. 2
( 4)


2.2 The “Taylor Principle” and Determinacy

As discussed in chapter 1, the well-known “Taylor rule” (Taylor, 1993) di¬ers from Wicksell™s
classic proposal in that it directs the central bank to respond to deviations of the in¬‚ation
rate from a target level, without any reference to the absolute level that prices may have
reached. Such concern with the in¬‚ation rate rather than the level of prices would seem
to characterize policy in all advanced nations, at least since the breakdown of the Bretton
Woods system in the early 1970™s. Thus there is greater relevance for contemporary policy
discussions in considering a Taylor rule of the form

it = φ(Πt /Π— ; νt ) (2.43)

for central bank policy, where Πt ≡ Pt /Pt’1 is the gross in¬‚ation rate, Π— is a (possibly

time-varying) target rate, νt again represents exogenous shifts in this relation, and φ(·; ν) is
an increasing function for each value of ν. Once again, we shall suppose that in a cashless
economy, the central bank™s interest-rate operating target is implemented by setting im

each period equal to the right-hand side of (2.43), so that (2.43) holds in equilibrium as a
consequence of (1.15). Once again, {Mts } is allowed to be an arbitrary process, and ¬scal
policy is speci¬ed by an exogenous process {Dt }.
Again we shall consider equilibria near a zero-in¬‚ation steady state. Assuming again
that φ(1; 0) = β ’1 ’ 1, such a steady state is an equilibrium in the case that Π— = 1,

Yt = Y > 0, and νt = ξt = 0 at all times. We look for equilibria in which Πt and it ¬‚uctuate
within neighborhoods of their steady-state values, assuming that the exogenous variables
{Yt , ξt , Π— , νt } all remain forever within neighborhoods of their steady-state values.

A log-linear approximation to (2.43) is given by

ˆt = φπ (πt ’ πt ) + νt ,

or equivalently by
ˆt = ¯t + φπ πt ,
i ± (2.44)

where now πt ≡ log Π— , φπ > 0 is the elasticity of φ with respect to its ¬rst argument,


evaluated at the steady state, and ¯t ≡ νt ’ φπ πt measures the total exogenous shift in
the central bank™s reaction function. Substitution of this into (1.32) yields an expectational
di¬erence equation for the in¬‚ation rate,

φπ πt = Et πt+1 + (ˆt ’ ¯t ).
r ± (2.45)

In the case of a rule that conforms to what we may call the “Taylor principle” ” that the
central bank should raise its interest-rate instrument more than one-for-one with increases in
in¬‚ation,25 so that φπ > 1 ” we can again solve (2.45) forward, yielding a unique bounded
solution of the form

φ’(j+1) Et [ˆt+j ’ ¯t+j ].
πt = r ± (2.46)

We thus obtain the following result.

Proposition 2.6. If monetary policy is characterized by an interest-rate feedback rule
of the form (2.43), with φπ > 1, then the rational-expectations equilibrium paths of in¬‚ation
and the nominal interest rate are (locally) determinate; that is, there exist open sets P
and I such that in the case of any tight enough bounds on the ¬‚uctuations in the exogenous
processes {ˆt , πt , νt }, there exists a unique rational-expectations equilibrium in which πt ∈ P
and it ∈ I at all times. Furthermore, equation (2.46) gives a log-linear (¬rst-order Taylor
series) approximation to the evolution of in¬‚ation in that equilibrium. If instead 0 ¤ φπ < 1,
rational-expectations equilibrium is indeterminate, as in Proposition 2.5.
The rule described in Taylor (1993) obviously conforms to this principle, as it speci¬es that φπ = 1.5.
Discussions of the general desirability of such a principle include Taylor (1995, 1999).

Here the proof follows exactly the same lines as our proofs of Propositions 2.3 and 2.5 above.
When the Taylor Principle is satis¬ed, we ¬nd once again that an interest-rate feedback
rule can be compatible with a determinate equilibrium price level.26 We also observe again
that equilibrium in¬‚ation is determined by interaction between the real determinants of the
equilibrium real rate of return, on the one hand, and the nature of the central bank feedback
rule for setting the nominal interest rate, on the other, quite independently of the associated
evolution of any monetary aggregate. Moreover, our theory of in¬‚ation determination once
again has a Wicksellian ¬‚avor: increases in the equilibrium real rate that are not o¬set
by su¬cient tightening of monetary policy result in in¬‚ation, as do loosenings of policy
(decreases in ¯t ) that are not warranted by an exogenous decline in the equilibrium real
rate. And once again, it is only current and expected future values of the “gap” variable
rt ’ ¯t that matter for the generation of in¬‚ationary or de¬‚ationary impulses. Finally, given
any stochastic process for the gap variable, the resulting equilibrium in¬‚ation variability is
smaller the larger is the response elasticity φπ .
The main qualitative di¬erence between this family of rules and the Wicksellian regimes
considered earlier is that transitory ¬‚uctuations in the gap variable rt ’ ¯t now give rise to
transitory ¬‚uctuations in in¬‚ation, which however permanently shift the absolute level of
prices. Thus such a regime almost inevitably results in price-level drift (a unit root in the
log price level), of the kind that has in fact been observed in all advanced countries in recent
decades. This contrasts with the stationarity of the ¬‚uctuations in Pt under the Wicksellian
regime; there, a deterministic trend for Pt— would su¬ce to imply trend-stationarity of the
equilibrium price level.
Note that this last result holds even in the case of very small positive values of φp , which
correspond to policies that stabilize nominal interest rates to an arbitrarily great extent.
Thus price-level drift is not a necessary consequence of policies that achieve a great deal
It should be observed that the price level, and not just the in¬‚ation rate, is determined in the solution
represented by (2.46). This is because the previous period™s price level Pt’1 is given at any date t as a
predetermined state variable; unique determination of Πt then implies a unique equilibrium price level Pt .
The “nominal anchor” that allows determination of the absolute price level is thus the dependence of (2.43)
on Pt’1 through its dependence upon Πt .

of nominal interest-rate smoothing. The result of Goodfriend (1987), according to which a
policy that aims to reduce interest-rate variability, among other objectives, results in price-
level drift, actually depends upon his speci¬cation of the central bank™s objectives other
than interest-rate smoothing. Because the central bank is assumed to care only about the
variability of in¬‚ation over a short horizon, and not about the size of cumulative changes
in the price level, almost any obstacle to complete in¬‚ation stabilization (whether due to
infeasibility of perfect control, or to the presence of a con¬‚icting objective such as Good-
friend™s assumed concern with interest-rate variability) will result in its choosing a path that
sacri¬ces price-level stability for a less variable in¬‚ation.27
Another important di¬erence between this family of rules and the Wicksellian rules is that
here a positive response of the interest-rate operating target to deviations of in¬‚ation from
its target level does not su¬ce for determinacy. If we assume instead that φπ < 1, (2.45) has
an in¬nity of bounded solutions, so that equilibrium in¬‚ation in this case is indeterminate,
just as in the case of pure interest-rate control.28
Interestingly, Taylor (1999) ¬nds that U.S. monetary policy during the 1960™s and 1970™s
did not conform to the “Taylor principle”, as discussed in chapter 1. At least in the case of
a ¬‚exible-price model of the kind considered in this chapter, a systematic policy of the kind
that Taylor estimates for the period 1960-79 would imply indeterminacy of the equilibrium
price level. Clarida, Gali and Gertler (19xx) reach a similar conclusion, on the basis of an
estimated forward-looking rule, and propose that the indeterminacy of equilibrium explains
the instability of U.S. in¬‚ation and real activity during the 1970™s.29
See, e.g., Van Hoose (1989).
This result was ¬rst obtained by Leeper (1991) using similar linearization methods to those employed
here. Leeper distinguishes between “active” monetary policies (rules of the form (2.44) with φπ > 1)
and “passive” policies (rules with φπ < 1). Our results correspond to Leeper™s case in which ¬scal policy is
“passive”, though our speci¬cation of ¬scal policy in terms of an exogenous path for {Dt } does not correspond
exactly to any member of his parametric family of ¬scal rules. See further discussion in chapter 4, section
xx. Kerr and King (1996) also contains an early discussion of the connection between the Taylor principle
and determinacy of equilibrium.
Indeterminacy of the equilibrium in¬‚ation rate implies indeterminacy of equilibrium real activity as well,
in the context of a model with sticky prices of the kind discussed in chapter 4 below. Chari, Christiano and
Eichenbaum (1997) also argue that the instability of the 1970™s can be attributed to self-ful¬lling expectations;
but in their analysis, the multiplicity of equilibria results from an absence of commitment on the part of the

