issue of policy interest, i. e. whether policy conducted with a single
instrument, such as the nominal interest rate, is su¢cient to determine a unique
As already mentioned, after Sargent and Wallace (1975) and McCallum (1981),
there is a large literature on multiplicity of equilibria when the government fol-
lows either an interest rate rule or a money supply rule. This includes the liter-
ature on local determinacy that identi…es conditions on preferences, technology,
timing of markets, and policy rules, under which there is a unique local equilib-
rium (see Bernanke and Woodford (1997), Clarida, Gali and Gertler (1999, 2000),
Carlstrom and Fuerst (2001, 2002), Benhabib, Schmit-Grohe and Uribe (2001a),
Dupor (2001) among others). This literature has in turn been criticized by recent
work on global stability that makes the point that the conditions for local deter-
minacy are also conditions for global indeterminacy (see Benhabib, Schmit-Grohe
and Uribe (2001b) and Christiano and Rostagno, 2002).
Our modelling approach is close to Adao, Correia and Teles (2003) for the case
with sticky prices. In this paper we show that even at the optimal zero interest
rate rule there is still room for policy to improve welfare since it is possible to use
money supply to implement the optimal allocation in a large set of implementable
allocations. This paper is also close to Adao, Correia and Teles (2004) where
we show that it is possible to implement unique equilibria in environments with
‡exible prices and prices set in advance by pegging state contingent interest rates
as well as the initial money supply. Bloise, Dreze and Polemarchakis (2004) and
Nakajima and Polemarchakis (2005) are also related research.
We assume that …scal policy is endogenous. Exogeneity of …scal policy could
be used, as in the …scal theory of the price level to determine unique equilibria.
The paper proceeds as follows: In Section 2, we consider a simple cash in
advance economy with ‡exible prices. In Section 3 we analyze a simple example
to discuss the properties of the equilibria obtained when a single monetary policy
instrument is used. In Section 4, we show that there are single instrument feedback
rules that implement a unique equilibrium. In Section 5 we show that in analogous
…nite horizon environments the single instrument rules would generate multiple
equilibria. In Section 6, we show that the results generalize to the case where
prices are set in advance. Section 7 contains concluding remarks.
2. A model with ‡exible prices
We …rst consider a simple cash in advance economy with ‡exible prices. The
economy consists of a representative household, a representative …rm behaving
competitively, and a government. The uncertainty in period t ? 0 is described
by the random variable st2 Stand the history of its realizations up to period t
(state or node at t), (s0;s1;:::;st), is denoted by st2 St. The initial realization s0
is given. We assume that the history of shocks has a discrete distribution. The
number of states in period t is ?t.
Production uses labor according to a linear technology. We impose a cash-
in-advance constraint on the households’ transactions with the timing structure
described in Lucas and Stokey (1983). That is, each period is divided into two
subperiods, with the assets market operational in the …rst subperiod and the goods
market in the second.
2.1. Competitive equilibria
Lt, described by the expected utility function:
The households have preferences over consumption Ct, and leisure
U = E0
where ? is a discount factor. The households start period t with nominal wealth
Wt: They decide to hold money, Mt, and to buy Btnominal bonds that pay RtBt
one period later. Rtis the gross nominal interest rate at date t. They also buy
Bt;t+1units of state contingent nominal securities. Each security pays one unit of
money at the beginning of period t + 1 in a particular state. Let Qt;t+1be the
beginning of period t price of these securities normalized by the probability of
the occurrence of the state. Therefore, households spend EtQt;t+1Bt;t+1in state
contingent nominal securities. Thus, in the assets market at the beginning of
period t they face the constraint
Mt+ Bt+ EtQt;t+1Bt;t+1? Wt
Consumption must be purchased with money according to the cash in advance
At the end of the period, the households receive the labor income WtNt; where
Nt= 1 ? Ltis labor and Wtis the nominal wage rate and pay lump sum taxes,
Tt. Thus, the nominal wealth households bring to period t + 1 is
Wt+1= Mt+ RtBt+ Bt;t+1? PtCt+ WtNt? Tt
The households’ problem is to maximize expected utility (2.1) subject to the
restrictions (2.2), (2.4), (3.4), together with a no-Ponzi games condition on the
holdings of assets.
