1 See DeLong and Summers(1992) and Summers (1991).Marty and Thornton (1995)provide an analysis of severalarguments asserting that theeconomy benefits frommoderate inflation.

2 See Howitt (1990).

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The Costs andBenefits ofPrice Stability:An Assessmentof Howitt’s Rule

Daniel L. Thornton

The central banks of New Zealand,Canada, and the United Kingdom haverecently decided to make price stability

the overriding goal of monetary policy.Similar proposals in the United States have received a lukewarm reception.Although some opponents have arguedthat moderate inflation is beneficial, manyconcede its effect on economic welfare is detrimental.1 Instead, they argue that a price stability policy is suboptimalbecause, once inflation is under way, it is better to tolerate some moderate infla-tion than to bear the cost necessary toachieve price stability. Howitt’s Rule is the clearest statement of the propositionthat the benefits resulting from reducinginflation must be weighed against the costof reducing it.2 (See the shaded insert,“Howitt’s Rule.”)

Assuming inflation affects both thelevel and growth rate of output, I stateHowitt’s Rule and explain how it argues fora continued policy of moderate inflation.Next, I analyze alternative estimates of thecosts of achieving price stability and com-pare these costs with estimates of the gainfrom achieving price stability when infla-tion reduces the level or growth rate ofoutput. Finally, I briefly review the cross-section and time-series evidence on theeffects of inflation on output growth. Inthe final section I offer a summary andsome concluding observations.

THE COST-BENEFITINFLATION TRADE-OFF:HOWITT’S RULE

Howitt (1990) argues that, althoughinflation is costly, once it is under way,society is better off to tolerate a little infla-tion than to bear the cost necessary toachieve price stability. Howitt (1990, p. 103 ) notes, “There are a host of reasonswhy the best average rate of inflation,ignoring costs of getting there, is zero . . .”(italics added). Arguing that the cost of achieving zero inflation may be sub-stantial, however, Howitt suggests his rulecan be used to determine the “optimal”inflation rate.

Howitt argues that the transitions cost“will not be negligible, so that it will beoptimal to stop disinflation, even when the gain from further reduction is still pos-itive.” Therefore, Howitt (1990, p. 104)concludes, “The optimal target is probablysomewhere above zero” (italics added).

The cost-benefit trade-off Howittalludes to is represented in Figure 1(under the assumption that inflationreduces the level of output). At time t0policymakers decide to pursue a policythat will reduce the steady-state inflationrate from its current level to a lower level,

Daniel L. Thornton is an assistant vice president at the Federal Reserve Bank of St. Louis. Jonathan Ahlbrecht provided research assistance.

HOWITT’S RULE

In order to estimate the optimal target rate ofinflation, one must somehow balance the gains fromreducing inflation against the costs of doing so. Thereduction in inflation should continue as long as thepresent discounted value of the benefits to society from a further small reduction exceeds the presentdiscounted value of the cost. The optimal target rate is the rate at which the benefit of further reduction justequals the cost of raising unemployment by the requiredamount above the natural rate. (Howitt, 1990, p. 104,italics added.)

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3 In Figure 1, once output hasreturned to its previous trendlevel, the time it takes to get to the new higher level is soshort that it is inconsequential.If this is not the case, theestimates of the gains fromreducing inflation (presentedlater) are overstated. More-over, if the period to achievethe higher level of output isvery long, it will be extremelydifficult empirically to differen-tiate between level and growthrate effects of inflation becauseoutput would grow at a rateabove its trend level for a longperiod before reaching itspermanently higher level.

4 See Barro (1995 and forth-coming); Bruno (1995); Clark(1993); Ericsson, Irons, andTryon (1993); Fischer (1993);Dotsey and Ireland (1993);Briault (1995); Grier andTullock (1989); Kormendi andMeguire (1985); Levine andRenelt (1991 and 1992);Logue and Sweeney (1981);Orphanides and Solow (1990);Ireland (1995); Jones andManuelli (1995); Chari, Jones,and Manuelli (1995); King andRebelo (1990); Golob (1993).

MA R C H/ A P R I L 1 9 9 6

perhaps zero. This change in policy causes output to fall below its steady-state path until time t1, when the economyachieves its new lower inflation rate and a permanently higher level of output.3

The permanent increase in the level ofoutput is taken to be some proportion, θ, of current period output, that is, theincrease is equal to θy0. The shaded areamarked G represents the permanent gainto output associated with achieving alower rate of inflation for a given timehorizon (T). The shaded area marked Ldenotes the temporary output loss associ-ated with reducing the steady-stateinflation rate.

Discounting Future OutputHowitt’s Rule states that disinflation

should continue until the present value of G equals the present value of L. Hence,the output levels in Figure 1 must bediscounted at some discount rate, β.Figure 1 can be easily modified to showthe effect of discounting. This is done inFigure 2, which shows the present (timet0) value of the output streams in Figure 1discounted at the rate β. Figure 2 assumesthat the discount rate is larger than thegrowth rate of output, α, so the presentvalue of future output always lies belowcurrent output.

Under this assumption, the presentvalue of future output approaches zero

as the time horizon approaches infinity, T → ∞. This is true for both the high- andlow-inflation output paths. Consequently,the present value of the gain from reducinginflation, pvg in Figure 2 (the shaded areabetween the present values of these altern a-t i v e output paths), is finite. Howitt’s Ruleis to cease the disinflation process whenpvg equals pvl. Hence, it is possible tochoose a positive inflation rate as the“optimal target inflation rate.”

