This allows us to identify the coe¬cients in equations (2.18) “ (2.19). Substitution of the

solutions for these variables into the policy rule then allows us to identify the coe¬cients in

(2.20) as well.

In the case of a rule such as (2.8) or (2.14), instead, the lagged nominal interest rate is a

predetermined endogenous variable that is relevant for equilibrium determination, because

of the way that it enters the policy rule. In cases of this sort, one instead obtains solutions

of the form

∞

π π

rn

πt = π + ω (ˆt’1 ’ ¯t’1 ) +

¯ ± ± ψj Et (ˆt+j ’ ¯t+j + π ),

± ¯

j=0

and similarly for the other endogenous variables.

Thus our model implies that for policy rules of these types, equilibrium in¬‚ation depends

ˆn

solely upon the path of the gap between the natural rate of interest rt and the intercept

term ¯t indicating the tightness of central-bank policy. In the case of the inertial interest-

±

rate rules, equilibrium in¬‚ation also depends upon a lagged interest rate (speci¬cally, upon

ˆt’1 ’¯t’1 ), but in equilibrium this variable will itself be a function of the history of the gaps

± ±

ˆn

rt’j ’ ¯t’j . As has already been noted in chapter 2, our theory of in¬‚ation determination

±

thus has a distinctively Wicksellian ¬‚avor: variations in the rate of in¬‚ation depend upon

the interaction between the real factors that determine the natural rate of interest on the

one hand, and the way in which the central bank adjusts short-term nominal interest rates

on the other. In¬‚ation will be stable insofar as the stance of monetary policy is varied to

keep up with the exogenous variations in the natural rate of interest that occur as a result of

real disturbances, and not varied otherwise; it will be variable insofar as either factor varies

other than in perfect tandem with the other. Our analysis here has a more fully Wicksellian

character than that presented in chapter 2, because we are now able to distinguish between

34 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

ψπ

j

0.1

φπ=1

0.08

φ =2

π

φ =4

π

0.06

0.04

0.02

0

0 1 2 3 4 5 6 7 8 9 10

ψx

j

2.5

φπ=1

2

φ =2

π

1.5 φ =4

π

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8 9 10

Figure 4.4: Consequences of varying the coe¬cient φπ in the Taylor rule.

the natural rate of interest (which would be the equilibrium real rate of return, in the

absence of nominal rigidities, and depends purely upon real factors) and the actual real rate

of return (which can di¬er from the natural rate as a result of short-run disequilibrium,

and is a¬ected by monetary policy among other factors). We are thus now able to explain

economic ¬‚uctuations in terms of the development of a gap between the natural and the

actual real rate of return, and to discuss the role of monetary policy in helping to minimize

such gaps.

π x

Examples of numerical solutions for the coe¬cients {ψj } and {ψj } in the case of the

Taylor rules of the form (1.10) are presented in Figures 4.4 and 4.5. Here the numerical

values assigned to the structural parameters β, σ, κ are again as in Table 4.1. Here we take

as our “baseline” policy rule a Taylor rule with coe¬cients φπ = 2, φx = 1; Figure 4.4

then illustrates the consequences of varying φπ around this baseline value, while Figure 4.5

2. INTEREST-RATE RULES AND PRICE STABILITY 35

illustrates the consequences of varying φx . We observe that for a range of parameter values

π

representing reaction functions similar to actual central-bank policies, the coe¬cients ψj and

x

ψj are positive for all small enough j, which are the coe¬cients of primary importance in

determining the equilibrium responses to typical shocks.28 Thus we ¬nd that higher output

gaps and in¬‚ation result from increases in the current or expected future natural rate of

interest, not o¬set by a su¬cient tightening of monetary policy, or by current or expected

future loosening of monetary policy, not justi¬ed by a decline in the natural rate of interest.

This is essentially a forward-looking variant of the traditional Wicksellian analysis. We also

observe that a higher response coe¬cient on in¬‚ation in the Taylor rule results in weaker

equilibrium responses of in¬‚ation to exogenous disturbances, especially to disturbances ex-

pected several quarters in the future; the response of output is also reduced, though less

dramatically. A higher response coe¬cient on the output gap in the Taylor rule instead sig-

ni¬cantly attenuates the equilibrium response of the output gap to news about the natural

rate or monetary policy in the current quarter or the next one, and this also weakens the

equilibrium response of in¬‚ation.

The consequences of interest-rate inertia in the Taylor rule are shown in Figure 4.6. Here

we assume a rule of the form (2.12), with values for φπ and φx as in the baseline case of

Figures 4.4 and 4.5, but with various positive values for ρ. We observe that for given φπ

and φx , a higher value of ρ reduces the equilibrium response of both in¬‚ation and output,

though the e¬ect is much more dramatic in the case of the in¬‚ation response. This should

be intuitive, since for given φπ and φx , a higher ρ implies a larger eventual interest-rate

response to a sustained increase in in¬‚ation or the output gap. When one considers super-

inertial rules (i.e., rules with ρ > 1), the response of in¬‚ation to an increase in the natural

rate (not o¬set by a corresponding tightening of monetary policy) actually becomes negative.

This is because the output gap still increases, as a result of which interest rates increase; the

strong interest-rate inertia then implies an expectation of much higher future interest rates

28

The coe¬cients for large j would dominate, in computing the e¬ects of news about the natural rate or

monetary policy, only if the news were to a¬ect expectations only about conditions many quarters in the

future.

36 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

ψπ

j

0.1

φ =.5

x

φ =1

0.08 x

φ =2

x

0.06

0.04

0.02

0 1 2 3 4 5 6 7 8 9 10

ψx

j

3

φ =.5

2.5 x

φ =1

x

2

φ =2

x

1.5

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8 9 10

Figure 4.5: Consequences of varying the coe¬cient φx in the Taylor rule.

as well, even if the output gap no longer exists. The expectation of future tightening leads to

an expectation of lower future output gaps, which in turn motivates an immediate reduction

in in¬‚ation, despite the initially higher output gap. (The reduction in in¬‚ation is insu¬cient

to prevent interest rates from rising under the policy rule, as there would otherwise be no

increase in expected future interest rates to generate the incentive to disin¬‚ation.)

These ¬gures indicate the immediate response of in¬‚ation and output to a disturbance

ˆn ±

that shifts the current and/or expected future values of rt or ¯t . The ¬gures do not, however,

indicate the dynamic response to such disturbances. In the case that ρ = 0, in¬‚ation and

the output gap are both purely forward-looking functions of the current and expected future

disturbances, as indicated in (2.18) “ (2.19). In this case, the dynamics of the response of

in¬‚ation and output to a shock are a straightforward consequence of the dynamics of the

disturbance itself. (A transitory disturbance must have a purely transitory e¬ect; a more

2. INTEREST-RATE RULES AND PRICE STABILITY 37

π

ψ

j

0.08

ρ=0

0.06

ρ=0.5

ρ=1

0.04

ρ=1.5

ρ=2

0.02

0

’0.02

’0.04

0 1 2 3 4 5 6 7 8 9 10

ψx

j

2.5

ρ=0

2

ρ=0.5

ρ=1

1.5

ρ=1.5

ρ=2

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8 9 10

Figure 4.6: Consequences of varying the coe¬cient ρ in the Taylor rule with interest-rate

inertia.

persistent disturbance has a correspondingly more persistent e¬ect, though the e¬ect is also

larger, owing the e¬ects of the anticipation of the continued disturbance in the future.)

