Michael Woodford

Princeton University

December 12, 2002

Preliminary and Incomplete

c Michael Woodford 2002

Contents

7 Commitment to a Policy Rule 1

1 The Optimal Long-Run In¬‚ation Target . . . . . . . . . . . . . . . . . . . . 7

1.1 The In¬‚ationary Bias of Discretionary Policy . . . . . . . . . . . . . . 7

1.2 Extensions of the Basic Analysis . . . . . . . . . . . . . . . . . . . . . 15

2 Optimal Responses to Disturbances . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Cost-Push Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Fluctuations in the Natural Rate of Interest . . . . . . . . . . . . . . 44

3 Optimal Simple Policy Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 The Optimal Non-Inertial Plan . . . . . . . . . . . . . . . . . . . . . 54

3.2 The Optimal Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . 58

4 The Optimal Instrument Path . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Commitment to an Optimal Targeting Rule . . . . . . . . . . . . . . . . . . 67

5.1 Robustly Optimal Target Criteria . . . . . . . . . . . . . . . . . . . . 69

5.2 Implementation of a Targeting Rule . . . . . . . . . . . . . . . . . . . 75

Chapter 7

Gains From Commitment to a Policy

Rule

We now turn to the characterization of optimal policy in the (realistic) case that not all of the

stabilization objectives discussed in the previous chapter can simultaneously be achieved, so

that it is not possible to fully eliminate all of the distortions that can be a¬ected by monetary

policy. It is in this case that the problem of optimal monetary policy becomes non-trivial,

in that the character of the optimal equilibrium cannot be read directly from the nature

of the loss function derived above. In this chapter we discuss how the constrained-optimal

equilibrium pattern of responses to disturbances can be characterized in such a case, and

consider in general terms the problem of the choice of a policy rule intended to bring about

the desired equilibrium.

The idea that it should be necessary to compromise among several stabilization goals,

each desirable in itself but not mutually attainable in an absolute sense, is likely to be

intuitive to most practical policymakers. It may be less obvious that it is desirable to

attempt to characterize the optimal compromise among these objectives in terms of a rule

for the systematic conduct of policy. One might think that it should su¬ce instead to clarify

the proper goals of policy on the one hand, and to develop a reliable quantitative model of

the e¬ects of alternative policy actions on the other, and then simply to charge the central

bank with the pursuit of the appropriate goals using an appropriate model to inform its

decisions. Economists are often willing to suppose that households and ¬rms do a fairly

1

2 CHAPTER 7. COMMITMENT TO A POLICY RULE

good job of pursuing their objectives without external guidance, even when the optimization

problems that they face are quite complex; why not similarly trust that central bankers can

be relied upon to behave in something close to an optimal fashion, once their objectives and

constraints have been made clear?

But the reason that rules are important in monetary policy is not that central bankers

cannot be relied upon to pursue the public interest, or that their highly trained sta¬s cannot

be expected to bring the latest knowledge to bear upon the analysis of the likely e¬ects

of alternative policies. It is rather that the central bank™s stabilization goals can be most

e¬ectively achieved only to the extent that the central bank not only acts appropriately, but

is also understood by the private sector to predictably act in a certain way. The ability to

successfully steer private-sector expectations is favored by a decision procedure that is based

on a rule, since in this case the systematic character of the central bank™s actions can be

most easily made apparent to the public.

Of course, one might imagine that market participants could become accurate predictors

of central-bank behavior without any articulation by the central bank of the principles of

its behavior, just as they predict the behavior of ¬rms that do not commit themselves to

rules of conduct. Yet while expertise of this kind certainly exists, the degree to which

central-bank behavior can be con¬dently predicted simply through extrapolation from past

actions is limited; the membership of monetary policy committees changes fairly often (and

invites speculation about the consequences for future decisions), and new circumstances are

constantly faced that are not too closely analogous to others confronted by the central bank

in the recent past. Articulation by the central bank of a rule that guides its decision process

(and commitment by the bank to actually follow it!) can greatly improve the predictability

of policy by the private sector, even when the rule is less explicit than a mechanical formula

that yields an unambiguous prescription on the basis of publicly available data.

Moreover, an optimal pattern of conduct by the central bank will generally not correspond

to what would result from discretionary optimization, as ¬rst stressed by Prescott (1977)

and Kydland and Prescott (1977, 1980). By discretionary optimization we mean a procedure

3

under which at each time that an action is to be taken, the central bank evaluates the

economy™s current state and hence its possible future paths from now on, and chooses the

optimal current action in the light of this analysis, with no advance commitment about

future actions, except that they will similarly be the ones that seem best in whatever state

may be reached in the future. This might seem an eminently sensible procedure; it allows

the bank to make use of all of its detailed knowledge about current conditions, rather than

having to classify the current state as one of the coarsely described possible future states

considered at some earlier time. It also avoids the necessity of making an explicit decision

about anything other than the action that must currently be taken (although a view about

likely future policy will be implicit in the evaluation of alternative possible current actions).

And it might seem to involve no loss of e¬ciency relative to a once-and-for-all optimal plan,

insofar as a large literature on dynamic programming and optimal control has stressed the

usefulness of recursive methods for the solution of dynamic optimization problems. Under

these methods, a dynamic problem is broken into a sequence of individual decision problems,

in which the optimal action at each stage is chosen given the state at that time, taking as

given the nature of optimizing behavior at stages yet to be reached.

But as Kydland and Prescott showed, these methods are appropriate only for the optimal

control of a system that evolves mechanically as a function of its past state, exogenous dis-

turbances, and the current action of the controller. They are not appropriate for the optimal

control of a forward-looking system, in which people™s expectations about future policy are

one of the determinants of current outcomes. This is because a dynamic-programming ap-

proach considers the optimal action at a given point in time considering only the discounted

current and future losses associated with alternative feasible continuation paths, given the

system™s current state. It thus neglects any e¬ects of the anticipation at earlier dates of

a di¬erent current action than the one that would be judged best by a discretionary opti-

mizer. By credibly committing itself in advance to behave di¬erently, a central bank can

steer expectations in a way that furthers its stabilization goals.

It follows, once again, that successful management of expectations is unlikely to be pos-

4 CHAPTER 7. COMMITMENT TO A POLICY RULE

sible except through commitment of the central bank to a fairly explicit rule for the determi-

nation of its appropriate action at any point in time. For conscious guidance by a rule is not

only an aid to the private sector™s understanding of policy; it also makes it more likely that

the central bank itself will act correctly. For the temptation to behave in a discretionary

fashion must be resisted; optimal policy requires that most of the time the central bank does

not set interest rates at the level that would be optimal from the point of view of its stabi-

lization objectives, taking as given both past and future policy, and past, present and future

private-sector policy expectations.1 This means that good will and a sound understanding of

the e¬ects of alternative policy actions and of the economic welfare associated with alterna-

tive possible paths for the economy is not enough; there must be a conscious commitment to

a criterion for action that will be counterintuitive for a discretionary policymaker, but that

actually serves (and can be understood to serve) the bank™s goals if pursued in a predictable

way.

The importance of creating the right sort of expectations regarding future policy has

another important lesson, of a somewhat subtler character, for the way that policy should

be conducted. This is that an optimal decision procedure will generally not be purely

forward-looking, in the sense of allowing the proper current action to be determined solely

from an analysis of the set of possible future paths for the economy given its current state.

The point seems to be contrary to the intuitions even of many who recognize the importance

of commitment to a rule in order to avoid the dangers of discretionary policy. Indeed,

many popular current proposals for policy rules are purely forward-looking in character.

For example, in¬‚ation-forecast targeting as currently practiced at the Bank of England (see,

e.g., the description in Vickers, 199xx), or as described in the early analytical literature (e.g.,

Svensson 1997, 1999; Leitemo, 199xx), selects a current interest-rate target on the basis of

the conformity of projections of the economy™s future evolution under that policy choice with

a criterion that is purely forward-looking (e.g., that RPIX in¬‚ation be predicted to equal 2.5

1

In the thought experiment proposed here, it is assumed that the private sector™s current and future

expectations about future policy are and will be correct, but that the current action need not be chosen in

a way consistent with past expectations regarding this decision.

5

percent per year at a horizon two years in the future). Similarly, Taylor™s classic formulation

(Taylor, 1993) of his policy rule prescribes a reaction to current in¬‚ation and output-gap

estimates that is independent of both past aggregate conditions and past monetary policy.2

And in discussions elsewhere (e.g., Taylor, 1999a), Taylor stresses the desirability of adjusting

the federal funds rate target immediately to changing aggregate conditions, without any of

the “partial adjustment” dynamics typically indicated by estimated Fed reaction functions,

such as those described in section xx of chapter 1.

