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)

0

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

3. THE CASE FOR PRICE STABILITY 35

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

29

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).

36 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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

d

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

here.

30

See, e.g., Roberts (1995) or Clarida et al. (1999).

3. THE CASE FOR PRICE STABILITY 37

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

31

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.

32

See Woodford (1990) for justi¬cation of this relation in a variety of alternative models of the demand

for money.

38 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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,

3. THE CASE FOR PRICE STABILITY 39

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

40 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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

below.

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. EXTENSIONS OF THE BASIC ANALYSIS 41

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

ˆ

mt = ·y Yt ’ ·i (ˆt ’ ˆm ) + m

ˆ ± ±t (4.1)

t

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)

where

≡ σ ’1 + ω ’ χ·y , (4.3)

mc

·i χ

•≡ , (4.4)

mc

and

σ ’1 gt + ωqt + χ m

ˆ t

Ytn ≡ . (4.5)

mc

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.

ˆ

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

42 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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)

mc

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.

33

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.

4. EXTENSIONS OF THE BASIC ANALYSIS 43

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

’1

ˆ2

ˆ

2 2

ˆˆ

+ χmt Yt + sm σ ’1 gt mt + sm (χ + σm ) m mt + t.i.p. + O(||ξ||3 )

’1

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

umξ

m

≡ (χ + σm )’1

’1

ξt ’ σ ’1 gt

t

um

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

34

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.

44 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

¯

’1

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

¯ mumm

¯

This allows us to substitute (·i v )’1 for sm σm and χ·i v for ·y in the coe¬cients of quadratic

’1

¯ ¯

¯ ¯¯

terms,35 where v ≡ Y /m is the steady-state “velocity of money”. With these substitutions,

(4.9) can be written as

1

¯ ˆ ˆ ˆ

u(Yt , mt ; ξt ) = Y uc Yt + (1 ’ σ ’1 )Yt2 + σ ’1 gt Yt + sm mt

ˆ

2

1

ˆˆ

+ χmt Yt ’ (·i v )’1 (mt ’ m )2 + t.i.p. + O(||ξ||3 ).

¯ ˆ (4.11)

t

2

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

ˆ

1

¯ 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

2

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

t

2 2

Subtracting (2.10) from (4.12), we obtain

1

¯ 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

2

1 1

ˆ

Yt2 ’ (θ’1 + ω)vari yt (i) + t.i.p. + O(||ξ||3 )

’ ˆ (4.13)

mc

2 2

¯

Y uc

±m ¯ 2

—2

¯’1 ±

=’ mc (xt ’ x ) + v ·i (ˆt ’ ˆt + ∆)

2

+(θ’1 + ω)vari yt (i) + t.i.p. + O(||ξ||3 )

ˆ (4.14)

35

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

below.

4. EXTENSIONS OF THE BASIC ANALYSIS 45

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

t

¯ ¯

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 ,

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

»i = •»x . (4.17)

·y

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

46 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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

mc

ˆ

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 ,

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

36

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.

4. EXTENSIONS OF THE BASIC ANALYSIS 47

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

±

t

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

written

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,

48 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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)

mc

Using this to substitute for x— , the discounted loss measure can be written as

∞ ∞

¦

[π0 ’ κiˆ0 ] + E0

t

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)

mc

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

4. EXTENSIONS OF THE BASIC ANALYSIS 49

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

variability.

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

37

More generally, the requirement is that it ≥ im , but here we shall suppose that zero interest is paid on

t

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.

50 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

¬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

t

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)

t=0

2

∞ ∞

β t i2 t

E0 (1 ’ β) ¤ K E0 (1 ’ β) β it . (4.24)

t

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)

t=0

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

t

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 ,

t=0

∞

β t i2

E0 (1 ’ β) ¤ m2 ,

t

t=0

since any policy consistent with both of these also satis¬es the weaker constraints (4.23) “

(4.24).

