. 2
( 3)


is no welfare gain, up to our order of approximation, from a commitment to in¬‚ation that
can be anticipated in advance. In particular, we ¬nd once again that except for transition
e¬ects, resulting from the di¬erent term in (3.3) for the initial period, it is again optimal to
commit to zero in¬‚ation, independent of the shocks to the economy.

Nonetheless, the term in (3.3) that is linear in π0 now a¬ects the optimal commitment
for periods later than π0 as well. That is because of the intertemporal linkage implied by
aggregate supply relation (2.19). The welfare gain from in¬‚ation at date zero can be obtained
with less increase in the period zero output gap (and hence less increase in the »x2 term)

if it is accompanied by an increase in expected in¬‚ation at date one; and since the welfare
loss from such in¬‚ation is merely quadratic, it is optimal to commit to some amount of such
in¬‚ation. Thus the in¬‚ation associated with the transition to the optimal regime lasts for
more than a single period in this case.

The optimal transition path is characterized in section xx of chapter 7. Here we content
ourselves with a few observations about the form of the solution to this problem. First,
because the only reason to plan a non-zero in¬‚ation rate in period 1 is for the sake of the
e¬ect of expected period 1 in¬‚ation on the location of the period 0 output-in¬‚ation tradeo¬,
there is no gain from planning on a period 1 in¬‚ation rate that is not deterministic. The
same is true of planned in¬‚ation in all later periods. Thus the optimal commitment from
date zero onward involves a deterministic path for in¬‚ation; it continues not to be optimal
for the in¬‚ation rate to respond at all to real disturbances of the various types considered
thus far. In addition, as shown in the next chapter, the deterministic path for planned
in¬‚ation should converge asymptotically to zero, the rate that would be optimal but for the
opportunity to achieve an output gain from unexpected in¬‚ation in the initial period.

Thus it is optimal (from the point of minimizing discounted losses from date zero onward)
to arrange an initial in¬‚ation, given that the decision to do so can have no e¬ect upon
expectations prior to date zero (if one is not bothered by the non-time-consistency of such a

principle of action). The optimal policy involves positive in¬‚ation in subsequent periods as
well, but there should be a commitment to reduce in¬‚ation to its optimal long-run value of
zero asymptotically. And the rate at which in¬‚ation is committed to decline to zero should
be completely una¬ected by random disturbances to the economy in the meantime.29 Thus
the assumption that ¦ > 0 makes no di¬erence for the conclusions of the previous section
with regard to the optimal response to shocks. And if one takes the view (as we shall argue
in the next chapter) that one should actually conduct policy as one would have optimally
committed to do far in the past, thus foregoing the temptation to exploit the private sector™s
failure to anticipate the new policy, then it is optimal simply to choose πt = 0 at all times “
i.e., to completely stabilize the price level “ just as in the previous section.
It is interesting to note that this result “ that the optimal commitment involves a long-
run in¬‚ation rate of zero, even when the natural rate of output is ine¬ciently low “ does not
depend upon the existence of a vertical “long-run Phillips curve” tradeo¬. For the aggregate
supply relation (2.19) in our baseline model implies an upward-sloping relation

xss = (1 ’ β)κ’1 π ss

between steady-state in¬‚ation π s s and the steady-state output gap xss . (This is because the
expected-in¬‚ation term has a coe¬cient β < 1, unlike that of the “New Classical” relation
(2.16).) It is sometimes supposed that the existence of a long-run Phillips-curve tradeo¬,
together with an ine¬cient natural rate, should imply that the Phillips curve should be
exploited to some extent, resulting in positive in¬‚ation forever, even under commitment. But
here that is not true, because the smaller coe¬cient on the expected-in¬‚ation term relative
to that on current in¬‚ation “ which results in the long-run tradeo¬ “ is exactly the size of
shift term in the short-run aggregate supply relation that is needed to precisely eliminate any
long-run incentive for non-zero in¬‚ation under an optimal commitment. If one were instead
to “simplify” the New Keynesian aggregate supply relation, putting a coe¬cient of one on
expected in¬‚ation (as is done in some presentations,30 presumably in order to conform to
These results agree with those of King and Wolman (1999) in the context of a model with two-period
overlapping price commitments in the style of Taylor (1980).

the conventional wisdom regarding the long-run Phillips curve), we would then fail to obtain
such a simple result. The optimal long-run in¬‚ation rate would actually be found to be
negative, as the stimulative e¬ects of lower expected in¬‚ation would be judged to be worth
more than the output cost of lower current in¬‚ation “ even though there would actually be
no long-run output increase as a result of the policy!
And once again, the character of optimal policy in the presence of in¬‚ation inertia due
to partial indexation to a lagged price index can be determined directly from our results
for the Calvo pricing model with no indexation; the optimal time path for πt in the case
of the model without indexation becomes the optimal time path for πt ≡ πt ’ γπt’1 in the
case with indexation. It then follows that once again the optimal path of in¬‚ation will be
completely deterministic. And while there will be initial in¬‚ation (even starting from an
initial condition with π’1 = 0) if the central bank allows itself to exploit initial expectations
d d
by choosing π0 > 0, optimal policy will involve a commitment to reduce πt asymptotically
to zero. In the case of any γ < 1, this will once again mean a commitment to eventual price
stability. And optimal policy has this character despite the fact that the level of output
associated with stable prices is ine¬ciently low, and despite the existence of a positively
sloped long-run Phillips curve trade-o¬, as this is ordinarily de¬ned.

3.3 Caveats

We have seen that, within the class of sticky-price models discussed above, the optimality of
a monetary policy that aims at complete price stability is surprisingly robust. Not only does
this conclusion not depend upon the ¬ne details of how many prices are set a particular time
in advance or left unchanged for a particular length of time, but it remains valid in the case
of a considerable range of types of stochastic disturbances, and in the case of an ine¬cient
natural rate of output. Nonetheless, it is likely that some degree of deviation from full price
stability is warranted in practice. Some of the more obvious reasons for this are sketched
See, e.g., Roberts (1995) or Clarida et al. (1999).

First of all, complete price stability may not be feasible. We have just argued, in section
3.2, that in our baseline model, it is feasible, because we are able to solve for the required
path of the central bank™s nominal interest-rate instrument. This is correct, as long as the
random disturbances are small enough in amplitude. But if they are larger, such a policy
might not be possible, because it might require the nominal interest rate to be negative at
some times, which, as explained in chapter 2, is not possible under any policy. Speci¬cally,
this will occur if it is ever the case that the natural rate of interest is negative. On average,
it does not seem that it should be, and thus zero in¬‚ation on average would seem to be
feasible; but it may be temporarily negative as a result of certain kinds of disturbances, and
this is enough to make complete price stability infeasible. As a result, a policy will have to
be pursued which involves less volatility of the short nominal interest rate in response to
shocks, and some amount of price stability will have to be sacri¬ced for the sake of this.31
The way in which optimal monetary policy is di¬erent in the presence of such a concern is
an important concern of chapter 7.

Varying nominal interest rates as much as the natural rate of interest varies may also be
desirable as a result of the “shoe-leather costs” involved in economizing on money balances.
As argued by Friedman (1969), the size of these distortions is measured by the level of
nominal interest rates, and they are eliminated only if nominal interest rates are zero at all
times.32 Taking account of these distortions “ from which we have abstracted thus far in our
welfare analysis “ provides another reason for the equilibrium with complete price stability,
even if feasible, not to be fully e¬cient; for as Friedman argues, a zero nominal interest rate
will typically require expected de¬‚ation at a rate of at least a few percent per year.