The sizes of pvg and pvl depend on the level of current output, y0, so it isconvenient to express the present value of the gain and present value of the lossfrom reducing inflation as a proportion of current output. That is,

PVG = pvg/y0

andPVL = pvl/y0,

respectively.

When Inflation Affects Output Growth

Figures 1 and 2 are drawn under the assumption that inflation affects only the level of output. Increasingly,economists have paid attention to thepossibility that inflation may affect thegrowth rate of output.4

The behavior of output associatedwith reducing the steady-state inflationrate when inflation reduces output growthis shown in Figure 3, as well as the outputpaths for the high- and low-inflation alter-natives and the present values of thesepaths. It is assumed that inflation reducesoutput growth, that is, the growth rate of output in the low-inflation state, µ, islarger than the growth rate of output in the high-inflation state, α. Figure 3 isdrawn under the assumption that the rateat which future output is discounted isgreater than the low-inflation outputgrowth rate, β > µ. In this case, thepresent value of future output approacheszero as T → ∞ for both the low- and high-inflation states. Consequently, PVG isfinite for any value of T.

Figure 1

α θ

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5 See Phelps (1979), Taylor(1983), Aiyagari (1990), and Ball (1994b). Aiyagari(1990, p. 2) discusses whyreal output is an appropriatemeasure of welfare for govern-mental policy analyses.

6 For example, during times ofwar, output typically increases;however, most people wouldconsider themselves to beworse off.

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AN ASSESSMENT OFHOWITT’S RULE

A number of conditions must befulfilled to apply Howitt’s Rule. Some, like determining the relative sizes of PVGand PVL, are obvious. Others are subtle,yet just as important. I begin my assess-ment of Howitt’s Rule with a discussion of the less obvious conditions upon whichit depends.

Output as the Appropriate Measureof the Benefits and Costs

In Figures 1, 2, and 3, I implicitlyassumed that the costs and benefits couldadequately be represented by output gainsand losses. This practice is common inmuch theoretical and virtually all appliedwork.5 Two important limitations on thispractice should be noted, however. One isthat measured output does not conformperfectly with the theoretical measures.For example, measured output does notinclude home production. Hence, applica-tions of Howitt’s Rule to real-world dataare problematic. This is a problem forvirtually all applied work, so I do notdiscuss it further.

A more important limitation is thatoutput is an imperfect measure of economicw e l f a re. In economics, welfare is measure dmore abstractly, but more correctly, by theconcept of utility. If there were a one-to-onec o rrespondence between output and utility,there would be no problem, but this is not the case.

In theory or in practice, output doesnot include the utility individuals obtainfrom leisure. When more time is spent inp roduction, less time is available for leisure .Consequently, it is possible for output toincrease while economic welfare declines.6

This consideration is particularly import a n tfor assessing the costs of inflation (and, con-s e q u e n t l y, the benefits from disinflation)because inflation is believed to distort theallocation of re s o u rces, in part i c u l a r, the dis-tribution of time between work and leisure .

The precise rationale for inflation’s dis-torting effect varies from model to model.

Many economists, however, believe thatmoney is held to facilitate trade. Money is costly to hold, with the annual marginalcost per real dollar held equal to the nom-inal interest rate. The nominal interestrate is equal to the real interest rate (deter-mined by real factors, such as productivityand thrift) and the expected rate of infla t i o n .In the long run, actual and expectedinterest rates should be equal, regardless of how expectations are formed. Inflationis costly because the nominal interest ratebecomes higher as the steady-state infla t i o nrate rises. Consequently, so too is the cost of holding money. Indeed, the nominal

Figure 2

α θ β

Figure 3

α β µ

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7 To argue that welfare is some-how higher is to argue thati n flation is beneficial. See Mart yand Thornton (1995) for ananalysis of four arguments thatmoderate inflation is beneficial.

8 Although it is possible thatHowitt’s Rule would choose anegative rate of inflation in thiscase, I will not consider it here.Friedman (1969) suggestedthat the optimal monetarypolicy was to set the inflationrate to the negative of the realinterest rate, so that the nomi-nal interest rate would be zero.Consistent with the aboveanalysis, a zero nominalinterest rate would induce indi-viduals to hold the maximumamount of real money balances(since the nominal interest ratecannot be negative) and, hence,derive the maximum benefitfrom holding money. Fried-m a n ’s analysis assumes, amongother things, a zero cost of main-taining the real money stock.

9 If the time horizon is infinite,for such problems to have asolution, utility must bediscounted at a positive rate.For example, in the case ofcontinuous time, this is necessi-tated by the fact that the objec-tive functional is of the generalform

0

∞F(t, y, dy/dt)dt. If this

objective functional is not finite,there may be several paths thathave infinite values. Althoughthere are methods for isolatingthe optimal path in caseswhere the objective functionaldoes not converge, thesemethods are difficult to apply.Therefore, it is common toexpress the objective functionalin discounted form, that is,

0

∞F(t, y, dy/dt)e– βtdt.

Discounting is necessary, butnot sufficient to guarantee theconvergence of this improperintegral, however. Discountingis sufficient only if the functionthat is being discounted is

M A R C H/ AP R I L 1 9 9 6

interest rate increases approximately pointfor point with the steady-state inflation rate.