Of course, such an interpretation depends both upon an assumption that the interest-
rate regressions of these authors correctly identify the character of systematic monetary
policy during the period. In fact, an estimated reaction function of this kind could easily
be misspeci¬ed. For example, consider the equilibrium described by (1.39) in the case of a
Wicksellian regime, and suppose that Pt— grows deterministically at a constant rate. Since
in equilibrium ˆt is equal to φp times Pt , Pt is a stationary series, and πt is equal to its ¬rst-
di¬erence (up to a constant), one can show that the (asymptotic) coe¬cient of a regression
of ˆt on πt (rather than upon the correct variable, Pt ) will equal φp /2. This coe¬cient
could easily be much less than one “ suggesting violation of the “Taylor principle”, and
that the price level should be indeterminate “ even though in fact, as φp > 0, the price
level is determinate. Thus Taylor™s interpretation of his ¬nding of positive coe¬cients much
less than one on in¬‚ation in estimated “Taylor rules” for the classical gold standard period
(for example, a coe¬cient of only 0.02 for the period 1879-91), as indicating an even more
extreme version of the kind of unduly passive interest-rate responses seen in the 1960™s and
1970™s, may well be incorrect.

A less dramatic case of the same problem may bias downwards Taylor™s estimate of the
in¬‚ation response coe¬cient in the period 1960-79 as well, as Orphanides (20xx) argues.30
Yet the suggestion that the policy mistakes of the period may have related to a failure to
understand the restrictions upon monetary policy required for price-level determinacy is an
intriguing one, that suggests that an improved theoretical understanding of this issue could
be of considerable practical importance.

Fed, rather than from a commitment to systematic policy of an unfortunate sort.
In Orphanides™ analysis, systematic over-estimation of potential output during the 1970s gave policy
an in¬‚ationary bias, even though the Fed followed a rule similar to the Taylor rule that describes its later
behavior. Regressions using corrected estimates of the output gap, rather than the real-time estimates on
which policy was actually based then lead to a downward-biased estimate of the in¬‚ation-response coe¬cient,
since the period in which policy was looser due to the omitted variable was a period of higher-than-average

2.3 Inertial Responses to In¬‚ation Variation

The simple class of “Taylor rules” (2.43) allows only the crudest sort of approximation
to actual central bank policies, for a number of reasons. One of the more obvious is the
allowance for feedback only from the current period™s rate of in¬‚ation. In practice, monetary
policy will never involve feedback from an instantaneous rate of in¬‚ation (as is sometimes
assumed in continuous-time treatments of our problem), because available in¬‚ation measures
will always be time-averaged over at least a period such as a month. In fact, Taylor™s (1993)
account of recent U.S. policy assumes that the operating target for the funds rate is a
function of in¬‚ation over the previous year. It is thus desirable to consider rules involving
feedback from the rate of in¬‚ation averaged over a time longer than one “period”. A case in
which the analysis remains quite simple (even if it is not realistic as a literal representation
of central bank procedures) is that in which the central bank responds to an exponential
moving average of past in¬‚ation rates of the form

δ j πt’j ,
πt ≡ (1 ’ δ)
¯ (2.47)

for some decay factor 0 < δ < 1. This case is simple to analyze because the relevant in¬‚ation
measure evolves according to a simple partial-adjustment formula

πt = (1 ’ δ)πt + δ¯t’1 .
¯ π

Let the central bank™s log-linear feedback rule be given by

ˆt = ¯t + ¦π πt ,
± ± ¯ (2.48)

instead of (2.44), where πt is de¬ned by (2.47). We then obtain the following result.

Proposition 2.7. Let monetary policy be described by a feedback rule of the form
(2.48), at least near the zero-in¬‚ation steady state, with ¦π ≥ 0. Then equilibrium is deter-
minate if and only if ¦π > 1. When this condition is satis¬ed, a log-linear approximation to
the equilibrium evolution of the smoothed in¬‚ation process is given by

(δ + (1 ’ δ)¦π )’(j+1) Et [ˆt+j ’ ¯t+j ].
πt = (1 ’ δ)
¯ r ± (2.49)

A corresponding approximation to the equilibrium evolution of the single-period in¬‚ation
rate πt is then obtained by substituting (2.49) into

πt ’ δ¯t’1
¯ π
πt = . (2.50)

The proof is given in the appendix. Note that determinacy is once again obtained if and
only if ¦π > 1, as required by the “Taylor principle”. Furthermore, in the determinate case,
the smoothed in¬‚ation rate πt bears the same qualitative relation as above to expectations
regarding the equilibrium real rate and future monetary policy shifts: it is rises when the
natural rate rises without an o¬setting tightening of policy, either currently or anticipated
to occur in the future.
Estimated central bank reaction functions also typically di¬er from the simple rule (1.30)
in that they incorporate some degree of partial-adjustment dynamics for the interest rate
itself: that is, the current setting of the operating target for the interest rate inevitably
depends positively upon one or more lags of itself, in addition to measures of current economic
conditions such as some measure of in¬‚ation. As a simple example, we may consider a rule
of the form
ˆt = ¯t + ρ(ˆt’1 ’ ¯t’1 ) + φπ πt ,
± ± ± ± (2.51)

where again φπ ≥ 0, and the coe¬cient ρ ≥ 0 measures the degree of intrinsic inertia in
the central bank™s adjustment of its operating target. Note that when ρ < 1, this rule can
be represented as a partial-adjustment rule, like those discussed in chapter 1. However, it
is also of interest to consider rules with ρ = 1 (i.e., rules in which it is the change in the
operating target that is a function of current in¬‚ation, as assumed in some estimated central
bank reaction functions, e.g., that of Fuhrer and Moore, 1995b), or even “super-inertial”
rules with ρ > 1, like those considered by Rotemberg and Woodford (1999).
In the case that ρ < 1, the feedback rule (2.51) can actually be equivalently expressed
in the form (2.48), simply by solving backwards to eliminate the dependence upon the
lagged interest rate. In this alternative representation, the response coe¬cient ¦π in (2.48)
corresponds to φπ /(1 ’ ρ) in the new notation, and the decay factor δ in (2.47) corresponds

to ρ in the new notation. Thus Proposition 2.7 applies, and equilibrium is determinate if and
only if ¦π > 1, again in accordance with the “Taylor principle”. (In a case of this kind, the
principle must be understood to require that the eventual increase in the nominal interest
rate as a result of a sustained increase in the in¬‚ation rate is more than one-for-one.) And
once again, an appropriate moving average of in¬‚ation is positively related to an average of
current and expected future values of the gap between the natural rate of interest and the
policy stance measure ¯t .
But well-behaved rational expectations equilibria can also exist in the case of rules with
ρ ≥ 1. In fact, we can show the following.

Proposition 2.8. Let monetary policy be described by a feedback rule of the form
(2.51), at least near the zero-in¬‚ation steady state, with φπ , rho ≥ 0. Then equilibrium is
determinate if and only if φπ > 0 and

φπ + ρ > 1 (2.52)

When these conditions are satis¬ed, a log-linear approximation to the equilibrium evolution
of in¬‚ation is given by

πt = ’ (ˆt’1 ’ ¯t’1 ) + (φπ + ρ)’(j+1) Et [ˆt+j ’ ¯t+j ],
± ± r ± (2.53)
φπ j=0

where the interest rate evolves according to

φπ (φπ + ρ)’(j+1) Et [ˆt+j ’ ¯t+j ].)
ˆt = ¯t +
± ± r ± (2.54)

The proof is again in the appendix. When ρ < 1, this condition is equivalent to the re-
quirement that ¦π > 1, just discussed. But (2.52) applies more generally. In fact, it shows
that if ρ ≥ 1, any positive value for φπ su¬ces for determinacy. And indeed, if ρ > 1, the
equilibrium in¬‚ation rate is determinate even in the case of moderate negative values of φπ ,
as is discussed further in the appendix. Determinacy, requires however, that φπ = 0, since
in the absence of any feedback from in¬‚ation, Proposition 2.5 applies.

