. 1
( 5)


Interest and Prices

Michael Woodford
Princeton University

September 2000
Revised September 2002

Preliminary and Incomplete

c Michael Woodford 2002

4 A Neo-Wicksellian Framework 1
1 A Basic Model of the E¬ects of Monetary Policy . . . . .......... . . 3
1.1 An Intertemporal IS Relation . . . . . . . . . . .......... . . 4
1.2 A Complete Model . . . . . . . . . . . . . . . . .......... . . 9
2 Interest-Rate Rules and Price Stability . . . . . . . . . .......... . . 12
2.1 The Natural Rate of Interest . . . . . . . . . . . .......... . . 13
2.2 Conditions for Determinacy of Equilibrium . . . .......... . . 18
2.3 Stability under Learning Dynamics . . . . . . . .......... . . 32
2.4 Determinants of In¬‚ation . . . . . . . . . . . . . .......... . . 32
2.5 Policy Rules for In¬‚ation Stabilization . . . . . .......... . . 43
3 Money and Aggregate Demand . . . . . . . . . . . . . .......... . . 48
3.1 An Optimizing IS-LM Model . . . . . . . . . . . .......... . . 48
3.2 Real-Balance E¬ects . . . . . . . . . . . . . . . .......... . . 53
3.3 Monetary Policy in a “Liquidity Trap” . . . . . .......... . . 65
4 Delayed E¬ects of Monetary Policy . . . . . . . . . . . .......... . . 65
4.1 Consequences of Predetermined Expenditure . . .......... . . 66
4.2 Small Quantitative Models of the E¬ects of U.S. Monetary Policy . . 77
4.3 Additional Sources of Delay . . . . . . . . . . . .......... . . 94
5 Monetary Policy and Investment Dynamics . . . . . . . .......... . . 94
5.1 Investment Demand with Sticky Prices . . . . . .......... . . 95
5.2 Optimal Price-Setting with Endogenous Capital .......... . . 101
5.3 Comparison with the Baseline Model . . . . . . .......... . . 106
5.4 Capital and the Natural Rate of Interest . . . . .......... . . 117
Chapter 4

A Neo-Wicksellian Framework for the
Analysis of Monetary Policy

We are now ready to consider the e¬ects of alternative interest-rate rules for monetary policy,
in a setting in which monetary policy has real e¬ects owing to the nominal rigidities discussed
in the previous chapter. As in chapter 2, a crucial ¬rst issue in the choice of an interest-
rate feedback rule is the selection of a rule that results in a determinate equilibrium. We
reconsider this issue (previously treated under the case of full price ¬‚exibility), and show
that the “Taylor principle” ” the requirement that interest rates be increased more than
one-for-one in response to sustained increases in the in¬‚ation rate ” continues to be essential
for determinacy.

We may then consider the nature of in¬‚ation and output determination in the case that a
determinate equilibrium exists. Once again, we shall show that the equilibrium evolution of
these variables can be understood without reference to the implied path of the money supply,
or to the determinants of money demand. When monetary policy is speci¬ed in terms of an
interest-rate feedback rule ” a speci¬cation which more directly matches the terms in which
monetary policy is discussed within actual central banks ” then it is possible to understand
the e¬ects of such policies by directly modeling the e¬ects of interest rates upon spending and
pricing decisions, without attaching any central importance to the question of how various
monetary aggregates may also happen to evolve. This is in fact the approach already taken


in many of the econometric models used for policy simulations within central banks.1 A
primary goal of the present exposition will be to show how models with this basic structure
” roughly speaking, models that consist of an “IS block,” and “aggregate supply (AS) block”
and an interest-rate feedback rule ” can be derived from explicit optimizing foundations.
In this way it is established that a non-monetarist analysis of the e¬ects of monetary policy
does not involve any theoretical inconsistency or departure from neoclassical orthodoxy.
Instead, we shall argue that in¬‚ation and output determination can be usefully explained
in Wicksellian terms ” as depending upon the relation between a “natural rate of interest”
determined primarily by real factors and the central bank™s rule for adjusting the short-
term nominal interest rate that serves as its operating target. Increases in output gaps
and in in¬‚ation result from increases in the natural rate of interest that are not o¬set by
a corresponding tightening of monetary policy (positive shift in the intercept term of the
interest-rate feedback rule), or alternatively from loosenings of monetary policy that are not
justi¬ed by declines in the natural rate of interest.2
While this basic approach to in¬‚ation determination has already been introduced in
chapter 2, it is only in an environment with sticky prices that we are able to introduce
the crucial Wicksellian distinction between the actual and the “natural” rate of interest, as
the discrepancy between the two arises only as a consequence of failure of prices to adjust
su¬ciently rapidly. Here we also discuss the underlying real determinants of variation in the
natural rate of interest, and discuss the way in which a central bank would respond to such
variations in order to maintain stable prices or a stable rate of in¬‚ation.
We ¬rst expound our neo-Wicksellian analysis in the context of a very simple intertem-
poral equilibrium model, in which we abstract from endogenous variation in the economy™s
capital stock. We then extend the model in section 3 to consider the consequences for the
See, e.g., Black et al. (1997), Brayton et al. (1997), and Coletti et al. (1996), for discussions of the
models currently used at the U.S. Federal Reserve Board, the Bank of Canada, and the Reserve Bank of
New Zealand. A similar approach is also already common in small macro-econometric models used for policy
evaluation in the academic literature as well (e.g., Fuhrer and Moore, 1995a), but such models are typically
not derived from explicit optimizing foundations.
For Wicksell™s views see Wicksell (1898, 1906, 1907). Recent discussions include Humphrey (1992),
Fuhrer and Moore (1995b), and Woodford (1998).

monetary transmission mechanism of endogenous capital accumulation. We show that many
aspects of the basic model are preserved by this extension. In particular, it is still true that
a useful approach to in¬‚ation stabilization involves commitment to a “Taylor rule” under
which the intercept term varies one-for-one with variation over time in the Wicksellian natu-
ral rate of interest. Finally, in section 4 we consider further extensions of the basic framework
that incorporate more realistic delays in the e¬ects of monetary policy upon in¬‚ation and
economic activity.

1 A Basic Model of the E¬ects of Monetary Policy

We here present a ¬rst complete general-equilibrium model of the monetary transmission
mechanism, of which we shall make frequent use in the remainder of this essay. (References
below to “the baseline model” refer to the model presented in this section.) This model
combines the relation between interest-rate targeting by the central bank and intertemporal
resource allocation developed in section 1 of chapter 2 with the relation between real activity
and in¬‚ation developed in section 2 of chapter 3. One should note that the assumptions made
in separately deriving these equilibrium relations are in fact mutually consistent, so that
our separate partial results can be combined to yield a complete, though highly stylized,
model. The resulting framework indicates how interest rates, in¬‚ation and real output
are jointly determined in a model that abstracts from endogenous variations in the capital
stock, and that assumes perfectly ¬‚exible wages (or some other mechanism for e¬cient labor
contracting), but monopolistic competition in goods markets, and sticky prices that are
adjusted at random intervals in the way assumed by Calvo (1983).3

The model expounded here was ¬rst presented as a simple example of an optimizing framework for the
analysis of alternative monetary policies in Woodford (1994a, 1996). Similar models have been extensively
used in the recent literature; see, e.g., Kerr and King (1996), Bernanke and Woodford (1997), Rotemberg
and Woodford (1997, 1999a), McCallum and Nelson (1999), and Clarida et al. (1999).

