k=0 j=0

for each m ≥ 0 under a t0 ’optimal policy chosen at any date t0 ¤ t, and again the same is

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

(In each of these characterizations of the paths of in¬‚ation and the output gap, the results

given are accurate up to an error term of order O(||ξ||2 .)

The proof is in the appendix; essentially, the result follows from the fact that in any of the

cases allowed for, the economy™s state-contingent evolution must satisfy (2.7).

Comparing Proposition 7.9 with Proposition 7.5, we again ¬nd, as in our deterministic

analysis in section 1, that there is an in¬‚ationary bias to discretionary policy, under the

assumption that x— > 0. (The size of the average-in¬‚ation bias is exactly the same as deter-

mined under the deterministic analysis, as a result of the well-known certainty-equivalence

property of both the equilibrium outcome under discretionary optimization and the opti-

mal plan under commitment, in the case of linear-quadratic policy problems of the kind

considered here.20 ) However, there is also a di¬erence in the responses to shocks under an

20

For discussions of certainty equivalence in the context of forward-looking models like the ones discussed

in this chapter, see, e.g., Backus and Dri¬ll (1986), xxxxxx.

2. OPTIMAL RESPONSES TO DISTURBANCES 35

inflation

4

= discretion

= optimal

2

0

’2

0 2 4 6 8 10 12

output

5

0

’5

0 2 4 6 8 10 12

price level

2

1.5

1

0.5

0

0 2 4 6 8 10 12

Figure 7.3: Optimal responses to a transitory cost-push shock compared with equilibrium

responses under discretionary policy.

optimal policy commitment from those that result from discretionary optimization. And

this “stabilization bias” is present even when x— = 0, so that there is no average-in¬‚ation

bias associated with discretionary policy.21

The di¬erence in the equilibrium responses to a “cost-push shock” under discretionary

policy and under an optimal commitment are contrasted in Figure 7.3, in the simple case

that the cost-push shock is purely transitory, and unforecastable before the period in which

it occurs (so that Et ut+j = 0 for all j ≥ 1). Here the assumed values of β, κ, and » are

again as in Figure 7.1, and the shock in period zero is of size u0 = 1, meaning that a one

21

Thus the result of King (1997), according to which discretionary policy achieves an optimal state-

contingent evolution for the economy as long as the output-gap target x— in the central bank™s loss function

is modi¬ed so as to be consistent with its in¬‚ation target, is special to the speci¬c simple model of aggregate

supply that he assumes. Early discussions of “stabilization bias” in the context of other forward-looking

models include xxxxx.

36 CHAPTER 7. COMMITMENT TO A POLICY RULE

percent increase in the general level of prices would be required to prevent any decline in

output relative to the natural rate (or more generally, its target level) from occurring, on

the assumption that prices will then be stabilized at the new higher level. Once again, the

periods represent quarters, and the in¬‚ation rate is plotted as an annualized rate, meaning

that what is plotted is actually 4πt .

Under the discretionary policy characterized in Proposition 7.5, the e¬ect of the dis-

turbance on equilibrium in¬‚ation and the equilibrium output gap last only as long as the

disturbance itself, so that both variables are expected to return to their normal levels by the

following quarter. Under an optimal commitment, instead, monetary policy remains tight

even after the disturbance has dissipated, so that the output gap returns to zero only much

more gradually. As a result of this, while in¬‚ation overshoots its long-run target value at the

time of the shock, it is held below its long-run target value for a time following the shock, so

that the unexpected increase in prices is subsequently undone. In fact, as the bottom panel

shows, under an optimal commitment, the price level eventually returns to exactly the same

path that it would have been expected to follow if the shock had not occurred.

This simple example illustrates a very general feature of optimal policy once one takes

account of forward-looking private-sector behavior: optimal policy is almost always history-

dependent in a way that discretionary policy would not be. Discretionary policy (in a

Markov equilibrium) is purely forward-looking, in the sense that the action chosen at any

date t depends solely upon the set of possible state-contingent paths for the target variables

(here, in¬‚ation and the output gap) from period t onward, given the economy™s state at

that time (the values of predetermined or exogenous variables at date t). In the present

example, the forward-looking aggregate-supply relation (2.1), which implies that the set of

possible equilibrium paths for in¬‚ation and output from period t onward are independent

of all lagged endogenous variables, together with the assumption that the disturbance ut

is completely unforecastable, imply that the only aspect of the economy™s state in period

t that a¬ect the set of possible paths for in¬‚ation and output from that period onward is

the current disturbance ut . Hence under any purely forward-looking decision procedure for

2. OPTIMAL RESPONSES TO DISTURBANCES 37

monetary policy, πt and xt will depend only on the current disturbance ut , and the e¬ects

of a shock on the paths of these variables will be as transitory as the e¬ect on ut itself.

(The same is true of the associated path of nominal interest rates, if the intertemporal IS

equation is of the form (2.23).) Instead, under an optimal policy, both πt and xt will depend

on disturbances in periods prior to t, in the case of any period t > t0 .

Optimal policy is history-dependent (and conventional optimal-control or dynamic-programming

methods lead to suboptimal policy) because the anticipation by the private sector that future

policy will di¬erent as a result of conditions at date t ” even if those conditions no longer

matter for the set of possible paths for the target variables at the later date ” can improve

stabilization outcomes at date t. Suppose that there is a positive cost-push shock at date

t, as illustrated in Figure 7.3. If the transitory disturbance is expected to have no e¬ect on

the conduct of policy in later periods (as under any purely forward-looking policy), then the

short-run tradeo¬ between in¬‚ation and the output gap in period t is shifted vertically by

the amount of the disturbance, ut , requiring the central bank to choose between an increase

in in¬‚ation, a negative output gap, or some of each. If instead the central bank is expected to

pursue a tighter policy in period t + 1 and later as a result of the shock in period t, as occurs

under the optimal policy depicted in the ¬gure, then the short-run tradeo¬ between in¬‚ation

and the output gap is shifted only by the amount of the total change in ut + βEt πt+1 , which

is smaller than the increase in ut . Hence greater stabilization (less of an increase in in¬‚ation,

less of a reduction in output, or both) is possible in period t. (The anticipation of tighter

policy later restrains price increases in period t, so that less contraction of output is needed

in period t to achieve a given degree of moderation of in¬‚ationary pressure.) Of course, to

achieve this bene¬cial shift in expectations in period t, it is necessary that the central bank

be committed to actually tightening policy later. This will mean less successful stabilization

in later periods than would otherwise have been possible; but nonetheless, the discounted

sum of expected stabilization losses can be reduced through some use of this tool.

It may not be obvious from Figure 7.3 that commitment to a history-dependent policy

can simultaneously improve the stabilization of in¬‚ation and of the output gap, since in

38 CHAPTER 7. COMMITMENT TO A POLICY RULE

this numerical example, the overall variability of the output gap is higher under the optimal

policy than under discretionary policy. (This need not have been true, but happens to be

true for the calibrated parameter values used here, which involve quite a low relative weight »

on output-gap stabilization, for reasons discussed in chapter 6.) The point can be illustrated

numerically by computing the in¬‚ation/output variance frontier associated with history-

dependent as opposed to purely forward-looking policies. These two variance frontiers are

shown in Figure 6.6 of the previous chapter, in the case that β and κ are calibrated as in

Figure 7.3, and ut is assumed to be an AR(1) process,

u

ut = ρu ut’1 + t, (2.18)

u

with ρu = 0.8. (Here is an i.i.d. mean-zero random variable.)

t

The variance frontier in the case of general linear policies (allowing arbitrary history-

dependence) is obtained by computing the optimal responses to disturbances, characterized

in Proposition 7.9, for values of » > 0 that are allowed to vary over the entire positive real

half-line. For each equilibrium in this one-parameter family, we compute the statistics V [π]

and V [x], where we again use the discounted measure of variability

∞

β t var(yt )

V [y] ≡ (1 ’ β)

t=0

for any random variable {yt }. Note that for each value of », the date zero-optimal policy

minimizes

∞

β t [πt + »(xt ’ x— )2 ]

2

E =

t=0

∞

β t (E[πt ]2 + »(E[xt ] ’ x— )2 + (1 ’ β)V [π] + (1 ’ β)»V [x].

t=0

Furthermore, the ¬rst term on the right-hand side depends only on the deterministic com-

ponent of policy, while the remaining two terms depend only on the prescribed responses

to shocks. As these two aspects of policy can be independently speci¬ed, the optimal equi-

librium responses to shocks minimize V [π] + »V [x]. It follows that the responses to shocks

2. OPTIMAL RESPONSES TO DISTURBANCES 39

characterized in Proposition 7.9 for alternative values of » describe the policies on the V [π]-

V [x] e¬cient frontier.

