Con Troll Ability of Economy

8/2/2019 Con Troll Ability of Economy

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When Can Policy Makers Anchor Expectations?

Dynamic controllability and the limits to time inconsistency*

Andrew Hughes Hallett

George Mason University, University of St Andrews and CEPR

Giovanni Di Bartolomeo

University of Teramo

Nicola Acocella

University of Rome La Sapienza

Abstract. Rational expectations are often used as an argument against policy activism, asthey may undermine or neutralize the policymakers actions. Although this sometimes

happens, rational expectations do not always imply policy invariance or ineffectiveness.In fact, in certain circumstances rational expectations can enhance our power to controlan economy over time. In those cases, policy announcements, properly communicated,

can be used to extend the impact of conventional policy instruments. In this paper we

present a general forward-looking policy framework and use it to provide a formalrationale for attempting to anchor expectations, and as a possible justification for

publishing interest rate forecasts or tax rate projections. This approach allows us to test

when policymakers can and cannot expect to be able to manage expectations.

JEL Classification: C61, C62, E52, E61, E62.Keywords: Rational expectations, controllability, policy neutrality, monetary policy, fiscal policy,

1. Introduction

Since the work of Barro (1974), Sargent and Wallace (1975) and Lucas (1976), rational

expectations have been regarded as placing severe limits on what can be achieved in a

world of policy conflicts; and as requiring strong policy commitments to get even that

far. Time inconsistency and the Lucas critique, coupled with rational expectations, are

often said to imply that such commitments cannot be considered credible and to lead

inevitably to Pareto inferior outcomes.

This argument however does not allow for policymakers who actively engage in

managing expectations by making policy announcements, alongside policy interventions,

*The authors are grateful to Reinhard Neck and Paolo Piacquadio for useful comments on an earlier draft. NicolaAcocella thanks the University of Rome La Sapienza for funding.

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for the express purpose of shifting the expectations path itself.1

If they can do that,

private expectations will be exactly consistent with what the private sector/policymakers

expect the outcomes to be; and no one will be required to move off their expected path

(make expectation errors) for the policies to work. The literature has often used this idea

formally and informally in debates over the feasibility and desirability of trying to anchor

inflation expectations for monetary policy, or in arguments over the desirability of

publishing interest rate forecasts.2

It is also an idea in the minds of the policy makers;

see, for example, the European Central Banks concern that long term policies introduced

to combat the current financial crisis (greater transparency, new regulation, reduced pro-

cyclicality, planned liquidity withdrawals) should have their effects now (Trichet, 2008);

equally the announcement of new fiscal stimulus or credit guarantee packages. But what

the literature has not done is identify the conditions under which we can expect to be able

to manage expectations in this way (and when we cannot), as opposed to pointing to the

possibility and importance of managing expectations.

This paper investigates the circumstances under which policy announcements, if

properly communicated, can be used to supplement or extend the impact of conventional

policy instruments. The idea is that rational expectations may, in certain cases, enhance

the power to control an economy over time. Hence, contrary to conventional wisdom,

rational expectations may, but do not always neutralize the policymakers action.

Specifically, we consider the design of economic policy within a general rational

expectations framework and show that policy invariance can only arise in specific cases

(where there is a unit root or the rank condition specified below fail). In all other cases

policy announcements can be used to help steer economic behaviour, and certain targets

become reachable in reduced time. The rationale for this result can be understood by

using the concept of controllability, introduced in the classical theory of economic policy

(Tinbergen, 1952), and its dynamic extensions. If a policymaker is able to achieve any

desired vector of targets given some exogenous expectations, then he will also be able to

1By actively manage we mean to make a distinction between cases where we study the outcomes and stability ofrationally expected policiesgiven the behaviour of the system (the conventional case: Blanchard and Khan 1980); andthe case where the policy authorities try to influence the behaviour of the system itself. In this paper, we are concerned

with the latter case. The distinction itself was made and analysed in Hughes Hallett et al (2010).2 See Woodford (2005), Blinderet al. (2008) or Rudebusch and Williams (2008); and more formal models will befound in Soderlind (1999), Woodford (2003).

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do it with endogenous expectations3. If nothing else, he could exploit the endogenous

expectations to achieve his targets in a shorter time.