1.1 An Intertemporal IS Relation

We recall from chapter 2 that the representative household™s optimal intertemporal allocation
of consumption spending must satisfy the Euler equation
uc (Ct+1 ; ξt+1 ) ’1
1 + it = β ’1 Et Πt+1 , (1.1)
uc (Ct ; ξt )

where it is the riskless one-period nominal interest rate controlled by the central bank. As
shown in chapter 2, the simple form (1.1) can be derived either in the case of a utility
function that is additively separable between consumption and real money balances, or
(our preferred interpretation) in the “cashless limit” discussed in section 3.3 of chapter 2.
(The consequences of allowing for non-trivial monetary frictions and a non-separable utility
function are taken up in section xx below.) In chapter 2, we neglected labor supply, as we
treated the economy™s supply of goods as a simple endowment; but even with endogenous
labor supply, condition (1.1) is una¬ected as long as utility is separable in consumption and
leisure, as assumed in chapter 3.4 Finally, in chapter 2 we assumed a single perishable good,
whereas we now assume the existence of a continuum of di¬erentiated goods, in order to
allow price-setting by monopolistically competitive producers, as in chapter 3. However, as
long as utility depends only upon the Dixit-Stiglitz aggregate of consumption of the various
di¬erentiated goods (de¬ned in equation xx of chapter 3), then the Euler equation (1.1)
continues to apply, where now Ct refers to the representative household™s demand for the
consumption aggregate, and Πt ≡ Pt /Pt’1 refers to the gross rate of increase in the Dixit-
Stiglitz price index Pt (de¬ned in equation xx of chapter 3).
Of course, equation (1.1) is not the only requirement that must be satis¬ed for the
household™s consumption plan to be optimal. Consumption spending must also be optimally
allocated each period across the various di¬erentiated goods; this requirement leads to the
constant-elasticity demand curve for each of the individual goods (equation xx of chapter
More precisely, we assume that utility is additively separable between a function of consumption and
real balances on the one hand, and a function of leisure or hours worked on the other. In the cashless limit,
the marginal utility of additional consumption is essentially independent of the level of real money balances,
as is assumed in (1.1).

3), assumed in our model of optimal pricing behavior. And (1.1) is not the only ¬rst-order
necessary condition for optimal intertemporal allocation of aggregate consumption expendi-
ture; in addition, the marginal rate of substitution between consumption at date t and any
possible state at date t + 1 must equal the relevant stochastic discount factor (equation xx
of chapter 2). However, this much more detailed set of ¬rst-order conditions turns out not
to be necessary for the derivation of a complete set of equilibrium conditions su¬cient to
determine interest rates, in¬‚ation and output, as long as the central bank™s reaction function
does not itself involve any asset prices other than the short riskless nominal interest rate.5
The additional equilibrium conditions are needed only if we wish also to determine the equi-
librium values of other asset prices, and we set that question aside here. Finally, optimality
also requires exhaustion of the household™s intertemporal budget constraint (conditions xx
or xx in chapter 2). But, just as in chapter 2, this additional equilibrium condition plays
no role in our analysis of equilibrium determination, as long as ¬scal policy is “locally Ri-
cardian” (so that the additional equilibrium condition is automatically satis¬ed in the case
of all paths involving small enough deviations of interest rates, in¬‚ation and output from
certain reference values) and as long as we are concerned solely with local analysis (i.e., with
the equilibrium responses to su¬ciently small shocks).

In a model where the only source of demand for produced goods is private consumption
demand, equilibrium requires that the Dixit-Stiglitz index of aggregate demand Yt (that
¬gures, for example, in the demand curve for an individual good, given by equation xx of
chapter 3) is equal to the representative household™s choice of the consumption aggregate
Ct . We can thus once again substitute6 Yt for Ct in the Euler equation (1.1), and obtain the

If, as is sometimes assumed, the central bank sets its short-rate instrument as a function of an observed
long bond rate or a term-structure spread, then a complete model of the consequences of such a rule would
have to include a model of the equilibrium term structure of interest rates as well. The statement in the text
also assumes that ¬scal policy is locally Ricardian (in the sense to be explained in chapter 5), as we shall
suppose throughout this chapter, or else that all outstanding government debt consists of riskless one-period
nominal bonds, so that again the short nominal rate is the only asset price that matters for equilibrium
determination. In the case of a non-Ricardian ¬scal policy and long-term government debt, a complete
model again must include a model of the term structure, as discussed in section 2.3 of chapter 5.
Recall that we have already made this same substitution in the derivation of our aggregate supply
relation in chapter 3. Thus the assumptions made there and those used here are mutually consistent.

equilibrium condition
uc (Yt+1 ; ξt+1 ) ’1
1 + it = β ’1 Et Πt+1 , (1.2)
uc (Yt ; ξt )
linking interest rates and the level of real activity. This condition is necessary for equi-
librium here, just as in chapter 2, since the derivation there depended in no way upon the
assumption that output was given exogenously. However, its interpretation is now somewhat
di¬erent. When we ¬rst encountered this relation (as equation xx in chapter 2), its natural
interpretation was as a relation that deternined the equilibrium real rate of return, given
the economy™s exogenous supply of goods. In the present context it is instead most usefully
viewed as the analog, in an intertemporal equilibrium model, of the Hicksian “IS curve”.7
That is, it determines the level of real aggregate demand associated with a given real interest
rate, and then since output is demand-determined in the present model, it determines the
equilibrium level of output associated with a given real interest rate.
It might seem that this intertemporal “IS relation” depends upon an extremely restrictive
conception of the demand for produced goods, namely that all demand is private demand
for non-durable consumption goods. However, the model can be understood to allow for
government purchases through a simple reinterpretation of the notation, as already noted
in chapter 2. The function u(Y ; ξ) should be understood to indicate the level of utility
from private consumption when aggregate demand is Y , even if that aggregate demand also
includes government purchases; as long as government purchases are an exogenous state
variable, we can regard them as simply another element of the vector ξ of exogenous random
disturbances to the functional relation between utility and Y .
Furthermore, one need not understand the model to assume that investment demand is
zero. (This point matters when it comes time to “calibrate” our model for use in quantitative
analysis.) A more generous view of our baseline model would be that it abstracts from the
e¬ects of variations in private spending (including those classi¬ed as investment expenditure
The analogy between this equilibrium relation and the “IS curve” is stressed in particular in Woodford
(1994a, 1996), Kerr and King (1996), Bernanke and Woodford (1997), and McCallum and Nelson (1999).
An early derivation of an “IS relation” from intertemporal optimization in the same spirit was provided by
Koenig (1987, 1993).

in the national income accounts) upon the economy™s productive capacity; the theory of
marginal supply cost that underlies our model of optimal pricing behavior assumes that the
capital stock in each sector of the economy evolves exogenously, so that any variations can
be subsumed under variation in the exogenous technology factor At . In addition, it assumes
that the marginal utility of additional real private expenditure at any point in time is a
function solely of the aggregate level of such expenditure, together with exogenous factors
” as if all forms of private expenditure (including those classi¬ed as investment expenditure)
were like non-durable consumer purchases. This is not a preposterous theory of “investment”
spending, since the existence of convex adjustment costs of the sort assumed in standard
neoclassical investment theory does imply that the marginal utility of additional investment
spending at a given point in time is decreasing in the real quantity of investment spending at
that time. However, neoclassical investment theory does imply in general that the marginal
utility of additional investment spending also depends upon other endogenous factors, such as
variations in expected future returns to capital, and our present model must be understood to
abstract from variations in these factors. (The implications of a fully-developed neoclassical
model of investment demand are presented in section 3 below, and compared to those of the
present model.)

This simple model of the e¬ects of real interest rates on aggregate demand is obviously
extremely stylized, and it might be wondered why we even bother to derive it from optimizing
foundations, if we intend to abstract from so many features of a more realistic equilibrium
model. The answer is that the model™s simplicity makes it useful as a source of insight into
basic issues; yet the consideration of intertemporal optimization introduces some subtleties,
even in this simple speci¬cation, that we believe are of considerable general importance. The
most important advantage of (1.2) over many simple “IS” speci¬cations (including those often
assumed in linear rational-expectations models with an IS-LM structure) is that it implies
that expected future real interest rates, and not just a current short real rate, matter for the
determination of aggregate demand.

