Monetary Policy when Interest Rates Are Bounded at Zero

MONETARY POLICY WHEN INTEREST RATES ARE BOUNDED AT ZERO

Jeffrey C. Fuhrer and Brian F. Madigan*

Abstract—This paper assesses the importance of the zero lower bound onnominal interest rates for the interest-rate channel of monetary policy. Wesimulate several interest-rate setting policy rules with either high or lowinflation targets. We determine the extent to which the zero bound preventsreal rates from falling, thus cushioning aggregate output in response tonegative spending shocks. For small temporary and large permanentshocks, the output path with zero inflation lies modestly below that forhigher inflation. For large shocks persisting a few quarters, differences inoutput paths across high- and low-inflation scenarios can be larger.

I. Introduction

THIS paper assesses the importance of the zero lowerbound on nominal interest rates for the conduct of

monetary policy. In the context of arguing that the optimalrate of inflation is positive, Summers (1991) stated that apossible drawback of aggregate price stability is that thecentral bank would be constrained in its ability to offsetadverse spending shocks because nominal interest ratescannot turn negative. Cushioning output appreciably in theface of a negative demand shock may require movinglong-term real rates down significantly. If short-term nomi-nal interest rates were already low before the shock becauseinflation was low, the central bank may not be able to reduceshort-term real rates much. The argument assumes implicitlythat the inability to lower short-term real rates significantlyimpedes the downward adjustment of long rates.

In this paper we assess this argument using a smallforward-looking model.1 The model was estimated byFuhrer and Moore (1995b). It incorporates multiperiodpricing contracts, a standard IS curve that depends onlong-term real interest rates, and a forward-looking bondmarket in which real long-term rates are set consistent withmarket participants’ expectations of future short-term realrates. This model and its characteristics are described insome detail in the next section.

We examine solution paths for the model under higher andlower rates of inflation and a variety of monetary policyreaction functions. We take the higher rate of inflation to be4% and the low rate to be zero. (We have ignored biases inprice indexes that may cause the desired measured rate ofinflation to be positive rather than zero.) We assess differ-ences in the high- and low-inflation scenarios by examiningrelative deviations of output from baseline.

We note that focusing on the differences in output gaps isonly an approximation to welfare analysis, in two respects.First, a zero aggregate output gap may still imply nonzerodeviations from optimal output for individual firms. Second,this approach ignores the possible welfare costs of positiveand/or variable inflation. We abstract from these consider-ations because the model that we use has no means ofreflecting either the deadweight loss to holders of non-interest-bearing money or the welfare loss associated withvariable (and presumably unpredictable) inflation. However,we feel that it is unlikely that the low inflation rates studiedin this paper correspond to those associated with highinflation variability (and low predictability), as documentedin the empirical work of Logue and Willett (1976) and Engle(1983).

The monetary policy reaction function specifies theresponse of the monetary policy instrument—nominal short-term interest rates—to deviations of nominal income ornominal income growth from target. Thus we do not studythe implications of the zero lower bound on interest ratesunder ‘‘optimal’’ monetary policy. Rather, we study theimplications of the zero lower bound for various interest-raterules of the type often used to characterize recent FederalReserve behavior (see, e.g., Taylor (1993)).

One limitation of this approach is that it focuses solely onthe interest-rate channel for monetary policy. Lebow (1993)and others have suggested that the Fed might circumvent thezero constraint by flooding the economy with reserves whennominal interest rates reach zero. The presumption is thatthis would raise inflation and lower real interest rates,providing the desired stimulus. We do not explore thispossibility in this paper for two reasons. First, in the modelwe use, inflation rises only when output exceeds potential.Thus reserves expansion will not increase inflation whennominal rates and inflation are zero. Second, we find noempirical support in U.S. data for the kind of ‘‘real balance’’effect that would admit a direct influence of reserves onaggregate demand (and thus on inflation).2

We consider two types of demand shocks:

● Permanent unanticipated● Temporary unanticipated

Simulations for anticipated shocks were also carried out, butshed little light on the issue addressed in this paper.3

In the case of permanent shocks, we reduce the naturalreal rate by 50 basis points from the model’s estimated

Received for publication July 27, 1995. Revision accepted for publica-tion April 15, 1996.

* Federal Reserve Bank of Boston and Board of Governors of the FederalReserve System, respectively.

The authors acknowledge George Moore’s role in initiating this researchand thank Thomas Sargent, Michael Woodford, and participants in theStanford/San Francisco Fed conference on ‘‘Monetary Policy in a LowInflation Regime’’ for helpful comments. The views expressed in this paperare the authors’ and do not necessarily reflect those of the Federal ReserveBank of Boston or the Board of Governors of the Federal Reserve System.

1 Lebow (1993) examines this as well as other arguments relating to thezero bound on nominal interest rates.

2 See Fuhrer (1994, pp. 288–291) for documentation of this empiricalfinding.

3 In the context of the current model, long real rates fall sharply inanticipation of a permanent expected shock. Output moves well above itsbaseline initially, and short rates rise to restrain it. Output then falls uponrealization of the shock, but does not move far below its baseline.Consequently, there is little need for policy to move short rates downaggressively, and the zero bound does not come into play.