But when the policy rule incorporates feedback from lagged endogenous variables, it is also

possible to obtain persistent e¬ects on in¬‚ation and output from even a purely transitory

disturbance. Since estimated central-bank reaction functions generally incorporate lagged

endogenous variables of several sorts, both lagged interest rates and lags of variables such as

in¬‚ation and the output gap as well (as discussed in chapter 1), it is not implausible to assume

such lags in seeking to account for the degree of persistence of the responses of output and

in¬‚ation to identi¬ed monetary policy shocks in historical data. Alternatively, of course, one

could simply assume that the monetary policy disturbance {¯t } is serially correlated. This

±

would su¬ce to allow our model to predict persistent responses to a monetary policy shock,

and indeed the two explanations are not even conceptually distinguishable. For example, a

38 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

policy rule of the form (1.10) where

¯t ’ π = ρ(¯t’1 ’ π ) +

± ¯ ± ¯ t

and { t } is an i.i.d. mean-zero shock is equivalent to a policy rule of the form

ˆt = (1 ’ ρ)¯ + ρˆt’1 ) + φπ (πt ’ ρπt’1 ’ (1 ’ ρ)¯ ) + φx (xt ’ ρxt’1 ’ (1 ’ ρ)¯)/4 + t , (2.22)

± π ± π x

which now has a serially uncorrelated disturbance term, but feedback from lagged endogenous

variables.

As an example of the kind of persistent response to a transitory shock that can result in

the case of feedback from lagged endogenous variables, Figure 4.7 presents impulse responses

to a monetary policy shock in the case of a policy rule of the form (2.22), where again { t } is

an i.i.d. mean-zero shock. In the ¬gure, the intertia coe¬cients are set equal to ρ = 0.6, 0.7,

or 0.8, while φπ and φx are chosen to imply the same long-run responses ¦π ≡ (1’ρ)’1 φπ = 2

and ¦x ≡ (1 ’ ρ)’1 φx = 1 in each case. Because these coe¬cients satisfy (2.7) in each

case, equilibrium is determinate. The ¬gure shows the dynamic response to an unexpected

monetary tightening (an unexpected increase in that raises the short-term interest rate by

t

one percentage point, for given values of the other arguments of the central-bank reaction

function).29 The baseline case is chosen to be ρ = 0.7, because this is approximately the

sum of the coe¬cients on lags of the federal funds rate in the rule estimated by Rotemberg

and Woodford (1997), as discussed below, and we wish to provide insight into the theoretical

responses obtained in their more complicated model.

One observes that the responses of both output and in¬‚ation to such a shock last for

many quarters; in the case of the present completely forward-looking model of in¬‚ation and

output determination, the degree of persistence of all four responses is determined directly

by the assumed value of ρ in the policy rule. The amplitude of the equilibrium responses, for

any given long-run responses to in¬‚ation and output in the policy rule, also depends on the

29

The responses plotted for the nominal and real interest rate and for in¬‚ation are all expressed in per-

centage points of the equivalent annualized rate, so that “in¬‚ation” actually means the variable 4πt , and so

on. The shock increases 4¯0 by one percentage point.

±

2. INTEREST-RATE RULES AND PRICE STABILITY 39

nominal interest inflation

0.1

0.05

0.05

0

’0.05

0

’0.1

’0.05 ’0.15

’0.2

’0.1

’0.25

0 5 10 15 20 0 5 10 15 20

real interest output

0.2 0.1

0

0.15

’0.1

’0.2

0.1

’0.3

ρ = 0.6

ρ = 0.7

0.05 ’0.4

ρ = 0.8

’0.5

0

0 5 10 15 20 0 5 10 15 20

Figure 4.7: Impulse responses to a contractionary monetary policy shock, for alternative

degrees of policy inertia.

value of ρ. The initial e¬ect on output is essentially the same regardless of ρ, but the e¬ect

is more persistent the larger is ρ; and a more persistent output contraction reduces in¬‚ation

more (and more persistently). Hence the reduction in in¬‚ation is greater, and the e¬ects on

both variables are more persistent, for larger values of ρ.

In the case of su¬ciently modest values of ρ, a contractionary monetary policy shock is

associated with a temporary increase in nominal interest rates; but for ρ = 0.7 or larger,

the predicted in¬‚ation reduction is strong enough that nominal interest rates are actually

predicted to decrease temporarily. Thus there is predicted to be no “liquidity e¬ect” in

the latter cases, a feature that has often been considered an embarrassment for calibrated

optimization-based models of the monetary transmission mechanism (see, e.g., Kimball, 1995,

or Edge, 2000). Figure 4.7 shows that a “liquidity e¬ect” is possible for some parameter

values. However, a more satisfactory resolution of the problem requires that additional

40 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

delays in the e¬ects of monetary policy be introduced, as discussed in section 4. There we

argue that the response shown in Figure 4.7 for the case ρ = 0.7 is not too di¬erent from the

empirically estimated responses for the second quarter following the shock and later. (By

that time, nominal interest rates are predicted to return nearly to the level that would have

been expected in the absence of the shock, as shown in Figure 4.xx.) It is the estimated

responses in the ¬rst two quarters that cannot be explained by this simple model; and here

the real puzzle is not that nominal interest rates temporarily increase, but rather that output

and in¬‚ation do not immediately fall, as indicated in Figure 4.7. Once that problem is solved,

the problem of obtaining a “liquidity e¬ect” is easily solved as well.

Our model can also be used to predict the response of the economy to real disturbances

of various sorts, under one or another monetary policy rule. This is important for the

explanation of business ¬‚uctuations, since it is widely agreed that the greater part of cyclical

variation in real activity is ultimately caused by real disturbances rather than by random

monetary policy.30 But it is also important for the choice of a monetary policy rule, for the

crucial question for the theory of monetary policy has to do with the choice of the systematic

component of monetary policy (and not the exogenous random component, which one plainly

wishes to eliminate to the extent possible), in the light of the implications of alternative

systematic policies for the way that the economy will respond to disturbances that, in their

origin, have nothing to do with monetary policy.

We shall not attempt a detailed treatment of the issue here. However, two general lessons

from our baseline model are worth pointing out. The ¬rst is that, insofar as we are concerned

solely with the responses of in¬‚ation, the output gap, and nominal interest rates to the real

disturbances (and in chapter 6 we shall explain why these are exactly the variables which

should be matter from the point of view of social welfare, under the assumptions that underly

the present model), and insofar as we restrict attention to policy rules of the general type

considered here (and in chapter 8 we shall show that optimal policy can be represented in

30

This is the implication, for example, of the variance decompositions implied by typical VAR studies.

Again see, e.g., Christiano et al. (1999).

2. INTEREST-RATE RULES AND PRICE STABILITY 41

this way), then the only feature of the real disturbances that matters is their e¬ect upon

the path of the natural rate of interest. The second is that the responses of in¬‚ation and

the output gap to a disturbance to the natural rate of interest are exactly the same as their

responses to a monetary policy shock (disturbance to the ¯t term in the policy rule) that has

±

the same serial correlation properties and the opposite sign. Both conclusions follow from

the fact that, in the case of the classes of policy rules considered above, equilibrium in¬‚ation

ˆn ±

and the output gap are functions solely of the path of the “gap” rt ’ ¯t . Thus Figure 4.7,

for example, also indicates the response of in¬‚ation and the output gap to an unexpected

reduction in the natural rate of interest, if the natural rate follows a ¬rst-order autoregressive

process with a coe¬cient of ρ, and the monetary policy rule is of the form (1.10).