The intuition of the proponents of purely forward-looking approaches, presumably, is that

it is not desirable for policy to depend on “irrelevant” state variables, that is, on something

that a¬ects neither the set of possible future paths for the central bank™s target variables

nor the proper ranking of alternative paths. Yet such an argument would be incorrect, for

the same reason that dynamic programming does not yield a truly optimal policy, as just

discussed. Optimal policy must take account of the advantages of the anticipation of the

policy at earlier dates; and for this reason it must generally be history-dependent rather

than purely forward-looking. Past conditions should be taken into account in choosing the

current policy setting, because it is desirable that people be able, at the earlier time, to

count on the fact that the central bank will subsequently do so. Alternative approaches to

the incorporation of history-dependence of an appropriate sort into policy deliberations are

discussed in sections xx below.

We begin our discussion of the advantages from policy commitment with a review in

section 1 of the most famous disadvantage of discretionary policy, namely the “in¬‚ation

bias” stressed by Kydland and Prescott (1977) and Barro and Gordon (1983). This issue

is one that can be treated in a purely deterministic analysis, and we consider the question

¬rst in such a setting. Once we allow in section 2 for random disturbances, the bias in

2

In the empirical implementation of the rule (Taylor 1993, 1999b), Taylor uses the cumulative increase in

the log of the GDP de¬‚ator over the previous four quarters as his proposed measure of current in¬‚ation; this

might appear to imply that Taylor™s rule is somewhat history-dependent, unlike the purely contemporaneous

“Taylor rules” analyzed in much of the subsequent theoretical literature. However, it appears from Taylor™s

discussion of the logic of his proposal that the use of a measure of in¬‚ation over a year-long period should

be understood as a simple attempt to estimate the current value of a target variable that is only imperfectly

measured by higher-frequency data, rather than genuine advocacy of history-dependent policy.

6 CHAPTER 7. COMMITMENT TO A POLICY RULE

the long-run average in¬‚ation rate continues to exist, but there is a further problem with

discretionary policy as well: suboptimal equilibrium responses to unexpected shocks. Unlike

the in¬‚ation bias, the distortion of the response to shocks cannot generally be cured by any

purely forward-looking policy; it is the emphasis on the in¬‚ation bias in many discussions

of the disadvantages of discretionary policy that probably accounts for the widespread as-

sumption that the advantages of commitment can be obtained within a decision framework

that remains purely forward-looking.

We then consider the problem of implementation of the desired equilibrium responses to

disturbances through commitment of the central bank to an appropriate policy rule. A mere

characterization of the desired state-contingent evolution of the economy does not su¬ce as

a policy prescription, for a number of reasons outlined in section 3. One of the simplest

of these is that a speci¬cation of the future evolution of policy (say, a speci¬cation of the

future path of overnight interest rates, to be concrete) that allows for enough contingencies

to represent a desirable policy will be too complex to actually write out in advance.

Instead, there are a variety of ways in which one may specify a decision procedure for

the central bank that su¬ces (in the context of an evaluation of current conditions and

possible future paths for the economy) to determine a policy action at each date, and such

that an understanding that the central bank will follow the procedure determines a rational

expectations equilibrium in which the economy™s state-contingent evolution is of the desired

sort. We compare several alternatives in sections 4 and 5, and argue in section 5 for the

particular desirability of a conception of rule-based policymaking under which the central

bank seeks in each decision cycle to determine the current interest-rate operating target that

is consistent with a projection of the economy™s subsequent path that satis¬es a “target

criterion”. Decision procedures of this kind are similar in form to the in¬‚ation-forecast

targeting currently practiced at the Bank of England and a number of other central banks.

But the content of the target criterion is likely to be somewhat di¬erent under an optimal

rule than under current U.K. practice; in particular, an optimal criterion will be history-

dependent, as we illustrate here through a simple example.

1. THE OPTIMAL LONG-RUN INFLATION TARGET 7

1 The Optimal Long-Run In¬‚ation Target

We begin our discussion of methods that can be used to characterize the economy™s optimal

evolution by considering the optimal rate of in¬‚ation in a purely deterministic setting. This

problem is non-trivial when it is not possible to simultaneously eliminate all of the distortions

that are a¬ected by monetary policy, for one of the reasons discussed in section xx of chapter

6; and the problem provides a simple ¬rst setting in which to analyze the distortions resulting

from discretionary policy. As it turns out, in the context of the linear-quadratic policy

problems that are mainly studied below, the optimal long-run average rate of in¬‚ation in the

presence of random disturbances continues to be the same one as in the deterministic analysis

presented here; this is a consequence of the familiar certainty-equivalence principle for such

problems. Hence our characterization here of the optimal rate of in¬‚ation also applies to the

stochastic settings considered in section 2.

1.1 The In¬‚ationary Bias of Discretionary Policy

Let us consider once again the basic neo-Wicksellian model of chapter 4 ” a purely cash-

less economy with exogenous capital, ¬‚exible wages (and/or e¬cient labor contracting), and

Calvo-style staggered pricing in goods markets ” and suppose that there are no real distur-

˜

bances (ξt = 0 for all t). Alternative possible perfect-foresight equilibrium paths for in¬‚ation

and output must satisfy the “new Keynesian” aggregate-supply relation

πt = κxt + βπt+1 (1.1)

for all dates t ≥ 0, where πt is the rate of in¬‚ation, xt is the output gap (here equivalent

simply to detrended output), and the coe¬cients satisfy κ > 0, 0 < β < 1. This is of course

only a log-linear approximation to the exact relation derived in section xx of chapter 3, valid

for the characterization of possible equilibrium paths in which in¬‚ation is always near zero.

But as we shall see, it is optimal in the present case for in¬‚ation to equal exactly zero, so

that a comparison of paths in the neighborhood of this particular steady state su¬ces for a

valid characterization of the optimal rate of in¬‚ation.

8 CHAPTER 7. COMMITMENT TO A POLICY RULE

Equilibrium paths must also satisfy another restriction, the intertemporal Euler equation

(or IS relation) that relates interest rates to the timing of expenditure. However, in the

present case, the optimal paths of in¬‚ation and output can be determined without reference

to that constraint, which simply determines the path of interest rates associated with any

given equilibrium path for in¬‚ation and output. We do need to verify that the implied path

for nominal interest rates is always non-negative; but we shall see that in the present case,

this is true for both the optimal commitment and the equilibrium resulting from discretionary

optimization.

We have shown in chapter 6 that a quadratic approximation to the utility of the repre-

sentative household in this model is a decreasing function of

∞

β t [πt + »(xt ’ x— )2 ]

2

(1.2)

t=0

where » > 0, x— ≥ 0 are functions of model parameters discussed in that chapter.3 As dis-

cussed in section 1 of chapter 6, a log-linear approximation to the model structural equations

su¬ces for a correct linear approximation to optimal policy only in the case that x— is small

enough. Speci¬cally, since in the present case there are no random disturbances, the solution

for the optimal paths for in¬‚ation and output obtained by minimizing (1.2) subject to the

constraint that (1.1) hold each period will be accurate up to a residual of order O(||x— ||2 ).

This su¬ces, however, for a characterization of the ¬rst-order e¬ects of allowing for x— > 0

(i.e., for ine¬ciency of the natural rate of output). And this, in turn, is enough to allow us

to understand the basic character of the in¬‚ation bias resulting from discretionary policy,

even if we cannot expect our analysis to yield an accurate estimate of the size of this bias

except when x— is small.4

We therefore consider the choice of monetary policy to minimize (1.2) subject to the se-

quence of constraints (1.1). Let us ¬rst consider the equilibrium outcome under discretionary

3

It does not matter for our characterization of optimal policy here whether we assume that the loss

function parameters are the ones that correctly re¬‚ect economic welfare as characterized in chapter 6 or not;

our results will be equally applicable if (1.2) is taken to represent an ad hoc policy objective of some other

kind.

4

Discuss Benigno-Woodford extension.....

1. THE OPTIMAL LONG-RUN INFLATION TARGET 9

optimization by the central bank. This is fairly simple to characterize, without any need to

fully specify the strategic interaction between the central bank and private sector. In the

model just recalled, the set of possible equilibrium paths for in¬‚ation and output from any

period T onward are independent of what in¬‚ation, output or interest rates have been in any

periods prior to T ; nor does the central bank™s evaluation of continuation paths from period

T onward depend on prior history. Hence in a Markov equilibrium,5 neither the equilibrium

behavior of the private sector nor that of the central bank from period T onward should

depend on what has happened earlier. This means that in such an equilibrium, the central

bank can (correctly) assume in period t that the action it chooses will have no e¬ect on

outcomes in any periods T ≥ t + 1; nor (since the private sector has rational expectations)

will it have any e¬ect on the private sector™s expectations in period t regarding πt+1 .