[ 573 ]r 1997 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

long-term real rate of 2.1%. (Given the estimated interestsensitivity of spending in this model, this is equivalent to adrop in aggregate demand of 0.4%.) This sort of event maybe comparable to the presumably permanent cutback infederal defense expenditures that has recently occurred inthe United States. In the case of temporary shocks, the shockis 0.4% of nominal income, for comparability with thesituation of permanent shocks, and lasts for one quarter.Lags in the model, of course, extend the impact of the shockover time. Given the extreme simplicity of the model, thereis no distinction between lowering government spendingand reducing the natural rate of interest.

We can trust the conclusions drawn from our simulationsto the extent that (1) the model is well specified, and (2) theshocks to which we subject the model are drawn from thesame distribution as the shocks that we identify in estimatingthe model. Evidence supporting (1) is provided in Fuhrerand Moore (1995b). As for the shocks, the estimated modelassumes a linear trend for potential output, so that permanentshocks to output are not identified. Thus we do not know ifthe permanent shocks entertained in this paper are consistentwith the shocks underlying the estimated model. Thetemporary output shocks that we simulate, however, fall wellwithin the estimated distribution of shocks to the outputprocess.4

We enforce the zero bound on nominal interest ratesthrough three alternative techniques involving the monetarypolicy reaction function, rather than through a nonlinearmoney demand equation.5 (In fact, the model includes nomoney demand equation or variable measuring the quantityof money.) The techniques are:

● The left-hand side of the reaction function is specifiedin terms of log differences of the short-term nominalrate.

● The left-hand side of the reaction function is specifiedinstead in terms of levels of short-term nominal rates,but the response to nominal income is adjusted to keepnominal rates from becoming negative.

● The left-hand side of the reaction function is specifiedin terms of levels of short-term nominal rates, but theresponse to nominal income is not adjusted as above.Instead, nominal rates are allowed to drop to the zerolower bound, and the model is solved conditional onnominal rates being at their bound.

An advantage of the first technique is that such a policyrule can be specified fully in advance; no negative nominalrate is arithmetically possible. Thus a reaction function withgiven weights on deviations of the targeted variable can beemployed in high- and low-inflation scenarios.

The second set of methods specifies the reaction functionin levels of interest rates but adjusts the parameters of thereaction functions on an ad hoc basis to prevent nominalinterest rates in both inflation scenarios from becomingnegative. This technique helps assess how much moreaggressive monetary policy can be in fighting recession interms of the numerical strength of the response to deviationsfrom target.

One advantage of the final method is that it allows thenominal rate to drop quickly to its lower bound, thusrepresenting the most aggressive interest-rate response avail-able in the face of zero inflation. A disadvantage of thismethod is that it probably implies more aggressive interest-rate responses than are realistic in a policy environment thatis complicated by uncertainty.

Most of the simulations were obtained using a techniquedubbed ‘‘resolver,’’ which was developed by Madigan(1996). Resolver is a general method for solving nonlinearforward-looking models. (The nonlinearities in the modelemployed in this paper are in the forward-looking consolequation, which links current short- and long-term rates tofuture long-term rates, and in the log-difference reactionfunctions.) Resolver essentially combines the AIM tech-nique of Anderson and Moore (1985) with a Newton–Raphson procedure. The technique is comparable in somerespects to relaxation methods for obtaining numericalsolutions to differential equations. Simulations of the purelylinear models were obtained using AIM.

II. The Model

The simple structural model that we use comprises threesectors: an IS curve that relates output to the ex antelong-term real interest rate, a monetary policy reactionfunction that moves the short-term nominal interest rate inresponse to deviations of target variables from desiredvalues, and a price contracting specification in whichnominal price contracts are negotiated in real terms. Themodel has been estimated on postwar quarterly data for the3-month Treasury bill, the deflator for nonfarm businessoutput, and a measure of the output gap for nonfarmbusiness output, defined as the residual from a regression oflog per-capita nonfarm output on a constant and a linear timetrend. Maximum-likelihood estimation yields significantestimates of all the structural parameters. The dynamicsimplied by the model, as summarized by the vector autocor-relation function, match the dynamics from an unrestrictedvector autoregression very well. At the estimated parametervalues, the model implies a sensible sacrifice ratio, about inline with the estimates in Gordon (1985). Overall, the modelbehaves similarly to a conventional macroeconometric model

4 A third concern is the stability of the estimated parameters acrossdifferent monetary policy regimes—a potential implication of the ‘‘Lucascritique.’’ Fuhrer (1997) tests the stability of the contracting and ISparameters across three different historical monetary regimes, and cannotreject the hypothesis that the contracting specification and the IS interestelasticity are stable across regimes. The lag coefficients in the IS curveappear to shift slightly after 1982.

5 That is, a possible alternative procedure would involve a demand forcentral bank money that went to infinity as nominal short rates asymptoti-cally approached zero and a reaction function specified in terms of moneyrather than in terms of short-term interest rates.

574 THE REVIEW OF ECONOMICS AND STATISTICS

such as the MPS model, despite its forward-looking assetand price sectors. Fuhrer and Moore (1995b) present a moreextensive discussion of the model and its properties.