Thus far we have considered in¬‚ation and output-gap determination only in the case of

a purely forward-looking model of in¬‚ation determination, namely, the basic Calvo pricing

model introduced in chapter 3. But as discussed in section xx of that chapter, there is a fair

amount of evidence suggesting that a model that allows for some degree of in¬‚ation inertia

can better explain observed in¬‚ation dynamics. To what extent does allowance for in¬‚ation

inertia require us to modify the neo-Wicksellian account just developed?

In fact, in¬‚ation inertia of the kind assumed by Christiano et al. (2001) makes only a

small di¬erence for our qualitative results, though the exact speci¬cation matters, of course,

for quantitative purposes. Let the aggregate supply relation (1.9) be replaced by

πt ’ γπt’1 = κxt + βEt [πt+1 ’ γπt ], (2.23)

where 0 ¤ γ ¤ 1 indicates the degree of indexation of individual prices to a lagged price

index, as in section xx of chapter 3. For simplicity let us again consider a policy rule of

the form (1.10). Our complete system of equations for the determination of the equilibrium

paths of in¬‚ation, output and the nominal interest rate then consists of equations (1.8),

(1.10), and (2.23). It is then easily seen that in the case of a policy rule that implies a

determinate equilibrium, this equilibrium is described by laws of motion of the form

∞

π

rn

πt = π + ωπ (πt’1 ’ π ) +

¯ ¯ ψj Et (ˆt+j + π ’ ¯t+j ),

¯± (2.24)

j=0

42 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

π

ψj

0.1

γ=0

0.08

γ = 0.5

γ = 1.0

0.06

0.04

0.02

0

0 1 2 3 4 5 6 7 8 9 10

ψx

j

2.5

γ=0

2

γ = 0.5

1.5 γ = 1.0

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8 9 10

Figure 4.8: In¬‚ation and output-gap responses under a contemporaneous Taylor rule, for

alternative degrees of in¬‚ation inertia.

∞

x

rn

xt = x + ωx (πt’1 ’ π ) +

¯ ¯ ψj Et (ˆt+j + π ’ ¯t+j ),

¯± (2.25)

j=0

∞

i

rn

ˆt = ¯t + ωi (πt’1 ’ π ) +

± ± ¯ ψj Et (ˆt+j + π ’ ¯t+j ).

¯± (2.26)

j=0

π x

Figure 4.8 plots the numerical values of the coe¬cients ψj and ψj as a function of the

horizon j, for three alternative values of γ. Here the assumed values of β, σ and κ are again

those given in Table 4.1, while the coe¬cients assumed in the policy rule are φπ = 2, φx = 1.

(Thus the case γ = 0 in this ¬gure corresponds once again to the baseline cases of Figures

4.4 and 4.5, and to the ρ = 0 case of Figure 4.6.) We observe once again that the qualitative

impact of news at date t on in¬‚ation and the output gap is the same as discussed earlier:

an increase in the expected natural rate of interest (now or in the near future) increases

both in¬‚ation and the output gap, while a tightening of monetary policy lowers both. The

2. INTEREST-RATE RULES AND PRICE STABILITY 43

main di¬erence made by a positive value of γ in this regard is that the in¬‚ation rate is

more sensitive to expectations regarding the natural rate and the policy-rule intercept many

quarters in the future.

The other di¬erence in the response of in¬‚ation and the output gap to these two types

of disturbances results from the presence of the πt’1 ’ π terms in each of equations (2.24)

¯

“ (2.25). When γ = 0, these terms are zero, but as γ increases, ωπ takes an increasingly

larger positive value, while ωx takes an increasingly larger negative value.31 In this model,

for any given expectations regarding current and future natural rates of interest and mon-

etary policy, the fact of a higher rate of in¬‚ation in the past acts as an adverse “supply

shock,” increasing current in¬‚ation while lowering the current output gap. This results in

an additional mechanism for the propagation of the e¬ects of ¬‚uctuations in the natural rate

and/or in the monetary policy rule applied by the central bank. Nonetheless, it continues

to be true that the natural rate of interest is a su¬cient statistic for the e¬ects of all real

disturbances on the evolution of in¬‚ation and the output gap; that it is only the gap between

the natural rate and the Taylor-rule intercept at each date that matters in this regard; and

that at least the immediate e¬ects of disturbances are have the same sign as in our earlier

analysis.

2.5 Policy Rules for In¬‚ation Stabilization

We now brie¬‚y consider the implications of the model of in¬‚ation determination just devel-

oped for the design of a monetary policy that would succeed at stabilizing the general level

of prices, or more generally at stabilizing the rate of in¬‚ation around some target rate.32

The theory of in¬‚ation implied by solutions such as (2.18) yields a simple prescription for

a policy under which (if the private sector regards the central bank™s policy commitment as

31

For the parameter values used in Figure 4.8, one obtains ωπ = 0.42, ωx = ’1.95 in the case γ = 0.5,

and ωπ = 0.73, ωx = ’3.17 in the case γ = 1.

32

Here we assume without further discussion a “locally Ricardian” ¬scal regime, in the sense discussed in

chapter 5; the role of ¬scal policy in the design of a regime conducive to price stability is taken up further in

that chapter. We also take it for granted that price stability is the goal of monetary policy, without seeking

to justify such an objective; the welfare-theoretic justi¬cation for such a goal is treated in chapter 6.

44 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

perfectly credible) in¬‚ation should never deviate from the target rate π . The central bank

¯

should commit itself to a policy rule belonging to one of the families discussed in the previous

section (or a generalization of these), with the further stipulations that (i) the time-varying

intercept term ¯t should track the exogenous variation in the natural rate of interest, so that

±

ˆn ¯

¯t = rt + π at all times, and (ii) response coe¬cients such as φπ , φx and ρ should be chosen so

±

as to imply a determinate rational expectations equilibrium. The latter proviso implies that

the rule should respect the Taylor principle (at least in the case of each of the simple families

considered above), but it may also place further restrictions upon the response coe¬cients

as well.

As shown in the previous section, as long as the policy rule involves feedback only from

current, lagged, or expected future values of the variables πt ’ π , xt ’ x and ˆt ’ ¯t , then if

¯ ¯ ± ±

equilibrium is determinate, the solution will make each of the variables just listed a function

ˆn ± ¯

solely of current and expected future values of the “gap” terms rt ’¯t + π , plus lagged values

of the endogenous variables that enter the central bank™s feedback rule and/or the aggregate

supply relation.33 This means that if ¯t = rt + π at all times, there will be no equilibrium

ˆn ¯

±

¬‚uctuations in the endogenous variables, except those due to non-zero initial values of those

variables; the latter variation will be purely deterministic, and will approach zero as time

passes, assuming a credible permanent commitment to the policy. Thus such a policy should

succeed in principle in complete stabilization of both in¬‚ation and the output gap (as here

de¬ned).34 Note that this conclusion holds equally in the case that we assume an aggregate

supply relation of the form (2.23), incorporating in¬‚ation inertia.

Such an approach to policy would make the natural rate of interest a key concept in

monetary policy-making. Furthermore, unlike some discussions (see, e.g., Monetary Policy

Committee, 1999) of the “neutral rate of interest”, which imply that this should not vary over

33

A similar conclusion holds if we allow for lagged endogenous terms in the IS relation as well, as discussed

in section xx below.