Let these equilibrium expectations be denoted π e . (In a Markov equilibrium, in¬‚ation

expectations are the same at all dates, owing to the time-invariant form of the continuation

game.) Then the central bank will perceive itself as being able to choose in period t among

in¬‚ation-output pairs that satisfy the constraint

πt = κxt + βπ e . (1.3)

As it can no longer a¬ect the contributions to (1.2) from periods prior to t and expects its

current decision to have no e¬ect on the contributions from later periods, a discretionary

optimizer will choose an action in period t intended to bring about the in¬‚ation-output pair

that minimizes πt + »(xt ’ x— )2 subject to this constraint. The ¬rst-order condition for this

2

static optimization problem is given by

»

πt + (xt ’ x— ) = 0. (1.4)

κ

Substitution of this into (1.3) implies that the bank will generate in¬‚ation satisfying

’1

κ2

[κx— + βπ e ]

πt = 1 + (1.5)

»

5

There may, of course, be additional “reputational” equilibria in a policy game of this kind, as analyzed,

for example, by Chari et al. (199xx) in the context of a sticky-price model with some prices ¬xed a period

in advance. But for present purposes, the demonstration that one possible equilibrium under discretion is

bad su¬ces to show that there is a potential gain from commitment to a suitable rule.

10 CHAPTER 7. COMMITMENT TO A POLICY RULE

in the case of any given expectations π e .

Rational expectations on the part of the private sector, of course, require that π e be such

that exactly that same in¬‚ation rate is generated. Thus the expected rate of in¬‚ation under

discretionary policy is

κ»

πe = x— > 0, (1.6)

2

(1 ’ β)» + κ

and this will also be the rate of in¬‚ation that the central bank chooses to generate each period,

given the (correctly) perceived in¬‚ation-output tradeo¬ (1.3). Hence we have obtained the

following result.

Proposition 7.1. Consider a cashless economy with ¬‚exible wages, Calvo pricing, and

no real disturbances. Assume that the initial dispersion of prices var{log p’1 (i)} is small

(of order O(||ξ||2 )),6 and suppose furthermore that real distortions are small (¦ = O(||ξ||)),

so that an approximation to the welfare of the representative household of the form (1.2) is

possible, with x— > 0 a small parameter (x— = O(||ξ||)). Then, at least among in¬‚ation paths

in which in¬‚ation remains forever near enough to zero, there is a unique Markov equilibrium

with discretionary optimization by the central bank. In this equilibrium, in¬‚ation is constant

at the value given by the right-hand side of (1.6), up to an error that is only of order O(||ξ||2 ).

But this outcome is not in fact the best possible rational-expectations equilibrium. Con-

sider instead the problem of choosing bounded deterministic paths for in¬‚ation and output

to minimize (1.2), subject to the constraint that the sequences must satisfy (1.1) each period.

We can write a Lagrangian for this problem of the form

∞

12

βt [πt + »(xt ’ x— )2 ] + •t [πt ’ κxt ’ βπt+1 ] ,

L=

2

t=0

where •t is a Lagrange multiplier associated with the period t aggregate-supply relation.

Di¬erentiation of the Lagrangian with respect to each of its arguments yields a pair of ¬rst-

6

One should recall that this assumption was used in chapter 6 in establishing that welfare is decreasing in

(1.2), neglecting terms independent of policy and a residual of only third order. In the equilibrium described

in this proposition, the condition continues to hold forever if it holds for the initial price distribution.

1. THE OPTIMAL LONG-RUN INFLATION TARGET 11

order conditions

πt + •t ’ •t’1 = 0, (1.7)

»(xt ’ x— ) ’ κ•t = 0, (1.8)

for each t ≥ 0, where in (1.7) for t = 0 we substitute the value

•’1 = 0, (1.9)

as there is in fact no constraint associated with ful¬llment of a period -1 aggregate-supply

relation.

Using (1.7) and (1.8) to substitute for πt and xt respectively in (1.1), we obtain a di¬erence

equation for the evolution of the multipliers,

κ2

•t + •t’1 = κx— ,

β•t+1 ’ 1 + β + (1.10)

»

that must hold for all t ≥ 0, along with the initial condition (1.9). The characteristic equation

κ2

2

βµ ’ 1 + β + µ+1=0 (1.11)

»

has two real roots

0 < µ1 < 1 < µ 2 ,

as a result of which (1.10) has a unique non-explosive solution consistent with the initial

condition (1.9), given by

»

•t = ’ x— (1 ’ µt+1 ) (1.12)

1

κ

for all t ≥ 0. This solution, which is the only one satisfying the relevant transversality con-

dition, represents the optimal perfect-foresight path from standpoint of period zero. Substi-

tuting this solution for the multipliers into (1.7), we ¬nd that the path of in¬‚ation under the

optimal commitment is given by

»

πt = (1 ’ µ1 ) x— µt (1.13)

1

κ

for all t ≥ 0.

12 CHAPTER 7. COMMITMENT TO A POLICY RULE

This result indicates that discretionary optimization leads to excessive in¬‚ation. Indeed,

under the optimal commitment, in¬‚ation should asymptotically approach zero, despite the

assumption that x— > 0. As has been mentioned in chapter 6, this results because the

aggregate-supply relation (1.1) implies that in any perfect-foresight equilibrium, the objective

(1.2) must equal

∞

x—

β t [πt + »x2 ]

2

’2 π0 + t

θ t=0

plus a positive constant. All terms in the above expression except the initial one are min-

imized by choosing πt = 0 each period, regardless of the value of π — ; the presence of the

initial term implies an advantage from an initial positive rate of in¬‚ation, but because the

additional term applies only to in¬‚ation in the initial period, it remains optimal to commit

to an in¬‚ation rate that is eventually zero.

Indeed, it is not obviously desirable to choose a positive in¬‚ation rate even initially. It is

true that (1.2) is minimized by choosing the in¬‚ation path (1.13) rather than zero in¬‚ation

from the initial date onward. This welfare gain, however, is obtained as a result of the fact

that the in¬‚ation rate that is chosen initially has no consequences for expectations prior to

date zero (that are taken as given at the time of the policy deliberations). This exploitation

of the fact that initially existing expectations need not be ful¬lled, however, is unattractive.

It implies that the optimal policy determined on the above grounds (optimality from the

standpoint of date zero) is not time consistent: if the same reasoning is used at any later date

t > 0 to determine the optimal policy commitment from that date onward, the policy chosen

will not be a continuation of the policy selected at date zero. (This can be seen from the fact

that the in¬‚ation rate πt varies with the date t in (1.13), even though the constraints de¬ning

the possible in¬‚ation paths from any date onward, and the social valuation of alternative

in¬‚ation paths looking forward from any date, are the same at all times.) This failure of

time consistency means that adherence to the policy must be due solely to a willingness to

conform to a commitment entered into at an earlier date. In practice, it is di¬cult to imagine

that a central bank would ever regard itself as being committed to a speci¬c sequence of

1. THE OPTIMAL LONG-RUN INFLATION TARGET 13

actions chosen at an earlier date, simply because they seemed desirable at the earlier time7

” whereas it is easier to imagine a bank being committed to a systematic decision procedure

in the light of which its current actions are always to be justi¬ed. Furthermore, the above

analysis assumes that it is possible to commmit to an arbitrary time path for in¬‚ation, and

have this be expected by the private sector; it is assumed to be possible to choose in¬‚ation

“just this time” while committing never to create in¬‚ation in the future. But there is reason

to fear that the public should observe the central bank™s method of reasoning, rather than

its announced future actions, and conclude instead that in the present it should always wish

to create in¬‚ation “just this time”.