38

By the expression it we here actually mean log(1 + it ), or ˆt + ¯.

± ±

4. EXTENSIONS OF THE BASIC ANALYSIS 51

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 ,

2

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)

µ1

i—— = ’ ¯ = Km1 ’ ¯.

± ±

2µ2

(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

ˆn

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

52 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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,

ˆ±

ˆ

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

2

± (4.27)

ˆ

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

ˆ

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

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.

4. EXTENSIONS OF THE BASIC ANALYSIS 53

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,

mc

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

ˆn

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

t

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

∞

ˆ 0

β t L0

E[L ] ≡ E (1 ’ β) (4.28)

t

t=0

40

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.

41

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.

42

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

derivations.

54 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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

0.63

mc

·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-

ˆ

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

t

ˆn

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

t=0

ˆ

We then consider the policies that minimize E[L0 ] subject to a constraint that V [i] not

43

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.

4. EXTENSIONS OF THE BASIC ANALYSIS 55

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

ˆ

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,

ˆ

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

ˆ

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

ˆ

shocks. Thus we can determine the optimal responses to shocks by minimizing E[L] with i—

ˆ

set equal to zero, which amounts to minimizing E[L0 ] + »i V [i]. This policy corresponds to

44

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.

56 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

2.5

2

1.5

^o

E[L ]

1

C

B

0.5

A

0

0 2 4 6 8 10 12 14

V[i]

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

lization.

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.

45

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.

46

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).

4. EXTENSIONS OF THE BASIC ANALYSIS 57

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.

ˆ˜

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

t

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.

58 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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

ˆ˜

E[L] when Lt is given by (4.15) rather than by L0 . As noted above, this policy minimizes a

t

ˆˆ ˆ

criterion of the form E[L], where Lt is given by (4.27). In the case of our assumed parameter

ˆ

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

ˆ

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

47

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.

4. EXTENSIONS OF THE BASIC ANALYSIS 59

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

48

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.

60 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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

1

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)

2

j

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).

ˆ

A similar second-order expansion of the index of sectoral demand is of the form

1

ˆ

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

ˆ iˆ (4.30)

2

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

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

1

¯

j

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)

2

generalizing (2.9), where now

˜j

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

ˆ

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)

j

˜ iˆ

2 2

Nj

+t.i.p. + O(||ξ||3 ).

4. EXTENSIONS OF THE BASIC ANALYSIS 61

ˆ

Then summing over the two sectors and using (4.29) to eliminate nj Yjt , we obtain

j

±

2

¯ uc (1 ’ ¦)Yt + 1 (1 + ω)Y 2 ’

1

ˆ ˆ ˆˆ

nj (ωqjt + · ’1 •jt )Yjt

v (yt (i); ξt )di = Y

˜ t

2

0 j=1

2

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

Ut

2 j=1

2

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

±

¯

2

Y uc j

’1 —2 ’1 2 ’1

= ’ (σ + ω)(xt ’ x ) + n1 n2 (· + ω)xRt + (θ + ω) vari yt (i)

ˆ

2

j=1

+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

2

ˆ

’1

ω)Ytn ’1

(σ + = σ gt + nj qjt ,

j=1

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

2

2

varj log pt (i) + t.i.p. + O(||ξ||3 ).

+θ(1 + ωθ) (4.34)

i

j=1

62 CHAPTER 6. INFLATION STABILIZATION AND WELFARE

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 ∆j + πjt + O(||ξ||3 ),

2

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

∞ ∞

±j

∆j

t

β t πjt + t.i.p. + O(||ξ||3 ).

2

β =

t

(1 ’ ±j )(1 ’ ±j β) t=0

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

form

2

wj πjt + »x (xt ’ x— )2 + »R (ˆRt ’ pn )2 ,

2

Lt = p ˆRt (4.35)

j=1

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

4. EXTENSIONS OF THE BASIC ANALYSIS 63

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

ˆRt

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

j

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

49

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