<<

. 3
( 3)



than the real exchange rate itself.
64 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

allocation of resources as would occur under complete price ¬‚exibility, since no suppliers
in the sticky-price sector have any desire to change their prices more frequently than they
already do. But in general, such a policy does not completely stabilize the broader price
index, despite the fact that an alternative policy exists that would do so. For if pRt is to
ˆ
track pn while the sticky-price index remains constant, there must be a variable in¬‚ation
ˆRt
rate in the ¬‚exible-price sector.
There is one case in which complete stabilization of πt continues to be optimal even in
the presence of relative-price disturbances; this is the case in which prices are equally sticky
in the two sectors (in addition to the sectors being symmetrical in the other ways assumed
in chapter 3). When ±1 = ±2 , so that κ1 = κ2 , w1 = w2 = 1/2, and the loss function can
alternatively be written
1
Lt = πt + (ˆRt ’ pR,t’1 )2 + »x (xt ’ x— )2 + »R (ˆRt ’ pn )2 .
2
p ˆ p ˆRt
4
As shown in chapter 3, in this symmetric case, stabilization of the aggregate in¬‚ation rate
πt is equivalent to stabilization of the aggregate output gap xt , while the relative price pRt
ˆ
evolves in the same way regardless of monetary policy. Thus while it is not possible for any
policy to reduce all terms in Lt to zero each period (since pRt = pn in general), a policy
ˆ ˆRt
that completely stabilizes πt reduces the value of each term to the greatest extent possible,
and so is optimal.
More generally, Benigno ¬nds that a policy that completely stabilizes an appropriately
weighted average of the sectoral in¬‚ation rates,
targ
πt ≡ φπ1t + (1 ’ φ)π2t , (4.36)

typically provides a reasonably good approximation to optimal policy, if the weight 0 ¤ φ ¤ 1
is properly chosen. (We have just described cases in which each of the values φ = 0, 1/2, or
1 is optimal, suggesting the interest of this general family of rules.) This can be illustrated
in a calibrated example.
Suppose that n1 = n2 = 1/2, · = 1, and let β, σ, κ, ω and θ take the values reported in
Table 4.1, derived from the study of Rotemberg and Woodford (1997).50 The implied values
4. EXTENSIONS OF THE BASIC ANALYSIS 65


Table 6.3: Calibrated parameter values for the quarterly model used for Figure 6.2.

Additional structural parameters
n1 , n2 0.5
· 1
Shock process
ρ(ˆn )
pR 0.8
sd(ˆn )
pR 1.67
Loss function
x— 0
»x .048
»R .028



of ±1 and ±2 are then derived from these coe¬cient values for an arbitrary choice of the
relative weight 0 ¤ w2 ¤ 1. (In the case that w2 = 0.5 is chosen, ±1 = ±2 , and the common
value of ± is the one reported in Table 4.1.) This allows us to vary the assumed relative
stickiness of prices in the two sectors between the two extremes of complete ¬‚exibility in
sector 2 (w2 = 0) and complete ¬‚exibility in sector 1 (w2 = 1), while assuming the same
overall degree of price stickiness (as measured by κ). Note that the assumed coe¬cients »x
and »R in the loss function (4.35) remain the same as we vary w2 ; the values implied by the
above calibration are indicated in Table 6.3.51 The tensions between alternative stabilization
objectives just discussed exist only insofar as the natural relative price pn is not constant; for
ˆRt
purposes of illustration, we assume that this follows an AR(1) process with an autoregressive
coe¬cient of 0.8. The assumed variance of the innovations in this process do not matter for
our results (all of our expected losses are proportional to this assumed variance), so it is set
equal to 1 without loss of generality.52
The solid line in Figure 6.2 plots the minimum attainable value for the expected dis-
50
Note that the model of Rotemberg and Woodford can be interpreted as a two-sector model in which
±1 = ±2 . Because no data on relative prices are used in that study, it provides no estimate of ·.
51
As in Table 6.1, the reported weights »x and »R are sixteen times as large as those implied by the
formulas given above, so that they correspond to the relative weights on these terms in the loss function
when the in¬‚ation rate is measured as an annual rather than a quarterly rate.
52
Note that an innovation variance of 1 implies a variance of 1/1 ’ (0.8)2 for the disturbance process, or
a standard deviation of 1.67.
66 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

5


4.5


4


3.5


3


2.5
>




E[L]

2


1.5


1 inf. target
gap target
opt. index
0.5 optimal


0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w2



Figure 6.2: Welfare losses under alternative policies with asymmetric disturbances.