The higher the nominal interest rate,the fewer real money balances individualswish to hold. In an attempt to hold lessm o n e y, individuals reallocate their re s o u rc e s .The result may be reduced output, lessleisure time, or both. Regardless of theoutcome, economic welfare is reduced.For example, if individuals have less timefor leisure, but output is unchanged, welfarefalls. Consequently, associating the costsof inflation with output tends to biasHowitt’s Rule toward a higher inflation ratethan would be optimal if the welfare costswere correctly identified and measured.M o re generally, unless inflation is somehowbeneficial, welfare must be lower at thehigher inflation rate—even if the measuredlevel of output is unchanged.7 Hence, whenoutput is the measure of welfare, an infla-tion bias is imparted to Howitt’s Rule.

Discounting Output or UtilityHowitt’s Rule calls for discounting the

cost and benefit at a positive discount rate.Discounting is critical to Howitt’s Rule.The reason is simple: Without discounting,Howitt’s Rule would always choose zero asthe optimal inflation target. The cost ofreducing inflation (the area re p resented byL in Figure 1) is finite. This is true whetheror not the cost of reducing inflation is discounted. In contrast, the b e n e fit fro mreducing inflation is finite only if it isdiscounted. (Altern a t i v e l y, the time horizon,T, is finite. I say more about this later.)C o n s e q u e n t l y, without discounting, Howitt’sRule will always choose zero inflation nomatter how small the effect of inflation oneither the level or growth rate of output.8

Discounting future income streams is such an integral part of financial decisionmaking that the need to discount futureincome is taken for granted. Discountingutility is fundamentally different, however.It is important to distinguish it fromdiscounting commonly used in finance.

Frequently, individuals are assumed to maximize the expected value of theirlifetime utility from consumption and

leisure, given their expected lifetime earn-ings (or resource endowments).9 Thediscount rate has a specific interpretation insuch analyses. The greater an individual’spreference for current consumption (rela-tive to future consumption), the larger therate at which future income is discounted.Individuals who are impatient to consumehave higher discount rates; those who areless anxious to consume have lower discountrates.10 Consequently, the discount ratecan be interpreted as an individual’s rate of time preference.

Although it may be reasonable toassume that individuals discount theirexpected future utility relative to theircurrent utility in making optimal plans,the role of discounting in evaluatingtransfers between individuals or betweengenerations is controversial. Indeed, themathematical economist Frank P. Ramsey(1928) considered it “ethically indefen-sible” for the current generation to discountthe utility of future generations. At issueis whether it is appropriate for the currentgeneration to assert that the welfare of futuregenerations is less important than its ownwhen making policy decisions that will aff e c tthe welfare of future generations forever.

Despite Ramsey’s moral indignation,the practice of discounting utility or outputis commonplace in theoretical, as well aspractical, intergenerational public policyanalyses. Perhaps intergenerationaldiscounting is nonchalantly invokedbecause it appears to be consistent withhuman behavior.11 Nevertheless, it isimportant to note that if you believe, assome economists do, that intergenerat i o n a ldiscounting is morally re p re h e n s i b l e , youmust conclude that the optimal inflationpolicy is price stability. If interg e n e r a t i o n a ldiscounting is inappropriate, so too isHowitt’s Rule. In this event, the only basisfor favoring a policy of moderate inflationis to argue that moderate inflation actuallyenhances economic welfare.

The Discount RateDiscounting utility or output is neces-

sary but not sufficient to cause the benefit

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from reducing inflation to be finite. Thediscount rate must also be larger than the growth rate of output under stableprices.12 If β is not larger than µ, thepresent value of the benefit of reducinginflation increases without bound as T → ∞. Hence, Howitt’s Rule will alwayschoose zero as the optimal inflation rateregardless of the cost of getting there.

Given the critical nature of thisrequirement, it is natural to ask, Is itreasonable to assume that β is larger thanµ? Unfortunately, there is no definitiveanswer. Nothing in economic theoryensures that this condition will hold.Likewise, nothing prevents it from holding.In simple models of intertemporal utilitymaximization, optimization requires thatthe rate of time preference equal the realinterest rate. Theoretical models of thissort are usually solved by imposing astrong form of rational expectations calledcertainty equivalence. For this reason, therate on very long-term, default-risk-freeTreasury securities averaged over a longperiod is frequently taken as a proxy forthe rate of time preference.

The ex post real rates on 10-year and30-year Treasury bonds over the period of available data are presented in Table 1,along with the growth rates of real grossdomestic product (GDP) for the sameperiods. These figures are generallyconsistent with the assumption that therate of time preference exceeds the growthrate of output. In the case of the 10-yearbond rate for the period 1954-94, however,the reverse is true. Moreover, the differ-ence by which the long-term Treasurybond rate exceeds the growth rate ofoutput is, at most, approximately 1.3 per-centage points.

Hurd’s (1989) attempts at estimatingthe rate of time preference directly havegenerally produced much smaller estimatesof the rate of time preference. AlthoughH u rd ’s estimates vary with the specific a t i o nof the model and the estimation technique,they are generally too small to justify theassumption that β > µ. Without this con-dition, however, Howitt’s Rule will alwayschoose price stability as being optimal.