The following are …rst order conditions of the households problem:
??uC(t + 1)
Qt;t+1= ?uC(t + 1)
Pt+1, t ? 0
Rt.Condition (2.5) sets the intratem-
From these conditions we get EtQt;t+1=
poral marginal rate of substitution between leisure and consumption equal to the
real wage adjusted for the cost of using money, Rt. Condition (2.6) is an in-
tertemporal marginal condition necessary for the optimal choice of risk-free nom-
inal bonds. Condition (2.7) determines the price of one unit of money at time
t + 1, for each state of nature st+1, normalized by the conditional probability of
occurrence of state st+1, in units of money at time t.
tion of the representative …rm is linear
The …rms are competitive and prices are ‡exible. The production func-
The equilibrium real wage is
plies, Mt, state noncontingent public debt, Bt. We can de…ne a policy as a map-
ping for the policy variables fTt;Rt;Mt;Bt, t ? 0, all stg, that maps sequences of
quantities, prices and policy variables into sets of sequences of the policy variables.
De…ning a policy as a correspondence allows for the case where the government
is not explicit about some of the policy variables. Lucas and Stokey (1983) de…ne
policy as sequences of numbers for some of the variables. Adao, Correia and Teles
(2003) de…ne policy as sequences of numbers for all the policy variables. Here
we allow for more generic functions (correspondences) for all the policy variables.
We do not allow for targeting rules that can be de…ned as mappings from prices,
quantities and policy variables to prices and quantities.
The period by period government budget constraints are
The policy variables are taxes, Tt, interest rates, Rt, money sup-
Mt+ Bt= Mt?1+ Rt?1Bt?1+ Pt?1Gt?1? Pt?1Tt?1, t ? 0
Let Qt+1? Q0;t+1, with Q0= 1. If limT!1EtQT+1WT+1= 0
EtQt;t+s+1Mt+s(Rt+s? 1) = Wt+
Market clearing in the goods and labor market requires
Ct+ Gt= AtNt,
Nt= 1 ? Lt.
We have already imposed market clearing in the money and debt markets.
tities and prices such that the private agents maximize given the sequences of
policy variables and prices, the budget constraint of the government is satis…ed
and the policy sequence is in the set de…ned by the policy.
The equilibrium conditions for the variables fCt;Lt;Rt;Mt;Bt;Tt;Qt;t+1g are
the resources constraint
A competitive equilibrium is a sequence of policy variables, quan-
Ct+ Gt= At(1 ? Lt), t ? 0
the intratemporal condition that is obtained from the households intratemporal
condition (2.11) and the …rms optimal condition (2.8)
At, t ? 0
as well as the cash in advance constraints (3.4), the intertemporal conditions (2.6)
and (2.7), and the budget constraints (2.9).
In this section we consider a particular utility function to discuss the properties
of equilibria when the central bank chooses either the interest rate or the money
supply as the sole instrument of monetary policy. We discuss the properties of the
equilibria, paying particular attention to the so called local determinacy property
of the equilibrium. Local determinacy means that in the neighborhood of an
equilibrium there is no other equilibrium.
We also consider an interest rate feedback rule as the literature is currently
dominated by a rule-based approach to monetary policy. We review what is meant
by an interest rate feedback rule guaranteeing local determinacy and show that
local determinacy is achieved if the interest rate feedback rule satis…es the Taylor
principle. The Taylor principle is veri…ed if in response to an increase in in‡ation
the increase in the nominal interest rate is higher.
To simplify the presentation we take Gt= 0 and the utility function u(Ct;Lt) =
Ct+v (Lt); with v (Lt) increasing in Lt, limLt!0v0(Lt) = 1 and limLt!1v0(Lt) =
0. We consider 3 monetary policies: a constant interest rate, a constant growth
rate for the money supply and an interest rate feedback rule. For the sake of sim-
plicity we consider the deterministic environment, i.e. st= st+1for all t. The
stochastic environment is considered in the appendix.
The equilibrium conditions for the variables fCt;Lt;Pt;Mt;Rtg are: the house-
hold’s intratemporal and intertemporal conditions
the feasibility condition
Ct= A(1 ? Lt);
and the cash in advance condition
Mt? PtCt; with equality if Rt> 1:
It will be useful for the discussion below to remember that from (3.1) and (3.3)
there is a positive relation between Ltand Rtand a negative relation between Ct
3.1. Constant interest rate
Here we assume that the central bank chooses to maintain a constant interest
rate equal to R ? 1: In this case Ct and Lt are pin down by (3.1) and (3.3).