One might think that this should make no more di¬erence to our analysis of optimal policy

In general, it will be optimal to back o¬ from complete price stability both by allowing in¬‚ation to
vary somewhat in response to disturbances, and by choosing an average rate of in¬‚ation that is somewhat
greater than zero, as suggested by Summers (1991), in order to allow more room for interest-rate ¬‚uctuations
consistent with the zero lower bound. However, the quantitative analysis undertaken below ¬nds that the
e¬ect of the interest-rate lower bound on the optimal response of in¬‚ation to shocks is more signi¬cant than
the e¬ect upon the optimal average rate of in¬‚ation.
See Woodford (1990) for justi¬cation of this relation in a variety of alternative models of the demand
for money.

than does the existence of an ine¬cient natural rate of output due to market power “ that it
may similarly a¬ect the deterministic part of the optimal path for in¬‚ation without creating
any reason for in¬‚ation to vary in response to random shocks. But monetary frictions do
not have implications only for the optimal average level of nominal interest rates. As with
distorting taxes, it is plausible that the deadweight loss is a convex function of the relative-
price distortion, so that temporary increases in nominal interest rates are more costly than
temporary decreases of the same size are bene¬cial. In short, monetary frictions provide a
further reason for it to be desirable to reduce the variability of nominal interest rates, even
if one cannot reduce their average level. (At the same time, reducing their average level will
require less variable rates, because of the zero ¬‚oor.) Insofar as these costs are important,
they too will justify a departure from complete price stability, in the case of any kinds of
real disturbances that cause ¬‚uctuations in the natural rate of interest, in order to allow
greater stability of nominal interest rates. This tradeo¬ is treated more explicitly in section
xx below.

Even apart from these grounds for concern with interest-rate volatility, it should be
recognized that the class of sticky-price models analyzed above are still quite special in
certain respects. One of the most obvious is that there are assumed to be no shocks as a
result of which the relative prices of any of the goods with sticky prices would vary over
time in an e¬cient equilibrium (i.e., the shadow prices that would decentralize the optimal
allocation of resources involve no variation in the relative prices of such goods). This is
because we have assumed that only goods prices are sticky, that all goods enter the model in
a perfectly symmetrical way, and that all random disturbances have perfectly symmetrical
e¬ects upon all sectors of the economy. These assumptions are convenient, but plainly an
idealization. Yet it should be clear that they are relied upon in our conclusion that stability
of the general price level su¬ces to eliminate the distortions due to price stickiness.

If an e¬cient allocation of resources requires relative price changes, due to asymmetries
in the way that di¬erent sticky-price commodities are a¬ected by shocks, this will not be
true. We show, however, in section xx below, that even in the presence of asymmetric shocks,

it is possible to de¬ne a symmetric case in which it is still optimal to completely stabilize
the general price level, even though this does not eliminate all of the distortions resulting
from price stickiness. But this holds exactly only in a special case, in which di¬erent goods
are similar, among other respects, in the degree of stickiness of their prices. If sectors of
the economy di¬er in their degree of price stickiness (as is surely realistic), then complete
stabilization of an aggregate price index will not be optimal. Stabilization of an appropriately
de¬ned asymmetric price index (that puts more weight on the stickier prices) is a better
policy, as argued by Aoki (2001) and Benigno (1999), though even the best policy of this
kind need not be fully optimal.

An especially important reason for disturbances to require relative price changes between
sticky commodities with sticky prices is that wages are probably as sticky as are prices.
Real disturbances almost inevitably require real wage adjustments in order for an e¬cient
allocation of resources to be decentralized, and if both wages and prices are sticky, it will
then not be possible to achieve all of the relative prices associated with e¬ciency simply by
stabilizing the price level “ speci¬cally, the real wage will frequently be misaligned, as will be
the relative wages of di¬erent types of labor if these are not set in perfect synchronization. In
such circumstances, complete price stability may not be a good approximation at all to the
optimal policy, as Erceg et al. (1999) show. As we show in section xx below, stabilization of
an appropriately weighted average of prices and wages may still be a good approximation to
optimal policy, and fully optimal in some cases. Thus concerns of this kind are not so much
reasons not to pursue price stability as they are reasons why care in the choice of the index
of prices (including wages) that one seeks to stabilize may be important.

Yet another quali¬cation to our results in this section is that we have assumed a frame-
work in which the ¬‚exible-price equilibrium rate of output is e¬cient, or at most di¬ers
from the e¬cient level by only a (small) constant factor. As we have seen, this assumption
is compatible with the existence of a variety of types of economic disturbances, including
technology shocks, preference shocks, and variations in government purchases. But it would
not hold in the case of other sorts of disturbances, that cause time variation in the degree

of ine¬ciency of the ¬‚exible-price equilibrium. These could include variation in the level of
distorting taxes, variation in the degree of market power of ¬rms or workers, or variation in
the size of the wage premium that must be paid on e¬ciency-wage grounds.

In the case of a time-varying gap between the ¬‚exible-price equilibrium level of output
and the e¬cient level, complete stabilization of in¬‚ation is no longer su¬cient for complete
stabilization of the welfare-relevant output gap. For while in¬‚ation stabilization may imply
a level of output at all times equal to the ¬‚exible-price equilibrium level, as discussed above,
this will no longer minimize the variability of the gap between actual output and the e¬cient
level of output. As a result, complete stabilization of in¬‚ation will not generally be optimal.
It is not obvious that stabilization of any alternative price index makes sense as a solution to
the problem in this case, either, whereas some degree of concern for stabilization of the (ap-
propriately measured) output gap is clearly appropriate, even if it should not wholly displace
a concern for in¬‚ation stabilization. This is an especially serious challenge to the view that
price stability should be the sole goal of monetary policy, if one believes that disturbances
of this kind are quantitatively important in practice. Their importance, however, remains
a matter of considerable controversy. Furthermore, even if disturbances of this kind are of
substantial magnitude, the degree of departure from price stability that can be justi¬ed on
welfare-theoretic grounds may well be less than is often supposed, as we show in section xx

4 Extensions of the Basic Analysis

Here we sketch extensions of our utility-based welfare criterion to incorporate several compli-
cations from which we have abstracted in the basic analysis presented in section 2. We shall
give particular attention to complications that illustrate some of the reasons just sketched
for complete stabilization of the price level not to be optimal.

4.1 Transactions Frictions

In section 2, we have abstracted from the welfare consequences of the transactions frictions
that account for the demand for the monetary base. Our results therefore apply to a “cash-
less” of the kind discussed in chapter 2. Here we consider the way in which they must be
modi¬ed in order to allow for non-negligible welfare e¬ects of transactions frictions.
As in chapter 2, we may represent the welfare consequences of variations in the degree to
which these frictions distort transactions by including real money balances as an additional
argument of the utility function of the representative household. This generalization makes
real marginal cost, and hence the equilibrium level of output under ¬‚exible prices a function of
the (endogenous) level of real balances, in addition to the exogenous state of preferences and
technology, in the case that the indirect utility function u(c, m) is not additively separable.
Substituting the equilibrium level of real balances

mt = ·y Yt ’ ·i (ˆt ’ ˆm ) + m
ˆ ± ±t (4.1)

into the household labor-supply relation, we have shown in chapter 4 that average real
marginal cost is given by
+ •(ˆt ’ ˆm )],
st =
ˆ mc [xt ± ±t (4.2)

≡ σ ’1 + ω ’ χ·y , (4.3)

·i χ
•≡ , (4.4)

σ ’1 gt + ωqt + χ m
ˆ t
Ytn ≡ . (4.5)

