<< стр. 3(всего 3)СОДЕРЖАНИЕ
and proposed that one ought instead to consider only candidate rules that explicitly specify the interest-rate
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 вЂ” 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
inп¬‚ation given by equation (2.12) is obtained by solving a system of equations consisting of
the п¬Ѓrst-order conditions (1.7) вЂ“ (1.8) together with the structural equation (2.1), under the
initial condition (2.11). Note furthermore that conditions (1.7) вЂ“ (1.8) must be satisп¬Ѓed in
the equilibrium associated with a timelessly optimal policy for all periods t в‰Ґ t0 (for some
value of the initial Lagrange multiplier П•t0 в€’1 ), regardless of the form of the initial constraint
(2.8).
If we use equation (1.8) to substitute for the Lagrange multiplier П•t in (1.7), we obtain
the relation
О»
ПЂt + (xt в€’ xtв€’1 ) = 0 (5.1)
Оє
that must hold for each t в‰Ґ 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 П•t0 в€’1 . Nonetheless, the fact that (5.1) must
be satisп¬Ѓed for all dates t в‰Ґ t0 + 1 under any timelessly optimal policy (and indeed, under
a t0 в€’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 inп¬‚a-
70 CHAPTER 7. COMMITMENT TO A POLICY RULE

tion and output must satisfy (2.1) for each date t в‰Ґ t0 . Let {ut } be a bounded exogenous
disturbance process, the statistical character of which is otherwise unspeciп¬Ѓed. Then if the
central bank commits itself to a policy that ensures that (5.1) will be satisп¬Ѓed at each date
t в‰Ґ t0 , there are unique bounded rational-expectations equilibrium processes {ПЂt , xt } for
dates t в‰Ґ 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 в‰Ґ 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 inп¬‚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) вЂ“ (1.8) together with (2.1),
that characterize the t0 в€’optimal plan. For equation (5.1) is equivalent to conditions (1.7) вЂ“
(1.8) plus the stipulation of an initial Lagrange multiplier38
О»
(xt0 в€’1 в€’ xв€— ).
П•t0 в€’1 = (5.2)
Оє
38
Note that (5.2) is just what the multiplier would have to have been equal to if the п¬Ѓrst-order condition
(1.8) also held at date t = t0 в€’ 1, as it would in the case of an optimal commitment chosen at date t0 в€’ 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 inп¬‚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 eп¬Ђect on the aggregate-
supply relation (2.1) is additive (in a log-linear approximation). The п¬Ѓrst-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 п¬‚exible inп¬‚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 inп¬‚ation rate is consistent at all times with the
target criterion. However, the acceptable inп¬‚ation rate depends on the projected path of the
output gap. An inп¬‚ation rate higher than the long-run target rate (here, zero) is acceptable
if the output gap is projected to decline, and a lower inп¬‚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 inп¬‚ation rate lower than the long-run target rate should be sought if even this
rate of inп¬‚ation requires a growing output gap, so that any higher current inп¬‚ation rate

39
To be more precise, the type of rule that we consider here corresponds to what Svensson calls a вЂњspeciп¬ЃcвЂќ
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) diп¬Ђers, however, from a common conception of вЂњп¬‚exible
inп¬‚ation targetingвЂќ 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 inп¬‚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 reп¬‚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 (ПЂt , xt ) depend
on the value of the lagged output gap xtв€’1 , even though the lagged output gap is irrelevant
both to the determination of current and future inп¬‚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 inп¬‚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 inп¬‚ation or a negative
output gap are necessary at that time.

The optimal target criterion (5.1) also diп¬Ђers from the kind of target criteria typically used
by inп¬‚ation-targeting central banks to justify their policy settings вЂ” such as the criterion
used by the Bank of England, mentioned above вЂ” in that it speciп¬Ѓes an acceptable near-
term inп¬‚ation rate (that is allowed to vary, under stated conditions) rather than a medium-
term inп¬‚ation objective (that should remain always the same, despite short-term inп¬‚ation
variability). It is important to note that this criterion does incorporate a long-run inп¬‚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 inп¬‚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 inп¬‚ation rate, and not simply by reference to whether policy continues to be consistent
with a projection that inп¬‚ation should eventually approach the long-term target value.

A п¬‚exible (but speciп¬Ѓc!) target criterion of this kind provides a clearer guide to short-run
policy decisions вЂ” that is, to the only kind of decisions that a central bank is actually called
upon to make вЂ” than does a mere speciп¬Ѓcation of the long-run inп¬‚ation target. After all, the
interest-rate decision made at any point in time is of little import for the expected long-run
inп¬‚ation rate, which should depend entirely on how policy is expected to be conducted in
the future. A commitment to return inп¬‚ation to its long-run target rate by a speciп¬Ѓed (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 speciп¬Ѓes the conditions under which near-term deviations from the long-run
target are justiп¬Ѓable, rather than merely specifying the long-run target.