ˆ
counted period loss E[L], again de¬ned as in (4.28), for each possible choice of the coe¬cient
w2 measuring the relative stickiness of sector 2.53 Only in the case that w2 takes one of the
extreme values (i.e., prices are completely ¬‚exible in one sector or the other) is the minimum
attainable value zero, for only in this case is it is possible for monetary policy to achieve the
allocation of resources associated with complete price ¬‚exibility. The expected loss under a
policy that strictly targets (completely stabilizes) aggregate in¬‚ation is instead shown by the
dashed line. The two lines coincide only when w2 = 0.5, the only case in which aggregate-
in¬‚ation targeting is optimal. Whenever the degrees of price stickiness in the two sectors
di¬er, aggregate-in¬‚ation targeting results in greater losses, and the losses associated with
this policy are greater the greater the degree of asymmetry, whereas the unavoidable losses
are smaller the greater the asymmetry. The expected losses resulting from strict targeting

53
The precise de¬nition that we assume of constrained-optimal policy in cases like this, as well as the
Lagrangian method that we use to characterize it, are explained in chapter 7.
4. EXTENSIONS OF THE BASIC ANALYSIS 67

1


0.9


0.8


0.7


0.6


0.5
φ




0.4


0.3


0.2
optimal index
gap target
0.1


0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w2



Figure 6.3: The optimal price index to stabilize as a function of the relative stickiness of prices
in the two sectors. The dotted line shows the price index that is equivalent to output-gap
targeting.


of the weighted index (4.36), where the weight φ is optimally chosen for each value of w2 , is
instead shown by the dotted line. This coincides with the solid line when w2 = 0, 0.5, or 1
(the three special cases already discussed), but not otherwise; thus these are the only cases
in which optimal policy is exactly described by a simple targeting rule of this kind. But even
in the other cases, the dotted line is only slightly above the solid line; thus a rule of this kind
is a reasonable approximation to optimal policy, if φ is properly chosen.

The optimal value of φ for each value of w2 is shown in Figure 6.3. The optimal value
are φ = 1 when w2 = 0, φ = 0.5 when w2 = 0.5, and φ = 0 when w2 = 1, for reasons
already discussed. More generally, the optimal φ is a decreasing function of w2 , as the
special cases had already suggested: the near-optimal policy stabilizes an in¬‚ation measure
that puts more weight on prices in the sector where prices are stickier. As Aoki suggests,
68 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

this provides theoretical justi¬cation for a policy that targets “core in¬‚ation” rather than
the growth of a broader price index, and o¬ers a theoretical criterion for the construction of
such an index. It also explains why it is not appropriate to target an in¬‚ation measure that
includes “asset price in¬‚ation” along with goods price increases, as is sometimes proposed;
even if asset prices are also prices, and also can be a¬ected by monetary policy, they are
among the prices that are most frequently adjusted in response to new market conditions,
and so their movements do not indicate the kind of distortions which we seek to minimize.
The choice of the right rule of the form (4.36) depends, however, upon on an accurate
estimate of the relative stickiness of prices in the two sectors. A simple rule that performs
relatively well regardless of the value of w2 is strict targeting of the output gap: using
monetary policy to ensure that xt = 0 at all times. The expected loss resulting from this
rule is shown in Figure 6.3 by the dotted solid line. This policy is only fully optimal in three
cases (w2 = 0, 0.5 or 1), and otherwise it is somewhat worse than the best weighted-in¬‚ation
targeting rule; but it is relatively good over the entire range of possible values of w2 (unlike
the equal-weighted in¬‚ation targeting rule, for example), despite involving no coe¬cients
that must be assigned values that vary with the changing value of w2 .
In fact, the output-gap targeting rule is reasonably successful regardless of the value of
w2 because it incorporates the principle of stabilizing more the prices that are most sticky.
If one multiplies the in¬‚ation equation for sector j (equation (xx) of chapter 3) by wj and
sums over j, one obtains

πt = κxt + βEt πt+1
¯ ¯

where

πt ≡ w1 π1t + w2 π2t .
¯

From this it follows that output-gap stabilization is equivalent to stabilization of πt . This is
¯
thus a policy of the form (4.36), with φ = w1 . Thus the weights are automatically adjusted
to place less weight on the prices in the sector with more ¬‚exible prices. This is not done in
precisely the optimal way (see the comparison of this function of w2 with the optimal one in
4. EXTENSIONS OF THE BASIC ANALYSIS 69

Figure 6.3), but this simple rule is not too di¬erent from the optimal member of the family
(4.36) for any value of w2 .