Note that if we are interested solely in characterizing equilibria involving small ¬‚uctu-

ations around a deterministic steady state, it su¬ces to use a log-linear approximation of
(1.2). As in chapter 2, this takes the form

ˆ ˆ
Yt = gt + Et (Yt+1 ’ gt+1 ) ’ σ(ˆt ’ Et πt+1 ),
± (1.3)

where once again the parameter σ > 0 represents the intertemporal elasticity of substitution
in private spending, and gt is a particular component of the exogenous disturbance ξt (which
may be interpreted, among other ways, as representing variation in government purchases).8
Furthermore, in the case of any solutions in which Yt and gt are both stationary variables,
it follows from (1.3) that

ˆ ˆ
Yt = Y∞ + gt ’ σ Et (ˆt+j ’ πt+j+1 ),
± (1.4)

using the fact that
ˆ ˆ
lim Et (YT ’ gT ) = Y∞ , (1.5)
T ’∞

ˆ ˆ
where Y∞ is the long-run average value of Yt under the policy regime in question.9
Thus aggregate demand in this model depends upon all expected future short real rates,
and not simply upon a current ex ante short real rate of return; and unless ¬‚uctuations
in short rates are both highly unforecastable and highly transitory, expectations of sl fu-
ture short rates will matter more than the current short rate.10 The exact way in which
See equations (xx) and (xx) of chapter 2. Here we have written the equilibrium relation somewhat
di¬erently. We no longer subsume all sources of variation in the equilibrium real rate of return under a
single term rt , because output is no longer an exogenous factor. And we now put Yt on the left-hand side,
to stress that the equation may now be viewed as determining aggregate demand.
The long-run average value of gt is assumed to be zero, by de¬nition. The long-run average value of
log Yt is not necessarily equal to log Y , the zero-in¬‚ation steady-state level around which we log-linearize. In
the case of the “New Keynesian” aggregate supply relation (1.6), a policy that results in a long-run average
rate of in¬‚ation π∞ di¬erent from zero will also imply a non-zero value for Y∞ , namely (1 ’ β)/κ times
π∞ ; and while our approximations assume that the in¬‚ation rate is always near zero, they do not require
that in¬‚ation be exactly zero on average. If instead we assume the “New Classical” AS relation discussed
in chapter 3, or we assume complete indexation of prices to a lagged price index (AS relation (2.23) below,
with γ = 1), then any policy that makes the in¬‚ation rate a stationary variable results in Y∞ = 0.
One way of interpreting (1.4) is as saying that it is a long-term real rate of interest, rather than a short
rate, that determines aggregate demand in this model. In fact, the part of the term structure that matters
according to (1.4) is the yield on a bond of in¬nite duration, i.e., the sort of “very long discount” bond
discussed by Kazemi (1992) and Fisher and Gilles (2000). These authors show that in an environment of
the kind assumed here, the yield on the VLD bond de¬nes the stochastic discount factor that can be used to

expectations of future short rates matter in (1.4) is undoubtedly special, and unlikely to
be precisely correct in reality. (We reconsider the question below in the context of a more
sophisticated model of investment dynamics.) Nonetheless, the conclusion that expected
future short rates matter a great deal is likely to be robust, and this general insight is of
considerable importance for the theory of monetary policy, as we shall see. It implies that
a central bank™s primary impact upon the economy comes about not through the level at
which it sets current overnight interest rates, but rather through the way it a¬ects private
sector expectations about the likely future path of overnight rates. This in turn implies that
the credibility of policy commitments must be a paramount concern, that discretionary opti-
mization will almost surely lead to a suboptimal outcome, and that interest-rate smoothing
is desirable, among other consequences, as we discuss below in chapters 7 and 8.

1.2 A Complete Model

We may now close our model by combining the above “IS relation” with any of several
aggregate supply relations derived in chapter 3. Alternative possible assumptions about the
timing of price changes, the information used in price-setting, or the degree of automatic
indexation of prices between revisions have no e¬ect upon the derivation above, which simply
depends on price-taking behavior by the buyers of goods. As our baseline case we shall
assume the “New Keynesian” model of staggered price-setting expounded in section 2 of
chapter 3.
In this model, prices are adjusted at random intervals, and remained ¬xed (in units of
the domestic currency) between the dates at which discrete adjustments occur, as proposed
by Calvo (1983). The model as expounded in chapter 3 is consistent with the assumptions
used in deriving our intertemporal “IS” relation above, in that the marginal utility of income

price all ¬nancial assets; as it happens, it also su¬ces to determine the optimal level of private expenditure,
given the value of the preference shock gt . The reason is that additive separability of preferences over time
allow one to de¬ne a “Frisch demand function” for consumption, in which desired consumption at any point
in time is a function of the marginal utility of income at that time. The marginal utility of income that
enters the Frisch demand function is in turn just the stochastic discount factor that is shown in the asset
pricing literature to equal the yield on a VLD bond.

(which matters for optimizing wage demands and hence for marginal supply costs) is assumed
to be a decreasing function of the current level of real activity Yt ; this relation is shifted by
various exogenous factors, but is independent of all other endogenous variables. (In section
3 below, we consider how both our “IS” and “AS” relations must be modi¬ed in order to
take account of endogenous capital accumulation.)
As we are here concerned solely with equilibria involving only small ¬‚uctuations in in-
¬‚ation and output, it su¬ces to recall the log-linear approximation to the “AS relation”
implied by this model. This is the so-called “New Keynesian Phillips Curve”,

ˆ ˆ
πt = κ(Yt ’ Ytn ) + βEt πt+1 , (1.6)

where κ > 0 is a coe¬cient that depends upon both the frequency of price adjustment and
the elasticity of real marginal supply cost with respect to the level of real activity, where
0 < β < 1 is again the discount factor of the representative household, and where Ytn
represents exogenous variation in the “natural rate of output” as a result of any of several
types of real disturbances. Let us combine equations (1.3) and (1.6) with an interest-rate
rule, such as a “Taylor rule” of the form

ˆt = ¯t + φπ (πt ’ π ) + φy Yt /4,
± ± ¯ (1.7)

where ¯t is an exogenous (possibly time-varying) intercept, and φπ , φy and the (implicit)
in¬‚ation target π are constant policy coe¬cients.11 We then obtain a complete system of
equations for determination of the three endogenous processes {ˆt , πt , Yt }, given the evolution
of the exogenous disturbances {gt , Ytn , ¯t }. As long as the only endogenous variables to
which the central bank™s reaction function responds are in¬‚ation and output (as in the
speci¬cation (1.7)), these three equations su¬ce for equilibrium determination under such a
policy rule. (Dependence upon additional lags of the interest rate instrument, in¬‚ation or
output, considered below, does not change this conclusion; nor does arbitrary dependence
upon exogenous state variables.)
Here we write the coe¬cient on the output term as φy /4 so that φy corresponds to the output coe¬cient
in a standard “Taylor rule”, written in terms of annualized interest and in¬‚ation rates. In terms of our
notation here, these annualized rates are 4ˆt and 4πt respectively.

It will often be useful to write our system of equilibrium conditions in terms of the output
ˆ ˆ
gap xt ≡ Yt ’ Ytn . This allows us (at least under our baseline assumptions) to write the AS
relation without any residual term; and it will be shown in chapter 6 that (under those same
assumptions) it is ¬‚uctuations in xt rather than in Yt that are relevant for welfare. “Taylor
rules” are also often speci¬ed in terms of a response to variations in the output gap, though a
question must raised as to whether the “output gap” measure that would be used in practice
corresponds to our theoretical de¬nition here. (We can in any event write our interest-rate
rule in terms of the gap, as a purely notational matter, by allowing the intercept to be a
function of the natural rate of output.) Our baseline model then consists of the equations

xt = Et xt+1 ’ σ(ˆt ’ Et πt+1 ’ rt ),
± (1.8)

πt = κxt + βEt πt+1 , (1.9)

together with an interest-rate rule such as

ˆt = ¯t + φπ (πt ’ π ) + φx (xt ’ x)/4.
± ± ¯ ¯ (1.10)

Note that this equation describes the same family of policy rules as (1.7), but that the
exogenous term ¯t is not the same under the two representations of any given rule. We
have also here written the “gap” term in our rule as xt ’ x, where x ≡ (1 ’ β)¯ /κ is the
¯ ¯ π
steady-state value of the output gap consistent with the in¬‚ation target π , so that in an
equilibrium in which the in¬‚ation target is achieved on average, the nominal interest rate ˆt
will on average equal ¯t .