The Time HorizonWe have implicitly assumed that the

time horizon, T, is infinite. But since it ispossible to make the output gain from dis-inflation finite simply by assuming that T is finite, this possibility should beconsidered. The analysis of the discountrate applies equally well to the choice oftime horizon. The assumption that thetime horizon is finite is tantamount tostating that, beyond some point, all outputor utility is discounted to zero. Hence,choosing a finite time horizon is analogousto imposing an arbitrarily high discountrate on all gains beyond some point. It isjust as difficult to rationalize a finite timehorizon as it is to rationalize an infinitelyhigh discount rate. Consequently, if youbelieve the time horizon should be finite,you might just as well argue that thediscount rate should be large.

APPLYING HOWITT’S RULEIf the conditions needed to yield a

nonzero optimal inflation target whenapplying Howitt’s Rule are accepted, it isstill necessary to calculate the loss associ-ated with the disinflation policy and thegain associated with achieving the lowerinflation rate. Neither task is easy.

The Costs of Disinflation PolicyThe costs of disinflation policy

are associated with the stickiness of prices and wages.13 In models whereprices and wages are completely flexible, it is difficult for disinflationary mone-

bounded, that is, has a finitelimit. If the function is notbounded, the objectivefunctional will only converge if the discount rate is suffi-ciently large.

10 Thus, some economists arguethat relative savings rates areevidence of relative rates oftime preference. Individuals or countries with high savingsrates have low rates of timepreference and vice versa.

11 If concern for future genera-tions is a strong motivation, weshould expect to find a strongbequest motive in individuals’consumption and savingbehavior. Hurd (1989) findsdesired bequests to be all butnonexistent, however. Heconcludes, “apparently mostbequests are accidental, theresult of uncertainty about thedate of death.”

12 There is also a question ofwhether output should be puton a per capita basis. Mosttheoretical work is done in percapita terms. Most appliedwork in this area has beendone using aggregate output.Since there is no compellingreason to prefer one over theother, the practical conventionis followed here, except whereexplicitly noted.

13 See, for example, Aiyagari(1990).

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Real Long-Te rm Tr e a s u ry Bond Rates andReal GDP Growth Rates

Period 10-Year Rate 30-Year Rate GDP

1954-94 2.66 NA1 2.88

1978-94 3.74 3.84 2.48

1 Data not available.

Table 1

tary policy to have adverse short-runoutput effects. Even with sticky prices,however, there is general agreement that, if the monetary authority’s disinflationpolicy is credible, the cost of disinflationcan be significantly reduced and perhapseliminated.14 For such reasons, there is no consensus about the cost of disinfla t i o n .Instead, I present estimates from a pro c e d u refor estimating the cost of disinflationfrequently used by advocates of a policy of moderate inflation.15

The Cost of Reducing InflationI estimate the present value of

the output loss that has accompaniedsignificant periods of disinflation. Theestimates are for the United States and are explicitly based on the work of Ball(1994a) and Howitt (1990), but theantecedent is Okun (1978).

The first step is to identify periods ofs i g n i ficant disinflation. Ball defines a periodof significant disinflation as a period whenthe long-run or trend inflation rate falls byone percentage point or more. The nextstep is to measure the output loss. BothBall and Howitt consider the output lostduring disinfla t i o n episodes to be thedifference between what output wouldhave been in the absence of the disinfla t i o nand actual output. The approaches differonly in the way they calculate what outputwould have been.

Ball does this by connecting outputwhen the trend inflation rate is at its peakwith output four quarters after the trendinflation rate reaches its trough. He doesthis on the assumption that the effects ofdisinflation on output continue for a while after disinflation ends.

Howitt uses the unemployment rate to gauge output loss. For Howitt, the disinflation period ends when theunemployment rate returns to the level it was when the disinflation episode began. The output loss is calculated byusing Okun’s Law. Howitt assumes thateach percentage point rise in the unemploy-m e n t rate for a year costs society theequivalent of 2 percent of real GDP.16

A significant difficulty with this generalapproach to measuring the output loss ofdisinflationary policy is that all outputlosses during these periods are attributedto disinflation. It is certainly possible—indeed, many would claim likely—thatsome part of the output decline is cyclical,that is, the economy’s dynamic response to past shocks—monetary or real.

Perhaps more important, thisapproach assumes that all disinflation is attributable to monetary policy actionsand not to the simultaneous occurrence of other shocks. This is particularlytroubling since no attempt is made toconnect the disinflation experience dire c t l yto policy actions. Consequently, there isno way to determine the extent to whichthe output loss during these disinflationexperiences may be a result of otherfactors.17 Because of these difficulties, it is safe to say that this approach overstatesthe output loss associated with disinflationto a greater or a lesser degree.18

The Benefit of Lower InflationEstimating the benefit of a lower

inflation rate is no easier than estimatingthe cost of disinflation policy. Part of thedifficulty stems from the fact that moneyaffects economic welfare in many andcomplex ways. No general model of thewelfare benefit of money and, hence, nomodel of the welfare cost of inflation, exists.Rather, a large body of literature outlinesthe potential benefits of price stability.19

Nevertheless, an approach toestimating the welfare cost of inflation,suggested by Bailey (1956) and Friedman(1969), is frequently used. This methodassumes that the benefit of money accruessolely to those who hold it. Consequently,Bailey and Friedman assume there is noexternal benefit from money’s use. Underthis assumption, the benefits from holdingmoney can be represented by what econo-mists refer to as consumer’s surplus or, moreesoterically, Harberger triangles. The ideais simply that money is barren in that itpays no explicit interest, so the value ofm o n e y ’s services, that is, the welfare

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14 See, for example, Taylor(1981 and 1983).

