# Unique equilibrium with single monetary instrument rules

**ABSTRACT** 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 economy has an infinite horizon or an arbitrarily large but finite horizon.

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**ABSTRACT:**One measure of the health of the Social Security system is the difference between the market value of the trust fund and the present value of benefits accrued to date. How should present values be computed for this calculation in light of future uncertainties? We think it is important to use market value. Since claims on accrued benefits are not currently traded in financial markets, we cannot directly observe a market value. In this paper, we use a model to estimate what the market price for these claims would be if they were traded. In valuing such claims, the key issue is properly adjusting for risk. The traditional actuarial approach â€“ the approach currently used by the Social Security Administration in generating its most widely cited numbers - ignores risk and instead simply discounts â€œexpectedâ€ future flows back to the present using a risk-free rate. If benefits are risky and this risk is priced by the market, then actuarial estimates will differ from market value. Effectively, market valuation uses a discount rate that incorporates a risk premium. Developing the proper adjustment for risk requires a careful examination of the stream of future benefits. The U.S. Social Security system is â€œwage-indexedâ€: future benefits depend directly on future realizations of the economy-wide average wage index. We assume that there is a positive long-run correlation between average labor earnings and the stock market. We then use derivative pricing methods standard in the finance literature to compute the market price of individual claims on future benefits, which depend on age and macro state variables. Finally, we aggregate the market value of benefits across all cohorts to arrive at an overall value of accrued benefits. We find that the difference between market valuation and â€œactuarialâ€ valuation is large, especially when valuing the benefits of younger cohorts. Overall, the market value of accrued benefitsAmerican Economic Review 01/2002; 92(2):79-84. · 2.69 Impact Factor - SourceAvailable from: Bennett Mccallum[Show abstract] [Hide abstract]

**ABSTRACT:**This paper reconsiders a result obtained by Sargent and Wallace namely, that price level indeterminacy obtains in their model if the monetary authorities adopt a feedback rule for the interest rate rather than the money stock. It is shown that the indeterminacy vanishes if the interest rate rule is chosen so as to have a desired effect on the expected quantity of money demanded. This holds even if considerable weight is given, in the choice of a rule, to the aim of smoothing interest rate fluctuations.Journal of Monetary Economics. 02/1982; - SourceAvailable from: Pedro Teles
##### Article: Gaps and Triangles

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**ABSTRACT:**We derive principles of optimal short run monetary policy in a real business cycles model, with money and with monopolistic firms that set prices one period in advance. The only distortionary policy intruments are the nominal interest rates and the money supplies. In this environment it is feasible to undo both the cash in advance and the price setting restrictions. We show that the optimal allocation is achieved under the Friedman rule. We also show that, in general, it is not optimal to undo the restriction that prices are set one period in advance. Sticky prices provide the planner with tools to improve upon a distorted flexible prices allocation.02/2001;

Page 1

BANCO DE PORTUGAL

Economic Research Department

UNIQUE EQUILIBRIUM WITH SINGLE MONETARY

INSTRUMENT RULES

Bernardino Adão

Isabel Correia

Pedro Teles

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: badao@bportugal.pt;

Isabel Correia, tel: # 351-21-3128385, email: mihcarvalho@bportugal.pt;

Pedro Teles, tel: # 351-21-3130035, email: pmpteles@bportugal.pt.

Page 2

Unique Equilibrium with Single Monetary

Instrument Rules.?

Bernardino Adão

Banco de Portugal

Isabel Correia

Banco de Portugal, Universidade Catolica Portuguesa and CEPR

Pedro Teles

Banco de Portugal, Universidade Catolica Portuguesa,

Federal Reserve Bank of Chicago, CEPR.

November, 2005

Abstract

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.

1. Introduction

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.

Page 3

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

paper.

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

competitive equilibrium.

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-

2

Page 4

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.

3

Page 5

2.1. Competitive equilibria

Households

Lt, described by the expected utility function:

The households have preferences over consumption Ct, and leisure

U = E0

(1

X

t=0

?tu(Ct;Lt)

)

(2.1)

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

(2.2)

Consumption must be purchased with money according to the cash in advance

constraint

PtCt? Mt:

(2.3)

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:

(2.4)

uL(t)

uC(t)=Wt

Pt

1

Rt

(2.5)

uC(t)

Pt

= RtEt

??uC(t + 1)

Pt+1

?

(2.6)

4

Page 6

Qt;t+1= ?uC(t + 1)

uC(t)

Pt

Pt+1, t ? 0

1

Rt.Condition (2.5) sets the intratem-

(2.7)

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.