34

That it is possible to completely stabilize both in¬‚ation and the output gap depends, of course, upon

the absence in our baseline model of ine¬cient supply disturbances of the kind discussed in section xx of

chapter 3. The nature of the trade-o¬ between in¬‚ation stabilization and output-gap stabilization goals that

arises in the presence of such disturbances is considered in chapters 6 through 8.

2. INTEREST-RATE RULES AND PRICE STABILITY 45

time, our theoretical analysis implies that the natural rate of interest should vary in response

to any of a wide range of types of real disturbances, and keeping track of its current value

would be an important (and far from trivial) task of central-bank sta¬ under such a regime.

Of course, insofar as the policy rule involves feedback from the output gap, as allowed for in

our discussion above, implementation of such a rule would also require a central bank to track

variations in the current natural rate of output (a topic that already receives a great deal

of discussion in most central banks). However, our analysis above implies that this concept

is less essential to the central bank™s task. For given an ability to adjust ¯t perfectly in

±

response to variations in the natural rate of interest, it would not matter what the responses

to endogenous variables are, as long as the response coe¬cients are large enough to imply

determinacy. As we have seen, the latter concern does not require any non-zero response to

the output gap. Thus it would be possible, in principle, to commit to a rule such as (1.10)

n

with ¯t = rt + π , φπ > 1, and φx = 0, and completely stabilize in¬‚ation and the output gap;

± ¯

but the implementation of such rule would not require the central bank to be aware of the

current natural rate of output.

On the other hand, simply producing an accurate estimate of the current natural rate of

interest and adjusting the bank™s operating target for an overnight interest rate accordingly

is not quite su¬cient for in¬‚ation stabilization; a commitment to the right sort of response

to variations in endogenous variables such as in¬‚ation is also necessary, in order to ensure

determinacy. It is true that, if the central bank™s adjustment of ¯t so as to track variations

±

in the natural rate of interest is exact, the precise degree of response to endogenous state

variables is irrelevant, as long as it su¬ces to put one in the range required for determinacy.

This is because, in equilibrium, variations in those variables never occur; the commitment

to a response to them matters not in order to change the nature of the desired equilibrium,

but simply in order to exclude other possible equilibria. However, one must recognize that

in practice, perfect tracking of the current natural rate of interest will be impossible, as

real-time information about the natural rate will inevitably be imprecise.35 In this case,

35

The way in which optimal policy is a¬ected by imperfect observability of the current state of the economy

46 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

ˆn ±

for any given degree of irreducible variation in the gap process rt ’ ¯t due to limits on the

central bank™s ability to track variations in the natural rate, the equilibrium ¬‚uctuations in

in¬‚ation implied by a solution such as (2.18) will also depend upon the policy rule™s response

coe¬cients.

If one™s goal is simply to stabilize in¬‚ation (and the output gap, as these goals are

equivalent in the present model) to the greatest extent possible, then one would want to

π x

choose response coe¬cients that make the coe¬cients {ψj } and {ψj } as small as possible.

In the case of the backward-looking Taylor rules, (1.10) or (2.8), it is possible to make all of

these coe¬cients simultaneously as close to zero as one likes, simply by making the response

coe¬cients φπ and φx large enough. (In fact, it su¬ces to make either of these coe¬cients

su¬ciently large; for an equilibrium with small ¬‚uctuations in the output gap must also have

small ¬‚uctuations in in¬‚ation, and vice versa.) Through this approach one can, in principle,

make the equilibrium ¬‚uctuations in in¬‚ation arbitrarily small, even without attempting

to track variations in the natural rate of interest at all. This represents an advantage

of backward-looking Taylor rules over the forward-looking variants; for in the latter case,

making the response coe¬cient on the in¬‚ation forecast too large results in indeterminacy

(see Figure 4.2), as is also found by Bernanke and Woodford (1997) in a related model. There

is thus a limit on the extent to which in¬‚ation ¬‚uctuations in response to real disturbances

can be reduced using a forward-looking rule, if one restricts attention to rules that result in

a determinate equilibrium.36

However, complete reliance upon the threat of extreme responses to in¬‚ation and/or

output-gap variations should they occur, as a means of preventing such variations from ever

occurring, is not obviously the most desirable approach. For under such a regime, there is

the obvious danger that random noise in the particular measure to which the central bank

responds might require violent adjustments of interest rates, that in turn create havoc in

is considered formally in chapter 8.

36

In the case of rules in the family (2.12), it is still possible to reduce in¬‚ation ¬‚uctuations to an arbitrary

extent without loss of determinacy by making the output-gap coe¬cient φx large. But this term in the rule

is the same as in a backward-looking (standard) Taylor rule, and a rule with the main weight on that term

is essentially equivalent to a backward-looking rule.

2. INTEREST-RATE RULES AND PRICE STABILITY 47

the economy. If we suppose that the in¬‚ation measure to which the central bank responds

represents the average of the prices set by optimizing suppliers, plus an exogenous measure-

ment error term, then a policy rule with a very large coe¬cient φπ would cause no great

trouble if price-setters were able to observe the quarter t measurement error disturbance

before choosing new prices for quarter t. In that case, the equilibrium rate of actual in¬‚a-

tion would adjust to (nearly perfectly) o¬set the measurement error, so that the in¬‚ation

measure used by the central bank would be nearly perfectly stabilized. This would result in

some variation in actual in¬‚ation, but this would be limited by the size of the measurement

error, and ¬‚uctuations in interest rates would be minimal. However, in the more realistic

case that quarter t prices are chosen prior to private-sector observation of the measurement

error in the government statistics, then it would not be possible for private-sector pricing

decisions to o¬set the measurement error, and large interest-rate variations would occur in

equilibrium. This could make too extreme a version of such a strategy highly undesirable.

Thus if one wishes to stabilize in¬‚ation to the greatest extent possible, it is fairly certain

that attempting to vary policy in response to estimates of the natural rate of interest will

improve policy.

On the other hand, as we show in chapters 6 and 7, there are good reasons why a central

bank may not wish to fully stabilize in¬‚ation ” for example, because the amount of interest-

rate variation required would be undesirable. In this case, a constrained-optimal equilibrium

will still involve non-trivial ¬‚uctuations in in¬‚ation and the output gap in response to real

disturbances, and as a result, feedback from these endogenous variables may substitute for

response to more direct measures of the real disturbances. It then may be possible, as we

show in chapter 8, to ¬nd an optimal policy rule in which there is no time-varying term ¯t

±

at all.

48 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

3 Money and Aggregate Demand

Thus far, our analysis of the e¬ects of monetary policy has made no reference to the evolution

of the money supply, which may seem to some a surprising omission. Here we discuss the way

in which the analysis can be extended to include analysis of the evolution of the money supply

under alternative policy rules, should that be desired, in the case that there exist monetary

frictions su¬cient for a well-de¬ned money demand function to exist. This extension of the

model also makes it possible to consider monetary targeting rules or rules that respond to

variations in money growth among the candidate policy proposals. We also consider the

extent to which allowance for “real balance e¬ects” would modify the conclusions reached

above about in¬‚ation determination under interest-rate rules.