A similar problem of time consistency is familiar in the context of ¬scal policy, and in that

context it is common to conclude that if, say, one wishes the public to be able to con¬dently

expect that capital will not be expropriated in the future, then one should adopt the rule

of refusing to expropriate already existing capital, even though the latter action should not

have the same kind of e¬ects on investment incentives (since the investment decisions in

question have already occurred). Similarly, if one wishes for the public to believe that a

non-in¬‚ationary policy will eventually be pursued, and there is no di¬erence between the

current situation and the one that is anticipated in the future (except that one currently has

an opportunity to create in¬‚ation without its having been expected), then it makes sense

that the central bank should be willing to choose a non-in¬‚ationary policy as well. Rather

than doing one thing now but promising to behave di¬erently in the future, one should follow

a time-invariant policy that is of the kind that one would always wish to have been expected

to follow. Woodford (1999) calls such a policy “optimal from a timeless perspective.”8

More speci¬cally, we shall say that a time-invariant policy is optimal from a timeless

perspective if the equilibrium evolution from any date t0 onward (at which date one may

consider the justi¬cation of the policy) is optimal subject to the constraint that the economy™s

7

Such an understanding of the meaning of policy commitment raises the question posed by Svensson

(1999xx): “What is special about date zero?”

8

For additional discussions of the selection of policy rules from a timeless perspective, see McCallum and

Nelson (199xx), Giannoni and Woodford (2002xx), Svensson and Woodford (2002xx), and section xx below.

14 CHAPTER 7. COMMITMENT TO A POLICY RULE

initial evolution be the one associated with the policy in question. (The presence of such a

constraint on the initial outcomes that may be contemplated is a way of committing oneself

to forswear the temptation to exploit the already given past expectations regarding those

initial outcomes.) In our present context, a time-invariant policy will imply a constant

in¬‚ation rate π . A constant in¬‚ation target π is optimal from a timeless perspective if the

¯ ¯

problem of maximizing (1.2) subject to the constraint that the bounded sequences {πt , xt }

satisfy (1.1) for each t ≥ 0, and the additional constraint that π0 = π , has a solution in which

¯

πt = π for all t.9 The ¬rst-order conditions for this latter problem are again given by (1.7)

¯

and (1.8) for each t ≥ 0, but now the initial condition (1.9) is replaced by the requirement

that π0 = π . One easily sees that this system of equations has a solution in which πt = π for

¯ ¯

all t if and only if π = 0. Hence this is the uniquely optimal in¬‚ation target from a timeless

¯

perspective.

Proposition 7.2. Consider a cashless economy with ¬‚exible wages, Calvo pricing, and

no real disturbances, and suppose that the initial dispersion of prices is small (of order

O(||ξ||2 )). Then a monetary policy under which in¬‚ation is zero for all t is optimal from a

timeless perspective, and is the unique policy with this property among all policies under

which in¬‚ation remains always in a certain interval around zero.

Note that for this proposition, unlike Proposition 7.1, it is not necessary for x— to be small,

since the optimal path is found to be near the zero-in¬‚ation steady state regardless of the

value of x— .

Under this analysis of the character of an optimal policy, there is a constant in¬‚ation bias

associated with discretionary policy, which is clearly positive when x— = 0. This is illustrated

in Figure 7.1, which plots the time paths of in¬‚ation under discretionary policy, under

the date-zero-optimal commitment, and under the policy that is optimal from a timeless

perspective.10 In¬‚ation is lowest under the last of these policies, since only in this last case

9

See section xx below for a de¬nition of optimal policy from a timeless perspective in a more general

context.

1. THE OPTIMAL LONG-RUN INFLATION TARGET 15

10

8

6 = discretion

= zero’optimal

= timeless

4

2

0

’2

0 2 4 6 8 10 12 14 16 18 20

Figure 7.1: Optimal policy from a timeless perspective compared to the result of discretionary

optimization, and to the policy commitment that is optimal from the standpoint of period

zero.

does the central bank refrain from the temptation to create in¬‚ation that cannot a¬ect prior

expectations, both in the short run and in the long run.

1.2 Extensions of the Basic Analysis

A striking conclusion of the previous section is that the optimal in¬‚ation rate is exactly zero

in our baseline model with Calvo pricing, regardless of parameter values, including those that

determine the size of the gap x— between the optimal level of output and that consistent with

zero in¬‚ation. However, this analysis does neglect various other factors that may bear on

the choice of a long-run average in¬‚ation target. One is the neglect of monetary frictions of

10

In this numerical example, the values of β κ are again those given in Table 6.1, » is assigned the value

given there for »x , and x— = 0.2. This last value follows from equation (xx) in chapter 6 for the optimal

output gap, under the assumptions that the elasticity of substitution among alternative di¬erentiated goods

(and hence the elasticity of demand faced by each ¬rm) is equal to 7.88, the value obtained by Rotemberg

and Woodford (1997) (see Table 4.1), and that there are no distorting taxes.

16 CHAPTER 7. COMMITMENT TO A POLICY RULE

the sort that led Friedman (1969) to argue for the optimality of de¬‚ation at the rate of time

preference.

As shown in the previous chapter, in the presence of non-negligible transactions frictions,

the welfare-theoretic loss function takes the form

∞

β t [πt + »x (xt ’ x— )2 + »i (it ’ im )2 ],

2

(1.14)

t

t=0

where »x > 0 is the coe¬cient called » in (1.2), »i > 0 as well, it is the short-term nominal

interest yield on non-monetary assets, and im is the interest rate paid on the monetary base.11

t

In the event that monetary policy is implemented by varying im so as to maintain a constant

t

interest di¬erential regardless of the desired level of nominal interest rates ” and that this

di¬erential is taken to be an institutional datum rather than an aspect of monetary policy

” then the additional term in (1.14) relative to (1.2) makes no di¬erence, as it is simply a

constant independent of the chosen target path for in¬‚ation. In this case, the conclusions of

the previous section continue to apply: the optimal policy from a timeless perspective will

involve a zero in¬‚ation rate at all times. But if, instead, the interest paid on the monetary

base is an institutional datum, and ¯m is lower than the rate consistent with zero in¬‚ation

±

(for example, if there is zero interest paid on money, as assumed by Friedman), then the

rate of in¬‚ation that would otherwise be optimal may not be because of its consequences for

nominal interest rates and hence the size of the last term in (1.14).

In the case that real balances enter the utility function in an additively separable way,

transactions frictions do not a¬ect the structural equations that determine equilibrium in-

¬‚ation and output, as discussed in chapter 4. (While additively separability is not realistic,

as discussed earlier, the quantitative magnitude of the real-balance e¬ects that are neglected

by such an assumption is likely to be small.) Feasible perfect-foresight equilibrium paths

11

In the present chapter and the next one, it refers to the continuously compounded rate of interest on

non-monetary assets, log(1 + it ) in the notation of the previous chapters, and similarly for im and the

t

natural rate of interest rt , appearing in equation (xx) below. Hence the interest di¬erential it ’ im is equal

n

t

¯ ¯

to ˆt ’ ˆm + ∆ in the notation of chapter 6, up to a term of order O(||∆||2 ) that can be neglected in our

± ±t

quadratic approximation to the welfare-theoretic loss function. The change in notation should not create

confusion, now that we need no longer discuss the exact nonlinear equations that are log-linearized in deriving

the linear structural relations assumed in this chapter and the next.

1. THE OPTIMAL LONG-RUN INFLATION TARGET 17

for in¬‚ation, output and nominal interest rates then must satisfy (1.1), together with the

corresponding deterministic IS equation

xt = xt+1 ’ σ(it ’ πt+1 ’ r), )

¯ (1.15)

for periods t ≥ 0, where r > 0 represents the constant natural rate of interest. A stationary

¯

policy commitment resulting in a constant in¬‚ation rate π , output gap x, and nominal interest

¯ ¯

rate ¯ will be optimal from a timeless perspective if the bounded sequences {πt , xt , ±t } that

±

minimize (1.14) subject to the constraints that (1.1) and (1.15) hold for each t ≥ 0, and that

π0 = π ,

¯ x0 = x,

¯

are given by πt = π , xt = x, it = ¯ for all t.

¯ ¯ ±

The Lagrangian for this generalization of our previous problem is of the form

∞

12

βt [πt + »x (xt ’ x— )2 + »i (it ’ im )2 ] + •1t [xt ’ xt+1 + σ(it ’ πt+1 ’ r)]

L= ¯

2

t=0

+•2t [πt ’ κxt ’ βπt+1 ]} ’ β ’1 •1,’1 [x0 + σπ0 ] ’ •2,’1 π0 ,

where there are now Lagrange multipliers •1t , •2t corresponding to constraints (1.15) and

(1.1) respectively, and multipliers β ’1 •1,’1 , •2,’1 corresponding to the constraints on the

values of x0 and π0 respectively. (The notation chosen for these last multipliers is selected

in order to give the ¬rst-order conditions below a time-invariant form.)