The variance frontier for the more restricted class of purely forward-looking (linear) poli-

cies can similarly be obtained by computing the policy from within this class that minimizes

V [π] + »V [x], for each possible value of » > 0. In the present context, this means that we

restrict consideration to state-contingent evolutions in which πt and xt are linear functions

of the current value of ut ,

π t = π + f π ut ,

¯ x t = x + f x ut .

¯ (2.19)

State-contingent evolutions within this family represent possible rational-expectations equi-

libria if and only if the coe¬cients satisfy

(1 ’ β)¯ = κ¯,

π x (2.20)

(1 ’ βρu )fπ = κfx + 1, (2.21)

as a consequence of (2.1). Finally, in the case of processes of the form (2.19),

2 2

V [π] + »V [x] = [fπ + »fx ]var(u).

2 2

Thus we choose the response coe¬cients fπ , fx so as to minimize fπ +»fx subject to constraint

(2.21). The optimal response coe¬cients are easily seen to be

κ»’1

1 ’ βρu

fπ = 2 ’1 > 0, fx = ’ 2 ’1 < 0. (2.22)

κ » + (1 ’ βρu )2 κ » + (1 ’ βρu )2

For each member of this one-parameter family of equilibria, the implied values of V [π]

and V [x] yield a point on the dotted frontier shown in Figure 6.6. (Point B on this frontier

indicates the e¬cient plan in the case of the value » = 0.05 corresponding to the welfare-

theoretic loss function.) We observe that this frontier is entirely inside the one computed

when policy is allowed to be history-dependent; thus commitment to a history-dependent

policy can simultaneously improve the stabilization of both in¬‚ation and the output gap.

A striking feature of the impulse responses under an optimal policy displayed in Figure

7.3 is the fact that following a cost-push shock, the unexpected in¬‚ation caused by the shock

40 CHAPTER 7. COMMITMENT TO A POLICY RULE

is entirely “undone”, so that the price level returns completely to its previously anticipated

path. This result is not special to the case of serially uncorrelated, completely unanticipated

disturbances assumed in the ¬gure.

Proposition 7.10. Under the same assumptions as in Proposition 7.9, under the

t0 ’optimal plan, there exists a well-de¬ned long-run expected price level,

lim Et pT = p∞ ,

T ’∞

that is the same in every period t ≥ t0 , regardless of the history of disturbances between

periods t0 and t. And the same is true under any policy followed from period t0 onward

that is optimal from a timeless perspective. In the particular case of the latter sort that is

optimal subject to the initial constraint (2.13) or (2.14), the long-run expected price level

corresponds to the constant p in the constraint.

¯

This is also shown in the appendix. Note that the long-run price level p∞ generally depends

on initial conditions (including the pre-existing price level pt0 ’1 ) at the time that the optimal

policy is adopted; it is furthermore di¬erent (for given initial conditions) under di¬erent

policies within the set to which the proposition applies. (In the case of initial constraints of

the form (2.13) or (2.14), p∞ is independent of initial conditions at the time that the policy

is adopted, as it is in fact determined solely by the constant p in the initial constraint; but

¯

any long-run price level is possible, under a suitable choice of p.) Nonetheless, under any

¯

policy of the kind speci¬ed in the proposition, the intitial conditions plus the form of optimal

policy adopted determine a long-run price level, which is not subsequently a¬ected by the

realization of any random disturbances. Hence equilibrium ¬‚uctuations in the price level are

stationary.

This feature of optimal policy may seem counter-intuitive. Indeed, it is often argued that

if one wishes to stabilize in¬‚ation, and does not care about the absolute level of prices, then

surprise deviations from the long-run average in¬‚ation rate should not have any e¬ect on

the in¬‚ation rate that policy aims for subsequently: one should “let bygones be bygones,”

2. OPTIMAL RESPONSES TO DISTURBANCES 41

inflation

4

2

0

’2

0 2 4 6 8 10 12

output

5

0

γ=0

γ = 0.5

γ = 0.8

’5

γ = 1.0

0 2 4 6 8 10 12

price level

10

5

0

0 2 4 6 8 10 12

Figure 7.4: Equilibrium responses to a transitory cost-push shock under discretionary policy,

for varying degrees of in¬‚ation inertia.

even though this means allowing the price level to drift to a permanently di¬erent level.

The claim is that “undoing” past deviations from the target rate of in¬‚ation simply creates

additional, unnecessary variability in in¬‚ation. And this would be correct if the commit-

ment to subsequently “undo” target misses had no e¬ect on the probability distribution of

the unexpected deviations from the in¬‚ation target. However, if price-setters are forward-

looking, the anticipation that a current increase in the general price level will predictably be

“undone” soon gives suppliers a reason not to increase their own prices currently as much

as they otherwise would, and so leads to smaller equilibrium deviations from the in¬‚ation

target in the ¬rst place. Hence such a policy can reduce equilibrium in¬‚ation variability, and

not simply the range over which the absolute price level varies.

Nonetheless, the result that optimal policy involves a stationary (or even trend-stationary)

price level depends on fairly special assumptions. For example, in the event of partial index-

42 CHAPTER 7. COMMITMENT TO A POLICY RULE

inflation

4

2

0

’2

0 2 4 6 8 10 12

output

2

0 γ=0

γ = 0.5

’2

γ = 0.8

γ = 1.0

’4

’6

0 2 4 6 8 10 12

price level

2

1.5

1

0.5

0

0 2 4 6 8 10 12

Figure 7.5: Responses to a transitory cost-push shock under optimal policy, again for varying

degrees of in¬‚ation inertia.

ation of prices to a lagged price index, it ceases to be optimal to completely restore the price

level to the path that it would have been expected to follow in the absence of a shock, even

though it continues to be optimal for positive in¬‚ation surprises to be predictably followed by

periods in which in¬‚ation is temporarily below its long-run average value. This is illustrated

in Figures 7.4 and 7.5, which display the impulse responses to the same kind of shock as in

Figure 7.3, but for economies with a range of values for the indexation parameter γ. (The

case γ = 0 corresponds to the basic Calvo pricing model, the case already shown in Figure

7.3.) In Figure 7.4, the responses are shown in the case that the central bank optimizes under

discretion, seeking to minimize the expected value of the loss function (1.22). In Figure 7.5,

the corresponding responses are shown in the case of an optimal policy commitment.

In the case of discretionary policy, we see that a positive cost-push shock results not only

in a burst of in¬‚ation during the period of the disturbance, but additional above-normal

2. OPTIMAL RESPONSES TO DISTURBANCES 43

price increases in subsequent periods, as the rate of in¬‚ation only gradually returns to its

normal level. As before, the perturbation of the target variable πt ’ γπt’1 is as transitory as

the disturbance ut ; but the in¬‚ation inertia resulting from the automatic indexation creates

additional in¬‚ation for several more quarters. In the limiting case of full indexation (γ = 1),

in¬‚ation remains permanently higher as a result of the transitory cost-push shock.

Instead, under optimal policy, the initial unexpected increase in prices is eventually

undone, as long as γ < 1; and this once again means that in¬‚ation eventually undershoots

its long-run level for a time. However, for any large enough value of γ, in¬‚ation remains

greater than its long-run level for a time even after the disturbance has ceased, and only

later undershoots its long-run level; the larger is γ, the longer this period of above-average

in¬‚ation persists. In the limiting case that γ = 1, the undershooting never occurs; in¬‚ation

is simply gradually brought back to the long-run target level.22 In this last case, a temporary

disturbance causes a permanent change in the price level, even under optimal policy. (Under

optimal policy, the price is an integrated process of order 1, while under discretion it is an

integrated process of order 2, since there is a unit root in the in¬‚ation rate.)

Even if there is not full indexation to a lagged price index, the result that the price level

is stationary under optimal policy is relatively fragile, given that welfare does not depend at

all on the range of variation in the absolute level of prices. Under many small perturbations

of the precise model considered here, optimal policy will involve a price-level process with a

unit root; this is true, for example, if there is even a small weight on interest-rate stabilization

in the loss function, as discussed in the next section, or if the zero lower bound on nominal

interest rates ever binds, as discussed in section xx. The more robust conclusion about

optimal policy is not that it is important for the price level to actually be stationary (or

trend-stationary); it is rather that it is desirable for an in¬‚ationary disturbance to be followed

22

The reason for this is easily seen. Optimal policy in the case that γ > 0 is the same as under the

characterization in Proposition 7.9, except that πt must be replaced by the quasi-di¬erenced in¬‚ation rate

πt ’ γπt’1 in each expression. In the case that γ = 1, the optimal evolution of the in¬‚ation rate πt is the

same as the optimal evolution of pt when γ = 0. Thus the impulse response of in¬‚ation (for γ = 1) in panel

1 of Figure 7.5 is the same as the impulse response of the price level (under optimal policy) in panel 3 of

Figure 7.3. The scales are di¬erent because the in¬‚ation rate plotted is an annualized rate, 4πt rather than

πt .