The intertemporal “IS” relation (1.8) now involves a composite exogenous disturbance

ˆ ˆn
rt ≡ σ ’1 [(gt ’ Ytn ) ’ Et (gt+1 ’ Yt+1 )].
ˆn (1.11)

This represents deviations of the Wicksellian “natural rate of interest” from the value con-
sistent with a zero-in¬‚ation steady state,13 a concept about which we shall have more to say
in the next section. Here it su¬ces to note that the only exogenous disturbance terms in
the system consisting of (1.8) “ (1.11) are the terms rt and ¯t . Hence insofar as our policy
rule implies a determinate rational expectations equilibrium, it must be one in which ¬‚uctu-
ations in both in¬‚ation and the output gap are due solely to variations in these two factors
” variations in the natural rate of interest due to real disturbances, on the one hand, and
variations in monetary policy (whether deliberate or accidental) on the other. The exact
way in which these factors a¬ect in¬‚ation and the output gap is explored further in the next

2 Interest-Rate Rules and Price Stability

We turn now to a brief consideration of implications of our baseline framework for the
explanation of economic ¬‚uctuations and the choice of a monetary policy rule. As in chapter
2, a ¬rst question to be addressed concerns the conditions under which an interest-rate rule
such as (1.11) implies a determinate rational expectations equilibrium. In the case that
equilibrium is determinate, we then inquire as to how equilibrium in¬‚ation and real activity
are a¬ected by both real disturbances and shifts in monetary policy. Finally, we use this
Note that the framework used by Clarida et al. (1999) includes an “IS” relation of exactly this form,
but with a disturbance term “gt ” that is described as a “demand shock”. This interpretation is somewhat
misleading, since it is apparent from (1.11) that any source of transitory variation in the natural rate of
output will also a¬ect the natural rate of interest. As a result, the conclusions of Clarida et al. about the
optimal policy response to “supply shocks” as opposed to “demand shocks” must be interpreted with care.
Real disturbances that have a transitory e¬ect upon the natural rate of output are not “supply shocks” in
the sense of Clarida et al., because they do not result in any disturbance term in equation (1.6), while they
are “demand shocks” in the sense of those authors, because they a¬ect the disturbance term in equation
The steady-state value of the natural rate is equal to the value of the nominal interest rate consistent
with that same zero-in¬‚ation steady state, so (1.11) takes the same form if we interpret ˆt as the (continuously
compounded) nominal interest rate itself and rt as the (continuously compounded) natural rate of interest.

analysis to consider the design of a monetary policy rule that should maintain stable prices.
The question of the extent to which price stability should be the goal of monetary policy is
deferred to chapter 6.

2.1 The Natural Rate of Interest

We ¬rst consider the relatively simple question of how interest rates must be adjusted in
order for monetary policy to be consistent with stable prices. To answer this question, we
simply solve our AS and IS relations for the equilibrium paths of output and interest rates,
under the assumption of zero in¬‚ation at all times. We ¬rst observe from the AS relation
that πt = 0 at all times requires that xt = 0 at all times, i.e., that output equal the natural
rate of output at all times. From the derivation of the AS relation in chapter 3, we observe
that this conclusion is exact, and not merely a property of the log-linear approximation. For
the natural rate of output is exactly the level of output in all sectors for which real marginal
cost of supplying each good will equal µ’1 , the reciprocal of the desired gross markup. (See
equation (xx) of chapter 3.) This latter quantity is equal to marginal revenue for a ¬rm
that adjusts its price, in the case that all ¬rms charge identical prices. Thus Yt = Ytn is
exactly the condition needed for no ¬rm to wish to charge a price di¬erent from the common
price charged by all other ¬rms, which is in turn the condition under which ¬rms that adjust
their prices will continue to charge the same price as ¬rms that do not, so that there is no
Substituting these paths for in¬‚ation and output into the intertemporal IS relation, we
obtain the required path of nominal interest rates. Substituting Πt = 1 and Yt = Ytn into
(1.2), we see that interest rates must satisfy it = rt at all times, where
uc (Yt+1 ; ξt+1 )
1 + rt ≡ β ’1 Et
. (2.1)
uc (Ytn ; ξt )

That is, the interest rate must at all times equal the Wicksellian natural rate of interest,
which may be de¬ned as the equilibrium real rate of return in the case of fully ¬‚exible prices.
Under this de¬nition, we observe a direct correspondence with our previously introduced

concept of the natural rate of output;14 indeed, the natural rate of interest is just the real
rate of interest required to keep aggregate demand equal at all times to the natural rate of
output.15 Log-linearizing (2.1), we observe that the exogenous term rt in (1.11) corresponds
to the percentage deviation of the natural rate of interest from its steady-state value,

1 + rt
ˆn n
rt ≡ log = log(1 + rt ) + log β.
1 + rn

We have thus far referred only to the conditions under which one could obtain complete
price stability, in the sense of a constant price level (and hence zero in¬‚ation). As we shall
see, there is a certain normative interest in this case, as, at least under the assumptions of
our baseline model, it would eliminate the distortions resulting from price stickiness. Yet
most “in¬‚ation targeting” countries instead seek to maintain in¬‚ation at a low positive level,
and so policies that stabilize in¬‚ation at some constant target level π are also of obvious
interest. In fact, in our log-linear approximation, our conclusions are exactly the same in
the case, up to certain constant terms. The required path for the output gap will still be a
constant (though not zero unless π = 0), and the required path for the nominal interest rate
will now be

ˆn ¯
ˆt = rt + π .

Though the average values of output and of the nominal interest rate depend upon the target
in¬‚ation rate, the way in which they should respond to shocks does not (up to a log-linear
Equation (2.1) (or equally usefully for most purposes, the log-linear version (1.11) pro-
vides us with a theory of how various types of real disturbances a¬ect the natural rate of
interest, and hence with a theory of how the interest rate controlled by the central bank
should respond to those disturbances, in an equilibrium characterized by price stability. To
Of course, Friedman (1968) originally proposed the concept of a “natural rate” of output (or of unem-
ployment) by analogy with Wicksell™s concept of a natural rate of interest, a notion that was at that time
more familiar!
The concept is thus closely related to Blinder™s (1998, chap. 2, sec. 3) notion of the “neutral” rate of

consider the e¬ects of individual disturbances, we need ¬rst to recall how various real dis-
turbances a¬ect the natural rate of output. Log-linearization of equation (xx) from chapter
3 implies that
σ ’1 gt + ωqt
Ytn ≡ , (2.2)
σ ’1 + ω
where gt denotes the variation in log output required to maintain a constant marginal utility
of real income as in (1.3), qt correspondingly denotes the variation in log output required
to maintain a constant marginal disutility of output supply as in chapter 3, σ > 0 is the
intertemporal elasticity of substitution of private expenditure as in (1.3), and ω > 0 is the
elasticity of real marginal cost with respect to a ¬rm™s own output as in chapter 3.
These composite disturbance terms can furthermore be expressed in terms of more fun-
damental disturbances as
ˆ ¯
gt = Gt + sC Ct , (2.3)

qt = (1 + ω ’1 )at + ω ’1 ν ht .

Here, as in chapter 2, Gt denotes the deviation of government purchases from their steady-
state level, measured as a percentage of steady-state output Y , which shifts the level of
ˆ ¯
private expenditure implied by any given level of aggregate demand Yt , and Ct denotes the
percentage shift in the Frisch (constant marginal utility of income) consumption demand,
due to a shift in the utility-of-consumption function. And as in chapter 3, at represents
variation in the log of the multiplicative technology factor that is common to all sectors,
and ht is the percentage shift in the Frisch labor supply, due to a shift in the disutility-of-
labor function v. (The exogenous shifts in the Frisch demand schedules are measured at
the steady-state values of their arguments.) In addition, 0 < sC ¤ 1 is the steady-state
share of private expenditure in total demand , and ν > 0 is the inverse of the Frisch (or
intertemporal) elasticity of labor supply. It then follows from (2.2) that

σ ’1 1 ¯
ˆ ˆ
Ytn = ’1 (Gt + (1 ’ sG )¯t ) + ’1
c ((1 + ω)at + ν ht ).
σ +ω σ +ω
We observe that each of the exogenous disturbances Gt , Ct , at , and ht increases the natural

rate of output, and thus, under a policy aimed at price stability, each of them must be
allowed to perturb the equilibrium level of economic activity Yt .
Substituting this solution into (1.11), and furthermore assuming (for simplicity) that each
of the exogenous disturbances follows an independent ¬rst-order autoregressive process, we
¬nd that the required interest-rate variations are given by