The ¬rst-order conditions are then

πt ’ β ’1 σ•1,t’1 + •2t ’ •2,t’1 = 0, (1.16)

»x (xt ’ x— ) + •1t ’ β ’1 •1,t’1 ’ κ•2t = 0, (1.17)

»i (it ’ im ) + σ•1t = 0 (1.18)

for each period t ≥ 0. Conditions (1.17) “ (1.18) have a solution with the output gap and

interest rate constant over time only if both Lagrange multipliers are also constant over time;

18 CHAPTER 7. COMMITMENT TO A POLICY RULE

but substituting constant values for the Lagrange multipliers in (1.16) and (1.18), one ¬nds

that these relations can be simultaneously satis¬ed only if

»i (¯ ’ im ) = ’β π .

± ¯

At the same time, equation (1.15) is satis¬ed by constant values if and only if ¯ = r + π .

± ¯¯

These two relations are jointly satis¬ed if and only if the in¬‚ation target is equal to

»i

(¯ ’ im ).

π=’

¯ r (1.19)

»i + β

We thus obtain the following result.

Proposition 7.3. Consider an economy with ¬‚exible wages, Calvo pricing, and no real

disturbances, with transactions frictions such that the quadratic approximation to welfare

be given by (1.14), while the log-linear approximate structural relations are of the form (1.1)

and (1.15). Suppose that the only feasible monetary policies are ones involving a constant

interest rate im on the monetary base, where im = r ’ O(||ξ||). Then the policy that is

¯

optimal from a timeless perspective involves a constant in¬‚ation rate equal to the right-hand

side of (1.19).

Note that in the case that »i > 0 and im < r, the optimal in¬‚ation target will be negative.

¯

In the limiting case of ¬‚exible prices, κ, and hence »i , becomes unboundedly large, and the

optimal rate of de¬‚ation is ’(¯ ’ im ), the rate required to make the real return on money as

r

high as r, as argued by Friedman (1969). When prices are sticky, and κ is ¬nite, the optimal

¯

rate of de¬‚ation is more moderate, and the optimal policy is one under which ¯ > im ; but

±

some de¬‚ation is still optimal, in order to reduce the distortions resulting from transactions

frictions. In the “cashless limit”, in which m is negligible, »i is much smaller than β, and

¯

the optimal rate of in¬‚ation is essentially zero, as found in the previous section.

In the case that real-balance e¬ects are non-negligible in the aggregate-supply and IS

equations, the same method can be employed as above, but (1.19) takes the more general

1. THE OPTIMAL LONG-RUN INFLATION TARGET 19

form12

»i (¯ ’ im ) + •»x x—

r

π=’

¯ , (1.20)

»i + •2 »x + β ’ (1 ’ β)•(θ’1 + ω)

where the coe¬cient • indexes the size of the real-balance e¬ects, as in chapter 4. If we use

the calibrated parameter values from Table 6.1, except that x— is computed from equation

(xx: xstaristar) of chapter 6 under the assumption that ¦/ = 0.2 as assumed in the

mc

previous section, and we assume that im = 0, this equation implies an optimal in¬‚ation rate

of -0.4 percent per year.13 This involves mild de¬‚ation, but much less de¬‚ation than would

be suggested by the arguments of Friedman (1969) or Lucas (2000), according to which the

optimal rate of in¬‚ation (under our parameter values and those of Lucas) would be -3 percent

per year. And we should recall that these parameter values have been chosen to exaggerate

the size of real-balance e¬ects; under the more plausible assumption that χ = 0.01, we would

obtain an optimal rate of de¬‚ation of only 0.2 percent per year.

It is perhaps surprising to observe that if • > 0 (the empirically realistic case, as argued

in chapter 2), and if the denominator of (1.20) is positive (as must be true if real-balance

e¬ects are not too strong), a higher value of x— actually lowers the optimal in¬‚ation target.

It is commonly supposed that a higher value of x— should justify higher in¬‚ation, in a model

(like the present one) that incorporates an upward-sloping “long-run Phillips curve” relation

between in¬‚ation and output. Yet this is not so. As shown in chapter 6, in this model,

commitment to a higher in¬‚ation rate in any period t > t0 (i.e., any period for which the

anticipation e¬ects must be considered) does not change the discounted sum of output-gap

terms implied by the aggregate-supply relation; for the increase in output in period t is

exactly o¬set by a reduction in output in period t ’ 1. The only e¬ects of policy that do

12

Equation (1.20) applies in the case that this in¬‚ation rate implies a nominal interest rate no lower

than im . In the case that this bound is not satis¬ed, the analysis must be modi¬ed to take account of the

constraint on policy resulting from the zero lower bound on nominal interest rates, assumed not to bind in

this derivation.

13

In this calculation, the values of »x , »i , r and β are taken directly from Table 6.1. The value used for •

¯

is 0.22 years, the value implicit in the Table 6.1 parameter values, as discussed there; similarly, we assume

the value θ = 7.88 as in the derivation of the values given in the table. The values for mc , σ ’1 , χ, and ·y

given in the table imply a value ω = 0.48. In using equation (xx) of chapter 6 to derive x— , we recall that in

the notation of that chapter i— = ’(¯ ’ im ). Finally, we note that in the denominator of (1.20), if we use an

r

annual measure for •, it is necessary to multiply (1 ’ β) by 4 to obtain an annual discount rate as well.

20 CHAPTER 7. COMMITMENT TO A POLICY RULE

not cancel in this way are the real-balance e¬ects on aggregate supply, when these exist. If

• > 0, a lower nominal interest rate causes a favorable shift in the aggregate-supply relation

each period, allowing a higher value for the discounted sum of output gaps. When x— > 0,

this channel creates a reason to prefer a lower nominal interest rate than the one consistent

with zero in¬‚ation, and hence to prefer de¬‚ation for reasons independent of Friedman™s.

Thus far we have discussed only reasons why the optimal in¬‚ation target might be even

lower than zero. Other considerations may instead justify a positive long-run average in¬‚a-

tion rate. These include a desire the prevent the zero lower bound on nominal interest rates

from being so tight a constraint on cyclical variation in real rates for stabilization purposes

(Summers, 1991), or a desire to make a social norm that prohibits nominal wage declines

less of a constraint upon the degree to which real wages can decline when necessary for an

e¬cient allocation of resources (Akerlof et al., 199xx). These latter considerations matter

only in the case of random disturbances, and cannot be addressed without considering the

optimal responses to such disturbances. It is worth noting, however, that they cast some

doubt on the desirability of deliberately aiming for de¬‚ation, as opposed to zero in¬‚ation

or even a very modest positive rate of in¬‚ation, of the sort typically aimed at by current

in¬‚ation targeting central banks.14

Our analyses thus far have also argued for the optimality of a policy that would bring

about a constant in¬‚ation rate in the absence of stochastic disturbances. Yet a number of

countries that have actually adopted in¬‚ation targets in the past decade have thought it

desirable to lower the target in¬‚ation rate from an (undesirably high) initial rate of in¬‚ation

only gradually, over several years, rather than announcing an intention to immediately jump

to the in¬‚ation rate that is regarded as optimal over the long run. The argument made for

the desirability of gradualism of this sort generally involves a belief that there is substantial

inertia in the in¬‚ationary process, though the models used above do not allow for this. We

can consider the e¬ect of in¬‚ation inertia on the optimal time path for in¬‚ation, still within a

14

The consequences of the zero lower bound have been considered in section xx of chapter 6, and are

considered further in section xx of this chapter. Both analyses display conditions under which a policy

would be chosen that involves a positive average rate of in¬‚ation.

1. THE OPTIMAL LONG-RUN INFLATION TARGET 21

purely deterministic context, by allowing for indexation of individual prices to a lagged price

index, as considered in section xx of chapter 3. In this case, the aggregate-supply relation

(1.1) generalizes to

πt ’ γπt’1 = κxt + β(πt+1 ’ γπt ), (1.21)

where 0 ¤ γ ¤ 1 indicates the degree of indexation, if we once again abstract from real-

balance e¬ects. Hence the possible perfect-foresight paths for in¬‚ation and output from any

date t0 onward depend upon the pre-existing rate of in¬‚ation t0 ’ 1.

As shown in chapter 6, the welfare-theoretic loss function (1.2) for a cashless economy

generalizes to

∞

β t [(πt ’ γπt’1 )2 + »(xt ’ x— )2 ] (1.22)

t=0

in the presence of in¬‚ation inertia. Because both the constraints (1.21) and the loss function

(1.22) are of the same form as before, but with πt ’ γπt’1 replacing πt in the previous

equations, the same calculations as before may be directly used to characterize optimal

policy (as well as the consequences of discretionary optimization), except that our previous

solutions for the path of πt now apply instead to the path of πt ’ γπt’1 .