15 See, for example, Fischer(1984) and Blinder (1989).

16 Some estimates of Okun’scoefficient put real GDP at 2.5 percent. Based on a study by Fortin and Bernier(1988), however, Howitt uses a lower figure (2 percent)to take account for the effecton leisure.

17 For a more detailed discussionof these and other problemswith Ball’s approach, seeCecchetti (1994).

18 A methodology that mitigatesagainst these problems isvector autoregession (VAR).Unfortunately VAR results arevery sensitive to the degree ofdifference in the model’s speci-fication (see Cecchetti, 1994,for an illustration of this point)and to the variables that areincluded in the specification,including the variable used to identify monetary policyshocks. VARs are subject toother criticisms as well (see, for example, Cecchetti, 1995,and Zellner, 1992).

19 See, for example, Aiyagari(1990), Briault (1995), Lucas(1994), Tatom (1976),Garfinkel (1989), Fischer(1981), and Howitt (1990).

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b e n e fit of money, is equal to the area underthe demand curve for real money.20 Thewelf a re cost of inflation is the amount byw h i c h this area shrinks when the inflationrate, and hence, the nominal interest rate,rises. Such an approach has been usedoften.21 Most estimates of the cost of infla-tion range from about 0.0001 to 0.0005 of GDP per percentage point reduction in the long-run inflation rate.

Lucas (1994) has recently presentedconvincing evidence that welfare cost ofinflation estimates based on Bailey’s andFriedman’s approach may be too low.A rguing that the data are better re p re s e n t e dby the double-log specification of moneydemand, rather than the semi-elasticityspecification commonly used and first sug-gested by Cagan (1956), Lucas estimatesthe annual welfare cost of inflation to bemuch higher than previous studies. Heestimates the benefits from reducing inflation to be about 0.002 of GDP per percentage point reduction in the inflation rate.22

Lucas’s estimates understate thewelfare cost of inflation if there are socialb e n e fits from money in addition to money’sprivate benefits. This possibility has beensuggested by Brunner and Meltzer (1971)and Laidler (1990). Elsewhere I explicitlyidentified the social benefits from money’suse.23 In addition, Briault (1995) and Iassert that if the social benefits of moneyare significant, disruptions of the monetarysystem because of inflation are potentiallymuch more serious and costly than suggestedby estimates based solely on money’sprivate benefits. If social benefits frommoney exist, the usual estimates of thecost of inflation understate the welfare cost of inflation.

Moreover, as Briault (1995) points out, arguments for why inflation and infla-tion uncertainty lead to a misallocation of resources suggest that inflation may not only affect the level of output, but mayaffect the growth rate of output as well.Indeed, there has been increasing intere s t —both theoretical and empirical—in thepossibility that inflation reduces out-put growth.

Comparing the Costs and Benefits

The estimates of the cost of disinfla t i o n ,PVL, and the benefit, PVG, are for the threed i s i n flation episodes Ball (1994a) identifie s .They are 1969:4-1971:4, 1974:1-1976:4,and 1980:1-1983:4. The average rate ofoutput growth for the United States was3.01 percent from 1960:1 to 1995:3.Consequently, the estimates presented in Tables 2, 3, and 4 are based on theassumption that the growth rate in the“high” inflation state, α, is 3 percent (0.03).The estimates of the effect of inflation onthe level of output as a proportion of cur-rent period income per percentage pointreduction in the inflation rate, θ, reflectestimates obtained in the literature andrange from θ = 0.0001 to 0.002. The esti-mates of PVG are calculated using thef o rmula presented in Appendix A. To makethe estimated cost and benefit as comparableas possible, PVG is based on the length of each disinflation episode and the corre-sponding reduction in the inflation rateidentified by Ball (1994a).

A comparison of PVL and PVG inTable 2 shows that PVL > PVG only if theeffect of inflation on the level of output isassumed to be relatively small and the dis-count rate is taken to be relatively large.Indeed, the importance of the discountrate in deciding that moderate inflation isoptimal is apparent. Table 2 assumes thereis no growth rate effect, µ = α. When thediscount rate is only slightly larger thanthe growth rate of output, PVG exceedsPVL even for an extremely small effect ofinflation on output. For example, if thediscount rate is assumed to be about half a perc e n t a g e point larger than the gro w t hrate of output, PVG > PVL for each of the disinflation episodes that Ball hasidentified, even if the percentage pointreduction in the level of output is only0.0001. When the likelihood that estimatesof PVL overstate the w e l f a re cost andestimates of PVG u n d e r s t a t e the welfarebenefits of disinflation policy a re taken into consideration, these estimates suggestthat a policy of moderate inflation isoptimal only if the effects of inflation on

20 Failure to understand thenature of the welfare costs of inflation has led Aiyagari(1990) to suggest that theycould be reduced simply byallowing “more forms ofmoney that are used in transac-tions to earn market rates ofinterest.” This recommenda-tion fails to recognize that anasset which provides multipleservices has a return thatreflects the marginal value ofeach of the services rendered.This insight is the basis for theDivisia monetary aggregates, or monetary services indexes,pioneered by Barnett (1980).Inflation would cause the rateson assets that provided rela-tively more monetary servicesto rise less than those whichprovide fewer monetaryservices. As a result, inflationwould continue to induce indi-viduals to reduce their use ofmonetary services.