A. The IS Curve

Let Rt be the yield to maturity on a coupon bond selling atpar, and letM be the maturity of the bond at the end ofquartert. Then the duration of the bond is given by

Dt51 2 e2RtM

Rt(1)

and the holding-period yield on the bond from quartert toquartert 1 1 is

Rt2 Dt(Rt11 2 Rt). (2)

Here we consider a real consol bond with yield to maturityrt. Maturity is infinite, so duration simplifies to 1/rt. Thus theintertemporal arbitrage condition that equalizes the expectedholding-period yields (up to a term premium) on realTreasury bills and the real consol bond is

rt21

rt11[rt11 2 rt ]5 it 2 pt11 (3)

where it denotes the Treasury bill rate,pt denotes theone-period rate of inflation, defined as log (Pt) 2 log (Pt21),andPt denotes the current price level. The abstraction from aterm premium could be important in the present context.Equation (3) implies that in the steady state, the shortnominal rate will settle atrn, the equilibrium long real rate,plus p*, the target inflation rate. If part ofrn is a termpremiumt, thenit should settle atrn 1 p* 2 t. Beginningfrom a steady state with low inflation rates and a modestterm premium, this implies even less room to lower thenominal rate. In the simulations conducted below, wegenerally ignore the possibility of a term premium on thereal long bond. The implications for this issue of a termpremium are discussed in Section IV A.

Because monetary policy in effect targets the inflation ratein the long run, both the short-term real nominal rate and theinflation rate are stationary in this model. This implies that atlong horizons, nominal rates and inflation are forecast to beat their means. As a result, the long real rate that isconstructed assuming a long duration will not exhibit muchvolatility (compared with, say, estimated ex post long-termreal rates). This observation has been made in the substantial‘‘variance bounds’’ literature. While thecorrelationof longreal rates with the output gap is unaffected by the scale oflong rate volatility, one should recognize that the volatilityof long rates depends on the particular monetary policy rule,as well as the parameters of the model. Thus the volatility oflong rates observed in this paper’s simulations may notcorrespond closely with that observed historically.

Given the definition of the expected long real rate, the realeconomy is represented as a simple IS curve that relates theoutput gapyt (the deviation of the log of output from the logof potential output) to its own lagged values and one lag ofthe long-term real interest rate,rt21,

yt 5 0.0171 1.254yt21 2 0.415yt22 2 0.798rt21 1 eyt (4)

where the parameters are taken from Fuhrer and Moore(1995b) andrt is the rate on consols defined previously.6 Thedemand shockeyt will be an unanticipated temporary orpermanent shock in the simulations below. In the steadystate, y 5 0, so the IS curve defines the equilibrium or‘‘natural’’ real rate of interestrn to be 2.1% (0.017/0.798).Note that the equilibrium real rate includes any real termpremium built into the long rate.

One potential concern over using such a simple IS curvewhen inflation and nominal rates are near zero is that thelinear representation will not capture an important nonlinearresponse of spending to interest rates when nominal rates arenear zero. However, the period of estimation for the IS curveincludes 1975–1979, during which short-term real ratesvaried from26% to 0% and the long-term real rate impliedby the model dropped well below its equilibrium. The IScurve shows no sign of misbehaving during this period,suggesting that if a nonlinear response exists, it is not ofprimary importance for total spending.

B. Contracting Specification

Agents negotiate nominal price contracts that remain ineffect for four quarters. The aggregate log price index inquartert, pt, is a weighted average of the log contract pricesxt2i that were negotiated in the current and the previous threequarters and are still in effect. The weightsfi are theproportions of the outstanding contracts that were negotiatedin quarterst 2 i,

pt5 oi50

3

fi xt2i (5)

wherefi $ 0 andS fi 5 1. We characterize the distribution ofcontract prices with a downward-sloping linear function ofcontract length,

fi 5 0.251 (1.52 i)s, 0 , s #1

6,

i 5 0, . . . , 3.

(6)

6 These parameters are consistent, but inefficient, partial-informationestimates. However, they differ insignificantly from the full-informationestimates presented in Fuhrer and Moore (1995b). The full-informationestimate of the interest elasticity parameter, for example, is20.746, with astandard error of 0.25.

575MONETARY POLICY WHEN INTEREST RATES ARE BOUNDED AT ZERO

This distribution depends on a single slope parametersand itis invertible. Whens5 0, it is the rectangular distribution ofTaylor, and whens5

16 it is the triangular distribution.

Let vt be the index of real contract prices that werenegotiated on the contracts currently in effect,7

vt 5 oi50

3

fi (xt2i 2 pt2i ). (7)

Agents set nominal contract prices so that the current realcontract price equals the average real contract price indexexpected to prevail over the life of the contract, adjusted forexcess demand conditions,

xt 2 pt 5 oi50

3

fiEt(vt1i1 gyt1i). (8)

Substituting equation (7) into equation (8) yields the realversion of Taylor’s contracting equation,8

xt 2 pt 5 oi51

3

bi (xt2i2 pt2i ) 1 oi51

3

biEt(xt1i2 pt1i)

1 g* oi50

3

fiEt( yt1i).

(9)

In their contract price decisions, agents compare the currentreal contract price with an average of the real contract pricesthat were negotiated in the recent past and those that areexpected to be negotiated in the near future. The weights inthe average measure the extent to which the past and futurecontracts overlap the current one. When output is expectedto be high, the current real contract price is high relative tothe real contract prices on overlapping contracts.