44 CHAPTER 7. COMMITMENT TO A POLICY RULE

by a period of tight monetary policy which keeps output below the natural rate for a time,

the anticipation of which helps to restrain price increases as a result of the disturbance.

2.2 Fluctuations in the Natural Rate of Interest

We turn now to the optimal response to real disturbances that cause temporary ¬‚uctuations

n

in the natural rate of interest rt . In the case that in¬‚ation stabilization and output-gap

stabilization are the only objectives of monetary policy (as in the case of loss function (1.2)),

and the zero interest-rate bound never binds (as assumed thus far), ¬‚uctuations in the

natural rate of interest do not prevent the central bank from completely stabilizing both

in¬‚ation and the output gap; it only e¬ects the kind of nominal interest-rate variations that

are required in order to achieve this. In such a case, optimal policy continues to involve zero

in¬‚ation and a zero output gap at all times, just as in the deterministic analysis in section

1.1. In this special case, there is no di¬erence between discretionary policy and an optimal

policy commitment, as regards the equilibrium responses to disturbances of this kind: in

either case, in¬‚ation and the output gap will not respond at all to the disturbance, while the

nominal interest-rate operating target will perfectly track the current value of the natural

rate of interest.23

However, our conclusion is di¬erent if the central bank is also concerned to minimize the

degree of variability of nominal interest rates, for either of the reasons discussed in section xx

of chapter 6, or perhaps for other reasons as well. In this case, as discussed in the previous

chapter, it is possible to reduce interest-rate variation at the price of increased variability

of in¬‚ation and the output gap, and optimal policy will do this to some extent. And once

simultaneous satisfaction of all of the stabilization objectives ceases to be possible, it is

almost inevitably the case that optimal policy will no longer coincide with discretionary

policy, and indeed that optimal policy will no longer be purely forward-looking.

In particular, there are important advantages to a more inertial adjustment of interest

23

These conclusions are easily obtained from the analysis of discretionary policy and optimal commitment

in the previous section, since it did not matter in that analysis what we assumed about ¬‚uctuations in the

natural rate of interest.

2. OPTIMAL RESPONSES TO DISTURBANCES 45

rates than would occur under a purely forward-looking policy. As discussed in chapter 4,

in an optimizing model, aggregate demand should depend on the expected path of (short-

term) real interest rates far into the future, and not simply on their current level. Hence a

commit to maintain real rates at a moderately high level for a longer period of time can be

as e¬ective a way of preventing a surge in aggregate demand due to real disturbances as a

sharper increase in real rates that is expected to be only temporary; but the former policy has

the advantage of requiring less variable interest rates. Of course, as in the previous section,

conditioning policy on past disturbances rather than current conditions causes distortions;

but the gains from anticipation of such behavior can make it nonetheless worthwhile to

engage in such behavior to an extent. It can therefore be desirable for the central bank to

raise interest rates only gradually in response to an increase in the natural rate of interest,

and similarly to lower them again only gradually once the disturbance has passed ” even

though the type of loss function proposed in chapter 6, section xx, penalizes deviations of

the level of nominal interest rates from the optimal level, and not large rates of change in

the interest rate. This is another example of the history-dependence of optimal policy.

Woodford (1999xx) shows that inertial interest-rate adjustment is optimal in the case of

the basic neo-Wicksellian model presented in chapter 4. If interest-rate variations matter

for welfare, the aggregate-supply relation (2.1) must be augmented by an intertemporal IS

relation

n

xt = Et xt+1 ’ σ[it ’ Et πt+1 ’ rt ], (2.23)

n

generalizing (1.15), where now {rt } is an exogenous disturbance process. We wish to choose

a policy to minimize a social loss function of the form

∞

β t’t0 [πt + »x (xt ’ x— )2 + »i (it ’ i— )2 ].

2

Et0 (2.24)

t=t0

Here i— = im , as in (1.14), if the interest-rate stabilization objective appears solely as a result

of transactions frictions; but we may wish to assign a higher value to i— (and to »i ) in order

to re¬‚ect the need to avoid negative nominal interest rates.

Once again, the model is one in which the possible rational-expectations equilibrium

46 CHAPTER 7. COMMITMENT TO A POLICY RULE

paths of in¬‚ation, the output gap, and the nominal interest from any date t onward depend

n

only on the real disturbances (rt , ut ) at date t, and upon information at date t about the

subsequent evolution of those disturbances. Hence if these disturbances are Markovian, so

that their current values contain all available information about their likely future evolution,

then any purely forward-looking policy will make all three of the target variables vary solely

in response to the current disturbances, and the e¬ects of any disturbance on any of these

variables will be only as persistent as the disturbance itself.

The optimal responses to such disturbances can instead be characterized using the same

Lagrangian method as in the previous section. We again obtain the ¬rst-order conditions

(1.16) “ (1.18) for each t ≥ t0 .24 In order to compute the t0 ’optimal commitment, these

conditions are solved under the initial conditions

•1,t0 ’1 = •2,t0 ’1 = 0. (2.25)

A policy that is optimal from a timeless perspective is instead only required to minimize

(2.24) subject to constraints of the form

π t0 = π t0 ,

¯ x t 0 = xt 0 .

¯ (2.26)

The equilibrium associated with such a policy will solve the same system of ¬rst-order condi-

tions, but with di¬erent initial values for the Lagrange multipliers •1,t0 ’1 , •2,t0 ’1 than those

given in (2.25). The following result, due to Giannoni and Woodford (2002b), guarantees

that these conditions determine a unique bounded solution.

Proposition 7.11. For any parameter values 0 < β < 1, κ, σ, »x , »i > 0, any bounded

n

processes for the exogenous disturbances {rt , ut }, and any speci¬cation of the initial lagged

Lagrange multipliers •1,t0 , •2,t0 , the system of equations consisting of (1.16) “ (1.18), (2.1)

and (2.23) has a unique bounded solution for the paths of the variables {πt , xt , it , •1t , •2t }

for periods t ≥ t0 .

24

In the general case, the coe¬cient im in (1.18) must be replaced by i— .

2. OPTIMAL RESPONSES TO DISTURBANCES 47

(a) Interest rate

1

optimal

0.8 non’inertial

nat. rate

0.6

0.4

0.2

0

0 1 2 3 4 5 6 7 8

(b) Inflation

0.15

0.1

0.05

0

’0.05

’0.1

0 1 2 3 4 5 6 7 8

(c) Output gap

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8

Figure 7.6: Optimal responses to a disturbance to the natural rate of interest.

This is proved in the appendix. It follows that the t0 ’optimal commitment involves bounded

¬‚uctuations in response to the disturbances, characterized by the unique bounded solution

to these equations. And once again, the responses under a timelessly optimal policy are of

the same kind.

The optimal responses to a disturbance to the natural rate of interest characterized by

these equations are history-dependent, as a result of the presence of the lagged Lagrange

n

multipliers in the ¬rst-order conditions. (For example, even in the case that {rt } is Marko-

vian, the period t endogenous variables will not be independent of the history of natural-rate

disturbances prior to period t.) These optimal responses are illustrated by the solid lines in

48 CHAPTER 7. COMMITMENT TO A POLICY RULE

Figure 7.6 for the case of an AR(1) process for the natural rate of interest,

rt = (1 ’ ρr )¯ + ρr rt’1 + r ,

n n

r (2.27)

t

with an autoregressive coe¬cient ρr = 0.35. (The numerical values for β, σ, κ, »x , and »i

used in this example are also again taken from Table 6.1.) The impulse response of the

natural rate of interest itself is shown by the dotted line in the ¬rst panel of the ¬gure.

The equilibrium responses of in¬‚ation, output, and the nominal interest rate under the best

possible purely forward-looking policy (the “optimal non-inertial plan,” characterized in the

next section) are also shown for purposes of comparison.

Under an optimal policy, the responses of these variables are not simple multiples of the

current deviation of the natural rate of interest from its long-run average value. The nominal

interest rate is raised much more gradually in response to an increase in the natural rate than

under the optimal purely forward-looking policy, but the higher interest rates are maintained

longer than the time for which the disturbance itself persists. This more persistent change in

the level of interest rates restrains the initial increase in the output gap to the same extent as

under the forward-looking policy, despite the more modest increase in interest rates; and it

returns the gap to zero (and even undershoots) quickly, as people come to foresee real rates

in the near future that will be even higher than the natural rate. Because the increase in

the output gap is much more transitory (and is even soon reversed), the immediate increase

in in¬‚ation is much smaller under the optimal policy. Yet this improved stabilization of

in¬‚ation and output requires less volatility of nominal interest rates as well.