Thus we conclude that even in the case of asymmetric disturbances, stabilization of a
price index provides a fairly good recipe for monetary policy, as long as the right price index
is chosen. On the other hand, this does not mean that seeking to stabilize the output gap
cannot be a sound approach as well, as long as the output gap is properly measured; in fact, in
the absence of information about which sector™s prices are more sticky, an output-gap target
is a more robustly desirable simple policy rule. The choice between the two approaches,
then, must turn on which sorts of information the central bank is able to rely upon with
more con¬dence. In practice, banks are likely to be more con¬dent that they can estimate
the relative stickiness of di¬erent prices with some con¬dence than that they can accurately
track the natural rate of output in real time; for the former question can be studied using past
data, while the latter depends upon correctly judging the economy™s current state despite
the possible occurrence of a vast number of di¬erent types of disturbances. For this reason,
one may still conclude that an appropriately chosen in¬‚ation target represents a sensible
approach to policy.



4.4 Sticky Wages and Prices

Similar issues arise if we assume that wages as well as prices are sticky. Once again, some
types of real disturbances will modify the “natural” relative price, i.e., the equilibrium real
wage under ¬‚exible wages and prices, so that no monetary policy can eliminate all of the
distortions resulting from wage and price stickiness. Here we analyze welfare-theoretic sta-
bilization goals for the model with sticky wages and prices set out in section xx of chapter
3; our results essentially recapitulate those of Erceg et al. (2000).

In this model, all ¬rms hire the same composite labor input; nonetheless, there exist
di¬erential demands for the labor supplied by di¬erent households j, owing to wage dispersion
(as a result of staggered wage adjustment). The demand for each di¬erentiated type of labor
70 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

is given by
’θw
wt (j)
ht (j) = Ht ,
Wt
where θw > 1 is the elasticity of substitution among di¬erent types of labor on the part of
¬rms. A quadratic expansion of v(ht (j); ξt ), integrated over the continuum of di¬erent types
of labor, yields
1
1
¯ˆ
¯ ˆ ˆ
v(ht (j); ξ)dj = Hvh Ht + (1 + ν)Ht2 ’ ν ht Ht
2
0

1
+ θw (1 + νθw )varj log wt (j) + t.i.p. + O(||ξ||3 ), (4.37)
2
where once again
¯
Hvhh
ν≡ > 0.
vh
The aggregate demand for the composite labor input Ht is in turn given by
1
f ’1 (yt (i)/At )di,
Ht = (4.38)
0

integrating over the demands of each of the ¬rms i. Using a quadratic approximation to an
individual ¬rm™s labor demand
1
¯
f ’1 (yt (i)) = H 1 + φˆt (i) + (1 + ωp )φˆt (i)2 + O(||ξ||3 ),
y y
2
where once again
¯
Y ff
φ ≡ ¯ > 1, ωp ≡ ’ > 0,
(f )2
Hf
we can expand (4.38) as
1
ˆ ˆ ˆ
H = φ (Ei yt (i) ’ At ) + (1 + ωp ’ φ)φ (Ei yt (i) ’ At )2
ˆ ˆ
2
1
+ (1 + ωp )φ vari yt (i) + O(||ξ||3 )
ˆ
2
1
ˆ ˆ ˆ ˆ
= φ (Yt ’ At ) + (1 + ωp ’ φ)φ (Yt ’ At )2
2
1
+ (1 + ωp θp )θp φ vari log pt (i) + O(||ξ||3 ). (4.39)
2
In the second line we have again used (2.11) to eliminate Ei yt (i) and (2.14) to write vari yt (i)
ˆ ˆ
as a function of the dispersion of individual goods prices, and adopted the notation θp for
4. EXTENSIONS OF THE BASIC ANALYSIS 71

ˆ
the elasticity of demand across di¬erentiated goods. Substituting (4.39) for Ht in (4.37),
and then combining this with (2.8), we obtain the expansion
¯
Y uc
(σ ’1 + ω)(xt ’ x— )2 + θp (1 + ωp θp )vari log pt (i)
Ut = ’
2

+θw φ’1 (1 + νθw )varj log wt (j) + t.i.p. + O(||ξ||3 ), (4.40)

where xt and x— are again de¬ned as in our basic (¬‚exible-wage) model.
This expression for the utility ¬‚ow each period holds regardless of our assumptions about
wage- and price-setting. If, following Erceg et al., we assume Calvo-style staggered adjust-
ment of both wages and prices, we again obtain a representation of the form (2.22), where
now the period loss function is of the form

Lt = »p πt + »w πwt + »x (xt ’ x— )2 .
2 2
(4.41)

Here the weights (normalized so that »p + »w = 1) are given by
’1
θw φ’1 ξw
’1
θp ξp
»p = > 0, »w = > 0,
θp ξp + θw φ’1 ξw
’1 ’1 θp ξp + θw φ’1 ξw
’1 ’1