ˆ ¯
rt = (σ + ω ’1 )’1 [(1 ’ ρG )Gt + sC (1 ’ ρc )Ct ’ (1 + ω ’1 )(1 ’ ρa )at ’ ω ’1 ν(1 ’ ρh )ht ], (2.4)

where ρG , ρc , ρa , and ρh are the coe¬cients of serial correlation of the four exogenous
disturbance processes. Since stationarity requires that ρi < 1 in each case, we observe
that under this assumption, interest rates must increase in response to temporary increases
in government purchases or in the impatience of households to consume, and decrease in
response to temporary increases in productivity or in the willingness of households to supply
labor. In each case, the e¬ects upon the natural rate of interest are larger the more temporary
the disturbance (i.e., the less positive the serial correlation).
This prescription may appear quite di¬erent from that of Clarida et al. (1999), who state
(in their “Result 4”) that optimal policy involves “adjusting the interest rate to perfectly
o¬set demand shocks,” while “perfectly accommodat[ing] shocks to potential output by
keeping the nominal interest rate constant”. In fact, the variable (their “gt ”) here referred
to as a “demand shock” corresponds to our natural rate of interest rt .16 What these authors

mean by “perfectly o¬setting” movements in this variable is that the central bank™s interest-
rate instrument should move one-for-one with variations in the natural rate of interest. (Thus
“perfectly o¬setting” the shocks does not mean that output is insulated from them, but that
the output gap is.) And what they mean by “perfectly accommodating shocks to potential
output” is that, given the value of the natural rate of interest, the interest rate should be
independent of the natural rate of output. That is, disturbances to the natural rate of output
that do not shift the natural rate of interest should not a¬ect nominal interest rates. Stated
The variable is evidently thought of as a “demand shock” because it is the disturbance term in the Euler
equation (1.8). But because this condition has been written in terms of the output gap xt rather than the
ˆ ˆn
level of output Yt , the composite disturbance rt , unlike our variable gt , cannot properly be regarded as a
pure demand shock, if one supposes that transitory disturbances to the natural rate of output occur.

this way, there is no di¬erence between their recommendation and our own.17 However, it
is not true, in general, that optimal policy involves no interest-rate response to shocks that
a¬ect the natural rate of output, because, as shown by (2.4) such shocks almost always do
a¬ect the natural rate of interest to some extent.

It is worth noting that the required interest-rate variations (2.4) in response to the various
types of shocks cannot be achieved, in general, through a simple “Taylor rule” under which
the nominal interest rate is a function solely of in¬‚ation and the deviation of output from
trend. In the equilibrium with completely stable prices, in¬‚ation does not vary in response
to the shocks at all, and so conveys no information about them. Output does vary in
response to each of the shocks, but the desired interest-rate response is not proportional to the
desired output response across the various types of shocks; indeed, one wants interest rates
to vary procyclically in the case of government-purchase or consumption-demand shocks, but
countercyclically in response to technology or labor-supply shocks. Thus the central bank
will need additional information in order to implement its policy, if complete price stability
is its aim.

Analysis of the sources of variation in the natural rate of interest is also important in
determining whether complete price stability is necessarily feasible. Our analysis above
suggests that it should be, insofar as we have been able to solve for paths of output and
interest rates that would imply that the IS and AS relations would be satis¬ed at all times by
a zero in¬‚ation rate. However, even supposing that the central bank possesses the information
required to adjust its interest-rate instrument as required by the above analysis, there is
another potential problem, and this is that the natural rate of interest may sometimes be
negative.18 If this occurs, then it is not possible for the nominal interest rate to perfectly

Actually, the results referred to in Clarida et al. are characterizations of optimizing central bank policy
under discretion, which is not in general optimal policy, in the sense of the policy that best achieves the central
bank™s assumed objectives, as we explain in chapter 7. However, in the case that complete stabilization of
both in¬‚ation and the output gap are possible, doing so corresponds both to optimal policy and to the result
of discretionary optimization, as we shall see.
Another possible problem is the existence of a non-Ricardian ¬scal policy, of a sort that makes a constant
price level inconsistent with the condition that households exhaust their intertemporal budget constraints.
This potential problem and its implications are taken up in chapter 5.

track the natural rate, owing to the zero lower bound on nominal interest rates. (Recall the
discussion of equation (xx) from chapter 2.) Whether the natural rate of interest is ever
negative is a topic of some debate, though Summers (1991) has suggested that it ¬‚uctuates
su¬ciently in the U.S. for a in¬‚ation target several percentage points above zero to be
desirable in order to allow more successful stabilization, and Krugman (1998) has argued
that it has recently been far below zero in Japan. Here we note simply that our theory
allows for variation over time in the natural rate for a variety of reasons, and no reason
why it should not sometimes be negative. (Our model does imply a positive average level of
the natural rate, determined by the rate of time preference of the representative household.)
Policy options when the natural rate of interest is temporarily negative are discussed further
in section xx below, and in chapters 6 and 7. We argue in the later chapters that in this
case it is appropriate not only to choose a non-zero in¬‚ation target, but to accept a small
amount of in¬‚ation variation in order to maintain a lower average rate of in¬‚ation despite
the constraint on interest-rate policy imposed by the zero bound.

2.2 Conditions for Determinacy of Equilibrium

We have thus far only considered how interest rates would have to vary, in order for there
to be an equilibrium with stable prices. Our answer to this question does not yet, in itself,
explain what sort of interest-rate rule would be suitable to bring about an equilibrium of this
kind. In particular, it should not be inferred from the above discussion that a suitable policy
rule would be simply to set the central bank™s interest-rate instrument to equal its estimate of
the current natural rate of interest. A policy rule of the form ˆt = rt would be consistent with
the desired equilibrium, but may allow many other, less desirable equilibria as well. Such a
rule makes the nominal interest rate a function of purely exogenous state variables, and just
as in the ¬‚exible-price analysis of chapter 2, all such rules imply indeterminacy of rational
expectations equilibrium. We thus must again take up the question of the determinacy of
equilibrium under alternative interest-rate rules, but now in the context of our model with
sticky prices and endogenous output variation.

We begin with a formal consideration of interest-rate rules, such as the one just proposed,
under which {ˆt } is an exogenous process. In this case we wish to solve the system (1.8)
rn ±
“ (1.9) for the endogenous variables {πt , xt }, given exogenous stationary processes {ˆt , ˆt }.
We observe that this system can be written in the form

rn ±
Et zt+1 = Azt + a(ˆt ’ ˆt ),

where the vector of endogenous variables is

zt ≡ ,

and the matrices of coe¬cients are

β ’1 ’β ’1 κ 0
A≡ , a≡ .
’β ’1 σ 1 + β ’1 κσ ’σ

The matrix A has characteristic equation

P(µ) = µ2 ’ [1 + β ’1 (1 + κσ)]µ + β ’1 = 0.

As our parameters satisfy κ, σ > 0 and 0 < β < 1, we observe that P(0) > 0, P(1) < 0, and
P(µ) > 0 again for large enough µ > 1. Hence A has two real eigenvalues, satisfying

0 < µ1 < 1 < µ 2 .

Since neither endogenous state variable is predetermined, the existence of an eigenvalue
|µ1 | < 1 implies that rational expectations equilibrium is indeterminate, just as in the
¬‚exible-price model of chapter 2 (and in the rational-expectations IS-LM-AS model of Sargent
and Wallace (1975)). Here the situation di¬ers from that in chapter 2 in that the alternative
stationary solutions include a large number of alternative stochastic processes for output
(and also for the expected component of in¬‚ation), rather than it being only the unexpected
component of in¬‚ation that fails to be uniquely determined. In the present context it is
also clearer that this indeterminacy is undesirable, since in the presence of staggered price-
setting, variations in in¬‚ation due to self-ful¬lling expectations create real distortions (of a
kind further characterized in chapter 6).