The in‡ation, ?t, is pin down by (3.2), ?t= R?. Any positive real number is an
equilibrium P0. Thus, there is a multiplicity of equilibrium price sequences and
as a consequence from (3.4) a multiplicity of equilibrium money sequences. The
literature has a jargon for this result, it is said that the outcome of setting the
interest rate is real determinacy and nominal indeterminacy. All the equilibria are
locally undetermined as for any equilibrium price level there is another equilibrium
price level in its neighborhood. In a stochastic environment with nominal frictions,
like sticky prices or sticky wages, the monetary policy of setting the interest rate
is less interesting since it leads to multiplicity of the real allocations. We clarify
this issue in the appendix.
3.2. Constant money growth
Here we study the equilibria when the central bank chooses M0and a constant
rate of money growth of the form Mt= ?tM0, where ? >
equilibria. In order to show that, we …nd it useful to de…ne real money as mt?Mt
and replace (3.1) in (3.2)
?: There are many
mt+1= ?(Lt)mt, where ?(Lt) =?v0(Lt)
There are two steady states: one with
that solve (3.1) and (3.3) for Rt=?
is another steady state with Rt= 1; Ctand Lt(=eL) that solve (3.1) and (3.3)
determined but the initial price level is not since (3.4) may not be binding when
The remaining equilibria can be divided according to the value of leisure in
period zero, L0: There are many equilibria with L0 > L. From (3.5) we get
m0= ?(L0) < 1. Thus, from (3.3) and the fact that (3.4) holds with equality
in period 1 we obtain L1 < L0 which implies ?(L1) < ?(L0): Proceeding in
this way we obtain mtand Ctapproaching zero and Ltapproaching 1: From (3.1)
and the fact that mtapproaches zero we obtain Rtand Ptapproaching in…nity.
There are also equilibria witheL < L0< L: By (3.1), R0> 1, which means
we get sequences fLt;Ct;mtg. Let t?be the …rst period such that Lt? obtained
from the process just described satis…es (3.1) with Rt? 1: The elements of the
sequence up to t?are part of the equilibrium, but the ones after are not. In period
t?(3.4) does not hold with equality which implies that Rt? = 1: This means that
in periods t?;t?+ 1;t?+ 2;::: the equilibrium Ltmust satisfy (3.1) for Rt = 1;
which we denoted byeL and the equilibrium Ct solves (3.3) for Lt =eL: Also
approaches in…nity and Ptapproaches zero.
= 1; Rt=
?> 1; Ctand Lt(= L)
?, and Ptsatisfying (3.4) with equality. There
for Rt= 1;
> 1 and
?(eL). In this steady state in‡ation is
?(Lt)approaches in…nity as the denominator approaches zero.
that (3.4) holds with equality. From (3.5), (3.3) and the assumption that mt= ct
> 1 for t ? t?; and in‡ation is constant,
?(eL)for t ? t?, mt
The steady state associated with Rt=?
the steady state associated with Rt= 1, for all t; is locally undetermined.
?, for all t; is locally determined and
3.3. Interest rate feedback rule
Now we study the equilibria when the central bank follows an interest rate feed-
back rule. Let R be a steady state equilibrium interest rate and let ? be the
corresponding steady state equilibrium in‡ation rate. Then, R =
is the real interest rate. Assume that the central bank conducts a pure current
nonlinear Taylor rule:1
where ?? ? 1 (the Taylor principle), and ?t?