Equation (2.53) also provides a direct generalization of our earlier solution (2.46) for the
equilibrium path of the in¬‚ation rate. Again we see that a weighted average of current and
expected future “gap” terms determines the current in¬‚ation rate, given the lagged interest
rate (which a¬ects current in¬‚ation in the same way as an exogenous shift in the current
stance of monetary policy). The pair of equations (2.53) “ (2.54) can be solved iteratively
for the entire paths of in¬‚ation and the nominal interest rate, given an initial lagged interest
rate and the paths of the exogenous disturbances.

For example, suppose that the equilibrium real rate follows an AR(1) process with au-
toregressive coe¬cient 0 < ρr < 1, and consider policies of the form (2.51) with ¯t = 0 at
all times (the central bank reacts only to in¬‚ation, with no target changes and no control
errors). Then in the solution described by (2.53), the nominal interest rate will be perfectly
correlated (positively) with ¬‚uctuations in the real rate. But in¬‚ation ¬‚uctuations are less
persistent than the real rate ¬‚uctuations, because the e¬ect of an innovation in the real rate
at date t upon Et πt+1 is (ρr ’ ρ) times its e¬ect upon πt , whereas the e¬ect upon Et rt+1 is
ρr times its e¬ect upon rt . If ρ = ρr , the in¬‚ation ¬‚uctuations become purely transitory, and
for larger values of ρ, they actually become anti-persistent, i.e., the e¬ect a period after the
shock is actually of the opposite sign to the initial e¬ect. Figure 2.1 plots impulse responses
to a one percent increase in the equilibrium real rate, assuming φπ = 0.7, ρr = 0.8, and a
variety of values for ρ consistent with (2.52). Panel (a) shows the impulse response of the
nominal interest rate for each value of ρ, and panel (b) shows the impulse response of the
in¬‚ation rate.

This simple example illustrates two desirable features of a more inertial response to
in¬‚ation variations, relative to the purely contemporaneous speci¬cation (2.43). First of all,
inertia allows a given degree of reduction of the variance of in¬‚ation to be achieved with less
interest-rate variability. This is an implication of the following result.

Proposition 2.9. Suppose that the equilibrium real rate {ˆt } follows an exogenously
given stationary AR(1) process, and let the monetary policy rule be of the form (2.51), with

(a) Interest rate
ρ = 0.4
ρ = 0.6
ρ = 0.8
ρ = 1.0
2 ρ = 1.2



0 1 2 3 4 5 6 7 8 9 10

(b) Inflation
ρ = 0.4
ρ = 0.6
ρ = 0.8
ρ = 1.0
2 ρ = 1.2



0 1 2 3 4 5 6 7 8 9 10

Figure 2.1: Responses to an increase in the natural rate of interest.

ρ ≥ 0, φπ > 0 and a constant intercept consistent with the zero-in¬‚ation steady state (i.e.,
¯t = 0). Consider the choice of a policy rule (ρ, φπ ) within this class so as to bring about
a certain desired unconditional variance of in¬‚ation var(π) > 0 around the mean in¬‚ation
rate of zero. For any large enough value of ρ, there exists a φπ satisfying (2.52) such that
the unconditional variance of in¬‚ation in the stationary rational-expectations equilibrium
associated with this rule is of the desired magnitude. Furthermore, the larger is ρ, the
smaller is the unconditional variance of interest-rate ¬‚uctuations var(ˆ) in this equilibrium.

The proof is in the appendix. We shall argue in chapter 6 that the reduction of interest-rate
variability, as well as the reduction of in¬‚ation variability, is an appropriate goal of monetary
policy. It follows that inertial rules, even super-inertial rules (with ρ > 1), have an advantage
over other members of the class (2.51).

Second, an inertial rule allows a greater degree of stabilization of the long-run price level,
by making in¬‚ation ¬‚uctuations less persistent (or even anti-persistent, so that increases in
the price level are subsequently o¬set). We shall argue in chapter 6 that the variability of the
rate of in¬‚ation over a short horizon is more directly related to welfare losses that monetary
policy should seek to minimize than is instability of the long-run price level. Nonetheless,
some have argued that stabilization of the long-run price level is also an appropriate goal of
policy, for example, in order to facilitate long-term contracting (see, e.g., Hall and Mankiw,
One possible measure of long-run price-level instability (proposed in Rotemberg and
Woodford, 1999) is the variance of innovations in the Beveridge-Nelson (198xx) price-level
trend. Let us de¬ne the long-run price level as

log Pt∞ ≡ lim Et [log Pt+j ’ jE(π)], (2.55)

where E(π) is the unconditional expectation of the rate of in¬‚ation.31 The innovation in this
variable at date t is then de¬ned as

log Pt∞ Et’1 [log Pt∞ ]
’ = [Et πt+j ’ Et’1 πt+j ].

In the equilibrium just described, this innovation is equal to (1 ’ ρ)(1 ’ ρr )’1 (φπ + ρ ’ ρr )’1
times the innovation in the equilibrium real rate of return. Thus the variability of such
innovations is minimized, and in fact reduced to zero, when ρ = 1. In this case, the log
price level is actually trend-stationary, so that the long-run price level de¬ned in (2.55) is
deterministic. For values of ρ near one, the price level still possesses a unit root, but the
long-run price level still evolves relatively smoothly.
The fact that a rule with ρ = 1 exactly stabilizes the price level is less mysterious once
one recalls the equivalence between rules of the form (2.51) and rules of the form (2.48). For
0 < ρ < 1, the rule (2.51) is equivalent to making the interest rate a function of a weighted
average of past rates of in¬‚ation. As ρ approaches one, the e¬ective weights on past rates
This de¬nition assumes that in¬‚ation is a stationary variable, i.e., that log Pt is di¬erence-stationary.
Note that all of the equilibria that we discuss here have this property.

of in¬‚ation cease to be discounted relative to more recent rates, so that the rule e¬ectively
responds to cumulative in¬‚ation over the entire past, which is to say, the price level. In the
limit, a rule of this kind thus becomes equivalent to a Wicksellian rule of the form (1.30),32
which as shown earlier stabilizes the price level.
It may be wondered why rules with ρ ≥ 1 do not lead to instrument instability, i.e.,
a non-stationary process for the central bank™s operating target. In fact, if an arbitrary
stationary stochastic process for in¬‚ation is fed into (2.51), a non-stationary interest-rate
process is almost inevitably implied. However, in a rational expectations equilibrium, the
only kind of in¬‚ation processes that occur are exactly the kind that do not cause the interest
rate to be non-stationary. Think again about the Wicksellian regime. For most stationary
in¬‚ation processes, the price level has a unit root, and so a policy that responds to deviations
of the price level from a deterministic path should result in a unit root in the interest rate
as well, under the reasoning just suggested. The reason this does not happen is that, in
equilibrium, a policy of responding to price level deviations makes the price level stationary,
and not just the rate of in¬‚ation. This explains why ρ = 1 does not create a problem, but
the logic is exactly the same in the case that ρ > 1.
Rules with ρ > 1 might seem implausible, because commitment to such a rule implies a
commitment to continue raising interest rates to higher and higher levels, at an explosive rate,
if in¬‚ation is ever even temporarily above its target level, as long as it never subsequently
falls below the target rate. Of course, in equilibrium, these extreme actions need never be
taken, as any temporary increase in in¬‚ation is followed by subsequent undershooting of
the target rate, as shown in Figure 2.1(b). But it might be suspected that they fail to be
triggered only because of the anticipation that they would be, a “threat” that might properly
be considered incredible.33 In fact, this is not so. Note that in the equilibrium calculated
above, equilibrium in¬‚ation ¬‚uctuates over a bounded interval, as does the nominal interest
rate. Hence only the de¬nition of the policy rule (2.51) for in¬‚ation rates and lagged interest
Note that when ρ = 1, (2.51) is just a ¬rst-di¬erenced version of (1.34).
At a conference, Bob Hall compared policies with ρ > 1 to the “Doomsday Machine” in Doctor

rates within those intervals matters for our conclusion that (2.53) describes a locally unique
equilibrium. We could change the speci¬cation of the rule in the case of more extreme values
of the endogenous variables ” in particular, change it to specify that interest rates would
never be raised above some ¬nite level, as seems more credible ” and still obtain exactly
the same conclusions as above. Such a change would at most a¬ect our conclusions about
the existence of other rational expectations equilibria, that do not remain forever near the
steady state, a topic not yet addressed.
A commitment to inde¬nitely follow a super-inertial instrument rule implies a certain
restriction upon the expected evolution of in¬‚ation, that must hold in any bounded rational-
expectations equilibrium. In the case of a rule of the form (2.51), the restriction takes the
following form.

