<<

. 2
( 5)



>>

j=0

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




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