The contracting specification is parameterized bys, theslope of the contract distribution, and byg, the response ofreal contract prices to expected excess demand conditions.The maximum-likelihood estimates of these parameters andtheir standard errors are given in table 1.9

C. Reaction Function

Monetary policy is represented as a policy reactionfunction that moves the short-term nominal rate in responseto deviations of its ultimate targets from their desired values.In this paper, while the operating instrument is always theshort nominal rate, we consider two basic variants of thereaction function:

log (it) 2 log (it21) 5 l(zt21 2 z*t21) (10)

it 5 rn 1 p* 1 l(zt21 2 z*t21). (11)

In both casesz is the ultimate target of monetary policy,and a largerl implies a more aggressive policy response todeviations ofz from its desired or target value. The firstvariant imposes the nonnegativity constraint on nominalrates and incorporates the incentive to smooth interest rates.In the second variant, the response to nominal incomel isadjusted on an ad hoc basis to keep nominal rates frombecoming negative. This technique effectively does awaywith interest-rate smoothing, and in a stronger sense than thefirst variant implies that the monetary authority knows theequilibrium long-term real rate. The ultimate target is eithernominal income or nominal income growth.

III. Simulations

A. Permanent Unanticipated Shocks

This section discusses simulations conducted under apermanent unanticipated shock. The shock increases theoutput gap initially by 0.4% by reducing the natural rate ofinterest by 50 basis points. In the post-shock steady state,real long rates will be 1.6% (2.1% minus 50 basis points);short nominal rates will be 5.6% with 4% inflation and 1.6%with zero inflation.

Operating Instrument—Log-Difference Nominal Rates:Fig-ure 1 illustrates a simulation using a log-difference reactionfunction and a nominal income target. The response ofinterest rate differences to deviations of the level of nominalincome from target—l—is set to a value of 60. This value isthe maximum at which a simulation could be obtained. Asshown by the solid line in the upper panel, the nominal shortrate in the zero-inflation case adjusts down over a period ofabout a year by a total of nearly 11⁄4 percentage points. Bycontrast, the nominal rate in the high-inflation scenario(shown by the dashed line) falls about 3 percentage points.As would be expected with the log-difference reaction

7 This is a convenient simplification from the theoretically preferablespecification that defines the real contract price as the difference betweenthe nominal contract price and the weighted average of price indexes thatare expected to prevail over the life of the contract. The simplificationyields an algebraically more straightforward model. The effects of thesimplification on the empirical properties of the model are relatively small.See Fuhrer and Moore (1995a) for details on the alternative specificationand associated empirical results.

8 Compare equation (9) with Taylor (1980, eq. (1)). The coefficients inequation (9) arebi 5 Sj fj fi1j /(1 2 Sj f j

2) andg* 5 g/(1 2 Sj f j2).

9 In a series of recent working papers, Roberts (1994) finds that theoriginal sticky price specifications of Taylor and others can be reconciledwith the persistent inflation data by using survey expectations, rather thanrational expectations, to close the model. With survey expectations, theadditional persistence imparted by the real contracting specificationdiscussed in this section is not necessary to reconcile the model with thedata. Under rational expectations, the maintained assumption in this paper,the early sticky price specifications imply far less inflation persistence thanis evidenced in the data.

TABLE 1.—PARAMETER ESTIMATES AND STANDARD ERRORS

Parameter Estimate Standard Error t-Statistic

s 0.0797 0.0116 6.9g 0.0045 0.0018 2.6Ljung–BoxQ(12) statistic 27.2


function, thepercentreduction in nominal interest rates issimilar in the two cases. The small difference reflects theslightly stronger nominal income in the high-inflation caseand the feedback through the reaction function to thenominal rate.

The middle panel of figure 1 shows that the long real rateovershoots in both cases—it initially falls by more than the50 basis point decline in the natural real rate, as marketsbring forward in time the lower short real rates in the futurethat will result from lower nominal short rates combinedwith only sluggishly declining inflation. In the zero-inflationcase, the long rate falls 56 basis points right away and thentrends up gradually. In the high-inflation scenario, the longrate initially drops a bit more—62 basis points. The long realrate then rises more steeply than in the low-inflation case,reflecting the anticipated need for monetary policy to leanmore heavily against the stronger upward burst of outputshown in the lower panel. The drop in real rates is obviouslysimilar across the two scenarios.

The drop in output in the first quarter is identical in thetwo cases—0.4%. The model incorporates a one-quarter lagin the response of demand to interest rates, so the drop inoutput in the first quarter represents solely the exogenousdecline in demand, and hence is identical in the twoscenarios. The slightly lower initial real rates of the high-inflation scenario cushion output in the second quarter,essentially preventing it from falling further as it does in thezero-inflation case. The difference, however, is slight—lessthan 0.1% of the level of output. Output subsequentlyrecovers a little more steeply in the high-inflation case. Theend of the recovery—defined as the point at which output‘‘recovers’’ its prerecession level—comes about a quarterearlier. The subsequent cycles are of greater amplitude inthis case.10

10 The thinner lines in the bottom two panels of figure 1 indicate 90%confidence intervals. These confidence intervals reflect the uncertainty inthe simulated paths arising from the sampling error in estimating themodel’s coefficients. The intervals depicted are the fiftieth highest and

FIGURE 1.—UNANTICIPATED PERMANENT SHOCK, NOMINAL INCOME TARGETING

LOG INTERESTRATE CHANGES; l 5 60


Figure 1 provides 90% confidence intervals for thesimulations (Computational details are provided in footnote10.) As the figure indicates, the time paths of the simulatedreal rate and output gap are significantly different from zerofor the early years of the simulation, especially for the 0%simulations. The long-term real rate falls significantly belowits long-run equilibrium, and the output gap turns signifi-cantly negative in the first two years. In the terminal years ofthe simulation, of course, the confidence intervals suggestthat the small deviations from the model equilibrium are notsignificantly different from zero. Because the amount ofuncertainty arising from sampling error is quite small forthis model, we do not depict confidence intervals in thefollowing charts.