Proposition 2.10. Consider a policy rule of the form (2.51), where ρ > 1 and {¯t } is
a bounded process, to be adopted beginning at some date t0 . Then any bounded processes
{πt , ˆt } that satisfy (2.51) for all t ≥ t0 must be such that the predicted path of in¬‚ation,
looking forward from any date t ≥ t0 , satis¬es

ρ’(j+1) φπ Et πt+j = ’(ˆt’1 ’ ¯t’1 ).
± ± (2.56)

Conversely, any bounded processes satisfying (2.56) for all t ≥ t0 also satisfy (2.51) for all
t ≥ t0 .

The proof is in the appendix. Condition (2.56) is a restriction upon the expected future
path of in¬‚ation that must be satis¬ed (in any non-explosive REE) given the commitment
to the rule (2.51), that is quite independent of any other assumptions about the conditions
required for an equilibrium.
In fact, the policy can alternatively be stated as a commitment to adjust interest rates
as necessary in order to ensure that the projected future path of in¬‚ation always satis¬es
the target criterion (2.56). This way of stating the rule is an example of an in¬‚ation-forecast
targeting rule, of the kind discussed by Svensson (1997, 1999, 2001), though it di¬ers from

the simpler examples discussed by Svensson in that the target for the weighted average of
future in¬‚ation forecasts is time-varying, both in response to variation in the lagged level of
the nominal interest rate and owing to the exogenous policy shifts represented by the Et¯t+j
terms. We have just shown that a commitment to the instrument rule (2.51) implies that the
target criterion (2.56) must be expected to be satis¬ed at each date. And conversely, a policy
that ensures that the target criterion is satis¬ed at each date must involve an interest-rate
path that satis¬es (2.51) at each date.
Thus these two super¬cially di¬erent forms of policy commitment are actually equivalent,
at least as far as the set of bounded rational-expectations equilibria associated with either of
them are concerned; and this equivalence is independent of the details of the structural model
of the monetary transmission mechanism that is used to predict the consequences of following
either rule. Which way of describing the policy commitment should be preferred will depend
on which is judged more e¬ective as a way of communicating the policy commitment to
the public. In chapter 8, we show that in the context of at least some plausible models of
the monetary transmission mechanism, an optimal policy rule can be represented as a more
complicated version of a rule of this kind. There we discuss the representation of optimal
policy both in terms of a super-inertial instrument rule and in terms of a forecast-targeting

3 Price-Level Determination with Monetary Frictions
We now take up the question of how the equilibrium prices of goods in terms of money are
determined in an economy where the monetary liabilities of the central bank do facilitate
transactions, contrary to our simplifying assumption in the previous two sections. In all
actually existing economies, we observe that positive quantities of base money are held by
private parties despite the fact that this asset yields a lower return than other very short-term
riskless assets; this indicates that there must be advantages to holding money not allowed
for in our model above. Our analysis above is only useful in understanding actual economies,
then, if we can show that even when such transactions services of money are allowed for,

our conclusions about price-level determination are fairly similar to those obtained when
abstracting from this complication. Introducing transactions frictions will also allow us to
compare our neo-Wicksellian theory of in¬‚ation determination with the implications of a
traditional quantity-theoretic analysis.

3.1 A Model with Transactions Frictions

We now extend the above analysis to allow for transactions frictions that can be ameliorated
through the use of central-bank monetary liabilities. For simplicity, we use a model that
has been extensively used in rational-expectations monetarist analyses, the representative-
household model of Sidrauski (1967) and Brock (1974, 1975).34 In this approach, the trans-
actions frictions are not explicitly modeled; instead, the transactions services supplied by
real money balances are directly represented as an argument of household utility functions.35
We again assume a representative-household economy, but now with a household objec-
tive of the form

β t u(Ct , Mt /Pt ; ξt ) ,
E0 (3.1)

where Mt again measures the household™s end-of-period money balances, and Pt is the price
of the single good in terms of money in period t. The function u is now an indirect utility
function, incorporating the costs of transacting with a given level of money balances; hence
the vector of exogenous disturbances ξt may now include random variation in the transactions
technology, as well as actual preference shocks.36 For any given realization of ξt , we assume
that the period utility function u(c, m; ξt ) satis¬es standard neoclassical assumptions: it is
For other recent expositions of the model, see, e.g., Obstfeld and Rogo¬ (1996, sec. 8.3) and Walsh
(1998, sec. 2.3).
In the alternative analysis presented in section xx of the appendix, the transactions frictions are more
represented by a cash-in-advance constraint. On the similarity between models with transactions frictions of
that sort and the Sidrauski-Brock model, see also Feenstra (1986), Lucas and Stokey (1987), and Woodford
It is important to note that we do not assume that ξt is a scalar process; for example, ξt might contain
two components, one of which represents variation in impatience, and the other of which represents variation
in the liquidity services provided by money balances. The use of the single symbol ξt to represent both
shocks does not imply anything about their assumed correlation; the two components of the vector might,
for example, be distributed independently of one another.

concave and strictly increasing in each of the arguments (c, m).37 We shall also suppose that
u implies that both consumption and real balances are (strict) normal goods; i.e., we shall
assume that income-expansion paths are upward-sloping in the case of any ¬nite positive
relative price for the two “goods”.
The household™s budget constraints remain of the form assumed in section 1; note that
we have already allowed there for a possible return di¬erential between money and other
riskless nominal claims, even though in equilibrium no such di¬erential turned out to exist
(in the cashless economy). Hence the household™s problem is to choose processes {Ct , Mt }
satisfying (1.12) given its initial wealth W0 , so as to maximize (3.1). Necessary and su¬cient
conditions for this problem are derived along the same lines as before.
Once again, (1.10) and (1.11) must hold at all times, since otherwise no optimal plan
exists. The ¬rst-order conditions for optimal choice of the household™s money balances now
require that Mt ≥ 0 and
um (Ct , Mt /Pt ; ξt )
¤ ∆t
uc (Ct , Mt /Pt ; ξt )
at each date, with at least one condition holding with equality at each date; thus at any date
at which Mt > 0, one must have

um (Ct , Mt /Pt ; ξt )
= ∆t . (3.2)
uc (Ct , Mt /Pt ; ξt )

(Note that this condition generalizes (1.15) to the case in which utility is increasing in
real balances.) Conditions (1.16), (1.17) and the exhaustion of the intertemporal budget
constraint are again necessary as before, with only the change that now the marginal utility
of consumption must be written uc (Ct , mt ; ξt ). And once again this set of conditions can be
shown to be both necessary and su¬cient for optimality of the household™s plan.
The existence of a ¬nite level of money balances at which there is satiation in money, which level will
typically be increasing in the level of consumption, creates only minor technical complications. And indeed,
many explicit models of transactions frictions, such as cash-in-advance models, imply that such a satiation
level should exist. But whether there is satiation does not matter, for present purposes, except for the
question of whether it is possible to reduce nominal interest rates all the way to zero. We shall suppose in
general that we do not wish to do so, though the reason for this (distortions associated with de¬‚ation) will
only be introduced after we extend our model to include nominal rigidities.