σ ’1 + ω
»x = > 0,
θp ξp + θw φ’1 ξw
’1 ’1

where ξw , ξp > 0 are again the elasticities appearing in equations (xx) “ (xx) of chapter 3.
Note that the relative weight on output-gap stabilization is again related to the slope
of the aggregate supply relation in a similar way as in the basic (¬‚exible-wage) model. In
particular, in the case that θw φ’1 = θp , the expression for »x reduces to κ/θ, where κ > 0
is the slope of the short-run Phillips-curve relation between a weighted average of price and
wage in¬‚ation and the output gap (equation (xx) of chapter 3) and θ > 1 is the common
value of θp and θw φ’1 . The weight on in¬‚ation stabilization is now, however, divided between
a price-in¬‚ation stabilization goal and a wage-in¬‚ation stabilization goal; the relative weights
on each depend on the relative stickiness of wages and prices (as indicated by the relative
’1 ’1
sizes of ξw and ξp ). If only prices are sticky, only price in¬‚ation matters (»w = 0), and
optimal policy involves complete stabilization of prices (as this also completely stabilizes
72 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

the output gap, in that case). If instead only wages are sticky, only wage in¬‚ation matters
(»p = 0), and optimal policy involves complete stabilization of wages (which also completely
stabilizes the output gap, in this case). In the intermediate case, both goals matter, and
complete stabilization is impossible in the presence of ¬‚uctuations in the “natural real wage”
n
wt .
An intermediate case in which optimal policy is nonetheless simple to characterize is the
special case in which θw φ’1 = θp and κw = κp , where κw , κp are the coe¬cients indicating
the e¬ects of output-gap variations on wage and price in¬‚ation respectively (see equations
(xx) “ (xx) of chapter 3). Note that (4.41) can alternatively be written

Lt = πt + »p »w (log wt ’ log wt’1 )2 + »x (xt ’ x— )2 ,
¯2 (4.42)

where
πt ≡ »p πt + »w πwt
¯

and wt is the real wage. In the case that κw = κp , the evolution of the real wage is independent
of monetary policy, as shown in chapter 3; hence the middle term in (4.42) is irrelevant. In
the case that θw φ’1 = θp , the weights »p , »w de¬ne a weighted average of price and wage
in¬‚ation that is stabilized if and only if the output gap is stabilized. (This follows from
equation (xx) of chapter 3.) Hence the two terms in (4.42) that can be a¬ected by monetary
policy are both minimized by the same policy, one that completely stabilizes πt .54
¯
This last result suggests that even when both wages and prices are sticky, and Lt cannot
be reduced to zero by any policy, a policy that stabilizes a weighted in¬‚ation measure of the
form
targ
πt ≡ φπ1t + (1 ’ φ)π2t , (4.43)

may be desirable. In fact, except in the special case just described, fully optimal policy
cannot be represented by such a simple rule, but numerical experimentation suggests that a
targeting rule of this kind can nonetheless be nearly optimal, if the weight φ is appropriately
chosen.
54
Even when x— = 0, one can show that this is the optimal policy from the “timeless perspective” that is
explained further in chapters 7 and 8. See in particular section xx of chapter 8.
4. EXTENSIONS OF THE BASIC ANALYSIS 73


Table 6.4: Calibrated parameter values for the quarterly model used for Figure 6.4.

Additional structural parameters
θp 7.88
θw φ’1 7.88
’1 ’1
ξw + ξp 26.7
κ .024
Shock process
ρ(wn )
ˆ 0.8
sd(wn )
ˆ 1.67
Loss function
x— 0
»x .048



This can be illustrated by a calibrated example. We again let the parameters β, σ, ωw , ωp
and θp take the values estimated by Rotemberg and Woodford (1997) and reported in Table
4.1,55 and in the absence of any direct evidence we assume the same value for θw φ’1 as
for θp . We wish to let the assumed relative degree of wage as opposed to price stickiness
vary between the two extremes of full wage ¬‚exibility and full price ¬‚exibility on the other;
but we assume a given degree of overall nominal rigidity by ¬xing the value of the slope
coe¬cient κ in the generalized Phillips-curve relation described by equation (xx) of chapter
3. The value assumed for κ is again the one shown in Table 4.1. (Note that in the case
of wage ¬‚exibility, as assumed by Rotemberg and Woodford, their estimated coe¬cient κ
corresponds to the κ of this model, rather than to κp .) This implies a ¬xed value for the sum
ξw + ξp , shown in Table 6.4,56 though we wish to vary the relative contributions of the two
’1 ’1