We compute a second simulation somewhat similar to thatof figure 1, except that the policy reacts to nominal incomegrowth rather than nominal income levels. Under nominalincomegrowth targeting, the price level will be lower in thepost-shock steady state than under its baseline rate, while theinflation rate will return to its baseline rate, which is equal tothe targeted growth rate of nominal income. By contrast,under nominal income targeting, both the price level and theinflation rate ultimately return to baseline after a shock. Therequirement that the price level return to baseline in thenominal income targeting case induces additional cycles inthe solution relative to the nominal income growth case,which only requires the inflation rate to return to its targetedlevel.

With policy reacting only to income growth rather thanincome levels, the short rate is reduced by less and is broughtup to the vicinity of its new equilibrium sooner. Real ratesconsequently drop a little less. In the low-inflation case, thereal long rate drops immediately to, but not below, the newnatural rate, while in the high-inflation case the long rateovershoots. The lower real rates of the high-inflation casebring the output gap to zero appreciably more quickly thanin the zero-inflation scenario.

Operating Instrument—Nominal Short-Rate Levels:Inthe preceding section the 0% floor on nominal interest rateswas enforced by considering reaction functions in log-difference form. In this section the operating instrument isconsidered to be levels of short-term nominal rates, as inequation (11). The policy responsiveness coefficientl, isadjusted on a case-by-case basis in such a way as to allowthe nominal short rate to fall to, but not below, zero.

Such a reaction function, that is, one specified in terms oflevels, seems most consistent with a view that the centralbank can determine the true level of the natural rate ofinterest with a high degree of precision. In such a situation,the central bank presumably would wish to move interestrates promptly to appropriate levels. By contrast, the previ-ous section’s log-difference reaction function, which embod-ies interest-rate smoothing, might better characterize policyas actually practiced, as monetary policy makers take intoaccount uncertainty, any costs of interest-rate variability, andperhaps an aversion to frequent reversals of course.

As shown by the dashed line in figure 2, in the 4% infla-tion case, policy lowers the short rate to zero in the secondperiod, after nominal income began to fall significantlybelow target in the first period. That is, nominal short ratesfall 6.1 percentage points almost immediately. This respon-siveness corresponds to al equal to 14. By contrast, thelower level of nominal rates in the low-inflation case permitsa much less aggressive policy response: short rates can fallonly 2.1 percentage points, corresponding to al of 3.5.

The lower panel shows that the larger move in nominalrates in the 4% inflation case results in a sharper initial dropin the real long rate, by about 9 basis points for two periods.Long rates subsequently move up more strongly in thehigh-inflation situation. The larger drop in real rates in thiscase causes the recession (defined as the period duringwhich the output gap is growing) to end after one quarter,whereas the recession in the low-inflation case lasts twoquarters. The recovery similarly ends sooner in the 4%inflation scenario. Output overshoots and cycles a little inthe high-inflation case. Although output also overshootsslightly in the low-inflation case, the approach to equilib-rium is more gradual.

As in the previous simulation pair, we compute a simula-tion for an interest-rate level operating target and a nominalincomegrowth ultimate target. The high- and low-inflationcases usel equal to 14 and 6.5, respectively. In bothsimulations, monetary policy drops nominal short ratessharply as an output gap opens and inflation falls belowtarget, leading to a shortfall in nominal income growth. Theeasing is reversed quickly, however, as a drop in the reallong rate prompts a rebound in real output that pushesnominal incomegrowth roughly back to target. In both thehigh- and low-inflation cases, after a few quarters thenominal short rate actually gets pushed a bit above its newlong-run equilibrium, and this overage is transferred to thereal long rate. Very slowly, the real long rate drifts towardthe new natural rate, bringing the output gap eventually tozero. Although output is a little higher in the second throughsixth quarters in the high-inflation case, the difference issmall.

B. Temporary Unanticipated Shocks

This section generally considers a temporary 0.4 percentshock to aggregate demand; in most simulations, the shockoccurs in the first period of the solution and lasts for one

fiftieth lowest simulated values from 1000 simulations based on randomnormal draws from the parameter distribution estimated in Fuhrer andMoore (1995b). Only the uncertainty in the nonpolicy parameters is takeninto account. Note that these confidence intervalscannot be used todetermine whether the simulated paths are significantly different from oneanother. For any coefficient draw (which would be used to simulate bothcases), the simulated path of output in figure 1 for the zero inflation casewill always initially lie below that of the 4% inflation case. This is aproperty of the model, and is not altered by taking into account coefficientsampling error.


quarter. Section 3.2.2 also considers a longer-lasting tempo-rary shock and a reaction function that includes forward-looking elements.