Substituting the relations implied by market-clearing into the conditions for household
optimization, and assuming a policy regime under which Mts > 0 at all times, we again obtain
equilibrium conditions (1.20), (1.21), and (1.22) for each date, together with the conditions
um (Yt , Mts /Pt ; ξt )
= ∆t (3.3)
uc (Yt , Mts /Pt ; ξt )

β T Et [uc (YT , MT /PT ; ξT )YT + um (YT , MT /PT ; ξT )MT /PT ] < ∞,
s s s
T =t
generalizing (1.15) and (1.23) respectively. These relations, together with a speci¬cation of
the policy regime, provide a complete description of a rational-expectations equilibrium.
Note that under the assumption that both consumption and real balances are normal
goods, um /uc is increasing in consumption and decreasing in real balances. It follows that
we can solve (3.3) for equilibrium real balances,38 obtaining a relation of the form
= L(Yt , ∆t ; ξt ). (3.5)
Here the liquidity preference function L is increasing in Yt and decreasing in ∆t , for any value
of the disturbance vector ξt . Note that equation (3.5) corresponds to the “LM equation”
of the Keynesian system, or to the “money market equilibrium” condition of a monetarist
model. (In the case that im = 0, as in standard treatments, we can alternatively write the

liquidity preference function in terms of yt and it .) From a quantity-theoretic point of view,
it is this equilibrium condition that is regarded as determining the price level at each point
in time, given the money supply Mts at that date.
Finally, we can show once again that condition (1.22) may equivalently be written in the
form (1.24).39 We thus obtain the following generalization of our previous de¬nition.
Technically, monotonicity of um /uc is not quite enough: it implies that there is a well-de¬ned level of
equilibrium real money balances for ∆t in some interval (that may depend on Yt and ξt ), but it does not
exclude the possibility that no solution exists for interest rates that are either too high or too low. We shall
simplify our analysis, when necessary, by assuming boundary conditions on preferences that imply a solution
for any ∆t > 0; speci¬cally, we suppose that um becomes unboundedly large as real balances are made small,
and arbitrarily small as real balances are made large enough, which may or may not involve satiation at a
¬nite level of real balances. Note that our primary concern here will be with stationary ¬‚uctuations around
an equilibrium steady state, and for these purposes such boundary conditions are irrelevant.
In the presence of monetary frictions, Proposition 2.2 must be restated as Proposition 2.2 , which is
stated and proved in section xx of the appendix.

Definition. A rational-expectations equilibrium of the Sidrauski-Brock model is a pair
of processes {Pt , it } that satisfy (1.21), (1.24),40 (3.4), and (3.5) at all dates t ≥ 0, given the
exogenous processes {Yt , ξt }, and evolution of the variables {im , Mts , Dt } consistent with the

monetary-¬scal policy regime.

3.2 Interest-Rate Rules Reconsidered

We now turn to the speci¬cation of the monetary-¬scal policy regime. In an economy with
monetary frictions, it is no longer necessary for equilibrium that either it = im or Mts = 0; this

increases the range of possible ways in which monetary policy may be speci¬ed. The central
bank may freely choose (within certain bounds) any two of the variables it , im , and Mts ,

leaving the third to be endogenously determined by the “LM relation” (3.5). In particular,
it might choose a target for the monetary base Mts while maintaining a ¬xed rate (zero) for
im , and let market nominal interest rates be endogenously determined, as in many textbook

analyses; but it might also choose a short-run operating target for it while maintaining a
¬xed rate (zero) for im , and let the monetary base be endogenously determined, as under

current Fed procedures. The central bank™s choice of a rate of an interest rate to pay
on central-bank balances now longer implies a particular operating target for short-term
market interest rates, and it is now important to distinguish between these two aspects of
the monetary policy regime.

In one case, however, the details of the way in which the central bank chooses to im-
plement its interest-rate operating targets will be irrelevant for price-level determination.
This is the case, often assumed for pedagogical purposes, in which u(C, m; ξ) is additively
separable between the arguments C and m for each possible vector of disturbances ξ.41 In
this familiar case, the marginal utility of consumption is independent of real money balances,
just as in the cashless economy, even though now um > 0 in the case of a low enough level

In writing (1.21) and (1.24) for this model, we must express the marginal utility of consumption as
uc (Yt , Mts /Pt ; ξt ).
See, for example, the presentation of the model by Obstfeld and Rogo¬ (1996, sec. 8.3) or Walsh (1998,
sec. 2.3).

of real balances.

Let monetary policy be speci¬ed by an interest-rate rule, such as the Wicksellian rule
(1.30), together with an additional equation that speci¬es either the supply of base money Mts
or the interest paid on base money im . If the additional equation speci¬es the supply of base

money, then the interest-rate target is implemented by adjusting im as necessary in order for

the target interest rate it to satisfy (3.5), given the supply of base money. (This generalizes
the method of policy implementation assumed above for the cashless economy, where (1.15)
played the role of equilibrium condition (3.5).) If instead the additional equation speci¬es the
interest paid on money (perhaps specifying im = 0 at all times), then the interest-rate target

is implemented by adjusting Mts so as to satisfy (3.5). (This is a stylized representation of
the current method of policy implementation in countries like the U.S.) We suppose once
again that ¬scal policy is speci¬ed by an exogenous process {Dt }.

We have shown earlier that in the cashless case, conditions (1.21) and (1.30) alone su¬ce
to determine locally unique rational-expectations equilibrium paths for the variables {Pt , it },
as long as φp > 0. The same argument continues to apply here, given that the marginal
utility of consumption in (1.21) is once again a function only of the exogenous states (Yt , ξt ).
Corresponding to these paths will be locally unique paths for {Mts , im }, obtained by solving

(3.5) together with the additional equation that speci¬es one or the other of these variables
each period. Thus the conditions under which the interest-rate rule implies a determinate
rational-expectations equilibrium are exactly the same as in a cashless economy, and (1.37)
and (1.38) continue to provide a log-linear approximation to the equilibrium evolution of
prices and interest rates in response to small enough exogenous disturbances.

The same is true if the interest-rate rule is of a form such as (2.43), (2.48), or (2.51)
rather than (1.30). The neo-Wicksellian account of price-level determination developed above
continues to apply even in the presence of transactions frictions that allow for a non-negligible
interest di¬erential ∆t in equilibrium. In our model with transactions frictions, the money-
demand or “LM” relation (3.5) is a requirement for equilibrium, yet it plays no role in
determining the equilibrium evolution of prices under a given interest-rate rule; this relation

is relevant only to the question of how the central bank must adjust the instruments under
its direct control (Mts and im ) so as to implement its interest-rate operating targets.

3.3 A Comparison with Money-Growth Targeting

The money-demand relation is, of course, a crucial element in the theory of price-level
determination in the case of a money growth rule, i.e., a monetary policy speci¬ed in terms
of exogenous paths for the monetary base {Mts } and the interest rate paid on money {im },

with the interest rate {it } left to be determined by the market. Let us consider the case in
which Mts /Mt’1 and im remain forever within bounded intervals containing the steady-state

values 1 and 0 ¤ ¯m < β ’1 ’ 1 respectively, and look for rational-expectations equilibria in
which mt ≡ Mts /Pt and it remain forever near the constant values

m ≡ L(Y , ∆; 0),

¯ = β ’1 ’ 1 > 0

associated with a zero-in¬‚ation steady state. (Here ∆ ≡ 1 ’ β(1 +¯m ) > 0 is the steady-state
interest di¬erential.)
Log-linearizing (3.5) around this steady-state, we obtain a relation of the form

mt = ·y Yt ’ ·i (ˆt ’ ˆm ) + m
ˆ ± ±t t, (3.6)

ˆm ≡ log(1 + im /1 + ¯m ).
mt ≡ log(mt /m),
ˆ ¯ ±t ±

Here the constant coe¬cients are
¯ ¯
Y ‚L 1 ’ ∆ ‚L
·y ≡ > 0, ·i ≡ ’ > 0,
m ‚y
¯ m ‚∆

with the partial derivatives evaluated at the steady-state values of the arguments of L, and
the exogenous disturbance term is
1 ‚L
≡ ξt .
m ‚ξ

(The signs asserted above for these coe¬cients follow from the assumptions regarding pref-
erences stated earlier.) Note that ·y measures the income elasticity of money demand, and
·i the interest semi-elasticity of money demand; numerical values for these coe¬cients, and
for the statistical properties of the disturbance term, can thus be obtained from standard
econometric studies of money demand.
We can then study the local determinacy of equilibrium under such a policy by considering
the bounded processes {mt , ˆt } that satisfy the log-linear equilibrium relations (1.32) and
(3.6) at all times. We obtain the following result.