’1 ’1
terms to this sum (between ξw = 0 when wages are fully ¬‚exible to ξp = 0 when prices
are). We parameterize our assumption about the relative degree of wage and price stickiness
’1 ’1 ’1
by the value of »w = ξw /(ξw + ξp ), which we allow to vary over the interval from zero to
’1 ’1
one. The assumed value of ξw + ξp implies a value for »x that is independent of the choice

55
Note that the parameter θp is referred to simply as θ in the previous table.
56
Note that the estimates of Amato and Laubach (2001), reported in Table 4.2, would instead imply a
value of 32.4, or a slightly greater overall degree of nominal rigidity than we assume here.
74 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

25




20 inf. target
gap target
opt.index
optimal


15
>




E[L]

10




5




0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
»
w



Figure 6.4: Welfare losses under alternative policies with sticky wages and prices.


of »w ; once again this is the same as in Table 6.1.

The only exogenous disturbance that matters for the optimal state-contingent evolution
ˆn
of wages and prices is the “natural real wage” process wt . For the sake of illustration, we
assume an AR(1) process with the properties listed in Table 6.4. (Note that once again the
assumed variance of the disturbance has no e¬ect on our results, other than to scale up the
expected value of each term of the loss function equally.)

ˆ
The solid line in Figure 6.4 then shows the minimum attainable value of E[L] associated
with these parameter values, for each possible value of »w . As in Figure 6.3, distortions can
be reduced to zero only in the two extreme cases (here corresponding to full wage ¬‚exibility
when »w = 0 and full price ¬‚exibility when »w = 1). The dashed line instead shows the
expected discounted loss associated with a policy of complete stabilization of the rate of
price in¬‚ation πt ; this policy is optimal only if »w = 0 (completely ¬‚exible wages), and grows
4. EXTENSIONS OF THE BASIC ANALYSIS 75

1


0.9


0.8


0.7


0.6


0.5
φ




0.4


0.3


optimal index
0.2
gap target

0.1


0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
»
w



Figure 6.5: The optimal index to stabilize as a function of the relative stickiness of wages
and prices. The dotted line shows the generalized in¬‚ation-targeting rule that is equivalent
to output-gap targeting.


worse the greater the relative stickiness of wages. The expected loss achievable by instead
targeting a weighted average of wage and price in¬‚ation, when the weight φ is optimally
chosen, is instead indicated by a dotted line. This only coincides exactly with the solid
line (indicating that the optimal policy belongs to this simple family) at three points, when
»w = 0, 0.48, or 1. (The intermediate special case is that in which »w = (σ ’1 +ωw )/(σ ’1 +ω),
so that κw = κp .) However, the dotted line can barely be distinguished from the solid line
over the entire interval; hence a well-chosen policy of this form o¬ers a good approximation
to optimal policy in all cases. The optimal choice of the weight φ is plotted as a function
of »w in Figure 6.5. Note that it is monotonically decreasing: the optimal weight to put on
price as opposed to wage in¬‚ation is lower the greater the relative stickiness of wages.

Once again, we also ¬nd that output-gap targeting is a fairly robust policy that does
76 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

not require knowledge of the exact degrees of wage and price stickiness. The expected losses
associated with this policy are shown by the dotted solid line in Figure 6.4. Output-gap
targeting is only fully optimal in the three special cases just mentioned, and otherwise it is
also not as good as the best policy within the generalized in¬‚ation-targeting family. In fact,
output-gap targeting is equivalent to a generalized in¬‚ation-targeting policy, corresponding
to an index with φ = »p = 1 ’ »w . But as shown in Figure 6.5, this choice of φ is not too
di¬erent from the optimal one, regardless of the relative stickiness of wages and prices; this is
exactly why output-gap targeting is a relatively robust policy prescription. Yet once again,
this observation is only of practical importance if direct measures of the output gap in real
time are fairly accurate; in practice, the best way to stabilize the output gap may well be to
seek to stabilize an appropriate weighted average of wage and price in¬‚ation.


4.5 Time-Varying Tax Wedges or Markups

Finally, as discussed above, complete stabilization of in¬‚ation ceases to be optimal, even when
this implies that aggregate output should perfectly track the equilibrium level of output
under ¬‚exible wages and prices, if the gap between this latter quantity and the e¬cient
level of output is not a constant. In such a case, unlike those considered in the previous
two subsections, there are time-varying distortions that do not result from delays in the
adjustment of any kind of prices; thus there exists no way of de¬ning the price index to be
stabilized that can make price stability an adequate proxy for welfare-maximizing policy.
Hence the challenge to price stability as a goal of policy is strongest in this case.
For simplicity, let us assume again a model in which wages are ¬‚exible, all prices are
equally sticky, and all real disturbances a¬ect the demand for and production costs of all
goods symmetrically, as in the models considered in section 2. Let us suppose now, however,
that the relative price pt (i)/Pt at which the supplier of a di¬erentiated good i would be
willing to supply a quantity yt (i) of that good, under conditions of full price ¬‚exibility, is
given by
˜
1 vy (yt (i); ξt )
˜
˜
p(yt (i), Yt ; ξt , ¦t ) ≡
˜ , (4.44)
˜
1 ’ ¦t uc (Yt ; ξt )
˜
4. EXTENSIONS OF THE BASIC ANALYSIS 77