To make use of this property of rational expectations, however, another ingredient

must be present. The policymakers must be able to communicate, in a clear and effective

way, the intent and purpose of their policies, and how exactly these policies will work.

This will be necessary to convince the private sector that the policy measures will in fact

be undertaken when it comes to the point; and that it is reasonable to expect that the

intended outcomes will be achieved. Otherwise there is no reason to suppose the private

sector would shift, or anchor, their expectations of future outcomes as a result of

announcements of future policy actions.

Our approach differs from the recent trend in the literature on communication. In

our framework, the crucial element is to reaffirm the targets and why the policies chosen

can be expected to reach them; emphasised by Eggertson and Pugsley (2006), Moessner

and Nelson (2008), Ferrero and Secchi (2009), Libich (2009), and by Woodfords (2003)

observation that policy trade-offs will be eased when expectations fall into line. By

contrast, much of the recent literature on communication has focused on the quality of

forecasts, on the degree of divergence or agreement among policymakers, and on

transcripts or voting records from policy committees (Ehrmann and Fratzscher, 2005,

2007a; Jansen and de Haan, 2006; Visser and Swank, 2007).

The rest of the paper is organized as follows. Section 2 poses the communications

problem by means of a simple example. Section 3 puts our example into a more general

framework, deriving the reduced and final form of a general model with a single policy

maker and rational expectations. In that part of the paper we deal with the conditions for

static and dynamic controllability; and demonstrate that, contrary to conventional

wisdom, dynamic controllability can be enhanced by rational expectations. Our purpose

is to identify the circumstances in which controllability is possible; and the conditions

when it is not. Section 4 then describes how announcements of future policies may help

in ensuring the static or dynamic control of the economy. Section 5 contains a specific

illustration of our main point in the context of monetary policy. Section 6 concludes.

3 Like most other papers with policy announcements, including those just cited, this paper supposes a singlepolicymaker. The extension to multiple policymakers is set out in Acocella et al (2009).

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2. Controllability in a simple macro-model.

Consider a standard example of an economy with a New Keynesian structure:

(1) ( ) 1 11t t t t t E x + = + + + +t tf v

t(2) ( )1t t t t t t t x E x i E f += + +

where t and are white noise, and wheretv t t ty y= is the output gap relative to a non-

market clearing trend; or relative to the natural rate of output arising from monopolistic

competition in the goods markets (Blanchard and Kiyotaki, 1987); or created by tax

distortions elsewhere in the economy (Alesina and Tabellini, 1987).

If we want to analyze the role of policy announcements, and the importance of

communication as a way to enhance the effectiveness of policy instruments, we must first

discuss the conditions under which the policymaker can control the system given by

equations (1) and (2). And to do that we need to make some generalisations.

3. A general model for analyzing controllability.

Dynamic models as (1)-(2) can be expressed in the following general form:

(3) for t= 1..T1 1/t t t t t y Ay By Cx u += + + + t

where Tis a finite, but possibly large number; and where )/( 1/1 tttt yEy = ++ denotes the

mathematical expectation of conditional on1+ty t (the information set available at t). In

our set up, is a vector ofn endogenous variables at time t; is a vector ofm potential

policy instruments; and is a vector of exogenous shocks or other influences which

have a known mean, but otherwise come from unspecified probability distributions.

ty tx

tu

4The

matrices A, B and Care constant and of ordern, n, and nm respectively, and have at

least some elements which are non-zero.

Model (1)-(2) can easily be written in the general form (3) by obtaining the

models reduced form

(4) .

+

+

+

=

+

+

t

t

t

t

t

t

t

t

t

t

t v

i

f

xxE

x

0

00

0

1

0)1(

10

1

1

1

1

1

1

4Note the generalization of the notation in this section. Note also that we restrict ourselves to additive uncertainty inorder to be able to obtain exact expressions for expectations of the target variables; there being no clear agreement onthe best way to proceed (or define controllability in expectation for this kind of RE model) otherwise.

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We can then define the parameter matrices in (3) to be:

(5) ; ; ; and

=

00

0A

+=

1

)1(

B

+=

C

1 0

1

t

t

v

where the last expression is the error term in (3).

Model (3) can now be solved from the perspective of any particular period, say

t= 1, by putting it into final form conditional on the information available in that period:5

(6)

1

1/1 1/1 1/1 0

/1 /1 /1 1/1

00 . 0 0 . . 0

: .. . 0 . . 0

: .0 . 0 . . . .