Proposition 2.11. In the context of a Sidrauski-Brock model with additively separable
preferences, consider the consequences of a monetary policy speci¬ed in terms of exogenous
paths {Mts , im }, together with a ¬scal policy speci¬ed by an exogenous path {Dt }. Under such

a regime, the rational-expectations equilibrium paths of prices and interest rates are (locally)
determinate; that is, there exist open sets P and I such that in the case of any tight enough
bounds on the ¬‚uctuations in the exogenous processes {Yt , ξt , Mts /Mt’1 , im , Dt /Dt’1 }, there

exists a unique rational-expectations equilibrium in which Pt /Mts ∈ P and it ∈ I at all
times. Furthermore, a log-linear approximation to the equilibrium path of the price level,
accurate up to a residual of order O(||ξ||2 ), takes the form

•j Et [log Mt+j ’ ·i log(1 + im ) ’ ut+j ] ’ log m,
log Pt = ¯ (3.7)

where the weights
•j ≡ >0
(1 + ·i )j+1
sum to one, and ut is a composite exogenous disturbance

ˆ m
’ ·i log(1 + ¯m ).
ut ≡ ·y Yt ’ ·i rt +
ˆ ±

The proof is given in the appendix.
We thus obtain a well-de¬ned rational expectations equilibrium price level under such a
policy, for arbitrary bounded ¬‚uctuations in the rate of money growth. This is the determi-

nacy result that Sargent and Wallace (1975) stress in their argument for the money supply
as the “optimal instrument of monetary policy.” However, we have seen that policy rules
need not take this form in order to imply a determinate equilibrium path for the price level;
interest-rate rules such as those advocated by Wicksell and Taylor also have this property.

Equation (3.7) also provides a simple theory of price-level determination; in the case that
one abstracts from disturbances other than the ¬‚uctuations in the rate of money growth,
it states that the log price level at any point in time is (up to a constant) just a weighted
average of current and expected future logs of the money supply.42 This appealingly simple
result may suggest that even in the analysis of other types of possible policy regimes, what
matters about any regime is the path of the money supply that it implies.44 It might then
seem natural that alternative strategies for policy should be considered in terms of how one
wishes to have the money supply evolve. But as we have seen, a straightforward analysis
of the consequences for in¬‚ation of alternative policy rules is equally possible without any
reference to either the evolution of the money supply or the determinants of money demand.

In fact, our conclusions above about the consequences of alternative interest-rate rules
can be viewed as more basic, for Proposition 2.11 is actually a consequence of our previ-
ous analysis of Wicksellian rules.45 For solving the money-demand relation (3.5) for the

This result is most often presented as the implications of a rational-expectations version of the Cagan
model of in¬‚ation determination.43 In Cagan™s (1956) model, desired log real money balances are a decreasing
linear function of the expected rate of growth of the log price level. A (discrete-time) relation of exactly
this kind is obtained by substituting (1.32) into (3.6) to eliminate it , except that in our case there is also a
time-varying intercept for this relation, equal to log m + ut + ·i log(1 + im ).
¯ t
The determinacy result in Proposition 2.11 applies only to policies that make the money supply an
exogenously speci¬ed process, and not to general feedback rules for determination of the money supply. Our
derivation of equation (3.7) as an implication of the requirements for rational-expectations equilibrium also
goes through more generally, assuming that the in¬nite sum on the right-hand side is well-de¬ned. But in
the case of feedback from endogenous variables (such as the price level) to the money supply, the expressions
on the right-hand sides of equations such as (3.7) need not be uniquely de¬ned. That is, there may be a large
number of equilibria, in each of which (3.7) is valid, but in each of which the expression on the right-hand
side takes a di¬erent value, as a result of di¬erent evolution of the endogenous determinants of the money
See Taylor (1999) for motivation in terms similar to these of the “Taylor rule”. Taylor™s discussion,
however, elides the distinction between rules such as (1.30), that respond to deviations of the price level)
from a target path, and rules such as (2.43) that respond to deviations of the in¬‚ation rate from its target.

equilibrium nominal interest rate, we obtain an equation of the form

it = ι(Pt /Mts ; im , Yt , ξt ), (3.8)

where ι is an increasing function of its ¬rst argument, for any values of im , Yt , ξt . This is

just Keynes™ (1936) “liquidity preference theory” of the interest rate; if it is graphed as a
function of Yt , suppressing the other arguments, this is the Hicksian “LM curve”. When Yt is
exogenous and prices are ¬‚exible, as here, it is more useful to think of this as an equilibrium
relation between it and Pt , as in some presentations of a “¬‚exible-price IS-LM” model.
This equilibrium relation between interest rates and prices, established through the cen-
tral bank™s control of its instruments Mts and im , is just an example of a interest-rate feedback

rule of the form (1.30), in which Pt— = Mts /m, νt = (im , Yt , ξt ),46 and
¯ t

φ(p; ν) = ι(p/m; ν).

It then follows that in the log-linear approximation (1.34), φp = ·i > 0 and

νt = ˆm + ·i (·y Yt +
’1 m
±t t ).

Because this is a policy rule under which φp > 0, Proposition 2.3 applies, and making the
above substitutions for φp , Pt— , and νt , we ¬nd that (3.7) is just the price-level path predicted
by our previous result (1.39) for a general rule of the Wicksellian type.
The prior formulation is the more general one, since it applies to Wicksellian rules in
which the elasticity of interest-rate response to price-level deviations need not equal exactly
·i . Furthermore, it is clear even from (3.7) ” which follows from a traditional quantity-
theoretic analysis, as shown in the appendix ” that the path of the money supply as such
is not important for price-level determination. What matters is the way in the central bank
chooses to adjust the composite variable log Mts ’ ·i log(1 + im ); it does not matter to what

extent this is achieved by varying the money supply as opposed to the interest rate paid on
base money (except, of course, that if there is to be trend growth in this variable it must
Here we extend our previous notation in (1.30) by allowing a vector νt of exogenous variables to a¬ect
the function φ.

occur through trend growth in the money supply, owing to the impossibility of reducing im

below zero). The reason for this is simple: it is the shift in this composite variable that
indicates the extent to which the central bank™s actions shift the equilibrium relation (3.8)
between interest rates and prices.
Of course, we could also develop a theory of price-level determination under money-
supply feedback rules, in which the path of the monetary base is not speci¬ed exogenously,
but is instead a speci¬ed function of the price level. Wicksellian rules (1.30) could then be
equivalently described as rules of this form,

Mts = M s (Pt ; ¶— , im , Yt , ξt , νt ), (3.9)

where the function M s is obtained by substituting (1.30) for it in (3.5), and solving for Mts
as a function of the other variables. But subsuming our theory of price-level determination
under interest-rate rules under such a theory of endogenous-money rules is not obviously
desirable. First of all, the theory of price-level determination under interest-rate rules ex-
pounded above continues to apply in a cashless economy (the form in which we have ¬rst
expounded it), whereas a theory of endogenous-money rules of the form (3.9) would not be
possible in that case.
Even when a well-de¬ned money-demand relation (3.5) exists, it is not obvious that
(1.30) and (3.9) are equally useful ways of specifying monetary policy in order to achieve
the equilibrium described by (1.39). For (1.30) su¬ces to specify the aspects of policy that
matter for the central bank™s stabilization objective (here assumed to be control of the price
level), while (3.9) does not ” the latter rule must be supplemented by a policy rule for
the control of im in order for the price level to be determined. (Of course, in traditional

accounts, it is taken for granted that im = 0.) And even granting a speci¬c, known policy

with regard to interest payments on money, (1.30) may be a more useful policy prescription.
This is because implementation of (3.9) requires that the central bank take account of the
current value of the disturbance to the money demand equation in setting its money-

supply instrument, while knowledge of this state is irrelevant for purposes of implementation

of the interest-rate rule (1.30).

Since the central bank will not be able to estimate this random disturbance with perfect
precision in practice, adoption of a monetary-base target necessarily results in interest-rate
variations in response to money-demand disturbances that could be avoided through the
use of an interest-rate instrument. Interest-rate variations in response to these shocks are
not desirable, from the point of view of price-level stabilization, unless they are positively
correlated with exogenous variations in the equilibrium real rate of interest. As the primary
source of money-demand variation is probably developments in the payments system that
have important e¬ects upon money demand without being of particular consequence for
equilibrium real rates of return, this seems an implausible line of argument.

The point that money-demand disturbances make an interest-rate operating target more
desirable than a money-supply target is of course a central argument in the celebrated
analysis of Poole (1970). Poole also argues that if, instead, “IS shocks” are a more important
source of instability than “LM shocks,” monetary control is superior to interest-rate control.
This might make it seem uncertain which concern, in practice, ought to dominate. But let
us recall the nature of the argument for a monetary instrument in the case of “IS shocks”. It
is desirable on stabilization grounds that interest rates rise in response to demand stimulus
(in Wicksellian terms, because these disturbances raise the natural rate of interest), but it
is assumed that with interest-rate control, interest rates will not rise. Instead, with control
of the money supply, such disturbances raise output (or in our ¬‚exible-price model, the
price level), and as a result increase money demand, leading to an automatic interest-rate
increase which aids stabilization. But this argument requires not only that the central bank
is not able to respond directly to the disturbance, but that it cannot condition its interest-
rate instrument upon output or the price level except indirectly, by letting interest rates be
a¬ected by an increase in money demand. If we assume instead that the central bank can
make its instrument a function of current prices and output directly, even though it cannot
make it a direct function of exogenous disturbances, then the interest-rate instrument is
unambiguously superior.