This result implies that even if the central bank has perfect information about the exoge-
nous ¬‚uctuations in the natural rate of interest, a desirable interest-rate rule will also have
to involve feedback from endogenous variables such as in¬‚ation and/or real activity, if only
to ensure determinacy of equilibrium. In fact, if one is seeking to ¬nd a rule that imple-
ments the equilibrium with completely stable prices (or more generally, a completely stable
in¬‚ation rate), then neither the variable πt nor xt will be useful as a source of information
about the real disturbances to the economy, for in the desired equilibrium neither variable
responds at all to any of the real disturbances.19 Nonetheless, it may be desirable for the
central bank to commit itself to respond to ¬‚uctuations in these variables, in addition to
its response to other sources of information about the real disturbances, in order to render
equilibrium determinate.
We illustrate this possibility by considering the determinacy of equilibrium under a Taylor
rule of the form (1.10). (Note that it is now necessary to write explicitly the dependence of
the interest-rate operating target upon the output gap, since output is here an endogenous
variable.) In this case, substitution of (1.10) into (1.8) to eliminate ˆt again yields a system
of the form
rn ±
Et zt+1 = Azt + a(ˆt ’ ¯t + π ),
¯ (2.5)

where now
πt ’ π
zt ≡ ,
xt ’ x
β ’1 ’β ’1 κ 0
A≡ , a≡ .
σ(φπ ’ β ’1 ) 1 + σ(φx /4 + β ’1 κ) ’σ
We observe that

trA = 1 + β ’1 (1 + κσ) + σφx /4, det A = β ’1 [1 + σ(φx /4 + κφπ )].
This is a common problem for an approach to stabilization policy based upon a commitment to respond
solely to deviations of one™s target variables from their (constant) target values, discussed in Bernanke and
Woodford (1997). It should be noted, however, that if complete stabilization of in¬‚ation and the output gap
is not desirable ” owing, say, to a desire to reduce the degree of interest-rate volatility ” then it may be
possible to implement an optimal equilibrium through commitment to a rule that responds directly to no
variables other than in¬‚ation and the output gap, as shown in chapter 8.

Let us furthermore restrict our attention to the case of rules for which φπ , φx ≥ 0. Then
necessarily det A > 1. We note that a 2 — 2 matrix with positive determinant has both
eigenvalues outside the unit circle (our condition for determinacy) if and only if20

det A > 1, det A ’ trA > ’1, det A + trA > ’1. (2.6)

Under our sign restrictions, the ¬rst and third of these inequalities necessarily hold, so that
both eigenvalues are outside the unit circle if and only if
φπ + φx > 1. (2.7)

Condition (2.7) for determinacy can be given a simple interpretation. We note that
the “New Keynesian Phillips Curve” implies that each percentage point of permanently
higher in¬‚ation (i.e., quarterly in¬‚ation πt permanently higher by 1/4 of a percent) implies a
permanently higher output gap of (1 ’ β)/4κ percentage points.21 Hence the left-hand side
of (2.7) represents the long-run increase in the nominal interest rate prescribed by (1.10)
for each unit permanent increase in the in¬‚ation rate. Our condition then corresponds once
more to the “Taylor principle”: at least in the long run, nominal interest rates should rise
by more than the increase in the in¬‚ation rate.
We note that contrary to our result in chapter 2, determinacy now depends upon the
output response coe¬cient φx , and not solely upon the in¬‚ation response coe¬cient φπ ; and
indeed, a large enough positive value of either coe¬cient su¬ces to guarantee determinacy.
This complicates slightly our interpretation of the Taylor (1999) contrast between pre-Volcker
and post-Volcker U.S. monetary policy. Taylor™s estimates (discussed above in section 2.3 of
chapter 2) imply that φπ < 1 in his pre-Volcker sample; but as they also imply that φx > 0
in that period, this does not in itself su¬ce to imply that equilibrium should have been
indeterminate under the earlier policy. Still, plausible numerical values for the parameters
of the NKPC imply this, at least if Taylor™s point estimates for the policy-rule coe¬cients
See Proposition 1 of the Appendix.
Thus the “long-run Phillips curve” is not perfectly vertical in this model. We show, however, in chapters
6 and 7 that this does not imply that the optimal long-run in¬‚ation rate is positive, even if the optimal
output level exceeds the natural rate.

are taken to be correct. For example, if one assumes the parameter values given in Table
1 below (based upon the estimates of Rotemberg and Woodford (1997)), then determinacy
would require that the in¬‚ation coe¬cient plus .1 times the output coe¬cient be greater than
one. Taylor™s estimates for the period 1960-79 would then imply an interest-rate increase
of only .81 + .1(.25) = .84 percentage points per percentage point long-run increase in
in¬‚ation. Thus just as we concluded in chapter 2, these estimates suggest that equilibrium
should have been indeterminate under the pre-Volcker regime, though clearly determinate
under the post-Volcker regime.
As discussed in chapter 1, most empirical estimates of Taylor rules incorporate some form
of partial adjustment of the short-term interest-rate instrument toward an implicit target
that depends upon the current in¬‚ation rate and output gap. (We shall also argue in chapter
8 that rules of that kind are desirable on normative grounds.) It is therefore of some interest
to consider the e¬ects of interest-rate inertia upon the question of determinacy. For the sake
of simplicity we restrict our analysis here to the family of generalized Taylor rules

ˆt = ¯t + ρ(ˆt’1 ’ ¯t’1 ) + φπ (πt ’ π ) + φx (xt ’ x)/4,
± ± ± ± ¯ ¯ (2.8)

where we assume that ρ, φπ , φx ≥ 0. Substituting (2.8) into (1.8), we again obtain a system
of equations that may be written in the form (2.5), but where now where the vector of
endogenous variables is ® 
πt ’ π¯

z t ≡ ° xt ’ x » ,
ˆt’1 ’ ¯t’1
± ±
and ®  ® 
β ’1 ’β ’1 κ 0 0
   
A ≡ ° σ(φπ ’ β ’1 ) 1 + σ(φx /4 + β ’1 κ) σρ » , a ≡ ° ’σ » .
φπ φx /4 ρ 0
As there is now a predetermined state variable (namely, ˆt’1 ’¯t’1 ), equilibrium is determinate
± ±
in this case if and only if the 3 — 3 matrix A has exactly two eigenvalues outside the unit
Necessary and su¬cient conditions for determinacy in a system of this form are given by
Proposition 2 of the Appendix. We note that in the present case, the characteristic equation

of matrix A is of the form

P(µ) = µ3 + A2 µ2 + A1 µ + A0 = 0,


A0 = ’β ’1 ρ < 0,

A1 = ρ + β ’1 (1 + ρ(1 + κσ)) + β ’1 σ(κφπ + φx /4) > 0,

A2 = ’β ’1 (1 + κσ) ’ 1 ’ ρ ’ σφx /4 < 0.

The proposition lists three possible sets of conditions under which there is determinacy.
Because of the signs of the coe¬cients Ai , we see immediately that condition (A.2) is violated
and that condition (A.4) must instead hold; thus we can exclude Case I of the proposition.
In the present case, the remaining conditions (in addition to (A.4) that, as we have just
noted, is necessarily satis¬ed) required for Case II of the proposition reduce to

φπ + φx > 1 ’ ρ, (2.9)

φx + (β ’1 ’ 1)[κ’1 σ ’1 (1 ’ ρ)(β ’ ρ) ’ ρ] > 0.
φπ + (2.10)

The remaining conditions required for Case III22 are instead (2.9) and

β ’1 (1 + κσ) + ρ + σφx /4 > 2. (2.11)

Equilibrium is determinate if and only if the coe¬cients of the policy rule (2.8) satisfy both
(2.9) and at least one of (2.10) and (2.11).
In fact, one can show that under our sign assumptions, (2.9) is necessary and su¬cient
for determinacy. We prove this by showing that any parameter values that satisfy (2.8) and
not (2.11) must necessarily satisfy (2.10). We ¬rst note that under our sign assumptions,
In the statement of the proposition in the Appendix, another condition listed is (A.6), which is the denial
that (2.10) holds. But this is not necessary, for if instead (2.10) holds, determinacy also obtains, as Case II
then applies. (The condition is listed in the statement of the proposition simply in order to make the three
cases disjoint.) Also condition (A.7) as written in the statement of the proposition allows the coe¬cient A2
to be either less than -3 or greater than 3. But as in the present case A2 is necessarily negative, it is only
the possibility that A2 may be less than -3 that is relevant; this is condition (2.11).