Operating Instrument—Log-Difference Nominal ShortRates: With a log-difference reaction function andl setequal to 30, short rates in the high-inflation case declineabout 4 percentage points over the span of a few quarters,while short rates fall 11⁄2 percentage points (to 65 basispoints) in the low-inflation case. (This simulation is shownin figure 3.) Real long rates drop considerably furtherinitially with 4% inflation—35 basis points, as opposed to 20basis points in the zero-inflation case. The lower real ratespermit a somewhat steeper recovery of output, but thepattern is not markedly different. The relatively modestvariation in output across the two cases reflects the smalldifference in long rates measured inpercentage points.

Operating Instrument—Nominal Short-Rate Levels:Fig-ure 4 shows results for temporary unanticipated shocks andan interest-rate-levels reaction function. Again, the high-inflation case permits a substantially more aggressive re-sponse measured in terms of percentage point movement innominal short rates. (The policy responsiveness parameterlis equal to 10 and 1.65, respectively, in the two cases.)Consequently the decline in real long rates is more thantwice as steep. But because the percentage point differencein long rates is relatively small, the trajectory of outputduring the recovery is only a little steeper, ending therecovery about a quarter earlier.

Figure 5 considers a more severe shock—an unantici-pated 1% drop in demand that persists for four quarters. Forthese simulations, we also modify the assumption concern-ing the recognition of the shock. We assume that althoughthe shock is not anticipated in advance, once it begins, the


INTERESTRATE LEVELS


magnitude and duration of the shock are recognized accu-rately. We assume further that the policy reaction functionincorporates a contemporaneous and forward-looking re-sponse to this shock, once it is recognized. Specifically, thereaction function is

it5 rtn 1 p*t 1 l(z*t21 2 zt21 1 Gt 1 Gt11

1 Gt12 1 Gt13)(12)

whereG denotes the shock, which may represent reducedgovernment spending or some other factor depressing aggre-gate demand.

In this simulation, nominal short and real long rates dropimmediately, but the nominal short rate can drop much moresharply—and stay low for longer—under the high-inflationcase. Over the first year, the real long rate averages about 25basis points lower in the high-inflation scenario. With realrates unable to fall as sharply in the low-inflation case as in

the high-inflation case, the decline in output is significantlysharper in the first case. The recoveries in the two casesproceed in parallel, with the level of output higher in the 4%inflation case, and the output gap is closed about one quartersooner in this case.

This shock pattern seems particularly informative becausethe depth of the ensuing recession, measured as the percent-age point shortfall of output below potential, matches thedepth of a ‘‘typical’’ recession in postwar U.S. data. In the4% inflation scenario, the output gap reaches a trough of2.8%, while in the zero inflation case, the gap troughs at3.6%. The 0.8 percentage point absolute difference in outputgaps in the two scenarios probably would be regarded bymany as economically meaningful.11

11 Recession depths may be estimated either from a log detrended outputseries (with the trend broken in 1973) or from an output gap series impliedby the unemployment rate and an inverted Okun’s law.


LOG INTERESTRATE CHANGES; l 5 100


Discontinuous Interest-Rate Response:Many of thepolicies discussed above are more vigorous than thosepursued historically by the monetary authority. However, thecentral bank could respond even more vigorously to anadverse demand shock than in the continuous responsesdepicted in figures 2 and 4. In those simulations, we use thelargest policy coefficient that yields a continuous path ofinterest rates that lies above zero. An alternative strategywould employ more vigorous policy coefficients that sendthe nominal rate quickly to the zero bound. Such a ‘‘bang-bang’’ policy might be vigorous enough to offset much of theconstraint imposed by the lower bound under the continuouspolicies explored above.12

Figure 6 depicts such a simulation. The simulation usesthe aggressive nominal income targeting parameter (l 5 1.5)of figure 5 for both the zero inflation and the 4% inflationcase. The simulation is conducted so that if the nominal ratewere to transgress its lower bound for a given period, it isconstrained for that period to remain at its lower bound, andthe other variables in the model are solved for conditional onthat constraint. Comparing figure 6 with figure 5, the abilityto drop the short rate immediately to its zero bound has arelatively small effect on the real rate and the output gap.Thus while the other interest-rate level simulations depictedin figures 2, 4, and 5 do not allow policy to discontinuouslydrop the short rate, it is likely that this restriction hasrelatively little effect on the simulated paths of real rates andoutput.12 The model used for this simulation substitutes the linear version of the

term structure equation. At each time period, if the solution value of theshort rate falls below zero, the short rate is set to zero and exogenized (forthat period only). We then re-solve for the other variables in the modelconditional on that exogenized value for the short rate. The solution then

proceeds on a period-by-period basis, imposing the zero lower bound inthis way.

FIGURE 4.—UNANTICIPATED TEMPORARYSHOCK, NOMINAL INCOME TARGETING

INTERESTRATE LEVELS


IV. Additional Considerations

The differences identified in the previous section betweenoutput under high- and low-inflation cases may depend inpart on certain aspects of the model’s specification andparameters. In this section we consider the followingmodifications:

● The existence of a term premium in long-term interestrates, which implies a lower steady-state level of realshort rates, may limit the ability of monetary policy tostimulate economic activity.