3.4 Consequences of Non-Separable Utility

The strong irrelevance result obtained in section xx ” accordance to which an interest-rate
rule such as (1.30) or (2.43) su¬ces to determine the equilibrium path of prices quite inde-
pendently of how the central bank implements its interest-rate operating targets through ad-
justment of the monetary base and/or the interest rate paid on money ” depends, of course,
on the special assumption of an additively-separable (indirect) utility function u(C, m; ξ).
And this assumption is not very realistic, despite its familiarity in textbook treatments; if
real balances supply a non-pecuniary yield owing to their usefulness in conducting trans-
actions, it makes sense that the marginal bene¬t of additional real balances should depend
on the volume of purchases that the household makes. (The most plausible assumption, on
intuitive grounds, would probably be that ucm > 0, so that consumption expenditure and
real balances are complements.)
Yet one can justify the neglect of real-balance e¬ects on the marginal utility of income in
equations such as (1.21) without assuming either additive separability or a genuinely cashless
economy (with the counter-factual implication that money must earn the same rate of return
as other riskless assets). In what Woodford (1998a) calls a “cashless limiting economy,” the
marginal utility of additional real balances becomes quite large as household real balances fall
to zero (assuming real purchases of a magnitude near Y ), so that it is possible in equilibrium
to have a non-trivial interest-rate di¬erential ∆t ; yet at the same time, the transactions that
use money are su¬ciently unimportant that variations in the level of real balances su¬cient
to require a substantial change in the interest-rate di¬erential have only a negligible e¬ect
on the marginal utility of real income (or of consumption).
The idea is that in such an economy money is used for transactions of only a very few
kinds, though it is essential for those. As a result, positive real balances are demanded even
in the case of a substantial interest-rate di¬erential (and hence, a substantial opportunity
cost of holding money); but equilibrium real balances are very small relative to national
income. Equation (3.3) requires that in such an equilibrium, an increase in real balances
equal in value to a substantial share of aggregate expenditure would have to increase utility

by as much as a substantial percentage increase in consumption; but if equilibrium real
balances are tiny in value relative to national income, a substantial percentage increase in
real balances may still have a negligible e¬ect on utility. As this e¬ect may continue to be
negligible for di¬erent levels of consumption, the e¬ect on the marginal utility of consumption
of a substantial percentage change in real balances may also be negligible. This then justi¬es
neglecting real-balance e¬ects in equations such as (1.21).
Formally, the elasticity of um , the marginal utility of additional real money balances,
with respect to changes in the level of real expenditure is equal to
Y umc
…≡ ,
a quantity that we have argued should plausibly have a non-trivial positive value. But what
matters for the extent to which variations in the level of real balances a¬ect the Fisher
relation (1.21) is the elasticity of uc , the marginal utility of additional expenditure, with
respect to changes in the level of real balances, or
χ≡ . (3.10)
We note that χ = sm …, where
¯ ¯m
sm ≡ ¯ =∆¯
Y uc Y
is the ¬‚ow rate of e¬ective expenditure by households on liquidity services (measured by
the interest foregone on the money balances that they hold) expressed as a proportion of
national income. It follows that χ and … must have the same sign. But in a “cashless lim-
iting economy,” sm is in¬nitesimally small (even though ∆ is not), as a result of which χ is
in¬nitesimally small (even though …) is not.47 Because χ is negligible, a log-linear approxi-
mation to (1.21) again takes the simple form (1.32); yet this is not equivalent to assuming
Woodford (1998xx) displays a parametric family of transactions technologies in which this limiting case
is approached as a certain parameter ±, measuring the fraction of goods that are purchased using cash, is
made arbitrarily small. The paper shows that the equilibrium solution for the state-contingent evolution of
prices is continuous in the parameter ±, so that the conclusions reached under the assumption that χ = 0,
as in a cashless model, provide an accurate approximation to the equilibrium dynamics in the case of any
economy with a small enough value of ±. At the same time, the interest-rate di¬erential remains bounded
away from zero as ± is made small; hence the equilibrium of the “cashless limiting economy” is in this respect
di¬erent from that of a genuinely cashless economy of the kind presented in section xx above.

additive separability, for one need not assume that … is negligible. (The latter quantity mat-
ters for the predictions of one™s analysis regarding the percentage ¬‚uctuations in the money
supply that should occur in equilibrium under a particular approach to implementation of
the interest-rate rule.)48
Thus the case of a “cashless limiting economy” is another in which one may legitimately
abstract from any e¬ects of variations in real balances on the equilibrium conditions that
determine the evolution of prices under an interest-rate rule. How accurately this limiting
case approximates the situation of an actual economy in which central-bank money still
provides some valuable services remains, of course, a question for quantitative analysis. We
turn, then, to the question of the extent to which our previous results must be modi¬ed if
we allow for a non-negligible value of the elasticity χ de¬ned in (3.10).
In the general case of the Sidrauski-Brock model, a log-linear approximation to (1.21)
takes the form
ˆt = rt + Et πt+1 ’ χEt (mt+1 ’ mt ),
± ˆ ˆ ˆ (3.11)

where rt continues to be de¬ned by (1.33). (Note, however, that rt no longer has the
ˆ ˆ
interpretation of being the equilibrium real rate of return; the latter quantity is no longer
completely exogenous.) Substituting (3.6) for the equilibrium level of real balances in this
equation, we obtain a relation of the form

(1 + ·i χ)ˆt = rt + Et πt+1 + ·i χEtˆt+1 + ·i χEt (ˆm ’ ˆm ),
± ˜ ± ±t+1 ±t (3.12)

where the composite exogenous disturbance in this relation is given by

ˆ ˆ m m
rt ≡ rt ’ ·y χEt (Yt+1 ’ Yt ) ’ χEt (
˜ ˆ ’ t ).

Then if one speci¬es the policy rule in a way that allows both it and im to be determined as

functions of the path of prices, equation (3.12) alone, together with the policy rule, su¬ces
The assumption of a “cashless limiting economy” leaves one free to assume elasticities ·y , ·i in the
money-demand relation (3.6) in accordance with empirical estimates, whereas the assumption of additive
separability would imply a restriction upon these elasticities that is not in accordance with typical estimates,
as discussed below.

to determine the equilibrium paths of interest rates and prices, just as in our analyses in
sections 1 and 2.
For example, consider a policy regime under which the central bank seeks to maintain
a constant interest-rate di¬erential ∆ between overnight market rates and the interest paid
on the monetary base (as is true of the channel systems described in chapter 1, at least in
the case of the interest paid on central-bank balances). In such a case, monetary policy may
be speci¬ed by an interest-rate rule, such as the Wicksellian rule (1.30), together with the
∆t = ∆ > 0 (3.13)

indicating the way in which the interest paid on money varies with the changes in the
interest-rate target required by the Wicksellian rule. (We assume that ∆ > 0 because of the
observed preference of central banks with channel systems for maintaining a small positive
spread.) Fiscal policy can again be speci¬ed by an exogenous sequence {Dt }.
In order to analyze local equilibrium determination under such a regime, it su¬ces that
we consider the log-linear relations (1.34) and (3.12), together with the equation

ˆt = ˆm ,
± ±t (3.14)

representing the log-linearization of (3.13). Using (3.14) to eliminate ˆm from (3.12), we
obtain a relation of the form
ˆt = rt + Et πt+1 .
± ˜ (3.15)

