operating target as a function of data available to the central bank at the time that the numerical value of

this target must be chosen. We do not accept this stricture here, because of our interest in the design of

robustly optimal rules.

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 69

certain that the central bank is precisely following the announced policy rule ” it has the

important advantage of making possible a commitment to a rule that is optimal under a much

broader range of circumstances, as discussed by Svensson and Woodford (1999) and Giannoni

and Woodford (2002a). This can be usefully illustrated through a further consideration of

optimal policy in the case treated in sections 2.1 and 4.

5.1 Robustly Optimal Target Criteria

Let us recall the characterization of timelessly optimal policy given in section 2.1. For

example, in the case to which Proposition 7.7 applies, the state-contingent evolution of

in¬‚ation given by equation (2.12) is obtained by solving a system of equations consisting of

the ¬rst-order conditions (1.7) “ (1.8) together with the structural equation (2.1), under the

initial condition (2.11). Note furthermore that conditions (1.7) “ (1.8) must be satis¬ed in

the equilibrium associated with a timelessly optimal policy for all periods t ≥ t0 (for some

value of the initial Lagrange multiplier •t0 ’1 ), regardless of the form of the initial constraint

(2.8).

If we use equation (1.8) to substitute for the Lagrange multiplier •t in (1.7), we obtain

the relation

»

πt + (xt ’ xt’1 ) = 0 (5.1)

κ

that must hold for each t ≥ t0 + 1 under any timelessly optimal policy. We cannot show in

the same way that the relation must hold at date t = t0 , for equation (1.8) need not hold for

dates t < t0 , and so cannot be used to eliminate •t0 ’1 . Nonetheless, the fact that (5.1) must

be satis¬ed for all dates t ≥ t0 + 1 under any timelessly optimal policy (and indeed, under

a t0 ’optimal commitment as well), makes a commitment to ensure that relation (5.1) holds

at all times a reasonable candidate for a timelessly optimal policy rule. The following result

shows that this guess is correct.

Proposition 7.16. Consider again the problem of choosing monetary policy from date

t0 onward so as to minimize the expected value of (1.2), where the joint evolution of in¬‚a-

70 CHAPTER 7. COMMITMENT TO A POLICY RULE

tion and output must satisfy (2.1) for each date t ≥ t0 . Let {ut } be a bounded exogenous

disturbance process, the statistical character of which is otherwise unspeci¬ed. Then if the

central bank commits itself to a policy that ensures that (5.1) will be satis¬ed at each date

t ≥ t0 , there are unique bounded rational-expectations equilibrium processes {πt , xt } for

dates t ≥ t0 consistent with this policy rule. Furthermore, the equilibrium determined by

this policy commitment is the same as the one characterized in Proposition 7.7. Thus the

proposed policy rule is optimal from a timeless perspective.

This result is proved in the appendix. Note that the result that the system of equations

consisting of (2.1) and (5.1) for each t ≥ t0 has a determinate rational-expectations solution

is important for two reasons. First, the existence of a solution is important, in order for

the proposed targeting rule to be feasible; Proposition 7.16 implies that there are in fact

equilibrium paths for in¬‚ation and output that would satisfy the target criterion at all times.

(We discuss the instrument settings required to implement such a rule in the next section.)

Second, the uniqueness of the solution implies that this policy rule, unlike the proposed rule

discussed in section 4, is not only consistent with an optimal equilibrium, but is furthermore

consistent with no other equilibria of a less desirable character. Hence a commitment to the

targeting rule can be said to implement the desired equilibrium, in a way that a commitment

to the associated state-contingent interest-rate path does not.

The determinacy result announced in the proposition is established directly in the ap-

pendix, through a consideration of the equation system consisting of (2.1) and (5.1). How-

ever, the result has a simple intuition; it is a consequence of the existence of a unique

bounded solution to the system of equations consisting of (1.7) “ (1.8) together with (2.1),

that characterize the t0 ’optimal plan. For equation (5.1) is equivalent to conditions (1.7) “

(1.8) plus the stipulation of an initial Lagrange multiplier38

»

(xt0 ’1 ’ x— ).

•t0 ’1 = (5.2)

κ

38

Note that (5.2) is just what the multiplier would have to have been equal to if the ¬rst-order condition

(1.8) also held at date t = t0 ’ 1, as it would in the case of an optimal commitment chosen at date t0 ’ 1 or

earlier.