21 See, for example, Cooley and Hansen (1989), Fischer(1981), and Lucas (1981 and 1994).

22 Lucas (1994) estimates that ata 10 percent nominal interestrate, the output loss per annumis equal to 1.3 percent ofcurrent output. Assuming thereal interest rate is 3 percent, a nominal interest rate of 10 percent is equivalent to asteady-state inflation rate of 7 percent. The proportionatedecline in output per percent-age point drop in the inflationrate is therefore equal to0.0018 [0.0130/7] ≈ 0.002.

23 See Thornton (1995).

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output are extremely small or the discountrate is large.

Estimates based on Howitt’s procedureare reported in Table 3. These estimatesare only presented for the third disinflationepisode. The reason is that Howitt’s proce-dure requires the unemployment rate toreturn to the level at the beginning of thedisinflation episode. The unemploymentrate was 3.6 percent in 1969:4 and 5.1 per-cent in 1974:1. The unemployment ratehas yet to return to either of these levels,so by Howitt’s criterion, these disinflationepisodes have yet to end. Of course, theproblem is that the unemployment rate isaffected by factors other than disinflation.The secular effects on the unemploymentrate clearly have been dominant. Because

of this, the unemployment rate has yet toreturn to levels reached in the late 1960s.This fact serves to underscore a problemwith both Howitt’s and Ball’s procedures.Namely, the methods assume that all thedecline in employment and output isbecause of disinflation and not because of other factors.

A comparison of PVL and PVG inTable 3 yields conclusions similar to thosereached in Table 2. The PVLs in Table 3are considerably larger than those of thecorresponding disinflation in Table 2primarily because the length of the disinfla-t i o n episode was much longer 7.25 years(compared with 3.75 years in Table 2), andoutput was assumed to be below its trendlevel during the entire period.

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Costs and Benefits of Disinflation: Ball’s Approach

PVL PVG1

1969:4- 1974:1- 1980:1- 1969:4- 1974:1- 1980:1-1971:4 1976:4 1983:4 1971:4 1976:4 1983:4

0.0001 0.0305 0.0118 0.0477 0.0973 0.4276 0.7989 1.76270.0001 0.0350 0.0118 0.0474 0.0964 0.0424 0.0789 0.17330.0001 0.0400 0.0117 0.0471 0.0955 0.0210 0.0389 0.08510.0001 0.0450 0.0117 0.0469 0.0945 0.0138 0.0256 0.0556

0.0002 0.0305 0.0118 0.0478 0.0973 0.8551 1.5978 3.52540.0002 0.0350 0.0118 0.0474 0.0964 0.0847 0.1578 0.34660.0002 0.0400 0.0117 0.0471 0.0955 0.0420 0.0778 0.17010.0002 0.0450 0.0117 0.0469 0.0945 0.0277 0.0512 0.1113

0.0010 0.0305 0.0118 0.0478 0.0973 4.2757 7.9890 17.62690.0010 0.0350 0.0118 0.0474 0.0964 0.4237 0.7891 1.73320.0010 0.0400 0.0117 0.0471 0.0955 0.2098 0.3891 0.85050.0010 0.0450 0.0117 0.0469 0.0945 0.1385 0.2559 0.5565

0.0020 0.0305 0.0118 0.0478 0.0973 8.5514 15.9780 35.25380.0020 0.0350 0.0118 0.0474 0.0964 0.8475 1.5782 3.46640.0020 0.0400 0.0117 0.0471 0.0955 0.4195 0.7783 1.70100.0020 0.0450 0.0117 0.0469 0.0945 0.2769 0.5118 1.1129

1 The estimates are based on the duration of the disinflation episode and the reduction in the inflation rates for each episode as reported by Ball(1994a). The length of the disinflation episode, (that is, t1–t0, are 2.00, 2.75, and 3.75, respectively). The decline in the inflation rates are2.14, 4.00, and 8.83 percent, respectively.

θ β

Table 2

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31

M A R C H/ AP R I L 1 9 9 6

If Inflation Affects Output Growth

The conclusion reached by applyingHowitt’s Rule changes dramatically ifinflation affects the growth rate of output.Estimates of PVG under the assumptionthat inflation reduces the growth rate ofoutput are presented in Table 4. Since thediscount rate, β, must be larger than thelow-inflation output growth rate, µ, therange of values considered for β is smallerthan that considered in Tables 2 and 3.Also, since estimates of PVG are notgreatly affected by changes in the length of the disinflation episode, the valuesreported are for a disinflation episode of 3.75 years.

These estimates show that the gainfrom reducing inflation is very large, evenif inflation reduces output growth by aslittle as one-thousandth of 1 percent(0.00001). Other things being equal,Howitt’s Rule moves decidedly in thedirection of choosing a lower inflationtarget if inflation affects the growth rate of output.