The way in which nominal interest rates evolve in an optimal equilibrium can be roughly

described as partial adjustment toward a “desired” level that is a function of current and

expected future levels of the natural rate; hence interest-rate inertia of the kind suggested by

the estimated Fed reaction functions discussed in chapter 1 is actually a feature of optimal

policy, rather than an indication of failure to react quickly enough to changing conditions.

The nature of the partial-adjustment dynamics can be seen especially clearly in a limiting

case of this model, in which κ = 0, so that the in¬‚ation rate never varies (or at any rate

2. OPTIMAL RESPONSES TO DISTURBANCES 49

evolves exogenously), regardless of the evolution of the output gap.

In this case, our problem reduces to one of choosing state-contingent paths for {xt , it } to

minimize the discounted sum of losses resulting from output-gap and interest-rate variation,

subject to the constraint that (2.23) be satis¬ed each period (given the exogenous path of

in¬‚ation). The ¬rst-order conditions characterizing optimal policy are then simply (1.17) “

(1.18), with the terms involving •2t omitted. Using the latter of these equations to eliminate

the Lagrange multiplier •1t from the former, we obtain25

σ»x (xt ’ x— ) ’ »i [(it ’ i— ) ’ β ’1 (it’1 ’ i— )] = 0.

Using this condition, in turn, to eliminate xt from (2.23), the latter relation becomes26

it ’ β ’1 it’1 = Et [it+1 ’ β ’1 it ] ’ »’1 »x σ 2 (it ’ rt ).

n

(2.28)

i

This equation in turn has a unique bounded solution of the form

it = µ1 it’1 + (1 ’ µ1 )¯t ,

± (2.29)

n

in the case of any bounded process for the disturbance {rt }, where

∞

’j n

¯t = (1 ’ µ2 )

± µ2 Et rt+j , ) (2.30)

j=0

and 0 < µ1 < 1 < µ2 are the two roots of the characteristic equation

»x σ 2

2 ’1

µ + β ’1 = 0.

µ ’ 1+β +

»i

Equation (2.29) takes the form of partial-adjustment dynamics toward the time-varying

desired level of short-term interest rates ¯t , with the root µ1 determining the speed of ad-

±

justment. The desired level ¯t is a weighted average of current and expected future natural

±

25

In eliminating the lagged multiplier using the same relation, we are in e¬ect choosing a particular time-

invariant way of selecting the initial lagged multiplier •1,t0 ’1 , using a relation between this multiplier and

it0 ’1 that would have been true in the case of any optimal commitment chosen at a date earlier than t0 .

Hence the solution obtained here represents at least one of the possible equilibria from date t0 onward that

are optimal from a timeless perspective. Since the equilibrium responses to natural-rate disturbances from

date t0 onward are the same in all such equilibria, the method used here gives the uniquely appropriate

characterization of the optimal dynamic response to such a disturbance.

26

Here we assume that the limiting case with κ = 0 is one in which prices are so sticky that in¬‚ation is

zero at all times. If instead in¬‚ation varies in response to exogenous “cost-push shocks”, then (2.28) again

n n

holds, but with the exogenous forcing process rt replaced by the exogenous variation in rt + Et πt+1 .

50 CHAPTER 7. COMMITMENT TO A POLICY RULE

rates of interest, with the root µ2 determining how far in the future the expected values of

the natural rate are averaged. In the case that the natural rate process is of the Markovian

n n

form (2.27), ¯t is an increasing linear function of rt (an average of rt and the long-run av-

±

erage natural rate r), with a slope less than one (though closer to one the more persistent

¯

are the ¬‚uctuations in the natural rate). The optimal inertia coe¬cient µ1 in these partial-

adjustment dynamics is an increasing function of »i /(»x σ 2 ); if interest-rate stabilization is

su¬ciently important relative to output-gap stabilization, the optimal rate of adjustment

of the central bank™s operating target in response to a shift in the current natural rate of

interest may be quite slow.

While this simple characterization of optimal interest-rate dynamics is exactly true only

in the limiting case in which κ = 0, the optimal interest-rate dynamics shown in Figure 7.6

are fairly similar to those described by (2.29), as is discussed further in Woodford (1999xx).

For while κ > 0 in a realistic model, the slope of the aggregate-supply relation that is

estimated econometrically is often fairly small; thus the limiting case just discussed provides

considerable insight into the character of optimal interest-rate dynamics.

3 Optimal Simple Policy Rules

We have thus far discussed the character of the state-contingent evolution that one would

ideally wish to arrange, and seen that this requires commitment to systematic behavior of a

kind that would not be chosen by a discretionary optimizer. We turn now to the question

of the type of policy rule to which a central bank should commit itself, in order to reap the

bene¬ts of policy commitment that we have discussed.

Much of the recent literature has addressed this question by asking which rule would be

best within some parametric family of relatively simple rules ” for example, the family of

contemporaneous Taylor rules

it = ¯ + φπ (πt ’ π ) + φx (xt ’ x)/4.

± ¯ ¯ (3.1)

Often it is supposed that commitment to a relatively simple rule of this sort represents the

3. OPTIMAL SIMPLE POLICY RULES 51

only feasible form of commitment, perhaps because of di¬culties of explaining to the public

the nature of the central bank™s commitment in some more complex case. Here we consider

the optimal choice of a policy rule from within such a restricted class, from the point of view

of the sort of forward-looking models of the e¬ects of policy commitments presented above.

We begin by discussing the optimality criterion that is appropriate in such an exercise.

It might seem that the obvious way in which to choose an optimal rule from within

a family such as (3.1) is to compute the rational-expectations equilibrium associated with

any given rule in the family, and evaluate a loss function such as (1.14) given the state-

contingent evolution of the target variables in that equilibrium. The optimal rule within the

family would then be the rule leading to the lowest expected loss. This is the de¬nition of

an “optimal simple rule” given, for example, in Currie and Levine (1991).

However, this criterion has undesirable features in the context of a forward-looking model

of the e¬ects of policy. As in our discussion of the t0 ’optimal commitment in the previous

two sections, one is evaluating alternative policies under a criterion which will favor policies

that exploit the fact that initial expectations are already given at the time that policy

is chosen; and in general, this will lead to a time-inconsistent policy choice, even when

policies are restricted to a simple family. The criterion favors policies that create in¬‚ation

in the initial periods following the policy choice, while committing to negligible average

in¬‚ation farther in the future. If a rule from within family (3.1) is chosen at a time when

negative output gaps are anticipated in the near term (while the long-run average output

gap is expected to be zero), a rule will be preferred under which a negative output gap

justi¬es a loosening of policy, exactly because this is expected to create the desired initial

in¬‚ation without implying long-run in¬‚ation. But if the question of the optimal simple rule is

reconsidered at a later date, at which a positive output gap is anticipated in the near term,

a rule will be preferred under which it is a positive output gap that justi¬es a loosening

of policy. The choice of an optimal simple rule on these grounds is time-inconsistent (as

Currie and Levine note), for essentially the same reason as in our earlier discussion of the

unconstrained policy problem. We wish to propose instead a criterion which will result in a

52 CHAPTER 7. COMMITMENT TO A POLICY RULE

time-consistent selection.

Another disadvantage of this criterion is that even when applied to a class of policies

¬‚exible enough to include a policy that is optimal from a timeless perspective, that policy

may well not be judged optimal within the restricted class of policies.27 This can be seen

from a reconsideration of the deterministic problem treated in section 1.1 above. There the

optimal policy from a timeless perspective was found to be one that resulted in zero in¬‚ation

each period. One might also consider the optimal policy within the restricted class of policies

that keep in¬‚ation constant at some rate π for all time. If one simply evaluates (1.2) for the

¯

paths of in¬‚ation and output associated with any such policy, one obtains a value

®

2

1’β

(1 ’ β)’1 °π 2 + » π ’ x— »,

¯ ¯

κ

which is not minimized at π = 0, in the case that x— > 0. One instead would prefer a

¯

somewhat positive in¬‚ation rate under this criterion, in order to take advantage of the gains

from unanticipated in¬‚ation in period t0 , even though the extent to which this is chosen is

limited by the restriction that one must choose the same in¬‚ation rate for all later periods

as well. In order to obtain a criterion that will result in choice of the policy that is optimal

from a timeless perspective when the class of simple policies is ¬‚exible enough to contain it,

it is necessary to evaluate the outcomes associated with alternative rules from a perspective

that penalizes a rule for taking advantage of pre-existing expectations at the time of policy

choice.

We accordingly propose to evaluate policy rules according to a criterion of the following

sort. A quadratic loss criterion such as (1.2), evaluated conditional on the economy™s state

at date t0 , can be expressed as the sum of two components, Ldet + Lstab , where Ldet depends

only on the deterministic component of the equilibrium paths of the target variables, and

Lstab depends only on the equilibrium responses to unexpected shocks in periods after t0 .

