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)

e˜

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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