(2.11) can fail to hold only if ρ < β. (Here we use the fact that β ’1 + β > 2.) We next
observe that the left-hand side of (2.10) is a decreasing function of ρ, for given values of all
the other parameters, for all values ρ < β. Thus the values of φπ required in order for (2.10)
not to hold become smaller, the smaller is ρ. On the other hand, the values of φπ consistent
with (2.9) become larger, the smaller is ρ. Thus if (2.9) is to be satis¬ed while (2.10) is not,
for any given values of β, κ, σ and φx , this must occur for the largest value of ρ consistent
with (2.11) being (weakly) violated. (Note that this last quantity is independent of φπ .)
Furthermore, the left-hand side of (2.10) is an increasing function of φπ . Thus if (2.9) is
to be satis¬ed while (2.10) is not, for any given values of β, κ, σ, φx , and ρ, this must occur
for the smallest value of φπ that is (weakly) consistent with (2.9). (The geometry of these
regions is illustrated in Figure 4.1.) It therefore su¬ces that we consider values of ρ and φπ
for which (2.9) and (2.11) hold as equalities, for given values of the other parameters. (This
is the point shown by the intersection of the solid line and the dashed line in Figure 4.1.) If
(2.10) is not violated in this case, it can never be.
The algebra required to check this is simplest if we solve (2.9) and (2.11) for φπ and φx
as functions of ρ, rather than for ρ and φπ as functions of φx . We obtain
1’β 1 + κσ
φπ = (1 ’ ρ) ’ 2’ρ’ ,
κσ β
4 1 + κσ
φx = 2’ρ’ .
σ β
Substituting these values into the left-hand side of (2.10), we obtain
(β ’ ρ)2 > 0,
which holds as a strict inequality because ρ < β. Thus (2.10) holds in this case, and so must
hold in any case where (2.9) holds but (2.11) does not. (This is illustrated for particular
numerical parameter values in Figure 1.23 ) It follows that condition (2.9) is necessary and
su¬cient for determinacy.
The values assumed for β, κ, and σ are given in Table 1 below; the value assumed for φx is .05. This last
value has no particular signi¬cance, except that the relative locations of the various regions are especially
easily seen for a small positive value of this order.







0 0.5 1 1.5

Figure 4.1: Regions in which each of three inequalities fail to hold. Grey region indicates
policy rules for which equilibrium is indeterminate; white region indicates determinacy.

Condition (2.9) will be recognized as a generalization of (2.7), and once again it can
be interpreted as requiring adherence to the Taylor principle. In the case that ρ < 1, the
rule (2.8) implies that a sustained increase in in¬‚ation of a certain size results in an eventual
cumulative increase in the nominal interest rate of ¦π ≡ (1’ρ)’1 φ times as much; similarly, a
sustained increase in the output gap results in an eventual cumulative increase in the interest
rate of (1/4 of) ¦x ≡ (1 ’ ρ)’1 φx times as much.24 In this case, (2.9) can equivalently be
written as
¦π + ¦x > 1,

which clearly has the same interpretation as (2.7) in the non-inertial case. Furthermore, if
ρ ≥ 1, the eventual cumulative increase in the nominal interest rate is in¬nite if at least
It may be recalled that the estimated Fed reaction functions described in chapter 1 are described in
terms of the values of these long-run response coe¬cients ¦π and ¦x rather than the immediate responses
φπ and φx .

one of φπ or φx is positive, so that the Taylor principle is necessarily satis¬ed; but (2.9) is
necessarily satis¬ed in this case as well. Thus (2.9) is equivalent to requiring conformity
with the Taylor principle.25 This result ” that the Taylor principle continues to be a crucial
condition for determinacy, once understood to refer to cumulative responses to a permanent
in¬‚ation increase, even in the case of an inertial interest-rate rule ” recalls our ¬nding in
chapter 2 in the case of a ¬‚exible-price model. The ¬nding that a determinate rational
expectations equilibrium necessarily exists for rules with ρ ≥ 1 (“super-inertial rules”) also
recalls an earlier result.
Some empirical papers (e.g., Clarida et al., 2000; Bernanke and Boivin, 2000) instead
estimate “forward-looking” variants of the Taylor rule, in which interest rates respond to
deviations of expected future in¬‚ation from its target level, instead of responding to the
amount that prices have already risen. As a simple example, let us consider the family of

ˆt = ¯t + φπ (Et πt+1 ’ π ) + φx (xt ’ x)/4,
± ± ¯ ¯ (2.12)

where we again assume that φπ , φx > 0. Substituting (2.12) into (1.8) to eliminate ˆt , we
again obtain an equation system of the form (2.5), but where now

πt ’ π
zt ≡ ,
xt ’ x

In fact, one ¬nds for a wide variety of types of simple interest-rate rules that the Taylor principle is
one of the conditions required for determinacy, even if it is not a su¬cient condition in itself, as is true here.
This should not be too surprising. One observes quite generally ” in the case of any family of policy rules
that involve feedback only from in¬‚ation and output, regardless of how many lags of these might be involved
” that the boundary between sets of coe¬cients that satisfy the Taylor principle and those that do not will
consist of coe¬cients for which there is an eigenvalue exactly equal to one. The eigenvalue of one exists for
any policy rule with the property that the long-run increase in the nominal interest rate is exactly equal to
the long-run increase in the in¬‚ation rate, for the associated right eigenvector is one with an element 1 for
each current or lagged value of in¬‚ation or the interest rate, and an element (1 ’ β)/4κ for each current or
lagged value of the output gap. This is because under the hypothesis about the policy rule, the IS relation,
the AS relation and the policy all share the property that the equation continues to be satis¬ed if in¬‚ation,
output and interest rates are increased at all dates by the constant factors just mentioned. It follows that
a real eigenvalue crosses the unit circle as the sign of the inequality corresponding to the Taylor principle
changes. This boundary is therefore one at which the number of unstable eigenvalues increases by one. Often
this results in moving from a situation of indeterminacy to determinacy, though we do not seek to establish
general conditions for this.

β ’1 ’β ’1 κ 0
A≡ , a≡ .
σβ ’1 (φπ ’ 1) 1 + σ(φx /4 ’ β ’1 κ(φπ ’ 1)) ’σ
Thus in this case we have

trA = 1 + β ’1 + σ(φx /4 ’ β ’1 κ(φπ ’ 1)),

det A = β ’1 (1 + σφx /4).

Determinacy again obtains if and only if conditions (2.6) are satis¬ed. In the present
case, the ¬rst condition takes a slightly di¬erent form than before, but it is again necessarily
satis¬ed by all rules satisfying our sign assumptions. The second condition again is equivalent
to (2.7). However, the third condition now takes a slightly di¬erent form than in the case
of conventional (backward-looking) Taylor rules, and is no longer necessarily satis¬ed by all
rules satisfying our sign restrictions. This condition now takes the form

(φx + 8σ ’1 ).
φπ < 1 + (2.13)

Equilibrium is then determinate if and only if the coe¬cients of the policy rule (2.12) satisfy
(2.7) and (2.13).
We note that condition (2.7) can once again be interpreted in terms of the Taylor princi-
ple. Thus yet again we ¬nd that conformity to the Taylor principle is a necessary condition
for determinacy. But as in the last case, we here ¬nd that it is not su¬cient. In particular,
condition (2.13) fails to hold for large enough values of φπ , even though the “Taylor princi-
ple” is satis¬ed. Thus adjusting interest rates in response to deviations of expected future
in¬‚ation from target can give rise to equilibrium ¬‚uctuations due purely to self-ful¬lling
expectations, as shown by Bernanke and Woodford (1997) for a closely related model; this
problem does not arise in the case of a strong response to the in¬‚ation that has already
occurred, no matter how large φπ is made. The region in which equilibrium is determinate
for rules of this family is shown in Figure 4.2, again assuming the same structural parameters
β, σ, κ as in Figure 4.1. As Clarida et al. (2000) also conclude, indeterminacy results from
too high a value of φπ only in the case of quite high values relative to empirical estimates











0 5 10 15 20 25 30 35 40 45 50

Figure 4.2: Regions of determinacy and indeterminacy for “forward-looking” Taylor rules.

for any central banks. However, this would be a problem if a central bank were to seriously
attempt to completely stabilize in¬‚ation through extremely tight “targeting” of an in¬‚ation
forecast, as we discuss below.
The interest-rate rule estimated by Clarida et al. (2000) for the Fed is more complicated
than rules of the family (2.12), most notably because it allows for interest-rate inertia. Thus
in order to interpret their results we need to consider forward-looking rules of the more
general family26