where ¦t is an exogenous time-varying composite distortion. Here the term ¦t includes the
distortions resulting from the market power of the supplier of each di¬erentiated good and
from the existence of distorting taxes on output, consumption, employment or wage income.
We have allowed for these distortions earlier, but assumed them to be constant over time
(and represented their composite e¬ect by a constant factor ¦ ≥ 0); we now allow them
to vary over time, but assume that their variation is exogenous. The composite distortion
¦t can also include the e¬ects of a wedge between the representative household™s marginal
rate of substitution between leisure (less supply of the labor used in producing good i) and
consumption and the real wage demanded from producers of good i, as a result of (possibly
time-varying) market power in labor supply.
The ¬‚exible-price equilibrium level of output for each good Ytf is then implicitly de¬ned
at each time by the relation
˜
p(Ytf , Ytf ; ξt , ¦t ) = 1.
˜ (4.45)

The time-varying e¬cient level of output Yte is instead implicitly de¬ned by the relation
˜
vy (Yte ; ξt )
˜
= 1. (4.46)

uc (Yt ; ξt )
˜
Note that Yte is a function solely of the exogenous disturbances to tastes, technology and
˜
government purchases at date t, re¬‚ected by the vector ξt . The ¬‚exible-price equilibrium level
of output Ytf is also a function solely of exogenous disturbances at date t; but the exogenous
disturbances that a¬ect it include all of the various disturbances that a¬ect the value of ¦t .
These latter disturbances result in ine¬cient variations in the ¬‚exible-price equilibrium. In
particular, log-linearizing both (4.45) and (4.46) and comparing terms, we ¬nd that

log Ytf = log Yte ’ (ω + σ ’1 )’1 ¦t + O(||ξ||2 ),

˜
where ||ξ|| is now a bound on the size of both disturbances ξt and ¦t .
Which of these ideal levels of output should be de¬ned as the “natural rate”? The
de¬nition that conforms most closely to standard usage would de¬ne the natural rate of
output as the ¬‚exible-price equilibrium rate of output in the case of certain constant values
78 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

for the tax wedges and the markups due to market power in price- and/or wage-setting, where
the constant values are the long-run average values around which these exogenous series
¬‚uctuate. Under this de¬nition, the average “output gap” associated with price ¬‚exibility
” or alternatively, with zero in¬‚ation ” will again be zero, as above. But price ¬‚exibility
would no longer imply a zero output gap at all times; for the ¬‚exible-price equilibrium level
of output varies in response to variations in ¦t , while our “natural rate” of output (like the
e¬cient level) does not.57
The natural rate of output de¬ned in this way also di¬ers from the e¬cient level of
output, although (up to our log-linear approximation) the di¬erence is a constant. For the
natural rate Ytn is implicitly de¬ned each period by the relation

˜¯
p(Ytn , Ytn ; ξt , ¦) = 1,
˜ (4.47)

¯
where ¦ represents the long-run average value of ¦t . Log-linearization of this indicates that

ˆ
log Ytn = log Ytf + (ω + σ ’1 )’1 ¦t + O(||ξ||2 ), (4.48)

ˆ ¯
where ¦t ≡ ¦t ’ ¦t , so that

¯
log Ytn = log Yte ’ (ω + σ ’1 )’1 ¦ + O(||ξ||2 ).

As a result, it continues to be desirable to stabilize the output gap xt ≡ log(Yt /Ytn ). At the
same this de¬nition (rather than de¬ning the output gap relative to Yte directly) has the
advantage that in terms of this output gap measure, our aggregate supply relation continues
to have a zero constant term.58
57
Note that de¬ning the natural rate in a way that implies that it does not vary in response to variation in
the size of the tax wedge is consistent with our de¬nition in the case of monetary frictions in section xx above.
There we de¬ned the natural rate as what equilibrium output would be, given current tastes, technology,
and government purchases, if wages and prices were ¬‚exible and the interest-rate di¬erential between non-
monetary and monetary assets were at its steady-state level. While time-variation in the distortion associated
with an interest-rate di¬erential may cause variation in the ¬‚exible-price equilibrium level of output, our
de¬nition implies no variation in the natural rate. Here we treat the consequences for aggregate supply of
time-variation in the tax wedge in a similar way.
58
This choice allows us to follow the literature, such as Clarida et al. (1999) in representing a non-zero
average gap between the level of output consistent with zero in¬‚ation and the e¬cient level by a non-zero
target value x— in the loss function, rather than by a constant in the aggregate-supply relation, while at the
same time representing time variation in this gap by a random term in the aggregate-supply relation, rather
than a time-varying target for the output gap that appears in the loss function.
4. EXTENSIONS OF THE BASIC ANALYSIS 79