In these expressions, we once again use the coe¬cient χ ≡ mucm /uc to measure the degree
of complementarity between private expenditure and real balances.
Here we de¬ne the natural rate of output Ytn as the ¬‚exible-price equilibrium level of
¯ ˆ
output when the interest rate di¬erential ∆t is ¬xed at its steady-state level ∆, so that Ytn is
again an exogenous process. This de¬nition also has the advantage that, up to a log-linear

approximation, the amount by which the (log) e¬cient level of output exceeds the (log)
natural rate is a constant,33 in the case that both the steady-state ine¬ciency wedge ¦ and
the steady-state interest-rate di¬erential ∆ are only of order O(||ξ||). This constant gap (up
to a residual of order O(||ξ||2 )) is given by
¦ + sm ·y
x— ≡ , (4.6)

where sm ≡ mum /¯uc ≥ 0 measures the interest cost of real balances as a fraction of the value
¯ c
> 0, so that if ¦ ≥ 0, x— > 0.
of private expenditure. We assume as in chapter 4 that mc

We observe that the signs of the e¬ects upon the natural rate of output of the various real
disturbances discussed in section 3.2 remain the same. Now, however (if χ = 0), disturbances
to the money demand function, possibly due to shifts in the transactions technology, also
a¬ect the natural rate of output.
It follows from (4.2) that in this more general case, the “New Classical” aggregate supply
relation takes the form
πt = κ[xt + •(ˆt ’ ˆm )] + Et’1 πt
± ±t (4.7)

ˆ ˆ
instead of (2.16), where again xt ≡ Yt ’ Ytn , and
ι mc
κ≡ > 0.
1 ’ ι 1 + ωθ
In the case of Calvo pricing, instead, the aggregate supply relation now takes the form

πt = κ[xt + •(ˆt ’ ˆm )] + βEt πt+1 ,
± ±t (4.8)

where again
(1 ’ ±)(1 ’ ±β) mc
κ≡ > 0.
± 1 + ωθ
Note that in either case the interest-rate di¬erential appears as a shift factor in the aggregate
supply relation, because of its (small) e¬ect on the real marginal cost of supply.
In the present case, the e¬cient level of output at any point in time is the solution to two equations,
stating that real marginal cost is equal to one and that there is satiation in real money balances. Because the
second condition implies a zero interest-rate di¬erential regardless of the real disturbances, both the natural
rate of output and the e¬cient level can be de¬ned as output variations in response to real disturbances that
maintain real marginal cost constant in the case of a constant interest di¬erential.

We can again approximate the utility from private expenditure using a second-order
Taylor series expansion, obtaining
1 1
¯ ˆ ˆ ˆ
u(Yt , mt ; ξt ) = Y uc Yt + (1 ’ σ ’1 )Yt2 + σ ’1 gt Yt + sm mt + sm (1 ’ σm )mt
2 2
+ χmt Yt + sm σ ’1 gt mt + sm (χ + σm ) m mt + t.i.p. + O(||ξ||3 )
ˆ tˆ (4.9)

as a generalization of (2.8). Here we again de¬ne the elasticity σm ≡ ’um /mumm > 0, the
exogenous disturbance term gt is de¬ned as in the cashless model, and
≡ (χ + σm )’1
ξt ’ σ ’1 gt
is the exogenous disturbance term in the money-demand relation (4.1).
Once again, we can legitimately substitute into this our log-linear approximate solution
to our structural equations only if the coe¬cients on the linear terms are at most of order
O(||ξ||). This means that we must assume that the economy is su¬ciently close to being
satiated in money balances. In order to contemplate a series of economies that come as close
as we like to this limit, without having to change the speci¬cation of preferences or technology
(including the transactions technology), it is important to allow for interest payments on the
monetary base, which we shall suppose are always close to a steady-state rate of ¯m . The
steady-state interest di¬erential ∆ ≡ (¯ ’ ¯m )/(1 + ¯) as a measure of the degree to which
±± ±
there is not complete satiation in money in the steady state around which we expand, we
shall now assume that both ∆ and ¦ are of order O(||ξ||).
As we consider economies with progressively smaller positive values for ∆, obtained by
raising the value of ¯m , we assume that the steady-state quantities Y , m approach ¬nite,
positive limiting values, and that the partial derivatives of utility also have well-de¬ned
limits from this direction. We furthermore assume that the limiting value of umm is negative
(so that σm is ¬nite), even though this requires that umm be discontinuous at the satiation
¯ ¯
level of real balances.34 Then in the limit of small ∆, we ¬nd that sm is of order O(∆), as
Note that our assumption that um = 0 in the case of all levels of real balances in excess of that required
for satiation implies that umm = 0 for all values of m higher than the limiting value m, in the case of an
income level Y . This sort of discontinuity typically occurs, for example, in the case of a cash-constraint
model of the transactions technology, like that considered in chapter 2, section xx.

is σm , though the ratio sm σm approaches a positive limit as ∆ ’ 0, since

m2 umm
sm ¯ ¯
=’ ¯ + O(∆).
σm Y uc

This allows us to drop some of the quadratic terms from (4.9), the coe¬cients of which are
only of order O(∆).
We note also that in the limit of small ∆, the elasticities of the money-demand relation
(4.1) reduce to
Y ucm uc
¯ ¯
·y = ’ + O(∆), ·i = ’ + O(∆). (4.10)
¯ mumm
This allows us to substitute (·i v )’1 for sm σm and χ·i v for ·y in the coe¬cients of quadratic
¯ ¯
¯ ¯¯
terms,35 where v ≡ Y /m is the steady-state “velocity of money”. With these substitutions,
(4.9) can be written as

¯ ˆ ˆ ˆ
u(Yt , mt ; ξt ) = Y uc Yt + (1 ’ σ ’1 )Yt2 + σ ’1 gt Yt + sm mt
+ χmt Yt ’ (·i v )’1 (mt ’ m )2 + t.i.p. + O(||ξ||3 ).
¯ ˆ (4.11)

We can then substitute (4.1) for equilibrium real balances mt , obtaining

¯ ˆ ˆ ˆ ˆ
u(Yt , mt ; ξt ) = Y uc Yt + (1 ’ σ ’1 )Yt2 + σ ’1 gt Yt + sm ·y Yt ’ sm ·i (ˆt ’ ˆm )
± ±t
1 1
ˆ ˆ
+ χ·y Yt2 + χ m Yt ’ v ’1 ·i (ˆt ’ ˆm )2 + t.i.p. + O(||ξ||3 ). (4.12)
¯ ± ±t
2 2

Subtracting (2.10) from (4.12), we obtain

¯ ˆ mˆ
Ut = Y uc (¦ + sm ·y )Yt ’ sm ·i (ˆt ’ ˆm ) + [σ ’1 gt + ωqt + χ ’ v ’1 ·i (ˆt ’ ˆm )2
± ±t t ]Yt ¯ ± ±t
1 1
Yt2 ’ (θ’1 + ω)vari yt (i) + t.i.p. + O(||ξ||3 )
’ ˆ (4.13)
2 2
Y uc
±m ¯ 2
¯’1 ±
=’ mc (xt ’ x ) + v ·i (ˆt ’ ˆt + ∆)
+(θ’1 + ω)vari yt (i) + t.i.p. + O(||ξ||3 )
ˆ (4.14)
The advantage of replacing ·y by χ·i v is that it is then clear what form our results take in the familiar
special case in which it is assumed that ucm = 0; we simply set χ equal to zero in the expressions derived

as a generalization of (2.12). Note that in (4.13), the linear terms both have coe¬cients that
are of order O(||ξ||), as is required for validity of welfare comparisons based on a log-linear
solution to our model, as long as both ¦ and ∆ are of order O(||ξ||). In (4.14), the optimal
level x— for the output gap is given by (4.6), while the optimal level for the interest-rate
di¬erential is zero, since the condition it = im , the Friedman (1969) condition for satiation
¯ ¯
in real money balances, corresponds to ˆt = ˆm ’ ∆ + O(∆2 ).
± ±t
Finally, substituting for output dispersion as a function of in¬‚ation as before, in the
case of our baseline (Calvo) model of price-setting, we again obtain an approximate welfare
criterion of the form (2.22), where now the normalized loss function is given by