3.1 An Optimizing IS-LM Model

We shall now again assume the existence of transactions frictions of the kind considered in

chapter 2, that can be represented by the inclusion of liquidity services from real money

balances in the household utility function. Once again, this results in a ¬rst-order condition

for the representative household™s optimal demand for money balances of the form

it ’ im

Um (Ct , mt ; ξt ) t

= , (3.1)

Uc (Ct , mt ; ξt ) 1 + it

where mt denotes the household™s end-of-period real money balances, and im denotes the

t

interest (if any) paid on such balances. As with our discussion of the intertemporal IS

relation above, we note that the ¬rst-order conditions describing the optimal behavior of a

price-taking household are the same in the case of the sticky-price model considered in this

chapter as in the ¬‚exible-price model of chapter 2. (The only di¬erence here is that Ct now

refers to an index of consumption of a large number of di¬erentiated goods, rather than the

single consumer good of chapter 2.) Imposing the requirements that the demands of the

representative household equal the economy™s aggregate supply of both goods and ¬nancial

3. MONEY AND AGGREGATE DEMAND 49

assets, we obtain the equilibrium condition

Mts

= L(Yt , ∆t ; ξt ), (3.2)

Pt

where ∆t ≡ (it ’ im )/(1 + it ) is the interest di¬erential between non-monetary and monetary

t

assets and the money demand function L(y, ∆; ξ) has the same properties as in chapter 2.

When log-linearized around the steady-state equilibrium with zero in¬‚ation, we again obtain

a relation of the form

ˆ

mt = ·y Yt ’ ·i (ˆt ’ ˆm ) + m

ˆ ± ±t t, (3.3)

m

where ·y , ·i > 0 and is an exogenous disturbance process.

t

The other equilibrium conditions used in the analysis thus far continue to apply in the

presence of monetary frictions. The only di¬erence between a cashless economy and one in

which central-bank money facilitates transactions is that in the latter case, the marginal util-

ity of additional real expenditure by the representative household is given by Uc (Ct , mt ; ξt ),

the value of which will in general depend on the level of real money balances in addition to

the level of real expenditure. However, in either the case of preferences additively separable

between consumption and real balances or the “cashless limit” discussed in section xx of

chapter 2, we may neglect the e¬ect of variations in real money balances on the value of

Uc , and the intertemporal IS relation and aggregate supply relation take exactly the form

assumed above. The complete system of equilibrium conditions for determination of the

nominal interest rate, the price level, and output (in the case of a money-growth rule) or

the price level, output and the money supply (in the case of an interest-rate rule) is then

given by the IS and AS relations considered above, together with (3.2) or (3.3). This system

of equations has essentially the structure of an IS-LM-AS system of the kind familiar from

undergraduate textbooks, though here the IS and AS relations are not purely static ones.

We note also that the “LM relation” (3.2), considered as an equilibrium relation between it

and Yt , is shifted by variations in either Mts or im , which appear as two separate instruments

t

through which monetary policy may be implemented, in addition to potentially being shifted

by the exogenous disturbances ξt .

50 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

In the case of an interest-rate rule of one of the forms considered above, the equilibrium

paths of in¬‚ation and output are determined by the IS and AS relations as above, but

the LM relation now allows us to solve for the implied path of the money supply as well,

under an assumption about the path of the interest rate on money (such as the conventional

speci¬cation ˆm = 0).37 In the case of a “cashless limiting economy” ” perhaps the most

±t

attractive justi¬cation for our neglect of real balance e¬ects in the analysis thus far ” the

absolute size of equilibrium real money balances is assumed to be negligible. However,

this does not mean that there may not be non-negligible (and well-de¬ned) ¬‚uctuations in

equilibrium real balances relative to their (extremely small) steady-state level, indicated by

the variable mt in (3.2). Similarly, there may be non-negligible, well-de¬ned ¬‚uctuations

ˆ

in the rate of growth of the nominal money supply, even in the absence of a well-de¬ned

absolute level of the equilibrium money supply.

For example, one may consider how money growth must respond to various kinds of real

disturbances, under a policy that succeeds in stabilizing in¬‚ation and the output gap. To

answer this question, we simply substitute the required interest-rate variations, discussed in

section xx above, into equation (3.3) to determine the implied path of the money supply.

We ¬nd that, in general, the money supply should be allowed to vary in response to all ¬ve

of the types of exogenous disturbances considered in section xx above, so that a constant

money growth rate is certainly not the best way to stabilize in¬‚ation .

Nor does the path of the money supply required for price stability necessarily involve

“leaning against the wind”. For example, procyclical variations in the money supply are

required in response to temporary ¬‚uctuations in productivity, as argued by Ireland (1996);38

ˆ

for an increase in at raises Ytn while lowering rt , thus warranting an increase in log Mts at the

ˆn

ˆ

same time as an increase in Yt . The same is true of temporary labor-supply shocks, and while

37

It is worth noting, however, that in the absence of a constraint on the evolution of im , the required path

t

of the money supply is not uniquely determined; for a given interest-rate policy may be implemented using

either quantity variations or variations in the interest paid on money, or some combination of the two.

38

Aiyagari and Braun (1998) reach a similar conclusion, in the case of their model with sticky prices,

though they assume convex costs of price changes, following Rotemberg (1995), rather than predetermined

prices as in the models considered here. These authors also reach a similar conclusion with regard to

government-purchase shocks, in the case of their numerical calibration of their model.

3. MONEY AND AGGREGATE DEMAND 51

the result depends upon parameter values, it is also true of government-purchase shocks and

consumption-demand shocks, at least if these are su¬ciently persistent. Furthermore, in the

case of technology or labor-supply shocks, it is actually desirable for the money supply to

be more procyclical than would be the case if interest rates were held unchanged; for one

actually wants nominal interest rates to decline in response to a positive shock. In the case

of the other two shocks, this is not true, but it is still possible that holding the nominal

interest rate ¬xed is closer to the optimal response that holding the money supply ¬xed; in

particular, this is necessarily true if the shocks are su¬ciently persistent, as in that case the

natural rate of interest is a¬ected very little.

Thus variations in the rate of money growth should not, in general, be a very accurate

indicator of whether interest rates have been allowed to adjust to the extent that they ought

to in response to disturbances. Monetary targeting amounts to an automatic mechanism

for bringing about procyclical interest rate variation (and interest-rate increases in response

to price-level increases as well); but the required change in the level of interest rates for

stabilization ” which depends upon the change in the natural rate of interest ” might not

even have the same sign as this. Monetary targeting also causes interest rates to vary in

response to money-demand disturbances, even though these should have negligible e¬ects

upon the natural rate of interest, so that it is not desirable for interest rates to respond to

them. One should therefore expect to be able to do better by adopting an interest-rate rule

that (i) achieves a desired degree of response of interest rates to price-level and/or output

gap increases through explicit feedback from measures of these variables, as called for in

the “Taylor rule,” and (ii) incorporates a direct response to changes in the central bank™s

estimate of the current natural rate of interest.

Our conclusions with regard to the usefulness of monetary targeting contrast with the

classic analysis of Poole (1970), according to which monetary targeting should be desirable

as long as money-demand disturbances are of less importance than real disturbances to

aggregate demand (“IS shocks”). Our government-purchase or consumption-demand shocks

presumably correspond to what Poole intends by “IS shocks”; yet even in the case of these

52 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

shocks, we have argued that some degree of accommodation (allowing the money supply to

vary in order to reduce the interest-rate response) is often desirable. And if the shocks are

su¬ciently persistent, the optimal degree of accommodation of the “IS shift” may be nearly

100 percent. The di¬erence, of course, is that Poole assumes that output stabilization should

be the goal of policy, whereas here we assume a goal of stabilizing the output gap instead,39

by which we mean output relative to a natural rate that is a¬ected by “IS shocks” among

other real disturbances. If we consider the possibility of technology or labor-supply shocks,

neglected by Poole altogether, our results are even more di¬erent, and even more strongly

support a presumption in favor of accommodation.