● If short real rates, in addition to long real rates, affectspending, the ability of long rates to jump down,cushioning the effects of an adverse spending shock,would be less relevant and the behavior of short rateswould be more relevant.

● If bond markets are partly backward looking, bondrates may be less apt to jump down when newsbecomes available about adverse spending shocks.

A. Term Premium in Interest Rates

As noted, the previous simulations assume that reallong-term interest rates contain no term premium. If there isa term premium in long rates, however, the ability ofmonetary policy to ease in response to an adverse spendingshock would be more constrained, because the equilibriumshort real rate would be lower than the equilibrium long rateby the amount of the term premium. Whitesell (1990)estimated that the equilibrium real rate on three-monthTreasury bills between 1978 and 1992 was between1

2 and 158percentage points below that on ten-year Treasury notes. Inthis section we assume a constant term premium in long-

FIGURE 5.—INITIALLY UNANTICIPATED TEMPORARY(FOUR-QUARTER) SHOCK, NOMINAL INCOME TARGETING

INTERESTRATE LEVELS (1% SHOCK)


term real rates of 1 percentage point. In contrast to theestimation results underlying the previous simulations, weassume that the equilibrium real short-term rate and theequilibrium real long rate excluding the term premium havealready been lowered by some exogenous event and are1.1% rather than 2.1%.

We compute a simulation that is comparable to that shownin figure 1, except for the term premium. Because nominalshort rates start out at 1.1%, rather than 2.1%, and becausethe reaction function is specified in log-difference form, theinitial percentage pointdrop in short rates in the low-inflation scenario is only about one-half that in the 4%inflation case. Consequently, real long rates cannot fall quiteas far initially with a term premium. Although the relativedifference in the behavior of long rates across the two casesseems small, it lasts for half a year and is large enough togenerate a relatively more robust recovery in the high-

inflation case with a term premium; the recovery is com-pleted noticeably more quickly with higher inflation. Acrossthe low-inflation scenarios, output runs about 0.1% lowerunder the term-premium case.

B. Shorter Term Real Rates in IS Curve

Our simple characterization of aggregate spending makesthe output gap a function only of long-term real rates.13 And

13 Indeed, our use of the nonlinear consol equation implies that interestrates on very long-duration instruments are relevant to spending decisions.Experimentation with instruments of shorter duration, by using the linearapproximation discussed in Fuhrer and Moore (1995b), indicated that theessential conclusions of this paper are unaffected, although the cyclicalproperties of the economy differ noticeably. In particular, the amplitude offluctuations in interest rates, output, and inflation with certain reactionfunctions is considerably larger with the shorter duration long rates thanwith the consol equation.

FIGURE 6.—INITIALLY UNANTICIPATED TEMPORARY(FOUR-QUARTER) SHOCK, NOMINAL INCOME TARGETING

INTERESTRATE LEVELS (1% SHOCK); l 5 1.5


yet it is likely that many important expenditure categories—auto purchases, business equipment expenditures, and evennew home purchases—depend at least in part on shorterterm real rates. If expenditures do depend on shorter termrates, then the dynamics of the monetary transmissionmechanism might be significantly altered, and our simula-tions might not be representative of the response of theeconomy to output shocks at low inflation. In particular, theability of the real long rate to jump down in both high- andlow-inflation scenarios would be less relevant, and thebehavior of the short rate would be more relevant, to thepotential disadvantage of a low-inflation policy if a con-straint on the inability of short nominal rates to decline alsoprevented short real rates from falling appreciably. On theother hand, monetary policy likely has more control over ashorter duration real rate than a longer duration one, despitethe zero bound, unless short real rates were quite close tozero before the shock. Consequently, it is not clear that thepresence of short rates in the IS curve necessarily disadvan-tages a low-inflation policy.

To test the importance of this modification to our specifi-cation, we simulate the model with an IS curve with both ashort-duration (eight quarters) and a long-duration (40quarters) real rate. For comparison with the simulationdepicted in figure 1, the operating instrument is a log-difference specification, the ultimate target is nominalincome, andl is set to 60. However, because the overallinterest elasticity of the IS curve is lower than in figure 1, theinitial decline in spending is smaller—about 0.3% instead of0.4%.14 The model behaves about as it does in the simula-tions in figures 1. Overall, for our specification, the exclu-sion of shorter duration real rates appears to be an unimpor-tant omission.

C. Backward-Looking Bond Markets

The expected long-term real rate as defined in equation(3) is completely forward-looking, satisfying period-by-period arbitrage. As a result, the real rate jumps immediatelyin response to an unanticipated shock, as is clear in figures1–6. This feature of the model’s long real rates may be a bitunrealistic; while holding period returns are unlikely todiverge over extended periods, they may not be equalizedperiod by period. Thus we explore the robustness of oursimulation results to a modified real rate specification that

combines both forward-looking and backward-looking be-havior. The ‘‘mixed’’ real rate is defined as

rtm5 wrt

b 1 (1 2 w)rtf (13)

wherertf is the purely forward-looking real rate defined in

equation (3), andrtb is a weighted average of past short real

rates, with weights that sum to 1 and decline geometricallyinto the past,

rtb5 drt21

b 1 (1 2 d)(it21 2 pt) (14)

which expands to

rtb5 (1 2 d) o

j50

`

d j (it212j2 pt2j ). (15)

By varying the degree of ‘‘backward lookingness’’w in thereal rate, and the rate at which the backward-looking weightsd j decay into the past, we can get an idea of the sensitivity ofour results to the real-rate specification.