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 71

However, the initial condition (5.2) is irrelevant for the question whether a determinate so-

lution exists; if a determinate solution exists in the case of the initial condition (2.5), then it

also should exist in the case of the initial condition (5.2). Hence a commitment to achieve-

ment of the target criterion (5.1) implies a determinate rational-expectations equilibrium.

We thus have an example of a policy rule that results in a determinate equilibrium that is

optimal from a timeless perspective. Moreover, the proposed rule is also robustly optimal in

the sense of Giannoni and Woodford (2002a). For Proposition 7.16 relies on no assumptions

about the nature of the exogenous disturbance process {ut }, except that it is bounded (as is

generally necessary in order for bounded equilibrium paths for in¬‚ation and the output gap

to be possible under any monetary policy, and hence for our approximate characterizations

of the equilibrium conditions and of welfare to be valid) and that its e¬ect on the aggregate-

supply relation (2.1) is additive (in a log-linear approximation). The ¬rst-order conditions

from which the target criterion (5.1) is derived are independent of any assumptions about the

statistical properties of the disturbances, and so the optimal policy rule obtained in this way

is optimal regardless of their character. Hence a commitment to bring about the optimality

condition (5.1) is equally sensible regardless of the particular type of shocks that the central

bank may believe to have most recently disturbed the economy.

The robustly optimal rule is an example of a ¬‚exible in¬‚ation targeting rule, in the sense

discussed by Svensson (1999, 20xx).39 The central bank commits itself to adjust the level of

nominal interest rates so that the projected in¬‚ation rate is consistent at all times with the

target criterion. However, the acceptable in¬‚ation rate depends on the projected path of the

output gap. An in¬‚ation rate higher than the long-run target rate (here, zero) is acceptable

if the output gap is projected to decline, and a lower in¬‚ation rate could be achieved only

by reducing the output gap even more sharply, making the left-hand side of (5.1) negative;

similarly, an in¬‚ation rate lower than the long-run target rate should be sought if even this

rate of in¬‚ation requires a growing output gap, so that any higher current in¬‚ation rate

39

To be more precise, the type of rule that we consider here corresponds to what Svensson calls a “speci¬c”

targeting rule.

72 CHAPTER 7. COMMITMENT TO A POLICY RULE

would be possible only with a positive value for the left-hand side of (5.1).

The optimal target criterion (5.1) di¬ers, however, from a common conception of “¬‚exible

in¬‚ation targeting” in that it is the projected rate of change of the output gap, rather than

the absolute level of the output gap, that should determine the acceptable deviation from

the long-run in¬‚ation target. This might seem paradoxical, in that it is the absolute level of

the output gap, rather than its rate of change, that one wishes to stabilize. But this is simply

a re¬‚ection, once again, of the fact that optimal policy is not purely forward-looking. The

target criterion is history-dependent in the sense that acceptable projections (πt , xt ) depend

on the value of the lagged output gap xt’1 , even though the lagged output gap is irrelevant

both to the determination of current and future in¬‚ation and output gap and to the welfare

evaluation of alternative possible paths for those variables from the present time onward.

But this sort of history-dependence is exactly what is necessary in order for the targeting

rule to bring about the kind of dynamic responses to a cost-push shock shown in Figure 7.3.

Once an adverse cost-push shock has caused a negative output gap, the history-dependent

target criterion requires that the output gap be restored only gradually to its long-run level,

and that in¬‚ation be kept below its long-run level during the period in which the output

gap is catching up to its long-run level. This kind of dynamic response to a transitory cost-

push shock implies that price-level increases due to the cost-push shock will subsequently

be undone, and as discussed above, the anticipation that this will be the case restrains price

increases at the time of the shock, reducing the extent to which either in¬‚ation or a negative

output gap are necessary at that time.

The optimal target criterion (5.1) also di¬ers from the kind of target criteria typically used

by in¬‚ation-targeting central banks to justify their policy settings ” such as the criterion

used by the Bank of England, mentioned above ” in that it speci¬es an acceptable near-

term in¬‚ation rate (that is allowed to vary, under stated conditions) rather than a medium-

term in¬‚ation objective (that should remain always the same, despite short-term in¬‚ation

variability). It is important to note that this criterion does incorporate a long-run in¬‚ation

target (namely, zero); because the change in the output gap from one quarter to the next must

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 73

be zero on average, ensuring that (5.1) holds each quarter will require a zero in¬‚ation rate

on average. Nonetheless, the rule requires the central bank to justify its instrument setting

in each decision cycle by reference to whether it is projected to result in an acceptable near-

term in¬‚ation rate, and not simply by reference to whether policy continues to be consistent

with a projection that in¬‚ation should eventually approach the long-term target value.