But the most important limitation of Poole™s analysis is that it assumes that monetary

targeting is the only alternative to a complete interest-rate peg. Yet the kind of policy that

we have proposed above (or that has been advocated by Wicksell or by Taylor) is not one

that ¬xes nominal interest rates in the face of changing macroeconomic conditions. Once

one recognizes that an interest-rate rule may specify interest-rate adjustments in response to

changes in the price level and/or in output, then Poole™s IS-LM analysis provides no ground

whatsoever for belief that it is desirable to respond to changes in monetary aggregates as

well.

One might, of course, make a case for responding to changes in the growth rate of

monetary aggregates if such statistics provide more up-to-date information about prices

and/or output than is otherwise available. This seems unlikely; at best, one might on these

grounds justify paying attention to monetary aggregates along with a large number of other

indicators.40 And even in this case, it would be most useful to understand the central bank™s

policy commitment in terms of its response to its estimates of in¬‚ation and the output gap,

rather than in terms of a commitment to respond to ” let alone to stabilize ” particular

indicators.

39

This objective is justi¬ed in terms of social welfare in chapter 6.

40

The optimal use of indicator variables given that in¬‚ation and the output gap are not perfectly observed

in real time is treated in chapter 8.

3. MONEY AND AGGREGATE DEMAND 53

3.2 Real-Balance E¬ects

Thus far we have considered only models in which monetary policy a¬ects aggregate demand

through the e¬ects of real interest rates on the desired timing of private expenditure. While

this is surely the most important single channel through which monetary policy matters, it is

sometimes argued that this channel alone is not the only one way in which monetary policy

matters, and that under at least some circumstances the neglect of other mechanisms may

lead to signi¬cantly misleading conclusions. One proposal of this kind with an especially

venerable history, is the argument often found in monetarist writing that the level of real

money balances held by the private sector should directly in¬‚uence aggregate demand, for

reasons that are independent of the reduction in equilibrium interest rates that will ordinarily

accompany a higher real money supply in order to induce the private sector to choose to

hold the higher money balances. Pigou (1943) argued for the existence of such a “real

balance e¬ect”, and proposed that its existence was especially important in guaranteeing

the e¬ectiveness of monetary policy even under the circumstances of a “liquidity trap” of

the kind posited by Keynes (1936). We take up the case of a liquidity trap below; but ¬rst

we consider the nature of real balance e¬ects under more ordinary circumstances.

It is ¬rst important to note that we have not omitted any e¬ect of real money bal-

ances upon aggregate demand that results from a simple (Hicksian) wealth e¬ect, as Pigou

supposed. The Euler equation (1.1) for the optimal timing of private expenditure, which

(when converted to the form (1.2) by imposing market-clearing) has proven to be the cru-

cial equilibrium condition of our Wicksellian framework, does not in any way contradict the

contribution of ¬nancial wealth to the intertemporal budget constraint of the representative

household (discussed in detail in chapter 2). Furthermore, solving our log-linearized Euler

equation forward to obtain a relation of the form (1.4) between the level of aggregate ex-

penditure at a given date and a very-long-term real rate (or equivalently, a distributed lead

of expected future short real rates) involves no neglect of any wealth e¬ects. It is true that,

for an individual household, optimal consumption demand is a function of the household™s

current ¬nancial wealth in addition to its future (after-tax) income expectations and its ex-

54 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

pectations regarding the available rate of return on its savings. But still, optimal current

expenditure relative to expected future expenditure depends only on the expected rates of

return on assets between now and the future date; for greater ¬nancial wealth should im-

ply greater expected future spending in the same proportion as it increases desired current

spending. And for households in aggregate, expected spending in the long-run future must

be tied down by the exogenous evolution (in the kind of model considered here) of the natural

rate of output. Given this anchor for the representative household™s expected long-run level

of expenditure, the sequence of expected future real interest rates does su¬ce to determine

what optimal expenditure must be in the present.

Nonetheless, there is a type of real-balance e¬ect from which the above analysis has

abstracted. Thus far, we have assumed the special class of utility functions described by

equation (xx) of chapter 2, that are additively separable between consumption and real

money balances. This familiar assumption makes the algebra simpler in many places, but

it can hardly be defended as realistic. If utility is obtained from holding money, this must

be because money balances facilitate transactions, and it is hardly sensible that the bene¬ts

of such balances should be independent of the real volume of transactions that a household

actually undertakes. In particular, as already argued in chapter 2, it is plausible that the

marginal bene¬t of additional real balances should be higher when real transactions are

greater, so that the cross-partial derivative Ucm > 0.

We accordingly here consider how our above results extend to the case of a more general

preference speci¬cation. We shall argue that the simple case treated above remains a good

guide to intuition, and indeed that it may be justi¬ed as an approximation without any

appeal to the plausibility of additive separability. But it is also useful to be able to see how

real balance e¬ects require our previous calculations to be modi¬ed, if we wish to be more

precise.

Let us recall from chapter 2 that in the general case, the equilibrium condition derived

3. MONEY AND AGGREGATE DEMAND 55

from the Euler equation for the optimal timing of private expenditure takes the form

’1

s

Uc (Yt+1 , Mt+1 /Pt+1 ; ξt+1 ) ’1

1 + it = β ’1 Et Πt+1 . (3.4)

Uc (Yt , Mts /Pt ; ξt )

(See equation (xx) of chapter 2.) This reduces to equation (1.2) above only under the

assumption of additively separable preferences (or the absence of non-negligible transactions

frictions). In the general case, a log-linear approximation to this relation takes the form

ˆ ˆ

ˆt = σ ’1 [Et (Yt+1 ’ gt+1 ) ’ (Yt ’ gt )] ’ χ(Et mt+1 ’ mt ) + Et πt+1 ,

± ˆ ˆ (3.5)

generalizing (1.3), where

mUcm

¯

χ≡

Uc

with partial derivatives evaluated at the steady state. Here gt is again the exogenous factor

¯

de¬ned in (2.3), where the shift in the marginal utility of consumption Ct is now evaluated

at the steady-state values of both Ct and mt . Note that (3.5) is the same as equation (xx)

of chapter 2, except that now we make the dependence on real output explicit, as output is

no longer exogenous.

Solving (3.5) forward again yields an “IS equation” of the form

∞

ˆ ˆ

Yt = Y∞ + gt + χσ mt ’ σ

ˆ Et [ˆt+j ’ πt+j+1 ].

± (3.6)

j=0

If χ > 0, then there is indeed a “real-balance e¬ect” upon aggregate demand. This results

not from a wealth e¬ect, but from the fact that even controlling for the path of real interest

rates, times when real balances are high are particularly convenient times to spend, due to

the way in which money balances facilitate transactions. We shall also see that, contrary to

Pigou™s suggestion, this e¬ect does not imply that increases in the money supply can increase

demand even in a liquidity trap; but this question is deferred until section xx below.

Let us ¬rst consider instead the consequences of the existence of a real-balance e¬ect

for in¬‚ation determination under an interest-rate rule, in the case of only small ¬‚uctuations

around the zero-in¬‚ation steady state (so that the zero lower bound on nominal interest rates

56 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

is never approached). As in section xx of chapter 2, it is convenient to use (3.3) to eliminate

real balances from the intertemporal IS equation (3.5), yielding

ˆ

(1+·i χ)ˆt = (σ ’1 ’·y χ)(Et Yt+1 ’Yt )+Et πt+1 +·i χEtˆt+1 ’σ ’1 (Et gt+1 ’gt )’χ(Et m m

±m ±m

± ± t+1 ’ t )’·i χ(Etˆt+1 ’ˆt ).