: .. . . . . 0 .

0 . 0 0 . . 0 0T T T T

y xI B C Ay

A I

B

y xA I C

+

= +

. .

. .

. 0

u

u By

+ +

In the above representation, is a known initial condition fort= 1; and is

some assumed or projected terminal condition most likely the one that describes the

economys expected long run equilibrium state as part of

0y 1/1+Ty

t. Although (3) has been

solved from the point of view of1, it must be understood that it could have been derived

for each t, t= 1....T, in turn (whereyj/t=Et(yj) ifjt, butyj/t=yj ifj < t; and similarly for

x and u). However, for simplicity, we will consider the 1 case only in what follows. The

generalisation to any other value oftis then obvious.

Second, the equation to which (6) is the solution makes it clear that neither the

policymakers, northe private sector are required to move off their expected paths (make

expectational errors) for the policies to work. In fact equations (6) and (7) below show

precisely the opposite; that those expectations are exactly consistent with what the private

sector/policymakers expect the outcomes to be. The only question is whether policies, or

policy announcements, can be found that will shift the expectations path by the required

amount. The task of the paper is to find the conditions under which that can be done; that

is, when it is possible to shift expectations in such a way that the economys anticipated

outcomes reach certain target values at certain points of time, and when it is not possible.

It is easy to see that (4) always exists if the matrix inverse, TT-1

in (6), exists. This

is demonstrated in Hughes Hallett et al. (2008); the condition forTT-1

to exist being that

the matrix productAB shall not contain a unit root.6

Given that, we rewrite (6) as:

5 Hughes Hallett and Fisher (1988), Hughes Hallett et al. (1996): (6) ties the expectations of yt/1 down as a function ofannounced policies, while those for each xt/1 are tied down by either (8) or proposition 1 below.

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(7) or

+

=

1/

1/1

1/

1/1

1

111

1/

1/1

.

.

.

.

.

.

...

...

...

...

...

.

.

.

TTTTT

T

T b

b

x

x

RR

RR

y

y

bRxy +=

where 1( ),R TT I C= 1 1 0 1/1{ ( / ) ( ' : 0) ' (0 : ') ' }T Tb T E u A y B y

+= + + , and denotes

a Kronecker product. In this representation, each

1/1/, / jtjt xyR = is an nm matrix of

policy multipliers for t, j = 1T. But notice that R is not block triangular. That is,

even ift


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exists since we already know that exists. But the instrument coefficient matrix, C, can

only possess an inverse ifn = m and C has full rank. Thus n = m is a necessary condition

for static controllability; and linear independence in the impacts of the instruments on the

targets is sufficient. There is therefore no generalisation or change to the traditional static

controllability conditions when there are rational forward looking expectations.

1TT

b) Next we considerdynamic controllability.8

An economy is said to controllable

dynamically if a sequence of instrument values can be found to reach any

arbitrary value, , for the target variables in period t, at least in expectation, given an

arbitrary starting point (Holly and Hughes Hallett, 1989). In that case, we are no longer

concerned with the period-by-period controllability of target variables between periods 1

and t 1. Viewed from period 1, dynamic controllability therefore requires a sequence of

intended instrument values that can guarantee is reached in period t.

Given (7), this will be possible only if the sequence of policy multipliers and anticipatory

effects, , is of full rank:

txx ,.....,1

d

ty

0y

1/1/1 ,....., Txxd

ty 1/

Ttt RR ,1, ..... nRRr Ttt =)....( ,1, , given an arbitrary initial state and

a specified terminal condition . This follows because is

reachable over (1, t), using a Moore-Penrose generalized left inverse, if .

0y

1/1+Ty 1//1,1/ )....( tTttd

t bxRRy +=

)....( ,1, nRRr Ttt =

But ifT

n, =)...( ,1, Ttt RRr nRRr ntt=

)...( ,1, which provides the result. Thus an economy

represented by (3) is dynamically controllable, over (1, t) with Tn, if ,1 ,( .... )t t nr R R n= .

These controllability conditions contain an important generalization over the

traditional case with a backward looking model. If n > t, which is entirely possible for

small values oft, then dynamic controllability will be available through the reactions of

to the implemented policy choices ; and through the anticipatory effects of

announced or anticipated policy interventions that still lie in the future, That

implies the policymaker can use policy announcements, in addition to policy inter-

ventions, to guide the course of the economy. In a conventional model that would not be

possible since for all j > t. In effect, the policymaker has a greater number of

policy instruments at his disposal than in an economy without anticipations.