Lt = πt + »x (xt ’ x— )2 + »i (ˆt ’ ˆm + ∆)2 ,
± ±t (4.15)

with weights
κ ·i
»x = > 0, »i = »x > 0. (4.16)
θ v
¯ mc

An alternative expression for the weight on the interest-rate term, equivalent under our
small-∆ approximation, though applicable only in the case that χ = 0, is

»i = •»x . (4.17)

Thus taking account of transactions frictions adds an additional term to the loss function,
with a positive weight on squared deviations of the interest-rate di¬erential from its optimal
size, which is zero.
Note that in the “cashless limit” discussed in chapter 2, v ’1 ’ 0, so that »i ’ 0,
and we recover our results above for the cashless model. However, it is important to note
that the interest-rate variability term does not vanish under the assumption that utility is
additively separable between consumption and real balances, so that ucm = 0. While this last
assumption (which implies that χ = 0) results in the disappearance of real-balance e¬ects
from both the aggregate-supply and IS relations of our model of the transmission mechanism,
and similarly implies that money-demand disturbances have no e¬ect on the natural rates of
output or of interest, it does not imply that »i = 0. Thus it makes a di¬erence whether one

assumes that χ is negligible in size because of approximate additive-separability, or instead
because equilibrium real balances are small (velocity is large).
One case in which our previous conclusions are largely una¬ected is that in which the
central bank™s interest-rate operating target is implemented through adjustments of the
interest paid on the monetary base, so that ∆t is equal to a ¬xed spread at all times,
regardless of how it varies.36 In this case, the (ˆt ’ ˆm + ∆)2 term in (4.15) is a constant,
± ±t
independent of how in¬‚ation, output and interest rates vary over time. Optimal policy will
then again be one that minimizes a loss function of the form (2.23); the only di¬erence that
monetary frictions make would be to the de¬nitions of (and hence of κ and »x ) and of

Ytn . In particular, we will ¬nd once again that optimal policy involves complete stabilization
of the price level, just as in our analysis of the cashless model in section xx.
If, instead, the interest paid on the monetary base is equal to a constant ¯m at all times
(perhaps zero, as in the U.S. at present), then the ¬nal term in (4.15) is not irrelevant. In
this case, the welfare-theoretic loss function reduces to

Lt = πt + »x (xt ’ x— )2 + »i (ˆt ’ i— )2 ,
± (4.18)

where now the optimal nominal interest rate is given by

1 + ¯m
± ¯ ¯

= ’∆ + O(∆2 );
i ≡ log
1 +¯ ±

that is, it is equal to the constant interest rate paid on the monetary base. Note that the ¬nal
term results in a loss function of the kind assumed by Williams (1999), where the additional
term is instead motivated by reference to “aversion to interest-rate variability”.
The additional term means that complete stabilization of the price level is no longer
optimal, for two reasons. The ¬rst is that, as long as im < ¯ ≡ β ’1 ’ 1, the steady-state
nominal interest rate that minimizes the last term in (4.15) requires expected de¬‚ation, as
As discussed in chapter 1, a number of central banks do already implement policy through channel
systems under which the interest rate paid on central bank balances is always equal to the current operating
target for the overnight cash rate minus a constant spread. However, even in these countries, no interest
is paid on currency, and currency balances continue to constitute most of the monetary base. Thus these
countries do not actually represent examples of the ideal system considered here.

argued by Friedman (1969). There is thus now a con¬‚ict between the steady-state rate of
in¬‚ation needed to minimize the ¬rst term and that needed to minimize the third. In fact,
the long-run in¬‚ation rate under an optimal policy commitment, in the absence of stochastic
disturbances, is generally intermediate between the two “ higher than the Friedman rate
(i.e., minus the rate of time preference), but still negative, as shown in chapter 7.
And second, there is now a con¬‚ict between the pattern of responses to shocks that
minimizes the ¬rst term (i.e., no in¬‚ation variation at all) and the pattern required to
minimize the third term (no interest-rate variation). Insofar as shocks a¬ect the natural rate
of interest (and we have shown that many di¬erent types of real disturbances all should),
nominal interest-rate variations are required to keep in¬‚ation stable, and vice versa. In
addition, it need not even be true any longer that complete in¬‚ation stabilization minimizes
the second term “ for if κi > 0, the interest-rate variations required to stabilize in¬‚ation will
result in at least a small amount of output-gap variation as well.
A special case is possible in which no such con¬‚ict arises. Suppose that we assume instead
the “New Classical” model of pricing, in which all prices are adjusted each period, though
some new prices are chosen a period in advance. Let us again suppose that im = ¯m at all

times. In this case, the corresponding normalized loss function is given by

Lt = (πt ’ Et’1 πt )2 + »x (xt ’ x— )2 + »i (ˆt ’ i— )2 ,
± (4.19)

where the weights are again given by (4.16), but using the de¬nition of κ in (4.7). If we
also assume that ¦ = 0, then the approximations (4.10) imply that x— = ’•i— > 0, up to
a residual of order O(||ξ||2 ). Then the aggregate-supply relation (4.7) can alternatively be
πt = κ[(xt ’ x— ) + •(ˆt ’ i— )] + Et’1 πt .
± (4.20)

Then there is no problem with simultaneously minimizing all three terms in (4.19). This
simply requires that one set ˆt = i— each period, and make in¬‚ation equal whatever value
was forecasted in the previous period, in which case (4.7) implies that xt = x— as well.
Minimization of the interest-rate variation term has implications only for expected in¬‚ation,

while minimization of the term representing the costs of price dispersion has implications
only for unexpected in¬‚ation, and so, in this special case, there is no con¬‚ict between fully
achieving both goals at all times.
Even if ¦ > 0, we note that
x— + •i— = > 0. (4.21)

Using this to substitute for x— , the discounted loss measure can be written as
∞ ∞
[π0 ’ κiˆ0 ] + E0
β t (πt ’ Et’1 πt )2
E0 β Lt = ’2 i
θ mc
t=0 t=0
»x •¦
+»x (xt + •i— )2 + 2 (ˆt ’ i— ) + »i (ˆt ’ i— )2
± ± , (4.22)

generalizing (3.2). All terms on the right-hand side except the ¬rst one are again minimized
by setting ˆt = i— and πt = Et’1 πt each period. (The third term inside the large square
brackets is necessarily non-negative each period, because of the equilibrium requirement
that it ≥ ¯m , or ˆt ≥ i — .) The ¬rst term indicates that there is an additional welfare gain
± ±
from unexpected in¬‚ation in period zero, because what is decided for this period cannot a¬ect
in¬‚ation expectations in the previous period; and if one allows oneself to take advantage of
that opportunity, the in¬‚ation rate in period zero should be chosen to be somewhat higher
than had been expected. But thereafter, one will make unexpected in¬‚ation equal zero
every period, as this is not inconsistent with setting ˆt = i— in every period. Furthermore,
under a policy that is optimal from a timeless perspective, one will simply arrange for zero
unexpected in¬‚ation and ˆt = i— each period.
But this case, in which the distortions resulting from price stickiness can be completely
eliminated without putting any restriction upon the process that expected in¬‚ation may
follow, is clearly a very special one. In general, variations in expected in¬‚ation as a re-
sult of ¬‚uctuations in the natural rate of interest (as will be required in order to maintain
ˆt = i— at all times) will result in relative-price distortions. Hence the goal of minimizing
the distortions associated with transactions frictions will con¬‚ict with that of minimizing
the distortions resulting from price stickiness. Before discussing further the nature of this

tension between alternative stabilization objectives, we shall argue that a similar concern
with nominal interest-rate stabilization can be justi¬ed on alternative grounds.