Firms

tion of the representative …rm is linear

The …rms are competitive and prices are ‡exible. The production func-

Yt? AtNt

The equilibrium real wage is

Wt

Pt

= At:

(2.8)

Government

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

1

X

s=0

EtQt;t+s+1Mt+s(Rt+s? 1) = Wt+

1

X

s=0

EtQt;t+s+1Pt+s[Gt+s? Tt+s]

(2.9)

5

Page 7

Market clearing

Market clearing in the goods and labor market requires

Ct+ Gt= AtNt,

and

Nt= 1 ? Lt.

We have already imposed market clearing in the money and debt markets.

Equilibrium

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

(2.10)

the intratemporal condition that is obtained from the households intratemporal

condition (2.11) and the …rms optimal condition (2.8)

uC(t)

uL(t)=Rt

At, t ? 0

(2.11)

as well as the cash in advance constraints (3.4), the intertemporal conditions (2.6)

and (2.7), and the budget constraints (2.9).

3. Example

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

6

Page 8

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

1

v0(Lt)=Rt

A

(3.1)

and

1

Pt

= Rt?

1

Pt+1;

(3.2)

the feasibility condition

Ct= A(1 ? Lt);

(3.3)

and the cash in advance condition

Mt? PtCt; with equality if Rt> 1:

(3.4)

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

and Lt:

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,

7

Page 9

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)

1

?: There are many

Pt;

mt+1= ?(Lt)mt, where ?(Lt) =?v0(Lt)

A?

:

(3.5)

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

Rt= 1.

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

m1

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.

In‡ation

Pt

=

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.

mt+1

mt

= 1; Rt=

?

?> 1; Ctand Lt(= L)

?, and Ptsatisfying (3.4) with equality. There

for Rt= 1;

mt+1

mt

= ?

?eL

?

> 1 and

Pt+1

Pt

=

?

?(eL). In this steady state in‡ation is

Pt+1

?

?(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

mt+1

mt

= ?

?eL

?

> 1 for t ? t?; and in‡ation is constant,

Pt+1

Pt

=

?

?(eL)for t ? t?, mt

8

Page 10

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

??t

where ?? ? 1 (the Taylor principle), and ?t?

rule in the intertemporal condition of the household, (3.2), we get

?

?, where

1

?

Rt= R

?

?

??;

Pt

Pt?1. After substituting the Taylor

zt+1= (zt)??;

where zt=?t

?: By recursive substitution we get

zt+k= (zt)k??; for all k and t:

(3.6)

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

version is,

Rt? R = ? (?t? ?):

9

Page 11

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

questions.

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

10

Page 12

conducted with the interest rate feedback rule

Rt=

?t

Et

?uC(t+1)

Pt+1

;

?tis an exogenous variable, there is a unique equilibrium.

Proof: Suppose policy is conducted with the interest rate feedback rule Rt=

?t

?uC(t+1)

Pt+1

can be written as

uC(t)

Pt

Et

. Then the intertemporal and intratemporal conditions, (2.6) and (2.11)

= ?t, t ? 0

(4.1)

uC(t)

uL(t)=

?t

?Et?t+1

At

, t ? 0

(4.2)

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=

1

?t, t ? 0, condition (4.2) becomes

uC(t)

uL(t)=

1

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.

uC(C(:);L(:))

Pt

=1

?t, t ? 0;

11

Page 13

and the money supply is obtained using the cash-in-advance constraint, Mt =

P(At;Gt)C(At;Gt).

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

Rt=

uC(C(At;Gt;Rt);L(At;Gt;Rt))

?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,

uC(C(At;Gt;Rt);L(At;Gt;Rt))

P

= ?t, t ? 0.

In an economy where the use of money is becomes negligible which corresponds

to a cash-in-advance condition

vtPtCt? Mt:

(4.3)

where vt! 0, there is a single path for the nominal interest rate consistent with

zero in‡ation,

Rt=

uC(C(At;Gt);L(At;Gt))

?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,

Mt=?Rt?1CtuC(t)

?t?1

there is a unique equilibrium.

12

Page 14

Proof: Suppose policy is conducted according to the money supply rule Mt=

?Rt?1CtuC(t)

?t?1

PtCt=?Rt?1CtuC(t)

. Then, the equilibrium conditions

?t?1

obtained using the cash in advance conditions (3.4),

uC(t)

Pt

= ?t

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

past variables.

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

?Rt+s? 1

Et

1

X

s=0

?suC(t + s)Ct+s

Rt+s

?

= uC(t)Wt

Pt

+ Et

1

X

s=0

?suC(t + s)[Gt+s? Tt+s]

Rt+s

(4.4)

using (2.7).

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

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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

dynamic equations:

"

Mt

uC(C(Rt);L(Rt))

Mt

C(Rt)

= ?RtEt

uC(C(Rt+1);L(Rt+1))

Mt+1

C(Rt+1)

#

, t ? 0

(4.5)

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.

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