1/ty 1/1/1 ... txx

..... 1/1/1 nt xx +

0, =jtR

8Dynamic controllability conditions are well known for conventional models; but not for rational expectations models.

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Thus itself is controllable from the first period, even if there are insufficient

instruments (m < n), provided that Tn and

1/1y

.)....( ,1, nRRr ntt = The astute policymaker

will therefore realise that good communication lies at the heart of the policy problem if

he/she wants to reach their policy targets in the early periods or at lower cost -- a fact that

has not been lost on central bank policymakers in their attempts to control or anchor priv-

ate sector expectations of future inflation in such a way as to make interest rate policies

more effective (Woodford 2005; Blinderet al. 2008; Rudebusch and Williams, 2008).

Second, dynamic controllability is possible with a much reduced instrument set

compared to static controllability. There are two parts to this reduction: a) the ability to

use one or more instruments sequentially rather than a group of several instruments used

once and in parallel; and b) the ability to augment or replace parts of an existing

instrument set with announcements of future policy changes.

Third, it is important to note that, while 1/1 will represent implemented decisions

when it comes to controlling , the1/ty 2 /1 /1.... nx x values, being policy announcements, may

never be carried out. However, because they are decisions that lie in the future, the

possibility of time inconsistency plays no role in the controllability of so long as they

aregenuinely held expectations at this point. And they will be genuine because they are

the values that deliver the first best outcomes. Policymakers have no incentive, strategicadvantage or interest in choosing to make themselves worse off than they need to be:

1/ty

Proposition 1: Forward looking rational expectations enhance the power to control an

economy over time in that: policy announcements may be used to supplement and extend

the impact of conventional policy instruments, and that controllability is now available

with a reduced instrument set from t = 1, if .)...( 111 nRRr n =

Corollary: all values, including the first period targets , are now dynamically

controllable if the rank condition in proposition 1 holds. This is an important extension

over conventional dynamic controllability where period t = n is the earliest date at which

we can guarantee controllability if there is a single policy instrument; ort = n/2if there

1/ty 1/1y

are two instruments, and so on. Thus is controllable from the first period, even if there

are insufficient instruments (m < n), provided that both Tn and proposition 1 holds.

1/1y

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Comments: i) It is important to see why time inconsistent behaviour will not appear here.

Controllability at period tmeans that, barring unforeseen shocks, the policymaker will be

ym

important to recognize that there is a class of

te

rs and the

able to reach his first best values forytin expectation. Hence,yt/t= yt/1= ytd

are fixed and

known quantities. Buty1/t= y1/1is fixed by history as pointed out below (6); and x1/t=x1/1

likewise. It is then easy to see, by (6), that if nothing else changes xt/t= xt/1. The policy-

maker is of course free to setxt/t xt/1. But he would never do so becaused

ty is a first best

value which can be reached given no further information changes or unforeseen shocks.

So to assert time inconsistency in this case is to claim that rational polic akers would

deliberately choose to make themselves worse off, given the chance, and that the private

sector should expect them to try to do so.

ii) It is important to examine the conditions that permit effective signaling and commit-

ment in many situations, but it is equally

problems for which they are neither necessary or relevant (although clear communication,

that the necessary policy changes will be made, is still required). And it will typically be

a large class, given that we get down to cases with just one instrument and many targets

if T n is large enough. Hence, time inconsistency is an exception rather than the rule.

iii) Put differently, our results explain when commitment is needed. So long as the no

unit root condition onAB and the rank condition in Proposition 1 are satisfied, the priva

sector knows that policymakers have controllability and can reach their desired target

values in all cases. There is no point in expecting anything else: the system is fully tied

down by (6) and proposition 1. But if the controllability conditions are not satisfied, or if

private expectations are capable of offsetting the policymakers action, then expectations

are not tied down and commitment by the policy authorities will be necessary.

iv) The importance of providing credible, convincing explanations of future policies and

target values, and the ability to reach them, is now obvious. If the policymake

private sector share information sets, or the policymakers are thought to possess superior

information, then the economy will reach its first best outcomes. Thus credibility requires

private agents to check if the policymakers announcement/explanations match their own

projections, in so far as they have firm projections. But if they do not share information,

or if the private sector has access to better information, then the policymakers announced

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outcomes will not be reached. Instead the outcomes will be what private agents expect to

happen given the announced policies. The policymakers then have an incentive to adjust

to regain their preferred target values. But that is the normal process of correcting an

implementation error (wrong information), not time inconsistency.