Our derivation of the aggregate supply relation under the assumption of Calvo pricing in
chapter 3 continues to hold under these assumptions, except that the output gap must be
replaced by
ˆ ˆ ˆ ˆ ˆ
Yt ’ Ytf = Yt ’ Ytn + (ω + σ ’1 )’1 ¦t ,

using (4.48), as it is the gap between actual output and the ¬‚exible-price level of output that
determines real marginal cost. Hence (2.19) becomes instead

πt = κxt + βEt πt+1 + ut , (4.49)

where
κ ˆ
ut ≡ ¦
’1 t
ω+σ
is a composite exogenous disturbance that Clarida et al. (1999) refer to as a “cost-push
shock”.59
Because under de¬nition (4.47), the natural rate of output continues to be the same
function of preferences, technology and government purchases as before, our derivation above
of the normalized utility-based loss function (2.23) still applies, with » > 0 and x— ≥ 0 de¬ned
as before. But the appearance of the random term ut in the constraint (4.49) implies that
it is no longer possible to simultaneously stabilize both in¬‚ation and the welfare-relevant
output gap.
The tradeo¬ that exists between these two stabilization goals in the presence of ine¬-
cient supply shocks can be illustrated, for a calibrated numerical example, using a ¬gure
of a kind popularized by Taylor (1979b). The coe¬cients β and κ of the AS relation are
again calibrated as in Table 6.1; the exogenous disturbance ut is assumed to be an AR(1)
process with serial correlation ρ(u) = 0.8 and an innovation variance of one.60 (Once again,
59
The terminology is not entirely satisfactory, however, as there is no necessary connection between shocks
that a¬ect in¬‚ation by increasing costs of production and time variation in the degree of ine¬ciency of the
natural rate of output. Technology shocks, energy price shocks, or variations in real wage demands may
all shift the aggregate supply curve in terms of the output relative to trend, without changing the relation
between in¬‚ation and the output gap as we have de¬ned it.
60
To be more precise, it is assumed that a one-standard-deviation innovation in the “cost-push shock”
raises the annualized in¬‚ation rate 4πt by one percentage point. In ¬gure 6.6, V [π] refers to the variability
of the annualized in¬‚ation rate, and the units on both axes are percentage points squared.
80 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

70




60 non’inertial
efficient


50




40
V[π]




30

D

20

C


10

B B™
A
0
0 50 100 150 200 250 300 350
V[x]


Figure 6.6: The tradeo¬ between in¬‚ation and output-gap stabilization in a model with
ine¬cient supply shocks. The solid line shows the e¬cient frontier, while the dotted line
corresponds to a class of simple rules that are e¬cient among purely forward-looking policies
(see chapter 7).


the assumed variance of the single disturbance is irrelevant for our conclusions, though it
determines the scale of the axes in Figure 6.6.) We then compute the e¬cient frontier in
ˆ
V [π] ’ V [x] space, by computing the policy commitment that minimizes E[L] for arbitrary
values of » > 0 in the loss function (2.23). The values of V [π] and V [x] associated with
each possible value of » are then plotted as the lower of the two convex curves (i.e., the
one closer to the origin) in Figure 6.6.61 Point A (optimal policy when » = 0) indicates
how much variability of the output gap must be accepted in order to completely stabilize

61
The other curve shows the possible combinations of V [π] and V [x] that are attainable using a particular
family of simple targeting rules, rules that prescribe stabilization of a weighted average of in¬‚ation and
the output gap. (The equilibria achieved using such rules correspond to the “optimal non-inertial plan”
characterized in chapter 7, for alternative assumed relative weights on in¬‚ation and output-gap stabilization.)
As shown, these rules are generally not on the e¬cient frontier. This indicates the suboptimality of purely
forward-looking policy, as is discussed further in the next chapter.
4. EXTENSIONS OF THE BASIC ANALYSIS 81

in¬‚ation, while point E (optimal policy when » is unboundedly large) instead indicates how
much variability of in¬‚ation must be accepted in order to completely stabilize the output
gap. Points B, C and D instead represent optimal policies for various ¬nite, positive values
of ».

