Michael Woodford

Princeton University

September 2000

Revised September 2002

Preliminary and Incomplete

c Michael Woodford 2002

Contents

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

1

2 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

1

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.

2

For Wicksell™s views see Wicksell (1898, 1906, 1907). Recent discussions include Humphrey (1992),

Fuhrer and Moore (1995b), and Woodford (1998).

1. A BASIC MODEL OF THE EFFECTS OF MONETARY POLICY 3

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

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).

4 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

’1

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

4

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).

1. A BASIC MODEL OF THE EFFECTS OF MONETARY POLICY 5

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

5

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.

6

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.

6 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

equilibrium condition

’1

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

7

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).

1. A BASIC MODEL OF THE EFFECTS OF MONETARY POLICY 7

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-

8 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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)

j=0

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

8

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

ˆ

single term rt , because output is no longer an exogenous factor. And we now put Yt on the left-hand side,

ˆ

to stress that the equation may now be viewed as determining aggregate demand.

9

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

ˆ

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,

ˆ

with γ = 1), then any policy that makes the in¬‚ation rate a stationary variable results in Y∞ = 0.

10

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

1. A BASIC MODEL OF THE EFFECTS OF MONETARY POLICY 9

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.

10 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

(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

ˆ

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

ˆ

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

¯

ˆ

equations for determination of the three endogenous processes {ˆt , πt , Yt }, given the evolution

±

ˆ±

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.)

11

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.

±

1. A BASIC MODEL OF THE EFFECTS OF MONETARY POLICY 11

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

ˆ

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

ˆn

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

12 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

term12

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

ˆn

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

section.

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

12

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

(1.8).

13

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

±

n

compounded) nominal interest rate itself and rt as the (continuously compounded) natural rate of interest.

ˆ

2. INTEREST-RATE RULES AND PRICE STABILITY 13

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

in¬‚ation.

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

n

(1.2), we see that interest rates must satisfy it = rt at all times, where

’1

n

uc (Yt+1 ; ξt+1 )

1 + rt ≡ β ’1 Et

n

. (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

14 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ˆn

to the percentage deviation of the natural rate of interest from its steady-state value,

n

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

approximation).

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

14

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!

15

The concept is thus closely related to Blinder™s (1998, chap. 2, sec. 3) notion of the “neutral” rate of

interest.

2. INTEREST-RATE RULES AND PRICE STABILITY 15

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

ˆ

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 .

ˆ

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

¯

ˆ¯

We observe that each of the exogenous disturbances Gt , Ct , at , and ht increases the natural

16 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

rate of output, and thus, under a policy aimed at price stability, each of them must be

ˆ

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)

ˆn

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

n

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

16

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 17

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

17

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.

18

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.

18 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

ˆn

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 19

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

πt

zt ≡ ,

xt

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).

20 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

¯

and

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 )].

19

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 21

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

1’β

φπ + φx > 1. (2.7)

4κ

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

20

See Proposition 1 of the Appendix.

21

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.

22 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

circle.

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

2. INTEREST-RATE RULES AND PRICE STABILITY 23

of matrix A is of the form

P(µ) = µ3 + A2 µ2 + A1 µ + A0 = 0,

where

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

1’β

φπ + φx > 1 ’ ρ, (2.9)

4κ

1’ρ

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

φπ + (2.10)

4κ

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,

22

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).

24 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

(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

1

(β ’ ρ)2 > 0,

βκσ

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.

23

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 25

1.5

(2.9)

fails

1

ρ

(2.10)

fails

0.5

(2.8)

fails

0

0 0.5 1 1.5

φπ

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

1’β

¦π + ¦x > 1,

4κ

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

24

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 .

26 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

rules

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

¯

25

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 27

and

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

1+β

(φx + 8σ ’1 ).

φπ < 1 + (2.13)

4κ

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

28 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

2

1.8

1.6

1.4

1.2

determinacy

x

1

φ

0.8

0.6

0.4

indeterminacy

indeterminacy

0.2

0

0 5 10 15 20 25 30 35 40 45 50

φπ

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)

26

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 29

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

now

® ®

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

1+β

(φx + 8σ ’1 (1 + ρ)).

φπ < 1 + ρ + (2.15)

4κ

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

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

30 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

1.4

1.2

1

determinacy

0.8

ρ

0.6

0.4

indeterminacy

indeterminacy

0.2

0

0 10 20 30 40 50 60 70

φπ

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.

2. INTEREST-RATE RULES AND PRICE STABILITY 31

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

p

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

4κ

φx > . (2.17)

1’β

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)

32 CHAPTER 4. A NEO-WICKSELLIAN FRAMEWORK

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

[TO BE ADDED]

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

n

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

∞

π

rn

πt = π +

¯ ψj Et (ˆt+j ’ ¯t+j + π ),

± ¯ (2.18)

j=0

∞

x

rn

xt = x +

¯ ψj Et (ˆt+j ’ ¯t+j + π ),

± ¯ (2.19)

j=0

∞

i

rn

ˆt = ¯t +

± ± ψj Et (ˆt+j ’ ¯t+j + π ).

± ¯ (2.20)

j=0

2. INTEREST-RATE RULES AND PRICE STABILITY 33

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 ).

rn

zt = ± ¯ (2.21)