Figure 7.2 shows the implied equilibrium paths for in¬‚ation under discretion, under an

unconstrained optimal commitment from the standpoint of period zero, and under a policy

that is optimal from a timeless perspective, when all parameters are the same as those

assumed in Figure 7.1, except that γ = 0.5, and (since the initial in¬‚ation rate now matters)

an in¬‚ation rate of 10 percent per year prior to period zero is assumed. In all three cases,

the fact that the economy starts from a condition of fairly high in¬‚ation makes the in¬‚ation

rate in the early periods higher than it would otherwise be; in fact, one can show that in

each of the three cases, the equilibrium value of πt is given by π’1 γ t+1 plus a term that is

independent of the initial condition. However, despite this e¬ect of inertial in¬‚ation, the

comparisons among the three paths remain of the kind discussed earlier: discretion leads to

an in¬‚ation rate that is permanently higher than the in¬‚ation that would occur under an

optimal policy, and unconstrained (time-inconsistent) optimization at date zero leads to the

choice of a higher in¬‚ation rate in the early periods, re¬‚ecting an attempt to exploit the

22 CHAPTER 7. COMMITMENT TO A POLICY RULE

20

18

16

14

12

= discretion

= zero’optimal

10

= timeless

8

6

4

2

0

0 2 4 6 8 10 12 14 16 18 20

Figure 7.2: Timelessly optimal policy, date-zero-optimal policy, and discretionary policy, in

the case of in¬‚ation inertia [γ = 0.5].

initially given expectations of in¬‚ation, though it leads to the same in¬‚ation rate as under a

timelessly optimal policy in the long run.

The proof of Proposition 7.2 above can be directly adapted to provide the following

characterization of optimal policy in this case.

Proposition 7.4. Consider an economy of the kind assumed in Proposition 7.2, except

that prices are (partially) indexed to a lagged aggregate price index, to a degree measured

by 0 ¤ γ ¤ 1, and suppose that the initial in¬‚ation rate satis¬es π’1 = O(||ξ||). Then the

policy that is optimal from a timeless perspective involves a path of in¬‚ation along which

πt = γπt’1 (1.23)

at each date t ≥ 0, up to a residual of order O(||ξ||).

1. THE OPTIMAL LONG-RUN INFLATION TARGET 23

This path has the property that the policy that minimizes discounted losses from any date

t0 onward, given the in¬‚ation history up through period t0 ’ 1 and subject to the constraint

that (1.23) hold in period t0 , is one in which (1.23) holds for all t ≥ t0 . Note that (1.23)

can easily be solved to show that πt = π’1 γ t+1 for any t ≥ 0. Once again we ¬nd (in the

case that γ < 1) that the optimal long-run in¬‚ation rate is zero; but it is now optimal to

approach this in¬‚ation rate only over a period of time, if the economy starts (for whatever

reason) from an initial situation in which in¬‚ation has been higher.

In the case of full indexation (γ = 1), this result implies that it is not optimal ever to

reduce in¬‚ation below whatever its initial level happens to be. Such a result should not be

acceptable as having too much practical relevance. It comes about because, in our simple

model, all prices are adjusted to re¬‚ect past in¬‚ation in a perfectly synchronized way, except

when they are reoptimized. When γ = 1, a policy that keeps in¬‚ation steady will result in all

prices being increased by exactly the same amount each period, so that no price dispersion

is created, regardless of the rate of in¬‚ation. But in practice, even in an economy with

substantial indexation (as one generally observes in economies with chronic high in¬‚ation),

in¬‚ation results in price dispersion, and higher in¬‚ation is generally associated with greater

dispersion. If our model were extended to allow for this ” for example, by supposing

that prices are not continuously adjusted in response to variations in the aggregate price

index between the occasions on which they are re-optimized, but that these mechanical price

adjustments also occur only at certain intervals ” then price dispersion would be minimized

only with zero in¬‚ation, though we would again ¬nd that changes in the rare of in¬‚ation

would increase price dispersion and hence lower welfare. We do not attempt to develop an

extended indexation model of this kind here. But it is fairly obvious that we can in this

way, or in various others, justify the assumption that there are at least some distortions

associated with a high rate of in¬‚ation, regardless of how steady it may be; and once we

introduce even a small amount of such a distortion, it will again be optimal to eventually

disin¬‚ate, regardless of the degree of in¬‚ation inertia.

For example, suppose that we allow again for transactions frictions and suppose that the

24 CHAPTER 7. COMMITMENT TO A POLICY RULE

interest paid on the monetary base is constant. For simplicity, let us consider only the case

in which χ = 0 though »i > 0. In this case, the structural equation (1.15) must be combined

with (1.21) as constraints on possible perfect-foresight paths, and the welfare-theoretic loss

function is of the form

∞

β t [(πt ’ γπt’1 )2 + »(xt ’ x— )2 + »i (it ’ im )2 ].

t=0

The ¬rst-order condition (1.16) generalizes to

(πt ’ γπt’1 ) ’ βγ(πt+1 ’ γπt ) ’ β ’1 σ•1,t’1 + •2t ’ •2,t’1 = 0,

while (1.17) “ (1.18) continue to apply as written above. The same analysis as above can

again be used to derive a unique long-run steady-state rate of in¬‚ation that is consistent

with these ¬rst order conditions, given by

»i

(¯ ’ im ) < 0,

π=’

¯ r

»i + β(1 ’ γ)(1 ’ βγ)

generalizing (1.19).

While in the case that »i = 0 (the cashless case just considered), this equation yields a

determinate outcome only when γ = 1, in the case that »i > 0, there is a uniquely optimal

long-run in¬‚ation rate even when γ = 1, namely

π = ’(¯ ’ im ),

¯ r

the rate of de¬‚ation called for by Friedman (1969). It is worth noting that the degree to

which it is eventually desirable to disin¬‚ate is independent of how large »i may be ” it

is simply necessary to allow for some frictions of this kind, that increase with the rate of

expected in¬‚ation. In fact, if we assume full indexation of prices to a lagged price index, this

indexation actually increases the degree to which it is optimal to eventually lower the rate

of in¬‚ation; for it is no longer true that steady de¬‚ation creates relative-price distortions

owing to the failure of prices to be re-optimized at perfectly synchronized times. This last

result, however, is again an example of an extreme conclusion that results from too simple

2. OPTIMAL RESPONSES TO DISTURBANCES 25

a model of indexation. In reality, even steady de¬‚ation is likely to create distortions, for

reasons identical to those mentioned in the case of steady in¬‚ation; and as a result, the

optimal long-run rate of in¬‚ation is likely not to be so low as the Friedman rate.

2 Optimal Responses to Disturbances

The in¬‚ationary bias resulting from discretionary policy has been much discussed. However,

emphasis on the problem of in¬‚ation bias has often led to a supposition that the problems

resulting from discretion can be cured through a simple adjustment of the targets and/or the

relative weights on alternative stabilization objectives assigned to the central bank, while

allowing the central bank™s decisionmaking framework to be otherwise one of unfettered

discretion. For example, our baseline analysis in section 1.1 above implied an in¬‚ation bias

equal to

κ»

x— ,

2

(1 ’ β)» + κ

when the central bank seeks to minimize the true social loss function, in which we have argued

that » > 0, x— > 0. But if the central bank instead seeks to minimize a loss function of the

form (1.2) with some other coe¬cients, the preceding analysis still applies; in particular, the

equilibrium in¬‚ation rate will be zero each period, even under discretion, as long as either

» = 0, x— = 0, or both. Thus one might suppose that the problem can be solved by appointing

a central banker with appropriate preferences, or by charging the central bank with the task

of minimizing a particular loss function that di¬ers from true social welfare. The choice

of a loss function with » = 0 corresponds to Rogo¬™s (1985) proposal that a “conservative

central banker” be chosen, while King (1997) and Blinder (1998) propose that the central

bank should have an objective under which x— = 0, i.e., an output target consistent with its

in¬‚ation target.