Settingd 5 0.98 (which implies weights decaying into thepast at the same rate as the weights decay into the future forthe forward-looking real rate),w 5 0.95,l 5 50, and theinterest elasticity to its benchmark value, the model is stablewith a log-difference reaction function.15 In this simulation,the real long rate still jumps down considerably, despite thevery large parameters on the backward-looking componentof real rates. But it takes three to four quarters for the rate toreach its trough, unlike the case in figure 1, where long ratesreach their low point essentially immediately. As in figure 1,real rates are lower in the high-inflation scenario than in thelow-inflation case for the first four quarters or so, but therelative differences between the purely forward-looking caseand the mixed backward/forward case are quite small.Consequently, the relative paths of output are similar.Overall, incorporating backward-looking behavior in thebond markets does not alter the qualitative properties of themodel simulations.16

V. Conclusion

This paper examined one argument that the optimal rate ofinflation is positive, namely, that the lower nominal rates ofinterest that would accompany zero inflation would limit theability of monetary policy to ease in response to an adversespending shock. To assess the argument, we utilized a smallmodel of the U.S. economy that captures forward-looking

14 The simulation uses an eight-quarter bond as an approximation to aneight-quarter duration bond. At the steady-state interest rate of 1.6%, aneight-quarter bond has a duration of 7.55 quarters. The jointly estimatedcoefficient on the short-duration real rate is zero. If we exclude thelong-duration real rate from the IS equation, the estimated coefficient onthe short-duration real rate is 0.079, ten times smaller than the estimatedcoefficient on the long real rate. (One would expect the coefficient to besmaller, since shortening the duration makes the long rate move moreclosely with the short rate, increasing its volatility.) We simulate the modelwith both long- and short-duration real rates in the IS curve by imposing aweight of 2⁄3 on the long rate and1⁄3 on the shorter duration rate, using thecoefficients estimated for each (0.79 for the long rate, 0.079 for the shorterrate).

15 Interestingly, the stability of the model is sensitive to the exactcombination ofw, d, and IS interest elasticity. For example, withw 5 0.5,d 5 0.9, and the interest elasticity in equation (4), the model does not havea stable, unique solution. The backward-looking long rate places too muchweight on the recent past, and thus exerts a destabilizing force on outputand inflation.

16 For more details on the simulations described in this section, seeFuhrer and Madigan (1994).


behavior both in financial markets and in product marketsand whose broad properties correspond with those oflarge-scale macroeconometric models. Our results indicatethat the argument is correct, qualitatively speaking. Al-though long-term real rates in forward-looking bond mar-kets do decline in response to news about adverse spendingshocks, thus cushioning the reduction in output, the declinein real rates can be constrained by the inability of nominalrates to fall below zero.

We find that for relatively small and short-lived spendingshocks, as well as for permanent and large shocks, the pathof output in the zero-inflation case is only modestly belowthat in the higher inflation case—on the order of a 0.1 or0.2%; the recession and the recovery tend to be completedone quarter later with higher inflation rates. But for largeshocks persisting a few quarters, differences in output pathsacross high- and low-inflation scenarios can be larger—onthe order of 1% of output. These results appear to hold forseveral types of monetary policy reaction functions whenmonetary policy responds quite vigorously to shocks. Moremeasured responses of monetary policy could reduce thedifferences across high- and low-inflation cases, dependingon the form of the monetary policy reaction function, andcould even eliminate them for small shocks. For example, areaction function that adjusted rates in terms of arithmeticdifferences rather than log differences and that respondedcautiously to deviations of target variables from their targetsmight not be constrained by the zero bound on interest ratesin cases of small shocks.

Consideration of several complications not included inthe basic simulations produced mixed results. Allowance fora term premium in long-term interest rates tends to augmentdifferences across inflation scenarios. Including shorter termrates in the IS curve also tends to be unfavorable to lowerinflation, although our simulations suggest that this effect isnot large. Assuming that bond markets are not fully forward-looking seems to have little effect on the conclusions.

The quantitative results of this paper clearly are no doubtmodel-specific. They depend on the specification and param-eterization of the model. In addition, their practical implica-tions depend on how quickly a central bank can recognizeshocks and how vigorously it can respond to them. Theassessment of these issues no doubt varies considerably withthe nature of the shock. Moreover, other economic policies—for example, fiscal policy—may be useful in fightingrecession even when conventional interest-rate monetary

policy is hamstrung by the zero bound on nominal interestrates. Conceivably, monetary policy actions other thanopen-market operations, such as lending through the dis-count window, could also provide stimulus by loweringspreads of private interest rates over yields on governmentsecurities. In any case, though, this paper has provided aninitial quantification and a starting point for future researchon the issue of the relevance of the zero bound.

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Fuhrer, Jeffrey C., and Brian F. Madigan, ‘‘Monetary Policy when InterestRates Are Bounded at Zero,’’ Federal Reserve Bank of BostonWorking Paper 94-1 (Mar. 1994).

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Monetary Policy when Interest Rates Are Bounded at Zero