4. Applications of the general theory to problems of controllability.

e now go back to the simple macro-model introduced in section 2, and examine its

oking

= 0 we have a recursive system in place of (1) and (2), i.e.:

W

controllability. We consider two cases. First we consider the simple backwards lo

case by assuming = 0.Second, we consider a more general case in which there are also

rational expectations and hence forward looking variables ( 0).

CASE 1: Controllability without rational expectations.

If

(9) ttttttt vffi +++++= )(1 ,

(10) tttt fix ++=

This system has static controllability in the classic sense since the policy

pliers in

+

ttf

ingular matrix so long as

multi

(11) ++ +

=

tttt

t

t vi

x

11

form a non-s Consequently any required values for t.0 and

can be reached in expectation. Howeverifonly one policy instrument, it,is available,tx

and no fiscal instruments )0( == , as has been the case in most of the monetary policy

erature since Barro and Gordon (1983) and Rogoff (1985), or if fiscal policy has just

one independent channel of influence )0(

lit

= , then static controllability is lost and the

desired values fort

andt

x cannot be reached in each (and hence the current) period.

The multiplier matrix in (11) is singul ither case. Nevertheless the system is still

dynamically controllable over two periods. Consider the first case: .0==

ar in e

Back-

substituting one period in (11), we get

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(12) +

+

+

=

1

1

1 00100 t

t

tt

v

i

allparameter values. We can reach

any desired values for

1

12 1

00

0 tt

t

t

t

t vi

xx

The policy multiplier matrix is now non-singular for

t and after two periods using monetary policy alone.

tx

The same is true if fiscal policy has only one channel of influence: .0= In this

case the multiplier matrix in (11) is singular and the static controllability property is lost.

But back-substituting one period implies

(13) +

+

+

11 00 tt fi

th multiplier matrices are non-singular for all nonzero parameter v

to reach desired values for

=

1

12

00

0 tt

t

t

t

t fi

xx

errors

in which bo alues. So,

t and after two periods, we can either use interest rates

it would still take two periods to reach our target values.

ctations.

back to (4); and hence to

odel has a policy

tx

twice; or fiscal policy twice; or first interest rates and then fiscal policy, or fiscal policy

and then interest rates in an asynchronous game. In the latter two cases, the multiplier

matrices would become

10 t

t

f

i

and

10 t

t

i

f

respectively. But

CASE 2: Controllability with rational, forward-looking expe

Allowing for the full effects of both policy instruments takes us

a model of the form of (3) with parameter matrices given by (5). That m

multiplier matrix C, which is non-singular if .0 Hence we have static controllability

given arbitrary initial conditions, and arbitrary values for the terminal conditions.

So far the story is the same as in the case with no rational expectations. However,

things change if we have only one instrument (monetary policy) available. To see that,

+

+

+

+

+ 1

1

1

1

1

)(

)(0 t

tt

tt

t

t w

iE

fECIA

x

we write our policy problem as a two period problem with both policy instruments:

(14)

+

=

0 tt

t

t

t

wi

f

CBIx

say,

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as a particular case of (6). We are interested in only the first two rows of (14) for static

llability. But we will need to use the whole system ic

controllability in the second period. The policy multiplier matrix implied by (14) is then:

+++

contro if we are to go to dynam

(15) 1.

00

)1()()(

)])1(([)])1(([][)(

++++++++++++

where 0])1([1 +=

]1/)1[(]1/)1[(

. The one period static controllability policy multiplier

matrix (the top left 2x2 sub-matrix) is non-singular as noted above. But if there is just one

policy (monetary policy) available, ,0== then instead of (15) we get

(16)0)(0

/)1(00

+

+

.

Alternatively, if fiscal policy is present but has one channel of influence

1

000

)1(00

]1)1([

+

+

.

)0( = , we get:

(17))1(

]1)1([)()(

/1(

+

+++

+

.