27

For example, Jensen and McCallum (2002) ¬nd that the targeting rule (5.1) below, that brings about

the equilibrium characterized by Proposition 7.7 and hence is optimal from a timeless perspective, is not

optimal (in the sense that they consider) within a family of simple linear rules that includes rule (5.1). This

is because they do not rank alternative policies according to the stabilization loss measure Lstab proposed

below. For further discussion of the Jensen-McCallum calculations, see xxxxxx.

3. OPTIMAL SIMPLE POLICY RULES 53

For example, in the case of (1.2), the deterministic component is given by

∞

det

β t’t0 [(Et0 πt )2 + »(Et0 xt ’ x— )2 ],

L =

t=t0

while the stabilization component is given by

∞

stab

β t’t0 [vart0 (πt ) + »vart0 (xt )].

L =

t=t0

Under any policy rule that is optimal from a timeless perspective (or, for that matter,

under a t’optimal commitment chosen at any date t ¤ t0 ), the equilibrium responses to

shocks in periods after t0 will be exactly those that minimize Lstab , among the set of possible

dynamic responses to shocks that are consistent with the (linear) structural equations. Hence

we wish to evaluate rules within more restricted families of linear rules according to the same

criterion: the coe¬cients that determine the equilibrium responses to shocks ” e.g., the

coe¬cients φπ , φx in the case of the family (3.1) ” should be chosen so as to minimize Lstab ,

to the extent that this is possible within the class of simple rules considered. This is a criterion

that will allow a choice of these coe¬cients that is independent of the economy™s state at

date t0 ; for that state may a¬ect the predicted deterministic component of the paths of the

target variables implied by given response coe¬cients, but it has no e¬ect on the predicted

variances that enter Lstab , given a linear model for the evolution of the disturbances. Thus

the proposed criterion leads to a time-consistent choice. And it is a criterion that implies

that if the family of rules considered is ¬‚exible enough to include one that is optimal from

a timeless perspective, the response coe¬cients in that rule will be judged optimal.

Given this choice of the response coe¬cients, the coe¬cients that determine the long-run

average values of the target variables with which the rule is consistent (the values ¯, π , x

±¯¯

in (3.1)) are not then chosen to minimize Ldet . Instead, in accordance with our discussion

of deterministic policy problems in section 1, a rule is chosen that is consistent with the

long-run average values that occur under the t0 ’ optimal commitment. (These are the same

values as would occur under an optimal commitment chosen at any other date, or under

a policy that is optimal from a timeless perspective.) Unlike the criterion of minimizing

54 CHAPTER 7. COMMITMENT TO A POLICY RULE

Ldet , this criterion leads to a choice that is independent of the state at t0 , and hence that

is time-consistent. And it leads to a rule that is consistent with the same long-run average

values of the target variables as any rule that is optimal from a timeless perspective; so if

the family of rules under consideration includes such a rule, it will be found optimal within

the restricted family. We turn now to some simple examples of policies that could be judged

optimal within a restricted family of simple alternatives, though they are not fully optimal.

3.1 The Optimal Non-Inertial Plan

A restricted class of policies of particular interest is the class of purely forward-looking

policies, under which policy (and hence equilibrium outcomes) at each date depend only on

the set of possible evolutions for the target variables that are possible from that date onward.

Basing policy solely on projections of the economy™s current and possible future states has

a certain intuitive appeal, and the forecast-targeting procedures of central banks often seem

to have this character; hence it may be of interest to know how di¬erent from fully optimal

policy the best possible rules of this kind are.

In order to consider this question, we may begin by considering which state-contingent

evolution we should wish to bring about, among all those consistent with any purely forward-

looking policy, and then subsequently ask which policy (or policies) can be used to implement

the desired equilibrium. The set of possible state-contingent evolutions to which we may re-

strict attention consists of those under which the current endogenous non-predetermined

state variables zt depend only on the vector of exogenous states st that contains all informa-

tion available in period t about the disturbances to the structural equations in period t or

later, and on the vector Zt of exogenous disturbances that matter for determination of the

variables zt . For the set of possible evolutions of the economy from date t onward depends

only on the values of st and Zt . It follows that if policy depends only on this set, it will also

depend only on those variables, and if the policy rule results in a determinate equilibrium,

it must be one in which the equilibrium values zt also depend only on (st , Zt ).

We shall call the optimal state-contingent evolution from within this restricted class the

3. OPTIMAL SIMPLE POLICY RULES 55

optimal non-inertial plan, following Woodford (1999xx). To be precise, this is the plan under

which (i) the long-run average values of the variables zt are those associated with a policy

that is optimal from a timeless perspective, and (ii) the ¬‚uctuations in response to shocks

are those that minimize the stabilization loss Lstab , subject to the constraint that zt depend

only on (st , Zt ).

As an example, consider again the model consisting of aggregate-supply relation (2.1),

and suppose once more that social welfare is measured by the expected value of (1.2). For

simplicity, suppose that the disturbance ut evolves according to (2.18), for some 0 ¤ ρu < 1.

As discussed above, in this case the set of variables (st , Zt ) reduces simply to the current

value of ut , and the only possible state-contingent paths that can be implemented by a purely

forward-looking (linear) rule are ones in which πt and xt are linear functions of the current

value of ut , as in (2.19). Plans of this form are consistent with the equilibrium relation (2.1)

if and only if

(1 ’ β)¯ = κ¯,

π x (3.2)

(1 ’ βρu )fπ = κfx + 1. (3.3)

Furthermore, in the case of any plan of this form, the stabilization loss is given by

β 1 ’ rho2 2

u

stab 22

L = [fπ + »fx ]σu , (3.4)

2

1 ’ β 1 ’ βρu

2

where σu is the unconditional variance of the disturbance process {ut }.

It follows from Proposition 7.2 that the long-run average values of in¬‚ation and the output

gap associated with a timelessly optimal policy are π = x = 0. The optimal values of the

¯ ¯

2 2

coe¬cients (fπ , fx ) are those that minimize (3.4), or equivalently, that minimize fπ + »fx ,

subject to constraint (3.3). The solution to this latter problem is given by

κ»’1

1 ’ βρu

oni oni

fπ = 2 ’1 , fx = ’ 2 ’1 . (3.5)

κ » + (1 ’ βρu )2 κ » + (1 ’ βρu )2

Thus we obtain the following.

Proposition 7.12. Consider the baseline (Calvo pricing) model, in which the aggregate-

supply relation is of the form (2.1), and abstract from any grounds for a concern with

56 CHAPTER 7. COMMITMENT TO A POLICY RULE

interest-rate stabilization, so that the period loss function is of the form (1.2). Let the cost-

push disturbance {ut } evolve according to (2.18), for some 0 ¤ ρu < 1. Then the optimal

non-inertial plan is a state-contingent evolution of the form (2.19) in which π = x = 0, and

¯¯

the coe¬cients fπ , fx indicating the response to cost-push shocks are given by (3.5).

Applying (2.3) to the case of an AR(1) disturbance process (2.18), we ¬nd that the

equilibrium responses under discretion are also of the form (2.19), but with

κ»

π disc = x— > 0

¯ 2

(1 ’ β)» + κ

and

»

disc oni

fπ = ≥ fπ ,

κ2 + (1 ’ βρu )»

where the last inequality is strict if ρu > 0. Thus discretionary policy is suboptimal, even

within the class of purely forward-looking policies, except in the special case that x— = 0 and

the cost-push shocks are serially uncorrelated. In general, discretionary policy leads both to

too high an average rate of in¬‚ation (if x— > 0), and to too large an in¬‚ation response to

cost-push shocks (if ρu > 0). It follows from (3.3) that too large a positive in¬‚ation response

fπ also means too small a negative output-gap response fx .

A similar analysis is possible in the case that welfare is reduced by variation in the level

of nominal interest rates, for reasons such as those discussed in chapter 6, section xx. If

the period loss function is of the form (2.24), then both (2.1) and (2.23) are constraints

on possible equilibrium paths of the target variables {πt , xt , it }. If we assume that the

disturbances to both structural equations are Markovian ” in particular, that {ut } evolves

n

according to (2.18) for some 0 ¤ ρu < 1, and {rt } similarly evolves according to a law of

motion (2.27) for some 0 ¤ ρr < 128 ” then under any purely forward-looking policy that

28

It is not actually necessary to assume that the two disturbances evolve according to independent AR(1)

processes, as posited here, in order for the form (3.6) to be correct; one might more generally assume that

n n

both rt and ut are linear functions of (rt’1 , ut’1 ) plus an unforecastable innovation term, where the two

innovation terms need not be uncorrelated with each other. Certainly there is no economic reason to assume

that these disturbance processes must be independent; in the case that steady-state distortions are large,

discussed in chapter 6, section xx, many types of disturbances, such as variations in government purchases,

n

will a¬ect rt and ut simultaneously. However, the case of two independent disturbance processes is easier

to treat, and allows us to conduct thought experiments such as the one considered in Figure 7.6.

