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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 ” 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]

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