The level and growth rate effects areadditive. Hence, if inflation reduces boththe level and growth rate of output, evenby relatively small amounts, the PVG fromreducing inflation could be very large,depending on the discount rate used. For example, assume that the inflation rate is cut from its current rate of about 3 percent to zero and this action raises thelevel of output by 0.0003 (a 0.0001 perpercentage point drop in the inflation rate)and increases output growth by 0.00001.Assuming disinflation takes 3.75 years to be achieved and that the discount rate is a fairly large 4 percent, the PVG fromachieving price stability would equal 0.125(the sum of a level effect of 0.0289 and agrowth rate effect of 0.0963). If we use the estimates of PVL per percentage pointreduction in inflation using Howitt’smethod from Table 3, the PVL would only be 0.0772. The difference is 0.0478.Annualized output in the second quarterof 1995 was approximately $5.5 trillion.Consequently, the net present value of the gain from reducing inflation from its

Table 3

Costs and Benefits of Inflation: H o w i t t ’s Approach1

θ β PVL 2 PVG3

0.0001 0.0305 0.2341 1.75960.0001 0.0350 0.2309 0.17030.0001 0.0400 0.2273 0.08210.0001 0.0450 0.2238 0.0528

0.0002 0.0305 0.2341 3.51920.0002 0.0350 0.2309 0.34060.0002 0.0400 0.2273 0.16420.0002 0.0450 0.2238 0.1056

0.0010 0.0305 0.2341 17.59610.0010 0.0350 0.2309 1.70310.0010 0.0400 0.2273 0.82120.0010 0.0450 0.2238 0.5280

0.0020 0.0305 0.2341 35.19220.0020 0.0350 0.2309 3.40630.0020 0.0400 0.2273 1.64250.0020 0.0450 0.2238 1.0560

1 The disinflation period is 1980:1–1987:2.2 The PVL does not change with θ.3 The length of the disinflation episode was 7.25; however, the decline in the steady-state

inflation rate was 8.83 percent (as assumed by Ball, 1994a).

Table 4

P V G When Inflation Affects Output Growth

µ β PVG 1

0.03001 0.035 0.39330.03001 0.040 0.09630.03001 0.045 0.0420

0.03010 0.035 4.00510.03010 0.040 0.97230.03010 0.045 0.4223

0.03100 0.035 49.06440.03100 0.040 10.69550.03100 0.045 4.4950

1 These estimates are based on the assumption that α = 0.03 and t1–t0 = 3.75.

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MA R C H/ AP R I L 1 9 9 6

current level of about 3 percent to zerowould be about $0.27 trillion, or about$1,000 per person.

EMPIRICAL EVIDENCE ONINFLATION AND GROWTH

The above analysis shows that animportant factor in determining whetherthe benefit of price stability exceeds thecost is whether inflation affects outputgrowth. If it does, it is very difficult to argue that a policy of maintainingmoderate inflation is socially optimal by appealing to Howitt’s Rule. Hence, it is not surprising that a relatively largeeffort has been devoted to this question.24

Although there is evidence that infla-tion reduces output growth, it appears tobe the case only at relatively high inflationrates. It is becoming increasingly clearthat the effect of inflation on growth isstatistically significant only at relativelyhigh rates of inflation.25 For example,Barro finds a statistically significant effectonly when inflation is 15 percent orhigher. Similarly, Bruno and Easterly find a statistically significant inflationeffect only when the inflation rate exceeds40 percent for two or more years.26

The evidence is controversial; how-ever, the important point is that thisevidence cannot support or reject the ideathat moderate inflation affects the growthrate of output. The reasons are twofold.

First, the data contain relatively fewobservations where the inflation rate is in the low to moderate range and evenfewer observations where inflation is zero.Extrapolating statistical results outside theregion for which the data are relevant isrisky. Consequently, using the results ofsuch data to infer the effect of moderateinflation on growth is tenuous at best.This is especially true if the effect of infla-tion on growth is believed to be nonlinear,that is, the effect of inflation on growth is disproportionately larger or smaller at lower rates of inflation.

It is commonly believed that theeffects of inflation will be proportionatelysmaller at lower rates of inflation than at

higher rates.27 Recently, however, Lucas(1994) presented evidence that the effectof inflation on output is proportionatelylarger at lower rates of inflation. It is safeto say that any statement about the exactnature of the nonlinearity at low rates ofinflation is conjectural. There is some evi-dence, however, that the effects at highrates of inflation are bounded. That is,beyond some point, higher inflation rateshave no further effect on either the level orgrowth rate of output. If the effects arebounded at high inflation rates and onebelieves that the effects of inflation areproportionately smaller at lower inflationrates, then there must be an inflectionpoint—the effects of inflation first increaseat an increasing rate and then increase at adecreasing rate.

Second, because an economically rele-vant growth rate effect can be very small, it may be difficult to find an economicallyrelevant growth rate effect that is alsostatistically significant. For example,Barro (1995) investigated the nonlinearityof the inflation effect by partitioning themedian inflation rates into those up to 15 percent, 15 percent to 40 percent, andabove 40 percent. His estimates of theeffect of inflation on growth for the threepartitions were –0.00016, –0.00037, and–0.00023, respectively. Only the last twocoefficients were significantly differentfrom zero. My previous analysis showedthat the estimate for the first partition(–0.00016) is large enough to be economi-cally relevant. Its absolute value, however,is less than one-half its estimated standarderror of 0.00035. Given the amount ofvariability in data like these, it may be very difficult to obtain an economicallyrelevant, statistically significant estimate of inflation’s effect on output growth.