3. OPTIMAL SIMPLE POLICY RULES 57

results in a determinate equilibrium, each of the target variables must evolve according to

equations of the form

n

yt = y + fy ut + gy rt ,

¯ (3.6)

where (¯, fy , gy ) are constant coe¬cients for each of the variables y = π, x, i. The equilibrium

y

relations (2.1) and (2.23) imply two linear restrictions on the coe¬cients fy , and another set

of two linear restrictions on the coe¬cients gy .

Under the assumption that the two disturbance processes are independent, the stabiliza-

tion loss can be decomposed into two parts,

Lstab = Lstab,r + Lstab,u ,

corresponding to the losses resulting from responses to unexpected shocks of the two types.

In the case of a plan of the form (3.6), these terms are equal to

β 1 ’ rho2 2

r

stab,r 2 22

L = [gπ + »x gx + »i gi ]σr , (3.7)

1 ’ β 1 ’ βρ2

r

β 1 ’ rho2 2

u

stab,u

[fπ + »x fx + »i fi2 ]σu ,

2 2

L = (3.8)

1 ’ β 1 ’ βρ2

u

2 2 n

where σr , σu are the unconditional variances of the disturbance processes {rt , ut } respec-

tively. Note that Lstab,r involves only the coe¬cients gy , while Lstab,u involves only the

coe¬cients fy .

The optimal non-inertial plan is then the state-contingent evolution of the form (3.6)such

that (i) the coe¬cients y are the long-run average values under a timelessly optimal policy

¯

(characterized in Proposition 7.3), (ii) the coe¬cients fy minimize Lstab,u subject to the two

constraints implied by (2.1) and (2.23), and (iii) the coe¬cients gy minimize Lstab,r subject to

the corresponding two constraints on these coe¬cients. The solution to this problem, derived

in Giannoni and Woodford (2002xx), is presented in section xx of the appendix.29 Here we

note simply that the optimal response coe¬cients to either of the real disturbances (for

example, the coe¬cients gy in the case of the natural-rate disturbance) are independent of

29

Note that the dashed lines plotted in Figure 7.6 indicate the responses implied by the coe¬cients gy in

the case of the calibrated parameter values assumed in that ¬gure.

58 CHAPTER 7. COMMITMENT TO A POLICY RULE

the properties of the other disturbance; indeed, they are the same regardless of whether other

(independent) disturbances are assumed to exist or not. The optimal response coe¬cients

to a given disturbance are also independent of the degree of variability of that disturbance,

though they do depend (as in Proposition 7.12) on the degree of serial correlation of the

disturbance.

3.2 The Optimal Taylor Rule

We now consider the optimal choice of a policy rule from within the simple family of Taylor

rules (3.1). Let us suppose that in¬‚ation and output determination under such a rule are

governed by equilibrium relations (2.1) and (2.23) of the basic neo-Wicksellian model intro-

duced in chapter 4. As shown in that chapter,30 a rule of this kind (with φπ , φx ≥ 0) implies

a determinate rational-expectations equilibrium if and only if it conforms to the “Taylor

principle,” i.e., its coe¬cients satisfy

1’β

φπ + φx > 1. (3.9)

4κ

In the case that the disturbance processes are of the form (2.18) and (2.27), this equilibrium

is one in which the equilibrium values of πt , xt , and it will all depend only on the current

n

disturbances rt and ut . The state-contingent evolution of the target variables will then be

of the form (3.6), for certain coe¬cients y , fy , gy given in the appendix.

¯

We are thus interested in choosing the coe¬cients of the policy rule (3.1), from within the

class of rules satisfying (3.9), so as to bring about a state-contingent evolution of the form

(3.6) that achieves as low as possible a value for Lstab , together with the long-run average

values y associated with a timelessly optimal policy. For at least a certain range of parameter

¯

values, it will be possible to choose the coe¬cients of the Taylor rule so as to implement the

30

In chapter 4, we considered a version of the “New Keynesian Phillips curve” in which no disturbance

term ut appeared. But the analysis of determinacy of equilibrium there would be una¬ected by the addition

of such a term. The solutions given in chapter 4 for the equilibrium evolution of in¬‚ation and the output gap

must be modi¬ed to take account of the cost-push disturbances. However, the equations given in chapter 4

remain applicable under the interpretation that the variable called “xt ” in chapter 4 corresponds to xt +κ’1 ut

in the notation used in equation (2.1). In the notation of chapter 4, then, the policy rule (3.1) would be one

with a time-varying intercept (equal to ¯ ’ (φx /4κ)ut ) in addition to the linear response to xt + κ’1 ut .

±

3. OPTIMAL SIMPLE POLICY RULES 59

best possible state-contingent evolution from within the restricted family (3.6), i.e., so as to

implement the optimal non-inertial plan. (Note that the 9 coe¬cients y , fy , gy must satisfy

¯

6 equalities in order to be consistent with the equilibrium conditions (2.1) and (2.23). Hence

a 3-parameter family of policy rules su¬ces, in principle, to support any plan within this

family; it is thus not fortuitous that, for a non-trivial range of parameter values, a Taylor

rule can be found that implements the optimal non-inertial plan.) An example of a case in

which is this is so is given by the following result of Giannoni and Woodford (2002xx).

Proposition 7.13. Suppose that the disturbance processes in the basic neo-Wicksellian

model are of the form (2.18) and (2.27), with a common degree of serial correlation ρr =

ρu = ρ. Then if ρ is in the range such that

(1 ’ βρ)(1 ’ ρ) (1 ’ β)(1 ’ βρ)

0< ’ ρ »i < »x + 1, (3.10)

κ2

κσ

the optimal non-inertial plan is consistent with a Taylor rule (3.1) with coe¬cients φπ , φx > 0

that also satisfy (3.9). Hence commitment to this rule implies a determinate equilibrium,

and implements the optimal non-inertial plan. It follows that this is the optimal Taylor rule.

The coe¬cients of this optimal rule are given by

»i 1’β

(i— ’ r),

π=

¯ ¯ x=

¯ π,

¯ ¯ = r + π,

±¯¯ (3.11)

»i + β κ

’1

(1 ’ βρ)(1 ’ ρ) φx »x

»’1 ,

φπ = ’ρ = (1 ’ βρ) φπ . (3.12)

i

κσ 4 κ

The proof is in the appendix. It is also shown there that the inequalities (3.10) are

necessarily satis¬ed for ρ in some non-empty interval ρ < ρ < ρ, where 0 < ρ < 1, so that

¯ ¯

this interval also contains some positive values of ρ (but not values of ρ that are too close

to one).31 Hence in an at least some cases, there exists a Taylor rule (with positive feedback

coe¬cients satisfying the Taylor principle) that represents optimal policy, at least among the

31

In the case of the calibrated parameter values given in Table 6.1 for β, κ, σ, »x , and »i , this interval

corresponds to 0.17 < ρ < 0.68.

60 CHAPTER 7. COMMITMENT TO A POLICY RULE

class of purely forward-looking policies. Nor is this result dependent upon the assumption

that ρr = ρu , made in Proposition 7.13 solely for the sake of algebraic simpli¬cation. As

shown in the appendix, there exists a pair of feedback coe¬cients (φπ , φx ) consistent with

the optimal non-inertial plan for almost all possible parameter values, even when ρr = ρu .

(Certain functions of the model parameters must be non-zero in order for a system of linear

equations to have a solution; but this excludes only certain extremely special parameter

values.) One must then check whether the required feedback coe¬cients satisfy the inequality

(3.9), so that the rule implies a determinate equilibrium.32 Proposition 7.13 shows that

some parameter values exist for which the implied feedback coe¬cients (3.12) satisfy this

inequality. Then, since the required feedback coe¬cients are continuous functions of the

model parameters (except at the degenerate parameter values where no solution exists), the

inequality is also satis¬ed for all model parameters (including values of ρr and ρu ) su¬ciently

close to these ones. For any values of the other parameters, there will thus be an open set of

non-negative values for (ρr , ρu ) for which the results announced in Proposition 7.13 obtain,

though the algebraic expressions for the optimal feedback coe¬cients are considerably more

complex in the general case.33

Optimality of the Taylor rule within the class of purely forward-looking policies implies in

particular that there is no gain from adopting a more forward-looking interest-rate feedback

rule, at least in the case of this simple model. For example, suppose that we consider rules

of the form

it = ¯ + φπ (Et πt+k ’ π ) + φx (Et xt+k ’ x)/4,

± ¯ ¯ (3.13)

for some forecast horizon k > 0. Such a rule is again purely forward-looking, and so can at

32

This condition is necessary and su¬cient for determinacy, as shown in chapter 4, only under the re-

striction that φπ , φx ≥ 0. Thus we must also verify that these two additional inequalities are satis¬ed, in

order to ensure that the conclusions of Proposition 7.13 hold. Alternatively, we could allow for rules with

negative feedback coe¬cients, as long as they imply a determinate equilibrium; but in that case, there are

again additional inequalities that must be checked. The presence of these additional inequalities does not

a¬ect the validity of the argument made in the text; the statement “the inequality is also satis¬ed” should

simply be modi¬ed to refer to “inequalities.”