Hence, for the first case, we can rewrite the one period, static controllability problem by

inserting (16) into (14) to yield

(18)

view of period t, despite having a single instrument and

argets. This is achieved by using itandpolicy announcements of what will happen

in period t+1. So, in this case we have static controllability even if there are insufficient

)

1.

00

t

tt

t

t

tw

iE

i

x+

++

+=

+

)(]1)1([)(

/)1(

1

1

whose multiplier matrix is easily seen to be non-singular. In other words, we have static

controllability from the point of

two t

policy instruments.

The same thing happens when fiscal policy is limited to one channel of influence.

If we choose to use monetary policy alone, then (18) applies again. But if we use fiscal

policy, (18) will be replaced by inserting (17) in (14) to yield:

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(19) ttt

t

t

tw

fE

f

x+

+=

+

)()( 1

1

whose multiplier matrix is also non-singular. So again we have static controllability at

d t, an option we didnt have in the non-rational expectations case. And for the same

reason: credible policy announcements become a second instrument. In addition, we can

anchor their expectations of the likely

onetary policy is whether central banks should

llow forecasts of future interest rates to be published. Rudebusch and Williams (2008)

ic

perio

easily generate examples which imply static controllability using mixed policies: for

example, monetary policy and announcements about future fiscal policy; or fiscal policy

plus announcements of future monetary policies.

The crucial point, however, is that case 2 requires the policymakers to exercise

good communication skills in order to make their announcements credible and have some

effect that is to make the private sector shift or

outcomes to values that suit the policymakers purpose. In other words, the private sector

has be made to believe that the necessary policy changes will actually take place and

achieve the said outcomes. Case 1 does not require the same communication skills since

there are no expectations to be anchored.

5. Communication: Should policymakers try to manage expectations?

Example 1: One of the great debates of m

a

and Eusepi and Preston (2007) argue that this can be used to strengthen econom

policy9. But others argue that to do so may imply a stronger consensus or more certainty

about future policies than actually exists; or that it may propagate errors and make it

more difficult to adjust policies again later in the face of unexpected shocks. In addition,

private agents may overreact to noisy public signals, but under-react to more accurate

private information (see, Faust and Svensson, 2002; Amato et al., 2003; Walsh, 2007).

To resolve this argument, consider an economy represented by the following well-

known model:

(20) ttttttttt EiEyy ++= ++ )()( 111

(21)t t0 1 2

( )t ti c c c y

= + + +

9See Archer (2005), Moessner and Nelson (2008), Ferrero and Secchi (2009) for some evidence to support this view.

13


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Equation (20) is an elaboration of the standard model which has been part of the theory of

monetary policy since the Barro-Gordon model was introduced in 1983. It consists of a

run Phillips curve with persistence (short 0 ), set within a standard forward looking

supply function (a long run P ips curve) and elaborated to include the effects ofLucas hill

real interest rate changes on output. It can therefore be interpreted as either a dynamic

open economy Phillips curve; or as a forward looking new Keynesian IS curve with

dynamics. In that context, ty is the deviation of output from its natural rate ; t is the

rate of inflation; 1t tE + the private sectors current expectation for the rate of inflation;

and * is the government or central banks target inflation rate; where ti is the nominal

rate of interest, t is a supply shock with mean zero and constant variance, and t is a

monetary policy shock with mean zero and constant variance. All parameters are positive.

The only p instrument in this example will be .ti Equation (21) is therefore a

Taylor rule: 0c is a constant term reflecting the equilibrium rate of interest t

olicy

; the possible

control errors; is the inflation target; and determinacy (the Taylor principle) suggests

11.c >

To obtain a reduced form for (20)-(21), corresponding to (3), we renormalize (21)

on t, set ( )* 11 0t tu c c v

= and solve (20) and (21). This transforms our system to:

(22) 1 1/11 0 0 1t t ty y y + = + +

1 1

1 1/12 1 2 10 0t t tc c c c

+

1

2 2 11

11 1

1 1

t

t

t

ci

uc c cc1

+ + + ,

where , which does not allow static controllability as it is. However,

n write the two period policy problem, as we did before

(23)

1

1 2(1 ) 0c c = +

we ca , using (6):