This is of exactly the same form as (1.32), except that the term rt is replaced by rt , a di¬erent
ˆ ˜
function of the exogenous disturbances. We observe from (3.15) that rt can be interpreted
as the equilibrium real rate of return in the case that a constant interest-rate di¬erential is
maintained. (It is only under the latter stipulation that we can de¬ne an equilibrium real
rate that is purely exogenous, in the case of non-separable utility.)
We now look for bounded solutions {Pt , ˆt } to the system of equations (1.34) and (3.15).
Because (3.15) is identical to (1.32), except for the replacement of rt by rt , our previous
ˆ ˜
results are directly applicable. Proposition 2.3 then implies that equilibrium is determinate

in the case of any rule with φp > 0, and a log-linear approximation to the equilibrium price
process is given by (1.39), with rt+j replacing rt+j in each term.49 Hence the same theory of
˜ ˆ
price-level determination as derived above continues to apply, with a small modi¬cation of
our interpretation of the exogenous disturbance term. The same is true for other families of
interest-rate rules, such as (2.43) or (2.51).
Suppose instead that policy is implemented in a way that involves a ¬xed rate of interest
on the monetary base (for example, a zero interest rate, as in the U.S. at present). In this
case, (3.13) is replaced by the condition im = ¯. We can again eliminate ˆm from (3.12),
± ±t

and obtain a system of two equations to solve for the equilibrium evolution of prices and
interest rates. Similar methods as have been used before can then be employed to derive
generalizations of our previous results.
Proposition 2.12. In a Sidrauski-Brock model where utility is not necessarily sepa-
rable, let monetary policy be speci¬ed by a Wicksellian rule (1.30) for the central bank™s
interest-rate operating target. Suppose that im = ¯ at all times, for some 0 ¤ ¯ < β ’1 ’ 1;
± ±

and let ¬scal policy again be speci¬ed by an exogenous process {Dt }. Finally, suppose that

χ>’ . (3.16)

Then equilibrium is determinate in the case of any policy rule with φp > 0. A log-linear
approximation to the locally unique equilibrium price process is given by
∞ ∞
φ’1 •j Et [log Pt+j ’ φ’1 νt+j ],

log Pt = •j Et rt+j +
˜ ˜ (3.17)
p p
j=0 j=0

where the weights are given by

(1 + ·i χφp )j φp
•j ≡ , (3.18)
[1 + (1 + ·i χ)φp ]j+1

(1 + ·i χφp )j’1 φp
(1 + ·i χ)φp
•0 ≡
˜ , •j ≡
˜ for j ≥ 1. (3.19)
[1 + (1 + ·i χ)φp ]j+1
1 + (1 + ·i χ)φp
In fact, our previous results for the Wicksellian regime in a cashless economy can be recognized as a
special case of the more general result obtained here. For that regime was also one in which (3.13) held at
all times, with the value ∆ = 0; and the composite disturbance rt is just the value of rt in the case of a
ˆ ˜
cashless economy, since χ = 0 in that case.

The proof is given in the appendix. Here the solution (3.17) generalizes our previous result
(1.39), to which it reduces in the case that χ = 0. As long as χ also satis¬es χ > ’(φp ·i )’1 ,
the weights {•j , •j } are all positive, and again both series of weights sum to one, though for
χ = 0, the weight •j no longer exactly equals •j .
Thus as long as χ ≥ 0 (the case of greatest empirical plausibility), and even for su¬ciently
small negative values of χ, the same condition as before su¬ces for determinacy. We also
obtain a qualitatively similar theory of price-level determination: again, the log price level
is equal to a weighted average of current and future price-level targets, plus a discrepancy
that involves current and expected future ¬‚uctuations in the (suitably quali¬ed) equilibrium
real rate of return rt and in the reaction-function shift factor νt in a manner similar to the
previous equation (1.39). The solution (3.17) is also observed to be continuous in χ, so that
for any economy in which χ is small enough, the equations derived for the cashless economy
provide a close approximation to the equilibrium evolution of prices.
We can similarly extend the results derived earlier for the case of a generalized Taylor
rule (2.51).

Proposition 2.13. Let monetary policy instead be speci¬ed by an interest-rate rule of
the form (2.51), with coe¬cients φπ , ρ ≥ 0, and again suppose that im = ¯m at all times.

Finally, suppose again that χ satis¬es (3.16).Then equilibrium is determinate if and only if
φπ > 0 and (2.52) holds. When these conditions are satis¬ed, a log-linear approximation to
the equilibrium evolution of in¬‚ation is given by
∞ ∞
πt = ’ (ˆt’1 ’ ¯t’1 ) +
± ± •j Et rt+j ’
˜ •j Et¯t+j ,
˜± (3.20)
φπ j=0 j=0

where the weights are given by
(1 + ·i χφπ )j
•j ≡ , (3.21)
[(1 + ·i χ)φπ + ρ]j+1
[1 + (1 ’ ρ)·i χ](1 + ·i χφπ )j’1
(1 + ·i χ)
•0 ≡
˜ , •j ≡
˜ for j ≥ 1. (3.22)
[(1 + ·i χ)φπ + ρ]j+1
(1 + ·i χ)φπ + ρ

The proof is in the appendix. Thus the Taylor principle is again necessary and su¬cient for
determinacy, as long as χ does not take a large negative value. The solution (3.20) generalizes
(2.53), to which it reduces when χ = 0. Again the weights •j are no longer exactly equal to
the •j weights when χ = 0. However, the weights continue all to be positive, if ρ ¤ 1 and χ
is not too negative.

We have found that our qualitative results are largely una¬ected by taking account of real-
balance e¬ects on the equilibrium real rate of interest. How much are such e¬ects likely to
matter quantitatively? The size of the required correction can be numerically calibrated from
estimates of money demand. Our simple model implies that the money demand elasticities
should be the same at both high and low frequencies, and if we expect the disturbances t

to be unimportant at low frequencies, low-frequency relations among the variables mt , Yt ,
and it will be most revealing about these elasticities. As a typical example, Lucas (1999)
¬nds that over the period xxxx, low-frequency variations in real M1 for the U.S. are fairly
well ¬t (up to a constant) by the variations in log Yt ’ .5 log it .50 Linearization of (3.3) in the
logs of Yt , mt , and it indicates that according to our model, mt should be proportional to

[σ ’1 + …]Yt ’ β log(it /¯).

(Here it should be recalled that we log-linearize around a steady state with zero in¬‚ation, and
that we assume zero interest on the monetary base, as is true for the U.S.) Thus β ’1 [σ ’1 + …]
should be the ratio of the income elasticity to the interest elasticity of money demand;
according to Lucas™ estimates, this ratio is approximately 2.51

Our model should really be interpreted as one of demand for the monetary base, but empirical studies
more often model the demand for a broader aggregate such as M1. At the low frequencies with which we
are here concerned, M1 and the base move roughly in proportion to one another, so we shall use Lucas™
estimated elasticities as estimates of the corresponding elasticities of demand for the monetary base.
Here we may observe the way in which the assumption of an additively separable utility function,
implying … = 0, would restrict the model™s implications regarding money demand in an undesirable way. We
would obtain a necessary relation between the degree of interest-sensitivity of private expenditure, on the
one hand, and the income-elasticity and interest-elasticity of money demand on the other. A negligible value
of χ is instead consistent with arbitrary values of these other elasticities, as long as one assumes a small
enough value for sm .

From this we can infer that

χ = sm … = sm (2β ’ σ ’1 ). (3.23)

If σ is greater than 0.5, as we shall assume,52 it follows that χ > 0, as argued above on
intuitive grounds. At the same time, no matter how large we assume σ to be, the factor
in parentheses in (3.23) cannot exceed 2, so that χ cannot realistically be assigned a value
larger than twice the size of sm . For the U.S., the value of the monetary base is about 25
percent of a quarter™s GDP, so that (using the value β = 0.99 for a quarterly model) sm
is approximately 0.0025. This suggests a value of χ no larger than 0.005. Alternatively, if
we use Lucas™ estimated coe¬cients for a semi-logarithmic speci¬cation of money demand,
namely ·y = 1 and ·i = 7 years (or 28 quarters), then the implied ratio of elasticities would
be ·y /(1 ’ β)·i = 1/0.28 = 3.6. This would allow χ to be a larger multiple of sm , but only
by a factor of less than two, so that χ should be no larger than 0.01. Since the monetary
base is equally small for most industrial economies, a similar conclusion as to the plausible
size of real-balance e¬ects would be reached for many economies.
As an illustration of how much allowing for χ > 0 would a¬ect our calculations, consider
the solution (3.20) for the case of an inertial Taylor rule. Let us consider a policy rule with
ρ = 0.8, a fairly typical degree of interest-rate inertia in estimated Fed reaction functions for
the U.S. using quarterly data (see Table 1.1), and φπ = 0.3, implying a long-run in¬‚ation
response coe¬cient of ¦π = 1.5, Taylor™s (1993) value for the Greenspan period. Then the


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