A ¬‚exible (but speci¬c!) target criterion of this kind provides a clearer guide to short-run

policy decisions ” that is, to the only kind of decisions that a central bank is actually called

upon to make ” than does a mere speci¬cation of the long-run in¬‚ation target. After all, the

interest-rate decision made at any point in time is of little import for the expected long-run

in¬‚ation rate, which should depend entirely on how policy is expected to be conducted in

the future. A commitment to return in¬‚ation to its long-run target rate by a speci¬ed (not

too distant) horizon may have less trivial implications for current policy, but a horizon that

is short enough for such a commitment to determine current policy is likely to result in too

rigid a criterion for such a commitment to be desirable. Hence the desirability of a target

criterion that speci¬es the conditions under which near-term deviations from the long-run

target are justi¬able, rather than merely specifying the long-run target.

Actual in¬‚ation-targeting central banks have probably avoided the articulation of a ¬‚ex-

ible nearer-term target criterion of this kind out of skepticism about whether it is possible

to specify in advance all of the conditions under which a given degree of temporary depar-

ture from the long-run in¬‚ation target should be justi¬able. But we have seen that it is

possible to derive a robustly optimal target criterion, that correctly determines whether a

given degree of departure of projected near-term in¬‚ation from the long-run target rate is

consistent with the optimal state-contingent in¬‚ation path, regardless of the size and nature

of the disturbances that have most recently a¬ected the economy. If the soundness of such a

criterion is accepted, then it ought to be possible for a central bank to commit itself to the

conduct of policy in accordance with a nearer-term target criterion of this kind.

While (5.1) represents an example of a robustly optimal target criterion, it is not the

only possible criterion with that property. Note that satisfaction of (5.1) each period implies

74 CHAPTER 7. COMMITMENT TO A POLICY RULE

that the quantity pt + (»/κ)xt never changes (since the left-hand side of (5.1) is just the

¬rst di¬erence of this quantity). Hence in any equilibrium that is optimal from a timeless

perspective, there exists some value of p such that

¯

»

p t + xt = p

¯ (5.3)

κ

at all times. This suggests an alternative policy rule, namely, that the central bank commit

to ensure that (5.3) holds each period. This too can be shown to be a robustly optimal

targeting rule.

Proposition 7.17. Under the same assumptions as in Proposition 7.16, suppose that

the central bank commits itself to a policy that ensures that (5.3) will be satis¬ed at each date

t ≥ t0 . Then there are unique bounded rational-expectations equilibrium processes {πt , xt }

for dates t ≥ t0 consistent with this policy rule. Furthermore, the equilibrium determined

by this policy commitment is the same as the one characterized in Proposition 7.8. Thus

the proposed policy rule is optimal from a timeless perspective.

The proof is in the appendix. This type of rule corresponds to the “¬‚exible price-level

target” advocated by Hall (1984).40 Note that our analysis provides a theoretical ground for

choosing a particular coe¬cient on the output gap in such a rule. As shown in chapter 6,

in the welfare-theoretic loss function, » = κ/θ, where θ > 1 is the elasticity of substitution

among alternative goods, and the elasticity of demand faced by each of the monopolistically

competitive suppliers. It then follows that the optimal ¬‚exible price-level targeting rule

stabilizes the value of pt + θ’1 xt . Since a reasonable calibration of θ must be much larger

than one (a valid on the order of 10 is most commonly assumed, in order for the model not

to imply an implausible degree of market power), this implies that the weight on the output

40

The observation that timelessly optimal policies bring about equilibria consistent with a rule of this kind

explains our comment, in section xx of chapter 6, that the e¬cient frontier in Figure 6.xx corresponds to

¬‚exible price-level targets with alternative weights on the output gap. The e¬cient frontier is constructed by

computing the optimal state-contingent evolution of in¬‚ation and of the output gap in the case of alternative

values of » ranging between 0 and +∞. For each value of », the optimal policy is a member of the family

(5.3), but with a di¬erent weight on the output gap in each case.