But while it is fairly simple to eliminate the bias in the average rate of in¬‚ation resulting

from discretionary policy through either of these means, this does not su¬ce to yield an

optimal policy framework, for in general the equilibrium responses to shocks that result

26 CHAPTER 7. COMMITMENT TO A POLICY RULE

will be sub-optimal, even if the long-run average values of the various state variables are the

optimal ones. It is perhaps obvious that the Rogo¬ proposal to alter the relative weight on the

two stabilization objectives will often result in incorrect responses to disturbances; indeed,

Rogo¬ discusses this as an important quali¬cation to his proposal (so that the optimal degree

of “conservativeness” is argued to be less than absolute, in the presence of certain kinds of

disturbances). But in the context of a linear-quadratic framework of the kind that we use

here to approximate the central bank™s problem, an adjustment of the target values of one or

more variables a¬ects the average equilibrium values of the endogenous variables resulting

from central-bank optimization without having any e¬ect on the equilibrium responses to

shocks; hence one might think that the proposal of King and Blinder should not result

in any distortion of stabilization policy.15 But in general, the equilibrium resulting from

discretionary optimization is sub-optimal, not only in the long-run average values of variables

such as in¬‚ation, but also in the equilibrium responses of these variables to random shocks,

for reasons discussed in the introduction; and an adjustment of the target values assigned to

the central bank does nothing to cure this problem.

In this section, we contrast the optimal responses to shocks to those resulting from

discretionary optimization in two simple examples, and then discuss the structure of the

problem more generally. We show not only that the optimal responses to shocks are generally

di¬erent from those resulting from discretionary optimization, but that they generally require

that equilibrium be history-dependent in a way that cannot result from any purely forward-

looking decision procedure for monetary policy. This implies, among other things, that an

approach to the implementation of optimal policy that charges the central bank with the

minimization of a loss function under discretion, which loss function involves the same target

variables as the true social loss function ” only with di¬erent target values and a di¬erent

matrix of weights in the quadratic form over deviations from those target values ” will

15

Indeed, King (1997) shows, in the context of a simple model in which an aggregate-supply relation of

the “New Classical” form discussed in chapter 3 is combined with an assumed stabilization objective of the

form (1.2), that under his proposed modi¬cation of the central bank™s loss function, the equilibrium resulting

from discretionary optimization is optimal. However, this result depends on extremely special features of

the example that he considers.

2. OPTIMAL RESPONSES TO DISTURBANCES 27

generally be inadequate, as such a decision procedure will be purely forward-looking.

2.1 Cost-Push Shocks

Probably the simplest example of this general problem arises in the case of “cost-push shocks”

that shift the aggregate-supply relation when written in terms of the welfare-relevant in¬‚ation

and output-gap measures.16 In the case of such disturbances, complete stabilization of both

in¬‚ation and the output gap is impossible; thus we need not assume any concern with

interest-rate stabilization, or other stabilization objectives, in order to conclude that there

is an essential tension among the stabilization objectives of monetary policy. In such a case,

discretionary policy does not generally lead to optimal responses to shocks.17

We can see this by considering the minimization of a social loss function given by the

expected value of (1.2), if the aggregate-supply relation each period is given by

πt = κxt + βEt πt+1 + ut , (2.1)

where ut is an exogenous cost-push shock. Let us ¬rst consider the case of discretionary

optimization by the central bank. Let st be the exogenous state at date t that contains all

information available at that time about current and future cost-push disturbance terms.

We observe that the set of possible equilibrium evolutions of in¬‚ation and the output gap

from period t onward depend only on st , and in particular are independent of the past values

of all endogenous variables. It follows that in a Markov equilibrium, πt and xt should be

functions only of st . Hence the central bank in period t believes that its policy action in that

16

Several interpretations of such disturbances are possible, as discussed in sections xx of chapter 6.

17

If complete stabilization of each of the target variables is simultaneously possible, then there is no

di¬erence in the responses to shocks under discretion and under an optimal commitment. A discretionary

optimizer ¬nds it optimal to completely stabilize each of the target variables in any given period, given the

assumption that they will be stabilized in the future; hence complete stabilization is an equilibrium under

discretionary policy, and is also optimal. This is obvious in the case that the target values of the various

state variables are mutually consistent with equilibrium in the absence of shocks, so that it is possible for

the loss function to equal zero at all times. But even when this is not so ” when, for example, x— is too high

to be consistent with zero in¬‚ation on average ” the same result is true, because the equilibrium responses

to shocks, both under discretion and under an optimal commitment, are independent of the assumed target

values, which matter only for the average values of the variables in equilibrium.

28 CHAPTER 7. COMMITMENT TO A POLICY RULE

period can have no e¬ect on terms in the loss function for periods T ≥ t + 1, and also that

its action will have no e¬ect on the private sector™s expectations Et πt+1 .

Then the central bank will perceive itself as being able to choose in period t among

in¬‚ation-output pairs that satisfy the constraint (2.1), for a given value of Et πt+1 (that

depends only on st ), and it will choose an action in period t intended to bring about the

in¬‚ation-output pair that minimizes πt + »(xt ’ x— )2 subject to this constraint. The ¬rst-

2

order condition for this static optimization problem is again given by (1.4). Substitution of

this into (2.1) implies that the bank will generate in¬‚ation satisfying

’1

κ2

[κx— + ut + βEt πt+1 ],

πt = 1 + (2.2)

»

generalizing (1.5). The Markov solution to this equation is then an in¬‚ation process

j+1

∞

κ» »

—

βj

πt = x+ Et ut+j . (2.3)

2 » + κ2

(1 ’ β)» + κ j=0

Proposition 7.5. Consider an economy of the same kind as in Proposition 7.1, except

that the aggregate-supply relation (2.1) is perturbed by an exogenous disturbance process

{ut } satisfying a uniform bound ||ξ||. Then there is a neighborhood of zero in which there

is a unique Markov equilibrium in¬‚ation process under discretionary optimization by the

central bank, for any small enough ||ξ||. In this equilibrium, in¬‚ation evolves according to

(2.3).

The corresponding Markov solution for output can be obtained by substituting this solution

for in¬‚ation into (2.1). Note that the long-run average rate of in¬‚ation in this equilibrium,

given by the constant term in (2.3), is the same as in the deterministic analysis. The

equilibrium responses of in¬‚ation and output are purely forward-looking; if ut is a Markov

process, with the value of ut being revealed only in period t, then equilibrium πt and xt

depend only on the current disturbance ut , and the e¬ects of a disturbance on the paths of

in¬‚ation and output will be only as persistent as the disturbance itself.

2. OPTIMAL RESPONSES TO DISTURBANCES 29

Now consider instead the nature of an optimal policy commitment.18 We ¬rst consider

the state-contingent evolution from some period t0 onward that minimizes the expected

discounted sum of losses from period t0 onward, conditioning upon the state of the world

in period t0 , and subject to the constraint that this evolution represent a possible rational-

expectations equilibrium, i.e., that it satisfy (2.1) for all periods t ≥ t0 . The Lagrangian

associated with this problem is of the form

∞

12

β t’t0 [πt + »x (xt ’ x— )2 ] + •t [πt ’ κxt ’ βπt+1 ] .

Lt0 = Et0 (2.4)

2

t=t0

Once again, if there is no welfare loss resulting from nominal interest-rate variation, we may

omit the constraint terms corresponding to the IS relation, as these constraints never bind.

And in writing the constraint term associated with the period t AS relation, it does not

matter that we substitute πt+1 for Et πt+1 ; for it is only the conditional expectation of the

term at date t0 that matters in (2.4), and the law of iterated expectations implies that

Et0 [•t Et πt+1 ] = Et0 [Et (•t πt+1 )] = Et0 [•t πt+1 ]

for any t ≥ t0 .

Di¬erentiating (2.4) with respect to the levels of in¬‚ation and output each period, we

obtain a pair of ¬rst-order conditions of exactly the form (1.7) “ (1.8) for each period t ≥ t0

(and for each possible state of the world at that date), together with the initial condition

•t0 ’1 = 0. (2.5)

Using (1.7) and (1.8) to substitute for πt and xt respectively in (2.1), we again obtain a

di¬erence equation for the evolution of the multipliers,

κ2

•t + •t’1 = κx— + ut ,

βEt •t+1 ’ 1 + β + (2.6)

»

which is a stochastic generalization of (1.10). Once again, the characteristic equation (1.11)

has two real roots 0 < µ1 < 1 < µ2 , as a result of which (2.6) has a unique bounded solution

18

Early treatments of this problem in the context of the present model with cost-push shocks include

Clarida et al., (1999), Woodford (1999xx), and Vestin (2000). Svensson and Woodford (2002xx) discuss a

similar problem, but with a one-period delay in the e¬ects of policy on both in¬‚ation and output.

30 CHAPTER 7. COMMITMENT TO A POLICY RULE

for {•t }, given by

∞

»—

µ’j’1 Et ut+j ,

’1

•t = µ1 •t’1 ’ (1 ’ µ1 ) x ’ β (2.7)

2

κ j=0

in the case of any bounded disturbance process {ut }. Note that this equation can be solved

recursively for the evolution of {•t } starting from any initial condition for •’1 . Substituting

this solution for the path of the Lagrange multipliers into (1.7) “ (1.8), one obtains unique

bounded solutions for the paths of in¬‚ation and the output gap.