rule in the intertemporal condition of the household, (3.2), we get
Pt?1. After substituting the Taylor
?: By recursive substitution we get
zt+k= (zt)k??; for all k and t:
There is no condition to pin down the initial value for in‡ation. Since the initial
in‡ation level can be any value there is an in…nity of equilibrium trajectories for
the in‡ation rate. Nevertheless, they can be typi…ed in 3 classes. Either in‡ation
is constant, ?t = ?, or there is an hyperin‡ation, ?t ?! 1, or in‡ation is
approaching zero, ?t?! 0. This is easy to verify. If ?0= ?; then (3.6) implies
that ?t= ? for all t: If ?0> ?; then (3.6) implies that ?t+1> ?tand ?t?! 1;
since ?? > 1: If ?0< ?; then (3.6) implies that ?t+1< ?tand ?t?! 0; since
?? > 1:
Thus, when the central bank follows a Taylor rule that obeys the Taylor prin-
ciple it is able to get local determinacy. In a neighborhood of the steady state
in‡ation ? there is no other equilibrium in‡ation trajectory. But we have just
seen that there is an in…nity of other equilibria for in‡ation which converge to zero
1Usually the Taylor rule is presented in its linearized form. As can be veri…ed the linearized
Rt? R = ? (?t? ?):
or to in…nity. These results beg two interrelated questions: Why is local deter-
minacy such an interesting property? Or why has most of the literature assumed
that undesirable equilibria do not happen? We do not know the answer to these
It is easy to verify, using an argument similar to the one above, that if the
Taylor rule did not obey the Taylor principle, i.e. ?? < 1, there would be just two
types of equilibrium. The steady state and an in…nity of equilibria converging to
the steady state. At …rst sight it would seem that it would be preferable that a
central bank would follow a Taylor rule that did not satisfy the Taylor principle, as
"undesirable" equilibria, hyperin‡ations or hyperde‡ations would not be possible.
This conclusion is not correct because whenever there is multiplicity of equilibria
it may be possible that sunspots can cause large ‡uctuations in in‡ation. In‡ation
can ‡uctuate randomly just because agents come to believe this will happen.
Why do we get so many equilibria? Is it possible that we are forgeting equilib-
rium conditions? There are no more equilibrium conditions over these variables.
The so called transversality conditions are satis…ed since in our economy there are
government bonds. Moreover, since our …scal authority has a Ricardian policy
the government’s in…nite-horizon budget constraint does not provide additional
information. In particular it cannot be used to obtain the initial price level as it
is done in the …scal theory of price level literature.
There may be institutions that we have ignored in the model, which can be used
to eliminate some of these "undesirable" equilibria. For instance, in some models
an hyperin‡ation can be eliminated if the central bank has su¢cient real resources
and can commit to buy back its currency if the price level exceeds a certain level.
This is known as fractional real backing of the currency (seeObstfeld and Rogo¤
(1983)). We are not going to pursue this issue here.
4. Single instrument feedback rules.
In this section we assume that policy is conducted with either interest rate or
money supply feedback rules. We show that there are single instrument feedback
rules that implement a unique equilibrium for the allocation and prices. The
proposition for an interest rate feedback rule follows:
Proposition 4.1. When the …scal policy is endogenous and monetary policy is
conducted with the interest rate feedback rule
?tis an exogenous variable, there is a unique equilibrium.
Proof: Suppose policy is conducted with the interest rate feedback rule Rt=
can be written as
. Then the intertemporal and intratemporal conditions, (2.6) and (2.11)
= ?t, t ? 0
, t ? 0
These conditions together with the cash in advance conditions, (3.4), and the
resource constraints, (2.10), determine uniquely the variables Ct, Lt, Ptand Mt.
The budget constraints (4.4) are satis…ed for multiple paths of the taxes and
state noncontingent debt levels?
The forward looking interest rate feedback rules that guarantee uniqueness of
the equilibrium resemble the rules that appear to be followed by central banks.
The nominal interest rate reacts positively both to the forecast of future consump-
tion and to the forecast of the future price level. In this there is a di¤erence to the
feedback rules that are usually considered in that it depends on the future price
level rather than in‡ation.
Depending on the exogenous process for ?t, with this feedback rule it is possible
to decentralize any feasible allocation distorted by the nominal interest rate. The
…rst best allocation, at the Friedman rule of a zero nominal interest rate, can also
be implemented. With ?t=
?t, t ? 0, condition (4.2) becomes
At, t ? 0
which, together with the resource constraint (2.10) gives the …rst best allocation
Ct= C(At;Gt), Lt= L(At;Gt). The price level Pt= P(At;Gt) can be obtained
using (4.1), i.e.
?t, t ? 0;
and the money supply is obtained using the cash-in-advance constraint, Mt =
Allocations where in‡ation is zero can also be implemented even if in this ‡ex-
ible price environment they are not desirable. There are multiple such allocations
with nominal interest rates satisfying
?EtuC(C(At+1;Gt+1;Rt+1);L(At+1;Gt+1;Rt+1)), t ? 0
where the functions C and L are the solution for Ct and Lt of the system of
equations given by (2.11) and (2.10).