01 1

0 2 1

1 1 1 12

1

1 1 22

0 t t t t t t

t t tt t

i uC E yB

u c cE

+ + ++

+

1 1

0 1

t t t

t t t t

t

yy uAi u c cI B C

y EA I

+

+ + +

+ = + + +

whereA, B and Care the first, second and third coefficient matrices in (22). The policy

multiplier matrix for this model is then:

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1

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )( ) ( ) ( )

(24)

( ) ( )12 1 21 1c c c c

2

2

12 2

2 1 2

2 1

1

1

1 1

1

11

1

1

11

1

c c

c c c cc c c

c

c c cc

c

+

+ + +

+ +

1

t

t t

i

E

+

i +

24) back into (23), this standard model tells us

that

where2 ( ) .c c = + Putting (1 2

),( tty and ),( 11 ++ tty are both statically (immediately) controllable using current

announcements or projections of future actions. That is because both the

upper and lower partitions of the multiplier matrix in equation (24) are non-singular.10

And since

policies and

),( tty and ),( 11 ++ tty are current and expected future target values from the

perspective of period t, this means that the policymakers can control not only current

n and output (in expec

asts for ext period

ectatio akes

inflatio tation at least); but also inflation expectations and growth

forec the n . That adds to their ability to control inflation (and output) by

anchoring those expectations.

This example is therefore in line with our theory of dynamic controllability under

rational exp ns, and m the case for publishing conditional forecasts of future

interest rates as a normal part of the monetary policy framework. Indeed it would become

more difficult to achieve successful outcomes if inflation expectations could not be tieddown at the same time as inflation itself. There may be other ways of controlling the out-

comes for ),( 11 ++ tty in period t+1 of course; for example using current and past interest

rates as implied by (24). But that is a conventional backwards looking use of dynamic

controllability, and may still fail to tie down/anchor expectations in the terminal period.

Example 2: It is not just in monetary policy that timing and expectations of policy

changes may be important for economic activity. House and Shapiro (2006) use a fairly

standard model to analyse the announcement and phasing in effects of the 2001 and 2003

tax cuts in the US, on output, investment, employment/hours worked and consumption.

10 Unless the underlying parameters satisfy =exactly in the upper sub-matrix (for current outcomes); or= c1 in thelower one (future expected outcomes). These are sufficient conditions; and they cannot both hold simultaneously since

we require c1>1 for the Taylor principle. So, even in these cases, we guaranteed to control either future expectations orcurrent outcomes through policy announcements/forecasts. Beyond that, there is one particular value in the dynamics,, which leads to singularity for the whole system corresponding to the unit root condition identified in section 3.

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Their model is a business cycle model with a government sector. Being nonlinear, we can

rewrite it in linearised form, i.e. (3), (6), (7) above, with partial derivatives replacing the

[ {(1 ) (1 )( /

t t t t t t t t t t

K K K

t t t t t t t t

q C R I K + + + + + += + + + 2

1

) / 2} (1 ) ]

[ /(1 ( / )(1 )]t

K

t t t t t

q

C q I K

+

+

= +

where it is understood that subscript t+1 means the expected value of that variable in

on an earlier information set. In obvious notation, I, K, Y, Nand Crepre

investment, capital, GDP, employment and consumption respectively; and W, R and q are

real wages, the rental price of capital, and the shadow cost of capital. Policy targets are

therefore Yt, It, Nt, and Ct; and the instruments

corresponding linear coefficients. The model itself contains 9 equations in 3 fiscal policy

instruments as follows (after renormalizing the equations forIt, Kt, Ttand Ct):

(25)

1

1/ (1 ) /

2

1 1/

2

(1 )

(1 ) /

/

( / ) / 2

[ (1 )]

[ ( / ) / 2]

t t t

t t t

t t t

t t t

t t t t t t t

N

t t t t

N K

I K K

K Y N

W Y N

R Y N

Y C I G K I K

N C W

T N W K R I K G

+

=

=

=

=

= + + +

=

= +

1/ 2 2

1 1 1 1 1 1

t+1 based sent

,N Kt t and : wage taxes, capital taxes and

government spending (which necessarily leaves Tt, lum sum transfers/subsidies, as the

tG

p

residual in the governments budget constraint). Thus the models structural form implies

an endogenous variable coefficient matrix of no particular pattern; hence of full rank and,

except for very particular parameter values, with an inverse that is full (all entries non-

zero). It is then easy to check that matrix C, of order 9x3 in (3) and (6), is also full with a

clear pattern that can be evaluated from the partial derivatives in (25). Likewise the 9x9

matrices A and B, in (3) and (6), are A=0 and a matrix of zeros except for the elements

b12=1,1/

84 1(1 )K

tb C += + and 88 (1 ).b = Consequently, (6) implies that matrix R

in (7) is given by

2 1...