4.2 The Zero Interest-Rate Lower Bound

Even in the case of a cashless economy, incomplete in¬‚ation stabilization may be optimal,
in order to reduce the variability of nominal interest rates in response to shocks. The reason
is the equilibrium requirement that it ≥ 0 at all times.37 If shocks are su¬ciently small,
this poses no obstacle to complete in¬‚ation stabilization, but if the natural rate of interest is
sometimes negative (and by this we mean the natural short rate, which is more volatile than
the associated natural longer rates), complete stabilization of in¬‚ation will be infeasible.
In that case, which seems reasonably likely, it is of some interest to consider the nature of
optimal policy subject to the constraint of respecting the zero lower bound. It is reasonably
clear that such policy will involve less variation in nominal interest rates than occurs in the
natural rate of interest; in particular, market rates will not fall as much as the natural rate
does in those states in which it becomes negative. Characterizing the optimal behavior of
market rates is a problem beyond the scope of the linear-quadratic optimization methods
used here; however, we can consider a related problem that gives some insight into the way
in which such a constraint should a¬ect optimal policy. This is to replace the constraint
that the nominal interest must be non-negative in every period with a constraint upon its
Speci¬cally, Rotemberg and Woodford (1997, 1999a) propose to approximate the e¬ects
of the lower bound by imposing instead a requirement that the mean federal funds rate be at
least k standard deviations above the theoretical lower bound, where the coe¬cient k is large
enough to imply that violations of the lower bound should be infrequent. The alternative
constraint, while inexact, has the advantage that checking it requires only computation of
More generally, the requirement is that it ≥ im , but here we shall suppose that zero interest is paid on
the monetary base, in order to make this constraint as weak as possible. We assume that the payment of
negative interest on the monetary base, as proposed by Gesell, Keynes (1936), and more recently, Buiter and
xxxx, is technically infeasible.

¬rst and second moments under alternative policy regimes, whereas checking whether the
funds rate is predicted to be negative in any state would depend upon ¬ne details of the
distribution of shocks. In addition, a constraint of this form has the advantage that, assuming
linear structural equations and a quadratic loss function, the constrained-optimal policy is a
linear rule, just like the unconstrained optimum. Hence our linear methods can still be used
to characterize optimal policy.
This can be demonstrated as follows. Note that the constraint can equivalently be ex-
pressed as a requirement that the average value of i2 be not more than K ≡ 1 + k ’2 times

the square of the average value of it ,38 which latter average must also be non-negative. If
we use discounted averages, for conformity with the other terms in our welfare measure, we
obtain constraints of the form

β t it ≥ 0,
E0 (1 ’ β) (4.23)

∞ ∞
β t i2 t
E0 (1 ’ β) ¤ K E0 (1 ’ β) β it . (4.24)
t=0 t=0

Now suppose that we wish to minimize an expected discounted sum of quadratic losses

β t Lt
E0 (1 ’ β) (4.25)

subject to (4.23) “ (4.24), and let m1 , m2 be the discounted average values of it and i2

associated with the optimal policy. Then this is also the policy that minimizes (4.25) subject
to the two constraints

β t it
E0 (1 ’ β) ≥ m1 ,

β t i2
E0 (1 ’ β) ¤ m2 ,

since any policy consistent with both of these also satis¬es the weaker constraints (4.23) “
By the expression it we here actually mean log(1 + it ), or ˆt + ¯.
± ±

Then by the Kuhn-Tucker theorem, the policy that minimizes the expected discounted
value of (4.25) subject to (4.23) “ (4.24) can be shown to also minimize an (unconstrained)
loss criterion of the form
∞ ∞ ∞
t t
β t rt ,
E0 (1 ’ β) β Lt ’ µ1 E0 (1 ’ β) β rt + µ2 E0 (1 ’ β)
t=0 t=0 t=0

where µ1 and µ2 are appropriately chosen Lagrange multipliers. (Both multipliers are non-
negative, and if the constraint (4.24) binds, µ1 = 2Km1 µ2 > 0.) Finally, the terms in this
expression can be rearranged to yield a discounted loss criterion of the form (4.25), but with
Lt replaced by
Lt ≡ Lt + »i (ˆt ’ i—— )2 , (4.26)

where »i = µ2 ≥ 0 and (if µ2 > 0)

i—— = ’ ¯ = Km1 ’ ¯.
± ±

(There is also a constant term involved in completing the square, but as usual we drop this
as it has no e¬ect upon our ranking of alternative policies. Note that we have written the
quadratic term in terms of a target value for ˆt ≡ it ’ ¯ rather than it , for consistency with
± ±
our previous results.)
Thus the optimal policy minimizes the expected discounted value of a quadratic loss
function (4.26), subject to the constraints imposed by the structural equations of our model.
If the latter are linear, the optimal policy will itself be linear. Note that the e¬ective loss
function (4.26) contains a quadratic penalty for interest-rate variations (in the case that
constraint (4.24) binds), even if the “direct” social loss function Lt is independent of the
path of the interest rate. For example, consider again our baseline model of Calvo pricing,
in the “cashless limit”. The direct loss function is then given by (2.23), which involves
only in¬‚ation and the output gap. But if the ¬‚uctuations in the natural rate of interest are
large enough for (4.24) to bind “ i.e., if (4.24) is violated by the solution ˆt = rt “ then
optimal policy actually minimizes a loss function of the form (4.15), exactly as we previously
concluded by taking account of transactions frictions. The particular type of departure from

price stability that is motivated by the need to respect the interest-rate lower bound is
exactly the same as the kind that results from taking account of transactions frictions.
The primary qualitative di¬erence between the loss functions motivated in the two ways is
that transactions frictions lead to a loss function (4.15) with a “target” interest rate i— < 0
(i.e., lower than the steady-state interest rate ¯ consistent with zero in¬‚ation), while the
interest-rate lower bound alone would suggest a “target” interest rate i—— > 0. For we have
shown above that when (4.24) does not bind, optimal policy in the cashless limit involves
a deterministic component of in¬‚ation that is non-negative (and converging asymptotically
to zero); hence average in¬‚ation is non-zero. If instead (4.24) does bind, the only reason
to choose a di¬erent deterministic component for in¬‚ation would be in order to relax the
constraint, which would involve making average in¬‚ation higher (so that the average funds
rate can be higher). Thus optimal policy should involve m1 > ¯. But since K > 1, this
implies that i—— > 0.
If transactions frictions are non-negligible and the interest-rate lower bound binds as
well, the quadratic interest-rate term in (4.26) is added to the quadratic interest-rate term
already present in (4.15). The result is a loss function that again has the same form,

Lt = πt + »x (xt ’ x— )2 + »i (ˆt ’ ˆ)2 ,
± (4.27)

where now »i = »i + µ2 is an even larger positive coe¬cient, while ˆ is intermediate be-
tween i— and i—— (and thus may have either sign). In fact, the value of ˆ, like the value of
x— , matters only for the deterministic component of optimal policy; the optimal responses
to shocks depend only upon the weights »x , »i of the loss function. Thus in this regard
both considerations point in the same direction, toward the likely importance of including a
quadratic interest-rate term in the loss function. Hence we shall give considerable attention
in chapter 7 to the consequences for optimal policy of including such a term.39
Note, however, that both considerations justify a concern to reduce the variability of the level of interest
rates, and not a concern with the variability of interest-rate changes. The latter sort of “interest-rate
smoothing” goal is often assumed to characterize the behavior of actual central banks. As we show in
chapter 7, it is possible to justify such the assignment of such a goal to the central bank as part of an
optimal delegation problem, even if it is not part of the social loss function with which are concerned here.