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 75

gap should be only a small fraction of the weight on the price level.

The targeting rule (5.3) is closely related to (5.1); indeed, committing to (5.1) from some

date t0 onward is equivalent to committing to a rule of the form (5.3) with a particular choice

of the price-level target, namely

»

p = pt0 ’1 + xt0 ’1 .

¯

κ

However, the choice of p is arbitrary, if one wishes only to ensure that the rule chosen is

¯

optimal from a timeless perspective. Di¬erent rules in the family lead to the same long-run

average in¬‚ation rate (though di¬erent long-run average price levels), the same long-run

average output gap, and the same equilibrium responses to shocks. The associated equilibria

di¬er only in a transitory, deterministic component, as to which we can make no choice from

a timeless perspective.

5.2 Implementation of a Targeting Rule

The results in the previous section (Propositions 7.16 and 7.17) describe the state-contingent

evolution of in¬‚ation and output that should result, in a rational-expectations equilibrium,

if the central bank succeeds in ensuring that the target criterion is satis¬ed at all times.

But can a central bank actually ensure this, and hence bring about such an equilibrium?

It is appropriate to discuss further what sort of adjustment of its interest-rate instrument

this would involve. This means describing the conduct of policy in accordance with such

a rule in terms of the associated reaction function for the nominal interest rate that is the

policy instrument. We can then consider whether such a reaction function determines a

unique (or at least a unique non-explosive) rational-expectations equilibrium, and whether

this equilibrium should be learnable, along the lines of our analysis of the determinacy and

learnability of equilibrium under simple interest-rate rules in chapter 4.

Let us ¬rst consider the implementation of the targeting rule (5.1). The policy rule

speci¬es that it should be set in period t in such a way that the central bank projects values of

πt and xt consistent with equation (5.1). The interest-rate decision that this implies depends

76 CHAPTER 7. COMMITMENT TO A POLICY RULE

on the way in which the central bank constructs the projections for current in¬‚ation and

output conditional on alternative interest-rate decisions.

We shall suppose that the central bank™s model of the e¬ects of alternative policies is

the correct one, i.e., that it consists of equations (2.1) and (2.23), together with a correct

understanding of the laws of motion of the exogenous disturbance processes. But this in

itself does not answer the question of what the central bank™s projection conditional on

its interest-rate decision should be, for according to the equations of the structural model,

current in¬‚ation and output, given the current nominal interest rate, depend on expectations

regarding the economy™s future evolution. One way of specifying these expectations would

be for the central bank to assume that the private sector expects the economy to evolve in

the future (i.e., in period t + 1 and later) according to the rational-expectations equilibrium

described in Proposition 7.16. That is, it assumes that the private sector expects it to succeed

in enforcing the target criterion in all periods from t + 1 onward, even though, for purposes

of constructing the conditional projection, the central bank contemplates the consequences

of deviation from the interest rate consistent with the policy rule in period t.

This means that the central bank expects the private sector to expect that in period t+1,

in¬‚ation and the output gap will be given by

∞

» 1 ’j’1

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

κ β j=0

∞

κ

µ’j’1 Et+1 ut+j+1 .

xt+1 = mu1 xt +

β» j=0 2

Taking the expectations of these two expressions conditional upon period t information, one

obtains solutions for Et πt+1 and Et xt+1 as linear functions of xt and terms of the form Et ut+j .

Substituting these solutions into the structural relations (2.1) and (2.23), one can then solve

n

those two relations for πt and xt as linear functions of it , rt , and terms of the form Et ut+j .

The bank™s projection of the current-period value of its target πt + (»/κ)xt , conditional on

its current instrument choice it and given its information about the exogenous disturbances,

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 77

would then equal

κ2 + » ∞ ’j

’σ 1 1

n

(it ’ rt ) + ut ’ µ Et ut+j .

β(1 ’ µ1 ) » j=1 2

»’1 κ ’ σ µ1 (1 ’ µ1 )

If the central bank equates this expression to (»/κ)xt’1 , as required in order for the

projection to satisfy (5.1), it obtains a relation that can be solved for it , yielding

±

(1 ’ µ )κ

∞

2

»σ κ κ 1 ’ µ1

1

µ’j Et ut+j ’

n

it = rt + µ1 1 ’ ut ’ 1+ xt’1 . (5.4)

2

κ »σ β»σ » σ

j=1

This is what Evans and Honkapohja (2002) call the “fundamentals-based reaction function”

for implementation of the target criterion (5.1);41 it gives a formula for the central bank™s

operating target purely in terms of exogenous and predetermined variables.