33

Proposition xx of Giannoni and Woodford (2002xx) gives an example of an open set of values for (ρr , ρu )

for which this is true, though the conditions established in that result are again only su¬cient conditions,

and not necessary for existence of an optimal Taylor rule.

3. OPTIMAL SIMPLE POLICY RULES 61

best implement the optimal non-inertial plan. However, in a case such as that treated in

Proposition 7.13, the Taylor rule already implements this plan, so that a consideration of

rules with a forecast horizon k > 0 yields no possible improvement. In fact, forward-looking

rules may be an inferior approach even to implementation of the optimal non-inertial plan.

It is true that it should be equally possible to ¬nd a rule of the form (3.13) that is consistent

with that state-contingent evolution. For example, in the case treated in Proposition 7.13,

there exists a rule of this form that is consistent with the optimal non-inertial plan, for any

k ≥ 0; it is the rule with coe¬cients y as in (3.11) and coe¬cients φy given by

¯

¯

φy = ρ’k φy , (3.14)

¯

for y = π, x, where φy refers to the coe¬cients given in (3.12). However, this alternative

policy rule, while equally consistent with the optimal non-inertial plan when k > 0, may not

also imply determinacy of equilibrium. Indeed, for large enough k, it necessarily does not,

as established by Giannoni and Woodford (2002xx).

Proposition 7.14. Consider an economy satisfying the assumptions of Proposition

7.13. Then for all forecast horizons k longer than some critical value, the rule of the form

(3.13) that is consistent with the optimal non-inertial plan implies indeterminacy of rational-

expectations equilibrium.

The proof is in the Appendix.34 Thus if the forecast horizon k is su¬ciently long, it is not

possible to implement the optimal non-inertial plan using a rule of the form (3.13).35 It

follows that, at least when model parameters satisfy (or are close enough to satisfying) the

conditions of Proposition 7.13, the best rule in this forward-looking family is not as desirable

as the best purely contemporaneous Taylor rule.

34

Batini and Pearlman (2002) establish a related result for a general family of interest-rate feedback rules

in which the current nominal interest rate operating target is a linear function of an in¬‚ation forecast and a

lagged nominal interest rate.

35

For example, in the case of the calibrated parameter values given in Table 6.1, the rule (3.13) with

coe¬cients (3.14) implies indeterminacy for all k ≥ 1.

62 CHAPTER 7. COMMITMENT TO A POLICY RULE

Yet even in this case, while the Taylor rule is optimal among the class of purely forward-

looking policies, it does not follow that one cannot do better; for optimal policy is history-

dependent, as shown in section 2.2. We show in chapter 8 that a generalized Taylor rule, that

includes dependence of the right kind on lagged variables (in particular, lags of the nominal

interest rate), can instead implement a fully optimal equilibrium (understood to mean one

that is optimal from a timeless perspective). And even the optimality of the simple Taylor

rule among purely forward-looking policies depends on fairly strong restrictions; for example,

it will not be true if the disturbances are not (at least jointly, if not individually) Markovian.

Moreover, even when one is willing to assume Markovian disturbances, the coe¬cients of

the optimal Taylor rule depend quite critically on the degree of serial correlation of the

disturbances, as indicated by (3.12). As shown in chapter 8, it is instead possible to choose

a generalized Taylor rule with the property that the same feedback coe¬cients are optimal

regardless of the serial correlation properties of the disturbances. Hence the more complex

rule is better not only in terms of the expected losses in the case of a particular speci¬cation

of the disturbance processes, but is also more robust.

4 The Optimal State-Contingent Instrument Path as

a Policy Rule

We turn now to the problem of the choosing a policy commitment that would be fully optimal

” that would lead not only to the optimal long-run average values of the target variables

characterized in section 1, but also to the optimal responses to disturbances characterized

in section 2. It might be thought that our characterization of the economy™s optimal state-

contingent evolution in section 2 has already given as complete a characterization of fully

optimal policy as may be desired. For we have shown how to compute the optimal state-

contingent paths of the various endogenous variables, including the optimal state-contingent

path for the central bank™s nominal interest-rate instrument. And it might be supposed that

a solution for the optimal state-contingent instrument path ” a formula that would tell

what the nominal interest-rate operating target should be at each date, as a function of the

4. THE OPTIMAL INSTRUMENT PATH 63

history of disturbances up to that time ” is itself a good example (perhaps even the canonical

example) of a fully optimal policy rule. That is, one might propose that commitment to a

fully optimal policy should mean a commitment by the central bank to choose its operating

target in each decision cycle according to this formula.

We shall argue that this is not a desirable way of deriving an optimal policy rule; but it is

¬rst useful to illustrate what such an approach would mean. Let us consider again the model

with cost-push shocks and no penalty for interest-rate variations treated in section 2.1. We

recall from Proposition 7.6 that equation (2.10) describes the state-contingent evolution of

in¬‚ation under one kind of policy that would be optimal from a timeless perspective.36 If

we substitute a speci¬c stochastic process for the disturbances {ut }, such as (2.18), into this

equation we obtain the solution

∞

1

µk’1 ut’k }

πt = {ut ’ (1 ’ µ1 ) (4.1)

1

β(µ2 ’ ρu ) k=1

for in¬‚ation as a function of the history of disturbances up through the current date. We

may substitute this in turn into (2.1) to obtain a similar solution for xt , and substitute both

of these solutions into (2.23) to obtain the following solution for the path of the nominal

interest rate.

Proposition 7.15. Consider again the policy problem treated in Proposition 7.6, and

suppose that the exogenous cost-push disturbances evolve according to (2.18). Then in

the timelessly optimal equilibrium characterized in Proposition 7.6, the state-contingent

evolution of the nominal interest rate is given by

∞

κ µ1 + ρ u ’ 1 1 ’ µ1

n

µk ut’k .

it = rt + 1’ ut ’ (4.2)

β(µ2 ’ ρu ) k=0 1

»σ β(µ2 ’ ρu )

36

While this is only one of several possible speci¬cations of the initial in¬‚ation constraint πt0 that is self-

¯

consistent in the sense discussed in section 2.1, one can also show that it is the only speci¬cation of the

initial constraint that leads to an expression (4.2) that is time-invariant, i.e., independent of the date t0

at which the policy rule is chosen. For example, the alternative speci¬cation (2.11) would lead instead to

a state-contingent path for the nominal interest rate in periods t ≥ t0 that would depend on the value of

xt0 ’1 , and on the number of periods t ’ t0 that had elapsed since the date at which the commitment was

chosen. Because we wish to choose a time-invariant policy rule, in order to address the problem of time

consistency discussed above, we accordingly assume that the speci¬cation of interest for present purposes is

the one discussed in Proposition 7.6.

64 CHAPTER 7. COMMITMENT TO A POLICY RULE

We might then take equation (4.2) to specify an optimal rule for setting the central

bank™s interest-rate operating target; indeed, some may suppose that this kind of description

of optimal policy, specifying the optimal instrument setting in each possible state of the

world (identi¬ed by the history of exogenous disturbances), should represent the canonical

speci¬cation of a policy rule. But this approach to the speci¬cation of optimal policy has

serious disadvantages. One is that a commitment to the rule by the central bank, even if

fully credible and correctly understood by the private sector, need not ensure that the desired

(optimal) equilibrium evolution of in¬‚ation and output is realized.

For a commitment to set interest rates according to (4.2), regardless of how in¬‚ation

and output may evolve, is an example of a policy that would specify an exogenous nominal

interest-rate path. It then follows from Proposition 4.xx that under the basic model (con-

sidered here), rational expectations equilibrium is indeterminate under such a policy. The

optimal in¬‚ation path (4.2) will be one possible equilibrium path for in¬‚ation under this

policy, but there will also be an uncountably in¬nite number of other non-explosive equi-

librium paths for in¬‚ation, includes ones in which in¬‚ation (and hence output) responds to

cost-push shocks in an entirely di¬erent way, and also ones in which in¬‚ation and output are

a¬ected by pure “sunspot” states. The fact that such a rule does not exclude these other,

quite undesirable, equilibria makes this an unattractive approach to the implementation of

optimal policy.