(3.7)

This gives us an equilibrium relation written solely in terms of the evolution of interest rates,

in¬‚ation and output, like (1.3).

Let us then de¬ne the natural rate of output Ytn as the equilibrium level of output at

each point in time that would obtain under ¬‚exible prices, given a monetary policy that

maintains a constant interest-rate spread ∆t between non-monetary and monetary riskless

short-term assets. This last stipulation in our de¬nition is now necessary, since when utility is

non-separable, equilibrium output under ¬‚exible prices is no longer independent of monetary

policy.41 Let us similarly de¬ne the natural rate of interest rt as the equilibrium real rate of

n

interest under the same hypothetical circumstances. Then we observe from (3.7), that must

hold equally whether prices are ¬‚exible or sticky, that

ˆn

rt = (σ ’1 ’ ·y χ)(Et Yt+1 ’ Ytn ) ’ σ ’1 (Et gt+1 ’ gt ) ’ χ(Et

ˆn ’ mt ). (3.8)

m,t+1

Note that this reduces to our previous de¬nition (1.11) in the case that χ = 0. Using this to

substitute for terms on the right-hand side of (3.7), and rearranging terms, we obtain

(1 ’ σ·y χ)xt = (1 ’ σ·y χ)Et xt+1 ’ σ(ˆt ’ Et πt+1 ’ rt ) + σ·i χ[Et (ˆt+1 ’ˆm ) ’ (ˆt ’ˆm )], (3.9)

ˆn

± ± ±t+1 ± ±t

ˆˆ

generalizing (1.8), where again xt ≡ Yt ’ Ytn . Thus we once more obtain an intertemporal IS

relation in terms of the output gap xt in which the only exogenous disturbance term is the

shift in the natural rate of interest rt .42

ˆn

Allowing for non-trivial monetary frictions and non-separable utility between consump-

tion and real balances also requires modi¬cation of our derivation of the aggregate supply

41

One might, of course, choose other de¬nitions of the natural rate, such as the ¬‚exible-price level of

output in the case of a monetary policy that results in a stable price level. The choice made here turns out

to be convenient in simplifying the welfare analysis of chapter 6.

42

Note that in the case of ¬‚exible prices and an exogenous goods supply, so that necessarily xt = 0 at

ˆn

all times, equation (3.9) reduces to equation (xx) of chapter 2, with rt corresponding to the composite

exogenous disturbance rt of that model.

ˆ

3. MONEY AND AGGREGATE DEMAND 57

relation (1.6). (Throughout chapter 3, we assumed either negligible monetary frictions or

separable utility.) In the case of non-separable utility, the marginal utility of income depends

upon real money balances as well as real expenditure, so that the household labor supply

equation becomes

vh (ht (i); ξt

wt (i) = .

Uc (Ct , mt ; ξt )

It follows that the real marginal cost of supplying good i, when log-linearized, is given by

ˆ

st (i) = ω(ˆt (i) ’ qt ) + σ ’1 (Yt ’ gt ) ’ χmt ,

ˆ y ˆ

where the exogenous disturbances qt and gt continue to be de¬ned as before. Substituting

(3.3) for real money balances mt , and recalling the de¬nition just proposed for the natural

ˆ

rate of output, we can write average real marginal cost in the form

ˆ ’ Ytn ) + ·i χ(ˆt ’ ˆm ),

ˆ

st =

ˆ mc (Yt ± ±t (3.10)

where the elasticity of average marginal cost with respect to aggregate output is now equal

to

= ω + σ ’1 ’ ·i χ, (3.11)

mc

and the natural rate of output is now given by

ωqt + σ ’1 gt + χ m

ˆ t

Ytn = , (3.12)

mc

generalizing (2.2).

Expression (3.10) can alternatively be written in the form

[xt + •(ˆt ’ ˆm )],

st =

ˆ ± ±t

mc

where

·i χ

•≡ .

mc

The case that would be seem to be of greatest empirical relevance is that in which χ satis¬es

the bounds

0 ¤ χ < ·y (ω + σ ’1 ),

’1

58 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

and • are positive coe¬cients.43 A calculation exactly parallel to

in which case both mc

that given in chapter 3 then allows us to derive an aggregate supply relation of the form

πt = κ[xt + •(ˆt ’ ˆm )] + βEt πt+1 ,

± ±t (3.13)

generalizing (1.6). Here

(1 ’ ±)(1 ’ ±β) mc

κ≡

± 1 + ωθ

as before, but is now given by (3.11). We ¬nd that in the non-separable case, in¬‚ationary

mc

pressure depends not only on the output gap, but also on the interest di¬erential between

non-monetary and monetary assets. The greater this return di¬erential is, the greater the

cost of holding the money balances needed to facilitate transactions; hence the greater is

the marginal cost of supplying a given level of output, and ultimately the incentive to raise

prices.

How much of a di¬erence is made by the inclusion of the additional terms in (3.9) and

(3.13)? If we assume an interest-rate rule of one of the kinds considered in section xx above,

and add the additional stipulation that the interest rate im paid on the monetary base is

t

adjusted along with all changes in the central bank™s interest-rate operating target, so as to

maintain a constant spread ∆t , then our previous conclusions are entirely unchanged, under

an appropriate calibration of the numerical coe¬cients. The reason is that in the case that

ˆm = ˆt at all times, equations (3.9) and (3.13) reduce to precisely our previous equations

±t ±

(1.8) and (1.6) respectively, given the new de¬nition (3.8) of the natural rate of interest, with

one exception: in our “IS relation” (3.4), the coe¬cient indicating the interest-sensitivity of

real expenditure is no longer σ, as in (1.8), but rather (σ ’1 ’·y χ)’1 .44 As long as χ < (·y σ)’1

” the empirically realistic case, under the calibration proposed below45 ” then there is no

43

As discussed in section xx below, Rotemberg and Woodford (1997) argue that a reasonable value for mc

is about .63 for the U.S. economy. We have argued in chapter 2 that a realistic value for χ·y for the U.S.

would be about .01, so this would imply a value for σ ’1 + ω of about .64. Even if these values are inaccurate,

σ ’1 + ω is likely to be much larger than χ·y .

44

In our analysis of price-level determination under this kind of policy in chapter 2, where we assumed ¬‚ex-

ible prices, this di¬erence did not matter, as no ¬‚uctuations in the output gap xt could occur in equilibrium.

In the case of price stickiness, the additional quali¬cation is needed.

45

In a more realistic extension of our model, where we distinguish among di¬erent categories of private

3. MONEY AND AGGREGATE DEMAND 59

qualitative change in our conclusions; we would obtain identical numerical results to those

presented in our earlier ¬gures under a suitably di¬erent calibration of the value of σ.

In¬‚ation and the output gap (the latter, rather than detrended output, being the welfare-

relevant quantity, as we shall argue in chapter 6) thus evolve according to exactly the formulas

derived earlier, once we take into account the modi¬cation of the e¬ects of real disturbances

on the natural rate of interest. (Computation of the implied path of detrended output

requires that one also take into account the modi¬ed expression for the e¬ects of real dis-

turbances on the natural rate of output.) Thus our previous results apply not only to fully

(or nearly) “cashless” economies, and to economies in preferences are additively separable

between consumption and real balances, but also to economies in which central-bank interest-

rate targets are implemented through adjustments of the interest paid on the monetary base,

rather than through adjustments of the supply of base money.46

If, however, interest-rate adjustments are implemented through variation in the supply

of base money, holding ¬xed the rate of interest paid on the monetary base ” as under

current U.S. monetary arrangements, for example ” then the additional terms in both the

IS relation and the AS relation matter, assuming non-trivial monetary frictions and non-

separable preferences. Nonetheless, for many purposes the additional terms do not seem

likely to have too large an e¬ect on our quantitative conclusions. We can illustrate this by

considering a numerical calibration of our model based on U.S. data.