How much are such considerations likely to matter? We investigate this numerically in
a calibrated example. The values of the parameters ±, β, σ, κ, and are as in Table 4.1,

based upon the estimates of Rotemberg and Woodford (1997) discussed in chapter 4.40 The
values for ·y , ·i , and χ given in the table are those implied by estimates of long-run money
demand for the U.S., as discussed in chapter 2.41
For present purposes, the only aspect of the exogenous disturbances that matters is the
implied evolution of the natural rate of interest rt ; for simplicity, we assume that this variable
follows a stationary ¬rst-order autoregressive process, with a mean of zero, and a standard
deviation and serial correlation coe¬cient as speci¬ed in the table. As is explained further
in chapter 7, these numerical values are motivated by aspects of the estimated model of
Rotemberg and Woodford (1997), though that model involves a more complex speci¬cation
of the shock processes.
Finally, the structural parameters given in the table imply a value for θ equal to 7.88.
Using this, we are able to obtain a theoretical value for »x in the utility-based loss function
(2.23), which value is also given in the table.42 We simplify our calculations by assuming
that ¦ = 0, so that x— = 0.
Using these parameter values, we can explore the tradeo¬ between minimization of the
deadweight losses measured by (2.23) and stabilization of the short-term nominal interest
rate. Letting L0 be this loss function (i.e., the loss function abstracting from any costs of

interest-rate variability), we can compute its expected discounted value

ˆ 0
β t L0
E[L ] ≡ E (1 ’ β) (4.28)

Note that the value of ω reported in Table 4.1 is not exactly the one implied by the values given here,
since in Rotemberg and Woodford the value of ω is inferred from the value of mc assuming that χ = 0.
The value for χ used here is actually slightly higher than that derived in chapter 2, as the value of χ
implied by the long-run money demand estimates depends upon the assumed value of σ. However, the values
used in both chapters agree to the ¬rst three decimal places.
Note that the value given here is equal to 16κ/θ, rather than κ/θ. This is because we report the
loss function weights »x , »i that are appropriate to use when in¬‚ation and interest rates are measured as
annualized percentage rates, despite the fact that our model is quarterly. The square of the annualized
2 2
percentage in¬‚ation rate is thus not πt but 16πt , in terms of the notation used in our earlier theoretical

Table 6.1: Calibrated parameter values for the quarterly model used for Figure 6.1.

Structural parameters
± 0.66
β 0.99
σ ’1 0.16
κ .024
·y 1
·i 28
χ 0.02
Shock process
ρ(ˆn )
r 0.35
sd(ˆn )
r 3.72
Loss function
x— 0
»x .048
»i .077
Interest-Rate Bound
k 2.26
r¯ 2.99

in the case of any stochastic processes for in¬‚ation and the output gap. Here we use the nota-
tion E to denote a discounted expectation rather than simply the unconditional expectation
of the random variable L0 ; and the operator E on the right-hand side indicates that we take

an unconditional expectation over possible initial states of the exogenous disturbance r0 .
We measure the degree of interest-rate variability43 associated with any possible equilibrium
in terms of the statistic

β tˆ2 .
V [i] ≡ E (1 ’ β) ±t

We then consider the policies that minimize E[L0 ] subject to a constraint that V [i] not
The statistic actually measures the average squared deviation of interest rates from the level ¯ consistent
with zero in¬‚ation in the absence of disturbances. However, all equilibria on the e¬cient frontier shown
in Figure 6.1 involve zero average values of ˆt , since any non-zero steady-state in¬‚ation rate would increase
ˆ 0
the value of both E[L ] and V [i], holding constant the way the variables deviate from their steady-state
values as a function of the history of disturbances. Among stochastic processes with that property, V [i] is a
discounted measure of the series™ variability about its mean.

exceed some ¬nite value.
The e¬cient frontier for these two statistics is shown in Figure 6.1. (The nature of the
constrained-e¬cient policies is discussed in the next chapter.) We observe that it is possible
to achieve the theoretical lower bound of zero for E[L0 ], by completely stabilizing in¬‚ation
and the output gap as discussed in section 3, only if we are willing to tolerate an interest-
rate variability of V [i] = 13.83, corresponding to a (discounted) standard deviation of 3.72
percentage points for the federal funds rate. (This is of course just the assumed standard
deviation of ¬‚uctuations in the natural rate of interest.) Lower interest-rate variability
requires that one accept less complete stabilization of in¬‚ation and the output gap, indicated
ˆ ˆ
by a positive value for E[L0 ]. Complete interest-rate stabilization would require that E[L0 ]
take a value of 2.10, a level of deadweight loss more than twice that associated with steady
in¬‚ation of one percent per year.44 Statistics relating to the variability of in¬‚ation, the
output gap, and the federal funds rate in these two extreme cases are given by the ¬rst and
last lines of Table 6.2, using measures analogous to V [i] in the case of the other two variables
as well. Note that in terms of these measures,

E[L0 ] = V [π] + »x V [x].

This frontier indicates how a concern with interest-rate variability, for whatever reason,
would a¬ect the degree to which it would be optimal to stabilize in¬‚ation and the output
gap. We are particularly interested, however, in the degree of attention to this goal that
would be justi¬ed by either of the two considerations treated above. Taking account of
transactions frictions, we would wish to minimize E[L], where now Lt is the loss function
(4.15) including an interest-rate variability term. The fact that i— = 0 in (4.15) a¬ects
only the deterministic part of the optimal paths of the endogenous variables, as with our
discussion of the case x— = 0 in section 3; it has no e¬ect upon the optimal responses to
shocks. Thus we can determine the optimal responses to shocks by minimizing E[L] with i—
set equal to zero, which amounts to minimizing E[L0 ] + »i V [i]. This policy corresponds to
A similar e¬cient frontier in the case of the more complicated model estimated by Rotemberg and
Woodford (1997) is shown in Figure 5 of that paper.




E[L ]





0 2 4 6 8 10 12 14

Figure 6.1: The tradeo¬ between in¬‚ation/output-gap stabilization and interest-rate stabi-

a point on the frontier in Figure 6.1, namely the point at which the slope is equal to ’»i ,
where »i is given by (4.17).
Assuming the structural parameters given in Table 6.1, equation (4.4) implies that for the
U.S., • should equal approximately .22 years. Then (4.17) implies that »i = .077, the value
also given in the table.45 This corresponds to point A on the frontier in the ¬gure.46 Line 2
of Table 6.2 reports the variability of each of the endogenous variables in this equilibrium.
This value, based on an assumption that χ = .02, is probably too large by a factor of two or more, for
reasons discussed in chapter 2. Williams (1999), for example, assumes a value (if the weight on the squared
in¬‚ation term is normalized as one) equal to .02.
In order to show the optimal equilibria under di¬ering assumptions on a single diagram, we use the same
structural model in each case “ namely, our baseline model, abstracting from real-balance e¬ects, that is also
used in the numerical analysis of chapter 7 “ simply varying the assumed welfare criterion in each case. This
means that in the case of point A, we are actually assuming parameter values in the structural equations
(χ = 0) that are not completely consistent with those used to derive the loss function (χ = .02). However,
this makes only a small di¬erence to our characterization of optimal policy when transactions frictions are
allowed for; the most important e¬ect is upon the loss function, rather than upon the structural equations.
Were we to assume additively separable preferences u(c, m), there would be no e¬ect upon the structural
equations at all, but the loss function would nonetheless be modi¬ed as indicated in (4.15).