These bounded solutions necessarily satisfy the relevant transversality condition; hence

the solution obtained starting from the initial condition (2.5) represents the state-contingent

evolution of in¬‚ation and output under the t0 ’optimal commitment. (Note that the in¬‚a-

tion solution thus obtained is a stochastic generalization of (1.13).) However, this solution

contains a deterministic component that depends on the time that has elapsed since the

date t0 at which the plan was chosen, and so is not time-consistent. Once again, we prefer a

time-invariant policy that is optimal from a timeless perspective, meaning that continuation

of the policy from any date t0 onward leads to an equilibrium from that date onward that

minimizes the expected discounted sum of losses, subject to a constraint of the form

πt0 = πt0 ,

¯ (2.8)

where the value of πt0 may depend on the state of the world at date t0 .

¯

Minimization of expected discounted losses subject to the constraint (2.8) leads to a

Lagrangian of the same form (2.4), except with the addition of a term representing the

additional constraint. This in turn leads to exactly the same system of ¬rst-order conditions,

except that initial condition (2.8) replaces (2.5). Hence there is a unique bounded solution

for the state-contingent evolution of in¬‚ation and output from date t0 onward, in the case of

any given speci¬cation of the initial condition. This solution will again be of the form (2.7),

for some choice of the initial value φt0 ’1 ; the proper choice of this initial multiplier depends

on the constraint value πt0 .

¯

In order for such a policy to be time-consistent, we need to select the constraint value

πt0 as a function of exogenous and predetermined variables at date t0 , according to some

¯

2. OPTIMAL RESPONSES TO DISTURBANCES 31

time-invariant rule that is satis¬ed by the constrained-optimal state-contingent in¬‚ation path

(i.e., the solution to the optimization problem with constraint (2.8)) at all dates t > t0 . One

example of such a speci¬cation would be

∞

µ’j’1 Et0 ut0 +j

’1

πt0 = β

¯ 2

j=0

∞ ∞

k’1 ’j’1

’1

’β (1 ’ µ1 ) µ1 µ2 Et0 ’k ut0 +j’k , (2.9)

k=1 j=0

as a consequence of the following result.

Proposition 7.6. The state-contingent evolution of in¬‚ation {πt } from some date t0

onward that minimizes the expected value of (1.2), taking as given the economy™s evolution

prior to date t0 and subject to the constraint (2.8), where πt0 is given by (2.9), is given by

¯

∞

µ’j’1 Et ut+j

’1

πt = β 2

j=0

∞ ∞

’j’1

’1

µk’1 µ2 Et’k ut+j’k

’β (1 ’ µ1 ) (2.10)

1

k=1 j=0

for all t ≥ t0 .

The proof is given in the appendix. Note that in the constrained-optimal evolution, in¬‚ation

in every period after t0 is chosen to be the same time-invariant function of past disturbances

as it has been constrained to be in period t0 . This makes the constraint self-consistent in the

desired sense.

One can easily show that (2.9) represents the only speci¬cation of πt0 as a function of the

¯

history of exogenous disturbances (only) that is self-consistent in this sense. For regardless

of the speci¬cation of πt0 , the constrained-optimal evolution from t0 onward must satisfy

¯

(2.7), for some choice of •t0 ’1 . Solving for πt as a function of •t0 ’1 and the history of

disturbances between dates t0 and t, and then taking the limit as t0 ’ ’∞ for some ¬xed

date t, one obtains (2.10), regardless of the assumed value of φt0 ’1 .19 Hence regardless of

19

The asymptotic independence of the initial condition •t0 ’1 results from the fact that |µ1 | < 1.

32 CHAPTER 7. COMMITMENT TO A POLICY RULE

the speci¬cation of πt0 , the dependence of in¬‚ation upon the history of disturbances must

¯

eventually be of the form (2.10). The only self-consistent speci¬cation of πt0 as a function

¯

of the history of disturbances must then be of that same form.

This does not mean, however, that in order for a policy to be optimal from a timeless

perspective, it must result in the state-contingent evolution of in¬‚ation indicated by (2.10).

For one can also ¬nd self-consistent speci¬cations of the initial in¬‚ation constraint that

involve predetermined endogenous variables. As a simple example, the speci¬cation

∞

» ’j’1

’1

πt0 = (1 ’ µ1 ) xt0 ’1 + β

¯ µ2 Et0 ut0 +j (2.11)

κ j=0

also results in a self-consistent constraint.

Proposition 7.7. Consider the same optimization problem as in Proposition 7.6, but

with πt0 given by (2.11). Then in the constrained-optimal state-contingent evolution of

¯

in¬‚ation and output, the in¬‚ation rate satis¬es

∞

» ’j’1

’1

πt = (1 ’ µ1 ) xt’1 + β µ2 Et ut+j (2.12)

κ j=0

at each date t ≥ t0 .

Again, the proof is in the appendix. Hence a time-invariant policy rule that results in a

determinate equilibrium in which the in¬‚ation rate always satis¬es (2.12) will be optimal

from a timeless perspective.

Yet another self-consistent constraint would be

∞

µ’j’1 Et0 ut0 +j ,

’1

πt0 = ’(1 ’ µ1 )(pt0 ’1 ’ p) + β

¯ ¯ (2.13)

2

j=0

where pt ≡ log Pt , and p is an arbitrary constant, that has the interpretation of a target price

¯

level. (It is the long-run average log price level in the equilibrium that is optimal subject

to this constraint, as discussed below.) In this case the initial constraint is perhaps most

intuitively expressed as a constraint on the initial log price level, pt0 = pt0 , where

¯

∞

’j’1

’1

pt0 = (1 ’ µ1 )¯ + µ1 pt0 ’1 + β

¯ p µ2 Et0 ut0 +j . (2.14)

j=0

2. OPTIMAL RESPONSES TO DISTURBANCES 33

For any choice of p, this is a self-consistent constraint, owing to the following result.

¯

Proposition 7.8. Consider the same optimization problem as in Proposition 7.6, but

with πt0 given by (2.13), or equivalently with a constraint on the initial price level given by

¯

(2.14). Then in the constrained-optimal state-contingent evolution of in¬‚ation and output,

the equilibrium price level each period satis¬es

∞

’j’1

’1

pt = (1 ’ µ1 )¯ + µ1 pt’1 + β

¯ p µ2 Et ut+j . (2.15)

j=0

Again, the proof is in the appendix. Note that this de¬nes an entire one-parameter family

of self-consistent constraints. For one particular choice of p, namely

¯

»

p = pt0 ’1 + xt0 ’1 ,

¯

κ

this constraint is identical to (2.11), and the implied state-contingent evolutions are the

same. But any other value for p would lead to a self-consistent constraint as well.

¯

We see that the state-contingent evolution of in¬‚ation under a policy that is optimal

from a timeless perspective is not uniquely determined. However, these alternative processes

for in¬‚ation and output di¬er only in transitory, deterministic components of the solutions

for in¬‚ation and the output gap; they agree both as to the long-run average values of both

in¬‚ation and the output gap, and as to the response of both variables to unexpected shocks

in any period from t0 onward.

Proposition 7.9. Consider again an economy of the kind assumed in Proposition 7.5.

Then in the case of any small enough ||ξ|| and any initial values of predetermined endogenous

variables that are close enough to the values associated with the zero-in¬‚ation steady state,

the long-run average values of in¬‚ation and the output gap satisfy

lim Et πT = 0,

T ’∞

lim Et xT = 0

T ’∞

34 CHAPTER 7. COMMITMENT TO A POLICY RULE

under the t0 ’optimal policy that would be chosen at any date t0 , and the same is true of

the equilibrium implemented by any policy that is optimal from a timeless perspective.

Furthermore, let the unexpected change in the forecast of any variable yt+m at date t be

denoted

It [yt+m ] ≡ Et yt+m ’ Et’1 yt+m .

Then the e¬ects of unanticipated shocks at any date t on the expected paths of in¬‚ation and

output are given by

∞

µ’j’1 It [ut+j ]

’1

It [πt+m ] = β 2

j=0

m ∞

’j’1

’1

µk’1 µ2 It [ut+m’k+j ],

’β (1 ’ µ1 ) (2.16)

1

k=1 j=0

m ∞

µk µ’j’1 It [ut+m’k+j ],

’1

It [xt+m ] = ’β (2.17)