As argued in section xx of chapter 2, evidence on long-run U.S. money demand suggests

money-demand elasticities on the order of ·y = 1, ·i = 28 quarters, and a value of χ = 0.02

expenditure with di¬ering degrees of interest-sensitivity, this would be even more clearly true. For one might

well calibrate a lower degree of interest-elasticity (larger σ ’1 ) for those expenditures that are complementary

with real money balances (the ones for which χ > 0).

46

Another case in which the previous results apply would be a hybrid of two of those just mentioned.

Suppose that utility each period is given by U (Ct , mcb ; ξt ) + V (mcu ; ξt ), where mcb indicates the real value

t t t

of the clearing balances held at the central bank to facilitate payments and mcu indicates the real value of

t

currency in circulation. (Here the liquidity services provided by clearing balances are assumed to depend on

the volume of aggregate transactions in the economy, while the usefulness of currency is assumed to be largely

una¬ected by variations in aggregate real expenditure.) Then it su¬ces for the validity of our previous results

that the interest rate icb paid on clearing balances be adjusted so as to maintain a constant spread between

t

this and the central bank™s interest-rate operating target, as is true under the channel systems mentioned in

chapter 1; it does not matter whether there is any interest paid on currency as well.

60 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

(or lower) for the elasticity of the marginal utility of expenditure with respect to additional

real money balances. Suppose that the other coe¬cients of our structural equations are

based on the parameter values used by Rotemberg and Woodford (1997), shown in Table 4.1

below, and discussed further there. Since Rotemberg and Woodford infer the value of ω from

their estimate of mc , rather than the reverse, allowing for χ = 0 does not imply any change

in our calibrated value of mc . Thus the value that we shall assume for • is (.02)(28)/(.63)

= 0.89 quarters.47 The values assumed for the parameters β, σ, and κ are those reported in

Table 4.1.

Using similar methods as above, we can show that in the case that a determinate equi-

librium exists,48 an interest-rate rule of the form (1.10)49 together with the stipulation that

ˆm = 0 implies equilibrium paths for the endogenous variables of the form

±t

∞ ∞

˜π ±

π

ˆn

πt = π +

¯ ψj (Et rt+j + π) ’

¯ ψj Et¯t+j , (3.14)

j=0 j=0

∞ ∞

˜x ±

x

ˆn

xt = x +

¯ ψj (Et rt+j + π ) ’

¯ ψj Et¯t+j , (3.15)

j=0 j=0

∞ ∞

˜i ±

i

ˆn

ˆt = ¯t +

± ± ψj (Et rt+j + π ) ’

¯ ψj Et¯t+j . (3.16)

j=0 j=0

˜π

In the case that χ = 0, the coe¬cients ψj and so on are no longer exactly equal to the

π

coe¬cients ψj and so on. The extent to which allowance for real-balance e¬ects a¬ects the

quantitative size of these coe¬cients is shown in Figures 4.10 and 4.11, for the calibrated

parameter values stated in the previous paragraph.

The ¬gures show the results in the case of a rule with φπ = 2, φx = 1, corresponding to

π

the baseline case in the previous Figures 4.4 and 4.5; Figure 4.10 shows the coe¬cients ψj

˜π ˜x

x

and ψj , while Figure 4.11 shows the coe¬cients ψj and ψj , for various future horizons j. In

47

This value multiplies a quarterly interest rate; if instead the interest rate is expressed as an annualized

rate, • should equal 0.22 years.

48

The conditions for determinacy of equilibrium can also be generalized using the same methods as above.

One again obtains a set of inequalities that the coe¬cients of the interest-rate rule must satisfy, and these

expressions are continuous functions of χ, so that the set of interest-rate rules that imply a determinate

equilibrium remain nearly the same when we assume a small positive value for χ.

49

Once again, we assume an output-gap target x consistent with the in¬‚ation target π . In the case that

¯ ¯

χ = 0, this requires that x = (1 ’ β)¯ /κ + •¯ .

¯ π π

3. MONEY AND AGGREGATE DEMAND 61

π

ψj

0.1

χ=0

0.08

χ = 0.02

0.06

0.04

0.02

0

0 1 2 3 4 5 6 7 8 9 10

ψx

j

2.5

χ=0

2

χ = 0.02

1.5

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8 9 10

Figure 4.10: E¬ects of anticipated natural-rate ¬‚uctuations under a simple Taylor rule,

allowing for real-balance e¬ects.

each panel, the coe¬cients are computed both for the parameter values stated above (the

dashed lines), and for the same values of β, σ, and κ, but under the assumption that χ and

• are equal to zero (as in the earlier ¬gures). We observe that the predicted e¬ects of both

types of disturbances on in¬‚ation and the output gap are not much a¬ected by allowing

for real-balance e¬ects. The main di¬erence is a greater in¬‚ationary impact of increases

in the natural rate of interest that are foreseen several quarters in advance, as a result of

the contribution of the resulting increase in the interest di¬erential to the marginal cost of

supply. But even this will matter for predicted in¬‚ation dynamics only to the extent that

¬‚uctuations in the natural rate of interest are predictable several quarters in advance.

In the case of an interest-rate rule of the form (2.8), again combined with the stipulation

62 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

π

tildeψj

0.1

χ=0

0.08

χ = 0.02

0.06

0.04

0.02

0

0 1 2 3 4 5 6 7 8 9 10

tildeψx

j

3

χ=0

2.5

χ = 0.02

2

1.5

1

0.5

0

’0.5

0 1 2 3 4 5 6 7 8 9 10

Figure 4.11: E¬ects of anticipated policy shifts under a simple Taylor rule, allowing for

real-balance e¬ects.

that ˆm = 0, we correspondingly obtain solutions of the form

±t

∞ ∞

˜π ±

π π

rn

πt = π + ω (ˆt’1 ’ ¯t’1 ) +

¯ ± ± ψj Et (ˆt+j + π) ’

¯ ψj Et¯t+j ,

j=0 j=0

˜π

and similarly for the other endogenous variables. Once again, the coe¬cients ψj and so on

π

are no longer exactly equal to the coe¬cients ψj when χ = 0. The quantitative signi¬cance of

the allowance for real-balance e¬ects is shown in Figures 12 and 13, using the same format as

in Figures 10 and 11. Here the assumed coe¬cients of the policy rule are φπ = 0.6, φx = 0.3,

and ρ = 0.7, as in the baseline case of Figure 4.6. Our conclusions in this case are essentially

the same.

Another way of considering the consequences of real-balance e¬ects for our previous

conclusions is to compute the predicted impulse responses of in¬‚ation, the output gap and

the nominal interest rate to a monetary policy shock using our modi¬ed structural equations.

3. MONEY AND AGGREGATE DEMAND 63

π

ψj

0.1

χ=0

0.08

χ = 0.02

0.06

0.04

0.02

0

0 1 2 3 4 5 6 7 8 9 10

ψx

j

4

χ=0

χ = 0.02

3

2

1

0

0 1 2 3 4 5 6 7 8 9 10