Table 6.2: Examples of Policies on the E¬cient Frontier.

ˆ ˆ
V[i] E[L0 ]
»i V[π] V[x]
0 0 0 13.83 0
.077 .037 4.015 4.961 .231
.236 .130 10.60 1.921 .643
.277 .151 11.75 1.623 .719
∞ .677 29.35 0 2.096

Now suppose instead that we abstract from transactions frictions, but take account of
the lower bound on nominal interest rates, or more precisely, that we impose the constraints
(4.23) “ (4.24). Following Rotemberg and Woodford (1997), we let k equal the ratio of
the standard deviation of the funds rate to its mean in the long-run stationary distribution
implied by their estimated VAR model of U.S. data. We also assume a value for ¯, the
steady-state real funds rate, equal to the mean real funds rate implied by this same long-run
stationary distribution. These values are indicated in Table 6.1.
As shown above, the optimal policy subject to these constraints will minimize E[L],
where Lt is de¬ned by (4.26) with Lt equal to L0 . Once again, the fact that i—— = 0 does not

a¬ect the optimal responses to shocks, and so these are as in one of the equilibria on the
frontier shown in Figure 6.1. As is explained further in section xx of chapter 7, under our
assumed parameter values the implied value of »i in (4.26), given by the Lagrange multiplier
on constraint (4.24), is equal to .236. (See Table 7.1.) The corresponding optimal responses
to shocks are those associated with point B on the frontier, which is the point at which
the frontier has this steeper slope. The third line of Table 6.2 reports the variability of
each of the endogenous variables in this equilibrium. (In the computations reported in the
table, the deterministic component of each variable is equal to zero in all periods, so that
these statistics refer only to the variations in the variables due to ¬‚uctuations in the natural
rate of interest.) The higher e¬ective penalty upon interest-rate variability results in less
equilibrium variation in nominal interest rates, at the cost of more variation in both in¬‚ation
and the output gap.

Finally, suppose that in our welfare analysis we take account both of transactions frictions
and of the lower bound on nominal interest rates. We then ¬nd the policy that minimizes
E[L] when Lt is given by (4.15) rather than by L0 . As noted above, this policy minimizes a
ˆˆ ˆ
criterion of the form E[L], where Lt is given by (4.27). In the case of our assumed parameter
values, »i is equal to .277, a larger value than we would obtain taking account of either of
these considerations individually. Once again, the optimal responses to shocks correspond
to a point on the frontier shown in Figure 6.1, the point labeled C. The variability of the
endogenous variables in this equilibrium is indicated on the fourth line of Table 6.2.
ˆ ˜
It is interesting to note that the value »i is not greatly larger than »i , so that point C
is not too much higher than point B on the e¬cient frontier. This indicates that taking
account of transactions frictions does not matter as much for our welfare analysis, once we
have already taken account of the interest-rate lower bound, though the reverse is not true.
There is a simple intuition for this result. Whether or not we allow for transactions frictions,
imposition of constraint (4.24) makes it optimal to choose a policy under which interest-
rate variability is roughly of the size that makes (4.24) consistent with an average in¬‚ation
rate of zero.47 The level of interest-rate variability that this would involve, the point on
the frontier corresponding to it, and the associated weight on interest-rate variations in the
e¬ective loss function (which is given by the negative of the slope of the e¬cient frontier at
that point) are all independent of whether the loss function directly penalizes interest-rate
variations (because of transactions frictions) or not. For this reason, increasing the weight
»i on interest-rate variations in the direct loss function Lt has relatively little e¬ect on the
weight »i in the e¬ective loss function. The increase in »i results in a smaller Lagrange
multiplier on constraint (4.24), as there is less desire to vary interest rates even in the
absence of the constraint; and so the sum of the two weights increases much less than the

When we take account of only the interest-rate lower bound, it is already optimal to reduce interest-rate
variability to a degree consistent with a long-run average in¬‚ation rate of only 14 basis points per year.
When one takes account of the welfare consequences of transactions frictions as well, it instead becomes
optimal to reduce interest-rate variability to an extent consistent with a long-run average in¬‚ation rate of
negative 11 basis points per year. This does not require a great deal of further reduction in the variability
of the short-term interest rate.

increase in »i . For this reason, there is not a great loss in accuracy involved in neglecting the
welfare consequences of transactions frictions, if the interest-rate lower bound has already
been taken account of (as in Rotemberg and Woodford, 1997, 1999a, or Woodford, 1999a).
Furthermore, the conclusions reached in those analyses are the same as would have been
obtained if transactions frictions were allowed for, but the lower bound were treated as
slightly less of a constraint ” as for example if one assumed slightly less variability or a
slightly higher average value of the natural rate of interest.

4.3 Asymmetric Disturbances

Next we consider the consequences of real disturbances that, unlike those considered in
section 2, do not have identical e¬ects upon demand and supply conditions for all goods.
Instead, we wish to allow for disturbances of kinds that would a¬ect equilibrium relative
prices even in the case that all prices were fully ¬‚exible. In the presence of such disturbances,
it is generally not possible to arrange for the equilibrium of an economy with sticky prices
to reproduce the state-contingent resource allocation of a ¬‚exible-price economy simply by
choosing a monetary policy that stabilizes an aggregate price index. We shall also allow
for asymmetries between sectors in that we shall no longer assume that the frequency of
price adjustments must be the same for all goods. (This introduces another reason for
di¬erent sectors of the economy to be di¬erentially a¬ected by shocks.) What are appropriate
stabilization goals in such a case?
The analysis here generalizes the work of Aoki (2001), and is essentially a closed-economy
interpretation of the analysis of Benigno (1999).48 We return again to the model with
asymmetric disturbances described in section xx of chapter 3. Our welfare measure is again
given by the utility of the representative household, that is once again of the form (2.2),
except that now Yt is a (time-varying) CES aggregate of the sectoral indices of aggregate
In the model of Benigno, there are two countries that specialize in the production of di¬erent sets of
goods, and real disturbances may have di¬erential e¬ects upon the markets for goods produced in a given
country. Here the two sets of goods are instead interpreted as simply two sectors of a single national economy.
However, our conclusions with regard to appropriate stabilization objectives directly follow from his analysis
of stabilization objectives for a two-country monetary union.

demand Y1t and Y2t , and the disturbances a¬ecting the disutility of output supply (2.4) are
now allowed to be di¬erent in the case of goods i in di¬erent sectors.
A second-order expansion of the equation de¬ning Yt as a function of the sectoral demand
indices is of the form

ˆ ˆˆ ˆ ˆ
= 12 nj (1 + · ’1 •jt )Yjt + n1 n2 (1 ’ · ’1 )(Y2t ’ Y1t )2 + t.i.p. + O(||ξ||3 ),
Yt = (4.29)

where nj is the fraction of the goods that belong to sector j, · is the elasticity of substitution
between the composite products of the two sectors (assumed equal to one by Benigno), and
•jt is the exogenous disturbance to the sectoral composition of demand (all as in chapter 3).
A similar second-order expansion of the index of sectoral demand is of the form

Yjt = Eij yt (i) + (1 ’ θ’1 )varj yt (i) + O(||ξ||3 ),
ˆ iˆ (4.30)

for each of the sectors j = 1, 2, generalizing (2.11). Here we introduce the notation Eij (·)
and varj (·) for the mean and variance of the distribution of values for the di¬erent goods i

belonging to a given sector j.
Similarly, in the case of any good i in sector j, a second-order expansion of the disutility
of output supply can be written in the form

v (yt (i); ξt ) = Y uc (1 ’ ¦)ˆt (i) + (1 + ω)ˆt (i)2 ’ ωqjt yt (i) + t.i.p. + O(||ξ||3 ),
˜ y y ˆ (4.31)

generalizing (2.9), where now
vyξ ξt
qjt ≡ ’ ¯
Y vyy
represents the sector-speci¬c variation in the level of output required to maintain a constant
marginal disutility of supply. Integrating (4.31) over the goods i belonging to sector j, and
using (4.30) to eliminate Eij yt (i), we obtain

1 1
¯ ˆ ˆ2 ˆ
v (yt (i); ξt )di = nj Y uc (1 ’ ¦)Yjt + (1 + ω)Yjt ’ ωqjt Yjt + (θ’1 + ω)varj yt (i)
˜ iˆ
2 2

+t.i.p. + O(||ξ||3 ).