Since » > 0 in the utility-based loss function, complete stabilization of in¬‚ation is seen
not to be optimal. However, this does not mean that the optimal degree of variability in
in¬‚ation need be very great. First of all, we observe from the ¬gure that the e¬cient frontier
is quite ¬‚at in the lower right region; thus it is possible to substantially reduce the variability
of the output gap without too much variability of in¬‚ation being required. Instead, on the
upper part of the frontier, where e¬cient policies involve substantial in¬‚ation variability, the
frontier is quite steep. Thus policies on that part of the frontier would be optimal only if the
relative weight » on output-gap stabilization were quite large. Second, the value of » that
can be justi¬ed on welfare-theoretic grounds is likely to be quite small. As shown in Table
4.1, our calibrated parameter values imply a value of » = 0.05. In Figure 6.6, the point on
the e¬cient frontier that minimizes this weighted average of the two criteria (and hence that
represents optimal policy) is point B. This policy involves quite a modest degree of in¬‚ation
variability, relative, for example, to the in¬‚ation variability that would be required to fully
stabilize the output gap.

The value of » in our utility-based loss function is much smaller than the relative weight on
output-gap stabilization that is often assumed in ad hoc policy objectives in the literature
on monetary policy evaluation. A commonly assumed value would be » = 1; this would
correspond to point D in the ¬gure. We see that optimal policy involves much more stable
in¬‚ation than would be chosen under an ad hoc criterion of that kind. As another example of
a familiar policy recommendation, it has sometimes been proposed that given the existence
of “supply shocks”, and hence a necessary tension between output stabilization and in¬‚ation
stabilization, nominal GDP targeting would represent a reasonable balance between the two
goals. In our model, in the case that there are no ¬‚uctuations in Yte (so that all variations in
the output level consistent with price stability are ine¬cient, as arguments for nominal GDP
82 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

targeting typically assume), nominal GDP targeting is a policy on the e¬cient frontier, as
shown in chapter 7.62 However, for our calibrated parameter values, nominal GDP targeting
would correspond to point C on the frontier; this still involves considerably more variation
in in¬‚ation than an optimal policy under the utility-based criterion. Thus according to our
utility-based analysis, the degree of departure from complete in¬‚ation stabilization that can
be justi¬ed even in the case of ine¬cient supply shocks of signi¬cant magnitude is quite
modest.

It should also be noted that the quantitative importance of shocks of this kind is far
from clear. The literature on monetary policy evaluation has given considerable emphasis
to the tension between the goals of in¬‚ation stabilization and output stabilization created
by the existence of “supply shocks”. It is taken as well established that such shocks are an
important factor in practical monetary policy. But the mere existence of “supply shocks,”
in the sense of real disturbances that shift the short-run Phillips curve, does not imply the
existence of ine¬cient supply shocks, the kind of shifts resulting from variation in ¦t as
˜
opposed to the elements of ξt . As we have seen, the vector ξt includes a variety of types
of real disturbances that are of clear importance in actual economies, and that can easily
cause substantial ¬‚uctuations in the ¬‚exible-price equilibrium level of output; but in our
model, these ¬‚uctuations are (to a ¬rst-order approximation) also shifts in the e¬cient level
of output. While one can also think of possible sources of variation in the ¬‚exible-price
equilibrium level of output that would clearly not be e¬cient, such as variations in tax
distortions or in market power, such factors have not been clearly established as important
sources of short-run ¬‚uctuations in economic activity. Thus while it is certainly possible that
substantial disturbances of this kind occur, the matter is far from having been established.

We shall nonetheless give considerable attention in the next two chapters to the design
of optimal policies in the case that ine¬cient supply shocks occur. One reason for this is

62
The policies on the e¬cient frontier correspond to complete stabilization of log Pt + φxt , for alternative
values of φ > 0. The value of » required to make one of these policies optimal is 16κφ. In the case that
ˆ
Yte (and hence Ytn ) is a constant, so that xt = Yt , nominal GDP targeting represents the e¬cient policy
corresponding to a weight » = 16κ. In our calibrated example, this corresponds to » = 0.38.
4. EXTENSIONS OF THE BASIC ANALYSIS 83

that, as explained in chapter 8, we wish to choose policy rules that are robust to alternative
beliefs about the nature of the real disturbances to the economy ” for example, to alternative
beliefs about how frequently particular types of shocks occur. Thus we wish to consider the
possibility of ine¬cient supply shocks, and to choose policy rules that would be optimal
whether or not disturbances of this kind are signi¬cant source of macroeconomic instability
in a given economy. In order to undertake this challenge, we must consider what optimal
policy would seek to achieve in the case that such shocks occur.
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