However, while this relation is consistent with the desired state-contingent evolution of

the interest rate and other variables, a commitment to set interest rates in this way does

not necessarily imply a determinate equilibrium; Evans and Honkapohja show that for many

parameter values, it does not.42 For while the right-hand side of (5.4) does not depend solely

on exogenous variables, all dependence on either current or expected future endogenous

variables has been eliminated, by substituting the values that these variables are expected

to take in the desired equilibrium. But a central bank that commits itself to act as if the

desired equilibrium is being realized regardless of whether this is observed to be the case

does not act su¬ciently decisively to ensure that this equilibrium is realized, rather than

some other one which is less desirable.

An alternative approach, recommended by Evans and Honkapohja, is for the central bank

not to substitute out for what the expectations Et πt+1 and Et xt+1 ought to be, given the

economy™s current state and the laws of motion that obtain in the desired equilibrium, but

41

They de¬ne this reaction function, like (xxx) below, only for the case of disturbance processes of the

special forms (2.18) and (2.27); but the logic of their derivation is the one given here. Note that the

determinacy of equilibrium when the central bank commits itself to a reaction function of this form does not

depend on the statistical properties of the exogenous disturbance processes; only the coe¬cient with which

the lagged endogenous variable xt’1 enters matters for that.

42

Svensson and Woodford (1999) reach a similar conclusion in the case of a closely related reaction function

in the case of a model in which the endogenous components of both in¬‚ation and output are predetermined

a period in advance.

78 CHAPTER 7. COMMITMENT TO A POLICY RULE

rather to condition its policy decision on what it actually observes current private-sector

expectations to be. Under this approach, the central bank produces its projections for

current-period in¬‚ation and output by solving the structural equations (2.1) and (2.23) for

πt and xt as functions of it , period t expectations, and the exogenous disturbances. In this

case, the bank™s projection for πt + (»/κ)xt conditional on its current instrument choice is

given by

κ2 + » κ2 + » κ2 + »

n

’σ (it ’ rt ) + ut + β + σ Et πt+1 + Et xt+1 .

κ κ κ

Equating this to (»/κ)xt’1 and solving for it , one obtains the alternative reaction function

κ βκ 1 »

n

it = rt + ut + 1 + Et πt+1 + Et xt+1 ’ xt’1 . (5.5)

σ(κ2 + ») σ(κ2 + ») σ(κ2 + »)

σ

Evans and Honkapohja call this an “expectations-based reaction function” intended to im-

plement (5.1).

If the central bank can commit itself to set interest rates in accordance with this reaction

function at all times, then rational-expectations equilibrium is necessarily determinate, due

to the following result.

Proposition 7.18. Consider an economy in which in¬‚ation and output are determined

n

by structural relations of the form (2.1) and (2.23), where the exogenous disturbances {ut , rt }

are bounded processes but otherwise unrestricted, and suppose that the central bank sets

its nominal interest-rate instrument in accordance with (5.5) in each period t ≥ t0 . Then

there is a determinate rational-expectations equilibrium evolution for in¬‚ation, output and

the nominal interest rate in periods t ≥ t0 , and the state-contingent paths of in¬‚ation and

output are the ones characterized in Proposition 7.7. Hence such a policy is optimal from a

timeless perspective.

The proof of this result is simple. Equation (5.5), together with (2.1) and (2.23), implies

that (5.1) must hold in each period t ≥ t0 . (This just reverses the steps in the derivation of

(5.5) sketched above.) But the system consisting of equations (2.1) and (5.1) has a unique

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 79

bounded solution for in¬‚ation and output, from Proposition 7.16. Equation (2.23) can then

be solved for the associated bounded solution for the path of the nominal interest rate.

Furthermore, we know from Proposition 7.16 that the equilibrium determined in this way is

the one characterized in Proposition 7.7.

We thus see that a commitment to achieving the ¬‚exible in¬‚ation target (5.1) is actually

an equivalent policy to one that results from commitment to an interest-rate rule of the form

(5.5).

[MORE TO BE ADDED]