Then summing over the two sectors and using (4.29) to eliminate nj Yjt , we obtain
 2
¯ uc (1 ’ ¦)Yt + 1 (1 + ω)Y 2 ’
ˆ ˆ ˆˆ
nj (ωqjt + · ’1 •jt )Yjt
v (yt (i); ξt )di = Y
˜ t
 2
0 j=1

1 1
ˆ ˆ nj varj yt (i) + t.i.p. + O(||ξ||3 ),
+ n1 n2 (· ’1 + ω)(Y2t ’ Y1t )2 + (θ’1 + ω) iˆ (4.32)

2 2 j=1

generalizing (2.10).
Combining (2.8) and (4.32), we ¬nally obtain
 2
1 ’1
¯ ˆ ˆ ˆˆ
= Y uc ¦Yt ’ (σ + ω)Yt + (σ ’1 gt + ωqjt + · ’1 •jt )Yjt
 2 j=1

1 1
ˆ ˆ varj yt (i) + t.i.p. + O(||ξ||3 )
’ n1 n2 (· ’1 + ω)(Y2t ’ Y1t )2 ’ (θ’1 + ω) iˆ 
2 2 j=1
± 
¯  
Y uc j
’1 —2 ’1 2 ’1
= ’ (σ + ω)(xt ’ x ) + n1 n2 (· + ω)xRt + (θ + ω) vari yt (i)
2 

+t.i.p. + O(||ξ||3 ), (4.33)

ˆ ˆ
generalizing (2.12). Here once again xt ≡ Yt ’ Ytn is the aggregate output gap and we
ˆ ˆn
correspondingly de¬ne the relative output gap xRt ≡ x2t ’ x1t , where xjt ≡ Yjt ’ Yjt is the
gap for sector j. In deriving (4.33) from the expression above, we use the fact that the
de¬nitions of the aggregate and sectoral natural rates of output in chapter 3 imply that
ω)Ytn ’1
(σ + = σ gt + nj qjt ,

ˆn ˆn
(· ’1 + ω)(Y2t ’ Y1t ) = · ’1 (•2t ’ •1t + +ω(q2t ’ q1t ).
ˆ ˆ

Using the sectoral demand equation to write relative sectoral demand as a function of relative
sectoral price, and the demand equation for an individual good to write sectoral output
dispersion as a function of sectoral price dispersion this can alternatively be written as
Y uc
(σ ’1 + ω)(xt ’ x— )2 + n1 n2 ·(1 + ω·)(ˆRt ’ pn )2
Ut = ’ p ˆRt

varj log pt (i) + t.i.p. + O(||ξ||3 ).
+θ(1 + ωθ) (4.34)

Here pRt ≡ log(P2t /P1t ) is again the (log) sectoral relative price and pn its “natural” value
ˆ ˆRt
(i.e., the equilibrium relative price under full price ¬‚exibility, a function solely of the asym-
metric exogenous disturbances).
This quadratic approximation to the utility ¬‚ow each period is valid regardless of our
assumptions about pricing. If we assume Calvo-style staggered price-setting in each sector,
then the measure of sectoral price dispersion

∆j ≡ varj log pt (i)
t i

evolves according to the approximate law of motion

∆j = ±j ∆j + πjt + O(||ξ||3 ),
t t’1
1 ’ ±j

where πjt is the rate of price in¬‚ation in sector j and ±j is the fraction of prices that remain
unchanged each period in sector j. Summing over time we then obtain
∞ ∞
β t πjt + t.i.p. + O(||ξ||3 ).
β =
(1 ’ ±j )(1 ’ ±j β) t=0

Using this substitution for the price dispersion terms in (4.34), we again ¬nd that discounted
lifetime utility can be approximated by (2.22), where now the period loss function is of the
wj πjt + »x (xt ’ x— )2 + »R (ˆRt ’ pn )2 ,
Lt = p ˆRt (4.35)

generalizing (2.23). Here the weights (normalized so that w1 + w2 = 1) are given by

nj κ κ n1 n2 ·(1 + ω·)
wj ≡ > 0, »x ≡ > 0, »R ≡ »x > 0,
σ ’1 + ω
κj θ

where the coe¬cients κj are de¬ned as in equation (xx) of chapter 3, and

κ ≡ (n1 κ’1 + n2 κ’1 )’1 > 0
1 2

is a geometric average of the two.
We now ¬nd that deadweight loss depends not only upon the economy-wide average rate
of in¬‚ation, but on the rate of in¬‚ation in each of the sectors individually; the relative weight

on in¬‚ation variations in sector j is greater the larger the relative size of this sector, and also
the smaller the relative value of κj , which measures the degree of price stickiness in sector j.
(A smaller value of κj indicates slower price adjustment in sector j; κj is unboundedly large
in the limit of perfectly ¬‚exible prices in sector j.) The relative weight on aggregate output-
gap variations depends as before on the measure κ of the overall degree of price stickiness in
the economy. Finally, misalignments of the relative price between the two sectors (relative
to what it would be under fully ¬‚exible prices) also distort the allocation of resources.49 The
relative weight on this stabilization objective is greater the larger the elasticity of substitution
· between the products of the two sectors.
In general, it is not possible to simultaneously satisfy all of these stabilization objectives.
In particular, if the natural relative price pn varies over time, it is not possible simultaneously
to stabilize in¬‚ation in both sectors and to eliminate gaps between the relative price and its
natural value. Accordingly, in this case we must consider second-best optimal policies. And
in general, complete stabilization of the aggregate in¬‚ation rate

πt ≡ n1 π1t + n2 π2t

is not the best available policy. This is most easily seen in the case considered by Aoki,
in which prices are fully ¬‚exible in one sector, but sticky in the other. If prices are fully
¬‚exible in sector j, κ’1 = 0, so that wj = 0. In this limiting case, we can also show that

the relative-price gap pRt ’ pn is a constant multiple of the output gap xt , regardless of
ˆ ˆRt
policy, so that the same policy completely stabilizes both gap variables; and that complete
stabilization of both gaps implies zero in¬‚ation in the sticky-price sector.
Hence all three terms in (4.35) with non-zero weights are minimized by the same policy,
one which completely stabilizes the price index for the sticky-price sector, and this is clearly
the optimal policy in this case. (Aoki interprets this policy as stabilization of an index
of “core in¬‚ation”.) Again the intuition is a simple one: such a policy achieves the same
In Benigno™s open-economy application, this corresponds to the real exchange rate, and we obtain a
welfare-theoretic justi¬cation for a real exchange-rate stabilization objective. However, it should be noted
that it is the gap between the real exchange rate and its “natural” level that should be stabilized, rather


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