For each path of the nominal interest rate, fRtg, associated with zero in‡ation,
there is a unique path for f?tg up to a constant term,
= ?t, t ? 0.
In an economy where the use of money is becomes negligible which corresponds
to a cash-in-advance condition
where vt! 0, there is a single path for the nominal interest rate consistent with
?EtuC(C(At+1;Gt+1);L(At+1;Gt+1)), t ? 0
An analogous proposition to Proposition 3.1 is obtained when policy is con-
ducted with a particular money supply feedback rule.
Proposition 4.2. When the …scal policy is endogenous and the policy is con-
ducted with the money supply feedback rule,
there is a unique equilibrium.
Proof: Suppose policy is conducted according to the money supply rule Mt=
. Then, the equilibrium conditions
obtained using the cash in advance conditions (3.4),
obtained from the intertemporal conditions (2.6), in addition to the resource con-
straints, (2.10) and the intratemporal conditions (2.11) determine uniquely the
four variables, Ct, ht, Pt, Rtin each period t ? 0 and state st.
The taxes and debt levels satisfy the budget constraint (4.4)?
The result that there are single instrument feedback rules that implement a
unique equilibrium is a surprising one. In fact it is well known that interest rate
rules may implement a determinate equilibrium, but not a unique global equilib-
rium. To illustrate this, consider the case where monetary policy is conducted
with constant functions for the policy variables. We will show that in that case
an interest rate policy generates multiple equilibria. That result is directly ex-
tended to the case where the interest rate is a function of contemporaneous or
4.1. Conducting policy with constant functions.
In this section, we show that in general when policy is conducted with constant
functions for the policy instruments, it is necessary to determine exogenously both
interest rates and money supplies.
The equilibrium conditions are the resources constraints, (2.10), the intratem-
poral conditions (2.11), the cash in advance constraints (3.4), the intertemporal
conditions (2.6) and the budget constraints (2.9) that can be written as
?suC(t + s)Ct+s
?suC(t + s)[Gt+s? Tt+s]
These conditions de…ne a set of equilibrium allocations, prices and policy vari-
ables. There are many equilibria. We want to determine conditions on the ex-
ogeneity of the policy variables such that there is a unique equilibrium in the
allocation and prices. We …rst consider the case in which a policy are sequences
of numbers for money supplies and interest rates.
From the resources constraints,(2.10), the intratemporal conditions (2.11), and
the cash in advance constraints, (3.4), we obtain the functions Ct= C(Rt) and
Lt = L(Rt) and Pt =
C(Rt), t ? 0. As long as uC(Ct;Lt)Ctdepends on Ctor
Lt, excluding therefore preferences that are additively separable and logarithmic
in consumption, the system of equations can be summarized by the following
, t ? 0
together with the budget constraints, (4.4).
Suppose the path of money supply is set exogenously in every date and state.
In addition, in period zero the interest rate, R0, is set exogenously and, for each
t ? 1, for each state st?1, the interest rates are set exogenously in #St?1 states.
In this case there is a single solution for the allocations and prices. Similarly,
there is also a unique equilibrium if the nominal interest rate is set exogenously in
every date and state, and so is the money supply in period 0, M0, as well as, for
each t ? 1, and for state st?1, the money supply in #St? 1 states. The budget
constraints restrict, not uniquely, the taxes and debt levels.
The proposition follows
Proposition 4.3. Suppose policy are constant functions. In general, if money
supply is determined exogenously in every date and state, and if interest rates are
also determined exogenously in the initial period, as well as in ?t??t?1states for
each t ? 1, then the allocations and prices can be determined uniquely. Similarly,
if the exogenous policy instruments are the interest rates in every state, the initial
money supply and the money supply, in ?t? ?t?1states, for t ? 1, then there is
in general a unique equilibrium.
The proposition states a general result. In the particular case where the prefer-
ences are additively separable and logarithmic in consumption, and money supply
is set exogenously in every state, there is a unique equilibrium in the allocations
and prices. There is no need to set exogenously the interest rates as well. This
example is helpful in understanding the main point of the paper, that the degrees
of freedom in conducting policy depend on how policy is conducted and on other
characteristics of the environment.