ˆt = ¯t + ρ(ˆt’1 ’ ¯t’1 ) + φπ (Et πt+1 ’ π ) + φx (xt ’ x)/4,
± ± ± ± ¯ ¯ (2.14)
Note that even this family is still too restricted to include any of the rules actually estimated by Clarida
et al., as they also replace the output gap term by a forecast of the future output gap. However, analysis of
the family of rules (2.14) does allow insight into how the three key parameters estimated by these authors
should be expected to a¬ect the determinacy of equilibrium, and indeed their own numerical examination of
the conditions under which equilibrium is determinate considers this family of rules, rather than anything
more general. Furthermore, for the reason explained in footnote 25, the Taylor principle has the same
importance for the eigenvalues of the equation system if the current output gap is replaced by an expected
future output gap; so the most important of our conditions for determinacy is likely to continue to apply.

where we now assume ρ, φπ , φx ≥ 0.
This again implies a system of equations that may be written in the form (2.5), but where
®  ® 
β ’1 ’β ’1 κ 0 0
   
A ≡ ° σβ ’1 (φπ ’ 1) 1 + σ(φx /4 ’ β ’1 κ(φπ ’ 1)) σρ » , a ≡ ° ’σ » .
β ’1 φπ φx /4 ’ β ’1 κφπ ρ 0

Once again there is a single predetermined state variable (namely, ˆt’1 ’ ¯t’1 ), so that
± ±
equilibrium is determinate if and only if A has exactly two eigenvalues outside the unit
circle, and once again necessary and su¬cient conditions for this are given by Proposition
2 in the appendix. We observe that conditions (A.1) and (A.2) imply that φx < 0, so that
Case I is once again impossible under our sign assumptions. It then follows that conditions
(A.3) and (A.4) are necessary for determinacy, as these are required by both cases II and III
of the proposition. In the present case, (A.3) corresponds once again to (2.9), while (A.4)
corresponds to
(φx + 8σ ’1 (1 + ρ)).
φπ < 1 + ρ + (2.15)

Note that the latter condition generalizes (2.13).
Once again condition (2.9) corresponds to the Taylor principle, and so we see that once
more conformity with that principle is necessary for determinacy. However (as we have
already concluded for the case ρ = 0), this principle does not su¬ce for determinacy in the
case of forward-looking policy rules. In particular, it is necessary that φπ satisfy the upper
bound expressed in (2.15). We observe that higher values of ρ relax this constraint, but do
not eliminate it. Thus it continues to be true that too large a degree of sensitivity of the
interest rate to the in¬‚ation forecast results in indeterminacy. This is illustrated in Figure
4.3, where the values of φπ and ρ consistent with a determinate equilibrium are indicated, in
the case of rules of the form (2.14) with φx = 0, and assuming the same model parameters
as in Figure 4.2.27
Note that conditions (2.9) and (2.15) are necessary for determinacy, but not su¬cient. Determinacy
requires in addition that at least one of a further set of three inequalities hold: either (A.5) must hold, or
one of the two cases allowed in (A.7) must hold, i.e., either A2 > 3 or A2 < ’3. However, for the parameter








0 10 20 30 40 50 60 70

Figure 4.3: Regions of determinacy and indeterminacy for “forward-looking” Taylor rules
with interest-rate inertia.

In the case of rules with coe¬cients in the range that is likely to be of practical interest,
however, the other requirements for determinacy are unlikely to be a problem. The require-
ment for determinacy that is of most practical interest thus remains the Taylor principle.
Like Taylor, Clarida et al. ¬nd that an estimated policy rule for the period 1960-79 in-
volves an insu¬cient response to in¬‚ation to be consistent with determinacy, whereas their
estimated rule for the period 1982-96 satis¬es the Taylor principle and would imply a deter-
minate equilibrium. For example, their baseline estimates for the earlier period are ρ = .68,
φπ = .27, φx = .09. In the absence of any increase in the output gap, these values imply
that a sustained increase in in¬‚ation of one percentage point would eventually raise nomi-
nal interest rates by only 83 basis points. Thus the Taylor principle is violated unless the

values used in this ¬gure, all policy-rule coe¬cients that satisfy (2.9) and (2.15) also satisfy at least one of the
other three inequalities. Thus these two conditions turn out to be necessary and su¬cient for determinacy,
at least in this case.

associated increase in the output gap is quite large. Condition (2.9) will be satis¬ed only if
κ < .4(1 ’ β), which is to say, if κ < .004; this is much smaller than any plausible value,
either from the point of view of the underlying microfoundations of price adjustment or of
estimated Phillips curves. On the other hand, their baseline estimates for the later period
are ρ = .79, φπ = .45, and φx = .20. Since in this case φπ > 1 ’ ρ, the Taylor principle is
satis¬ed regardless of the assumed slope of the long-run Phillips curve. On the other hand,
because φπ < 1, condition (2.15) is necessarily satis¬ed as well, and such a policy rule implies
a determinate equilibrium.
Another class of possible policy rules that is of at least theoretical interest is that of rules
which incorporate a price level target ” the “Wicksellian” rules considered in chapter 2.
In the case of a model with endogenous output, one must also consider the consequences of
possible feedback from the level of output, and so the class of rules previously considered is
generalized to include all rules of the form

ˆt = ¯t + φp (pt ’ pt ) + φx (xt ’ x)/4,
± ± ¯ ¯ (2.16)

where now pt is the log price level, and {¯t } is a target path for the log price level, growing
deterministically at some rate π (to which x again corresponds). Once again we shall restrict
¯ ¯
attention to rules with φp , φx ≥ 0. Giannoni (2000) shows that in the case of rules of this
sort, a su¬cient condition for determinacy is that φp > 0. On the other hand, when φp = 0,
rules of this sort correspond to rules of the form (1.10) with φπ = 0, and so we already know
that determinacy obtains if and only if

φx > . (2.17)

Thus necessary and su¬cient conditions are that either φp > 0 or φx exceeds the bound
(2.17). These conditions correspond once again to the Taylor principle. For sustained in¬‚a-
tion above the rate π must eventually make pt ’ pt arbitrarily large, and then if φp takes
¯ ¯
any positive value at all, a rule of the form (2.16) must eventually imply arbitrarily large
increases in the level of nominal interest rates. If on the other hand, φp = 0, then (2.17)

corresponds to the Taylor principle, as discussed earlier. Thus the Taylor principle is yet
again determinative. Note also that the weakness of the condition required for determinacy
in the case of a rule of the form (2.16) is one of the appealing features of rules of this kind,
as discussed by Giannoni.

2.3 Stability under Learning Dynamics


2.4 Determinants of In¬‚ation

Having established that interest-rate rules can result in a determinate equilibrium, we turn
now to the further characterization of that equilibrium. We are particularly interested in
the determinants of equilibrium in¬‚ation under the kinds of policies just considered, and the
conditions under which ¬‚uctuations in the price level can be minimized.
Each of the classes of interest-rate rules considered above has the property that the
equilibrium conditions can be written entirely in terms of πt ’ π , xt ’ x, ˆt ’¯t , and rt ’¯t + π .
¯ ¯± ± ±¯
This implies that when equilibrium is determinate, it is possible to solve for the endogenous
variables in this list (the ¬rst three) as a function of initial conditions and the current and
expected future values of the exogenous variable (the last one). In the case of policy rules
(1.10) or (2.12), the equation system involves no predetermined endogenous variables, so
that there are no relevant initial conditions (other than those relating to the path of the
exogenous variables). One therefore obtains in these cases a solution of the form

πt = π +
¯ ψj Et (ˆt+j ’ ¯t+j + π ),
± ¯ (2.18)

xt = x +
¯ ψj Et (ˆt+j ’ ¯t+j + π ),
± ¯ (2.19)

ˆt = ¯t +
± ± ψj Et (ˆt+j ’ ¯t+j + π ).
± ¯ (2.20)

In particular, if both eigenvalues of A are outside the unit circle (the condition for determi-
nacy), then A’1 is a stable matrix, and one can obtain a unique bounded solution to (2.5)
by “solving forward”, namely

A’j’1 aEt (ˆt+j ’ ¯t+j + π ).
zt = ± ¯ (2.21)

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