0 . .

. . . .

. . .

0 . . 0

TC BC B C B C

C BC

R C

BC

C

=

. HenceRt,j=0 all t>j, butRt,j=Bt-j

C if

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jt.11

controllable and e specified target values in period t. We

In the general case, even if we do not assign specific numerical values to the

parameters/partial derivatives in the model, we can still determine if the policy targets are

xpectations managed to achieve

can do that by evaluating the appropriate row in R and using proposition 1. Evidently

matrix C, when condensed onto 4 policy targets, has rank 3 at most. Consequently static

lability never was available for these tax reductions, however convincing the

kage that is phased in over

ter-

is paper shows that, because policy invariance can only emerge when there is

control

policy announcements may have been. On the other hand, matrixBChas nonzero entries

in rows 1 and 8, and matrix BiC nonzero entries in row 8 only (all i 2).

Thus , ,( .... ) ( , ) 4t t t nr R R r C BC = = , except for the case where specific numerical parameter

values reduce r(C) below 3 orr(BC) below 1. In other words, these policy targets are not

controllable by prior announcement (i.e. in periodj, using announcements of actions to be

taken in period tor later, wherej


18/21

a conflict between the public and the private sector, this will not always be what happens.

Policymakers may be able to induce private agents to shift their expectations.

Second, rational expectations are also a powerful mechanism, in combination with

the chosen policy values, for influencing the natural dynamics of the economy. There has

been much less interest in this second aspect of rational expectations which implies that

policy announcements may often be used to systematically increase the power of the

policymakers interventions. A notable exception is Woodfords timeless perspective, and

tions,

because it would

s still required so that agents can check the consistency of the announced

the use of interest rate forecasts or inflation targets to anchor inflation expectations.

In proposition 1, we have obtained a rank condition that defines the circumstances

under which rational expectations can be used to increase a policymakers power to

control an economy over time. It shows how communication and policy announcements

can be exploited to supplement and extend the impact of conventional policy instruments.

This gives us a formal justification for using policies designed to manage expecta

and also the circumstances in which expectations cannot be anchored.

One implication of this rank condition is that, in the absence of changes in inform-

ation, once a policy sequence has been announced there can be no question of revising an

announcement for strategic reasons, or of expecting it to be revised, because the policy

maker can and is known to be able to reach his first best outcomes. So they have no

incentive or interest in not following through on their announcements

make them worse off than they need to be. In that sense, these policy announcements are

implementable.

A second implication is that the quality and credibility of communication by policy

makers is a key part of the problem. It is important to examine the conditions that permit

effective signalling and commitment, but it is equally important to recognize that there is

a whole class of problems for which they are neither necessary or relevant (although clear

communication i

policies and target values). And it will typically be a large class given that we can get

down to one instrument and still have controllability if the policy horizon is long enough.

This last point then leads four corollaries:

a) Successful communication decomposes into two parts; consistency with the announced

targets, and clear priorities to signal credibility when there are insufficient instruments or

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time periods. In the past the literature has concentrated on the second element, neglecting

the first.

b) Policymakers who are patient can achieve, and be expected to achieve, what goals they

too short a time.

t perhaps

d Theil meet Nash:Controllability in policy games,Economics Letters, 90: 213-218.

cocella, N., G. Di Bartolomeo, A. Hughes Hallett and P. Piacquado (2009), Announce-

ment wars as an equilibrium selection device, Discussion paper 2009/77, CORE,

vain-la-Neuve.

G. Tabellini (1987), Rules and discretion with noncoordinated monetary

Am monetary policy,

Arr International Settlements, Basel, mimeo.

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American Economic Review, 77: 647-666.

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c) Time inconsistent behaviour is therefore limited, but will become a problem if policy

makers find themselves under pressure and want a quick fix by trying to reach too many

targets in

d) Time inconsistency is not a general or widespread problem. That is a useful conclusion

because policymakers appear not make extensive use of it in practice, excep

during elections (point c)), and we need to be able to explain why.

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