Actual inп¬‚ation-targeting central banks have probably avoided the articulation of a п¬‚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 inп¬‚ation target should be justiп¬Ѓable. 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 inп¬‚ation from the long-run target rate is
consistent with the optimal state-contingent inп¬‚ation path, regardless of the size and nature
of the disturbances that have most recently aп¬Ђected 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 + (О»/Оє)xt never changes (since the left-hand side of (5.1) is just the
п¬Ѓrst diп¬Ђerence of this quantity). Hence in any equilibrium that is optimal from a timeless
perspective, there exists some value of p such that
ВЇ

О»
p t + xt = p
ВЇ (5.3)
Оє

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 satisп¬Ѓed at each date
t в‰Ґ t0 . Then there are unique bounded rational-expectations equilibrium processes {ПЂt , xt }
for dates t в‰Ґ 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 вЂњп¬‚exible price-level
targetвЂќ advocated by Hall (1984).40 Note that our analysis provides a theoretical ground for
choosing a particular coeп¬ѓcient on the output gap in such a rule. As shown in chapter 6,
in the welfare-theoretic loss function, О» = Оє/Оё, where Оё > 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 п¬‚exible price-level targeting rule
stabilizes the value of pt + Оёв€’1 xt . Since a reasonable calibration of Оё 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 eп¬ѓcient frontier in Figure 6.xx corresponds to
п¬‚exible price-level targets with alternative weights on the output gap. The eп¬ѓcient frontier is constructed by
computing the optimal state-contingent evolution of inп¬‚ation and of the output gap in the case of alternative
values of О» ranging between 0 and +в€ћ. For each value of О», the optimal policy is a member of the family
(5.3), but with a diп¬Ђerent 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

О»
p = pt0 в€’1 + xt0 в€’1 .
ВЇ
Оє

However, the choice of p is arbitrary, if one wishes only to ensure that the rule chosen is
ВЇ
optimal from a timeless perspective. Diп¬Ђerent rules in the family lead to the same long-run
average inп¬‚ation rate (though diп¬Ђerent long-run average price levels), the same long-run
average output gap, and the same equilibrium responses to shocks. The associated equilibria
diп¬Ђer 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 inп¬‚ation and output that should result, in a rational-expectations equilibrium,
if the central bank succeeds in ensuring that the target criterion is satisп¬Ѓed 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 п¬Ѓrst consider the implementation of the targeting rule (5.1). The policy rule
speciп¬Ѓes that it should be set in period t in such a way that the central bank projects values of
ПЂt 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 inп¬‚ation and
output conditional on alternative interest-rate decisions.

We shall suppose that the central bankвЂ™s model of the eп¬Ђects 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 bankвЂ™s projection conditional on
its interest-rate decision should be, for according to the equations of the structural model,
current inп¬‚ation and output, given the current nominal interest rate, depend on expectations
regarding the economyвЂ™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,
inп¬‚ation and the output gap will be given by

в€ћ
О» 1 в€’jв€’1
ПЂt+1 = (1 в€’ Вµ1 ) xt + Вµ2 Et+1 ut+j+1 ,
Оє ОІ j=0
в€ћ
Оє
Вµв€’jв€’1 Et+1 ut+j+1 .
xt+1 = mu1 xt +
ОІО» j=0 2

Taking the expectations of these two expressions conditional upon period t information, one
obtains solutions for Et ПЂt+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 ПЂt and xt as linear functions of it , rt , and terms of the form Et ut+j .
The bankвЂ™s projection of the current-period value of its target ПЂt + (О»/Оє)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

Оє2 + О» в€ћ в€’j
в€’Пѓ 1 1
n
(it в€’ rt ) + ut в€’ Вµ Et ut+j .
ОІ(1 в€’ Вµ1 ) О» j=1 2
О»в€’1 Оє в€’ Пѓ Вµ1 (1 в€’ Вµ1 )

If the central bank equates this expression to (О»/Оє)xtв€’1 , as required in order for the
projection to satisfy (5.1), it obtains a relation that can be solved for it , yielding
пЈ± пЈј
пЈІ (1 в€’ Вµ )Оє пЈЅ
в€ћ
2
О»Пѓ Оє Оє 1 в€’ Вµ1
1
Вµв€’j Et ut+j в€’
n
it = rt + Вµ1 1 в€’ ut в€’ 1+ xtв€’1 пЈѕ . (5.4)
2
пЈі
Оє О»Пѓ ОІО»Пѓ О» Пѓ
j=1

This is what Evans and Honkapohja (2002) call the вЂњfundamentals-based reaction functionвЂќ
for implementation of the target criterion (5.1);41 it gives a formula for the central bankвЂ™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 suп¬ѓciently 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 ПЂt+1 and Et xt+1 ought to be, given the
economyвЂ™s current state and the laws of motion that obtain in the desired equilibrium, but
41
They deп¬Ѓne 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 coeп¬ѓcient with which
the lagged endogenous variable xtв€’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 inп¬‚ation and output are predetermined
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 inп¬‚ation and output by solving the structural equations (2.1) and (2.23) for
ПЂt and xt as functions of it , period t expectations, and the exogenous disturbances. In this
case, the bankвЂ™s projection for ПЂt + (О»/Оє)xt conditional on its current instrument choice is
given by
Оє2 + О» Оє2 + О» Оє2 + О»
n
в€’Пѓ (it в€’ rt ) + ut + ОІ + Пѓ Et ПЂt+1 + Et xt+1 .
Оє Оє Оє
Equating this to (О»/Оє)xtв€’1 and solving for it , one obtains the alternative reaction function
Оє ОІОє 1 О»
n
it = rt + ut + 1 + Et ПЂt+1 + Et xt+1 в€’ xtв€’1 . (5.5)
Пѓ(Оє2 + О») Пѓ(Оє2 + О») Пѓ(Оє2 + О»)
Пѓ
Evans and Honkapohja call this an вЂњexpectations-based reaction functionвЂќ 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 inп¬‚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 в‰Ґ t0 . Then
there is a determinate rational-expectations equilibrium evolution for inп¬‚ation, output and
the nominal interest rate in periods t в‰Ґ t0 , and the state-contingent paths of inп¬‚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 в‰Ґ 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 inп¬‚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 п¬‚exible inп¬‚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).