We have similarly seen in chapter 4 that commitment to a rule of this kind, that speci¬es

an exogenous path for the nominal interest rate, implies that the minimum-state-variable

equilibrium (which in the present case corresponds to the optimal in¬‚ation evolution (4.1))

will not be learnable through least-squares regression techniques. Hence from this point of

view as well, a rule such as (4.2) represents an undesirable approach to the implementation

of optimal policy.

Addressing both of these problems requires that the central bank™s policy commitment be

speci¬ed in a di¬erent way, so that the implied interest-rate path will depend on the observed

(or projected) paths of in¬‚ation and output, and not simply on the bank™s evaluation of the

4. THE OPTIMAL INSTRUMENT PATH 65

history of exogenous disturbances. The Taylor rule is an example of a policy rule with this

latter property, and we have seen in chapter 4 that a rule of this kind results both in a

determinate equilibrium and in one that is stable under least-squares learning dynamics.

However, in the present context, we have seen that a simple (contemporaneous) Taylor rule

does not result in an equilibrium that is optimal, as it fails to bring about the history-

dependence required for an optimal equilibrium. The question that remains to be addressed,

then, is whether we can ¬nd a rule that introduces feedback from in¬‚ation and/or output

to the central bank™s interest-rate target of the kind needed to ensure determinacy and

learnability, and that at the same time involves the sort of history-dependence needed to

implement an optimal equilibrium. We show in the next section how this is possible.

Equation (4.2) is also unappealing as a policy rule, however, for a quite independent

reason. The speci¬c formula (4.2) has been shown to be consistent with an optimal state-

contingent evolution of in¬‚ation and output only under a single very speci¬c assumption

about the statistical properties of the cost-push shocks ” that they evolve according to

u

a process of the form (2.18) with serial correlation coe¬cient ρu , with the innovation t

completely unforecastable before period t. Were we to assume any other stochastic process

for the cost-push shock ” something more complex than an AR(1) process, or even an AR(1)

process with a di¬erent degree of persistence, or even an AR(1) process with the same degree

of persistence but with innovations revealed some number of periods in advance ” then the

solution for the optimal interest rate as a function of the history of the shocks would be

given by a di¬erent formula.

Yet there is little practical interest in a characterization of optimal policy such as (4.2),

which is valid only under the assumption that the real disturbances are of one speci¬c type,

no matter how that type is chosen. Suppose that we have estimated the coe¬cients of the

aggregate supply relation (2.1), and that we have an accurate historical series for both in¬‚a-

tion and the output gap, and thus can construct a historical series for the disturbance term

ut in this equation. We might then propose to estimate ρu using the historical disturbance

series, and could then compute the numerical coe¬cients of a rule of the form (4.2). But a

66 CHAPTER 7. COMMITMENT TO A POLICY RULE

central bank would be highly unlikely to be willing to commit itself to follow the rule even

in that case.

For central bankers always have a great deal of highly speci¬c information about the kind

of disturbances that have just occurred, which are always somewhat di¬erent than those that

have been faced at other times. Hence even if it is understood that “typically” cost-push

disturbances have had a coe¬cient of serial correlation of 0.7, there will often be grounds to

suppose that the particular shock that has just occurred is likely to be either more persistent

or less persistent than a “typical” disturbance. And it is unlikely that central bankers will

be willing to commit themselves to stick rigidly to a rule that is believed to lead to outcomes

that would be optimal in the case of “typical” disturbances, even in the case that they are

aware of the economy™s instead being subjected to “atypical” disturbances. In order for an

a proposed policy rule to be of practical interest, it must instead be believed that the rule is

compatible with optimal (or at least fairly good) outcomes in the case of any of the extremely

large number of possible types of disturbances that might be faced on di¬erent occasions.

Of course, the sort of analysis that we have used to derive (4.2) can be extended to deal

with the case in which there are many di¬erent types of real disturbances that may shift the

aggregate-supply relation. That is, we may suppose that the residual in equation (2.1) is

actually of the form

j

ut = ψjk t’k , (4.3)

j

where the { j } are large set of di¬erent types of shocks that may occur in period t, that

t

a¬ect the aggregate-supply relation in that period or later to varying degrees, with varying

degrees of persistence, and in ways that are forecastable in advance to varying extents. Given

j

a speci¬cation of the dynamic e¬ects on the AS relation of a given type of shock t, we can

compute the response of the nominal interest rate to this particular type of shock in an

optimal equilibrium; and we can do this in principle for each of the types of shocks indexed

by j, and thus obtain an optimal state-contingent path for the nominal interest rate, where

the state in period t is now speci¬ed by the histories of realizations of each of the di¬erent

shocks. But in this case, the formula corresponding to (4.2) will contain separate terms, with

5. COMMITMENT TO AN OPTIMAL TARGETING RULE 67

di¬erent numerical coe¬cients, for each of the possible types of shocks. Such a description of

optimal policy will thus become completely unwieldy in the case of any attempt to capture

even in very coarse terms the sorts of di¬ering situations that central banks actually confront

at di¬erent times.

Giannoni and Woodford (2002a) instead show that if the central bank™s policy com-

mitment is described in terms of a relation among endogenous variables that the bank is

committed to bring about ” rather than in terms of a mapping from exogenous states to

the instrument setting, as in (4.2) ” it is possible, in a large class of policy problems, to

¬nd a rule that is robustly optimal, in the sense that the same rule (with given numerical

coe¬cients) continues to be optimal regardless of the assumed statistical properties of the

(additive) disturbance terms such as ut . Indeed, the rule is optimal even if the disturbance

terms in the model structural equations are actually composites of an extremely large (not

necessarily ¬nite) number of di¬erent types of real disturbances, as in (4.3). We illustrate

how this is possible in the next section, and discuss the kinds of policy rules to which this

approach leads one in greater detail in the next chapter. A rule of this kind represents a

policy commitment that a central bank could reasonably make, despite its awareness that it

will constantly be receiving quite ¬ne-grained information about current conditions. For a

belief that the rule represents a good criterion for judging whether policy is on track does not

require the central bank to believe that all shocks are alike, or even that all of the possible

types of disturbances to which it may have to respond can all be listed in advance.

5 Commitment to an Optimal Targeting Rule

We now consider an alternative approach to the speci¬cation of a policy rule that can im-

plement an optimal equilibrium, and show that this approach can avoid the problems just

discussed with a speci¬cation of optimal policy in terms of a state-contingent instrument

path. The alternative is what Svensson (1999, 20xx) calls a targeting rule. Under such a

rule, the central bank is committed to adjust its instrument as necessary in order to ensure

that a certain target criterion is satis¬ed at all points in time, or more precisely (as this is

68 CHAPTER 7. COMMITMENT TO A POLICY RULE

all that is possible in practice), so that the criterion is projected to be satis¬ed, according

to the central bank™s forecast of the economy™s evolution. The target criterion speci¬es a

condition that the projected evolution of the bank™s target variables ” such as in¬‚ation, the

output gap, and possibly interest rates as well ” must be projected to satisfy if policy is to

be regarded as “on track.” A simple example would be the criterion that RPIX in¬‚ation two

years in the future be expected to equal 2.5 per cent per annum; this is the criterion used

to explain the policy decisions of the Bank of England under current procedures (Vickers,

1998).

A rule of this kind represents a “higher-level” description of policy than an explicit

speci¬cation of the instrument setting in each possible state of the world, such as (4.2). The

instrument setting that is implied by such a rule at any point in time can only be determined

through the use of a quantitative model of the e¬ects of monetary policy on the economy.

In each decision cycle, the central bank must use its model (and, of course, the judgment

of policymakers) to determine what interest-rate operating target will result in projections

that satisfy the target criterion. But a targeting rule is not di¬erent, in this respect, from

commitment to a rule like the Taylor rule discussed in section 3.2. For in our basic neo-

Wicksellian model (the model for which the Taylor rule was shown to constitute an optimal

purely forward-looking policy), both in¬‚ation and the output gap in period t depend on

period t interest rates; hence the policy rule (3.1) does not indicate what the level of interest

rates in period t should be, without a calculation of what πt and xt are projected to equal in

the case of one level of interest rates or another. Our discussion above (as in chapter 4) of

the consequences of commitment to such a rule assumed that implementation of an “implicit

instrument rule” of this kind is possible.37

While such incompleteness of the speci¬cation of prescribed central-bank behavior has

some disadvantages ” for example, it makes it more di¬cult for the private sector to be

37

McCallum (19xx) has instead criticized the Taylor rule for not being an “operational” policy proposal,