Time-Series EvidenceRecently, Ericsson, Irons, and Tryon

(1993) criticized cross-sectional studies ofthe effects of inflation on growth, arguingthat the results are very sensitive to a fewvery high inflation countries and thattime-averaged data can give misleading

24 See, for example, Wallich(1969); Barro (1995 andforthcoming); Bruno (1995);Jones and Manuelli (1995);Mankiw, Romer, and Weil(1992); Logue and Sweeney(1981); Kormendi andMeguire (1985); Grier andTullock (1989); Bruno andEasterly (forthcoming); Levineand Renelt (1992); Levine,Renelt, and Zervos (1993);and Fischer (1993). All theseempirical studies use cross-sectional or panel data.

25 See Bruno and Easterly (forth-coming) and Barro (1995 and forthcoming).

26 Although these results arebroadly consistent with those of Levine and Renelt (1992);Levine, Renelt, and Zervos(1993); and Fischer (1993),who showed that the effect ofinflation on growth is due tothe presence of a few highinflation countries, Barro andBruno and Easterly presentevidence that their results arenot sensitive to a few highinflation countries.

27 See, for example, Howitt(1990) and DeLong andSummers (1992).

results about the causal relationshipbetween inflation and output growth.Instead they use time-series data for theUnited States for the period 1953-90.Using a statistical procedure called co-integration analysis, Ericsson, Irons, and Tryon find evidence that inflation does not affect the long-run rate of outputgrowth.28 Instead, they find that the levelof output is positively related to the long-run inflation rate. Their result is circ u m-spect, however. Taken literally, it impliesthat not only is a little inflation good forthe economy, but that a lot of inflation is even better.

More important, their finding dependscritically on their assumption about thetime-series properties of the price level. It can be shown that different assumptionsabout the time-series properties of theprice level yield considerably differentresults. Appendix B gives the details. Inparticular, it can be shown that long-runinflation and output growth are negativelyrelated. The estimated long-run relation-ship between per capita output and theprice level is:

(1) Xt = –0.0695Pt + 0.0217T,

where X is the natural log of real per capitaoutput, P is the natural log of the implicitprice deflator, and T is a deterministic timetrend. Equation 1 suggests that the long-run growth rate of output is equal to0.0217 less 0.0695 times the inflation rate.The inflation rate over this period was 4.37 percent. Using this figure, per capitaoutput growth is estimated to be 1.87 per-cent, somewhat larger than the actualgrowth rate of per capita output of 1.68 percent over this period.

This analysis is very simple, includingonly output and the price level. Moreover,co-integration analysis cannot indicatewhether the negative long-run relationshipbetween the price level and output is aresult of the effects of inflation on outputgrowth, the effect of output growth oninflation, or the effect of some othervariable(s) on both output and prices.Nevertheless, it is interesting to note that

a very reasonable change in the specifica-tion of Ericsson, Irons, and Tryon’s modeldramatically alters their result. The datasuggest that, for the United States (overthis period), inflation could have had a detrimental effect on output growth.Hence, the time-series evidence is at leastconsistent with the proposition that infla-tion reduces the growth rate of output.

IS PRICE STABILITY THE BEST POLICY?

Most policymakers and economistsagree that very high rates of inflation are detrimental. So much so, that there is broad agreement that the gain fromreducing inflation from “high” levels to“moderate” levels warrants the temporaryloss of employment and output that manybelieve is necessary to achieve this goal.Despite the fact that most economistsbelieve inflation is detrimental to economicwelfare, many believe that, once underway, it is better to live with some moderateinflation than to bear the costs necessaryto achieve stable prices. Peter Howitt(1990) succinctly states the propositionthat the cost of going from moderate tozero inflation does not warrant the benefitsfrom price stability. Howitt’s Rule statesthat a policy of disinflation should be con-tinued until an inflation rate is reachedwhere the present value of the costs of further disinflation are just equal to thepresent value of the gains from a furtherreduction in the inflation rate.

Does economic analysis support or refute the idea that it is desirable toestablish a policy of price stability? Unfor-tunately, there is no definitive answer tothis question. The analysis presented here,however, suggests that a popular and (formany) a compelling economic argumentfor a policy of positive inflation requiresseveral controversial conditions. If onebelieves that the welfare gains of futuregenerations should not be discounted orthat the discount rate is low, it is difficultto argue for a policy of moderate inflation.

Moreover, this argument is weakenedsignificantly if inflation reduces the growth

28 See Dickey, Jansen, andThornton (1991) for adiscussion of co-integration and unit roots.

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33

rate and not simply the level of output.The economic case for a policy of pricestability based on the effects of inflation onoutput growth may come down to whetherone finds compelling various theoreticalarguments of why even moderate inflationshould affect output growth.29 If youbelieve it is likely that inflation reduces the growth rate of output, you are muchmore likely to believe that price stability is the best monetary policy.

REFERENCESAiyagari, S. Rao. “Deflating the Case for Zero Inflation,” Federal Reserve

Bank of Minneapolis Quarterly Review (Summer 1990), pp. 2-11.

Bailey, Martin J. “The Welfare Cost of Inflationary Finance,” Journal of Political Economy (April 1956), pp. 93-110.

Ball, Laurence. “What Determines the Sacrifice Ratio?” in MonetaryPolicy, N. Gregory Mankiw, ed. University of Chicago Press, 1994a,pp. 155-82.

_______. “Credible Disinflation With Staggered Price-Setting,”American Economic Review (March 1994b), pp. 282-89.

Barnett, William A. “Economic Monetary Aggregates: An Application of Index Number and Aggregation Theory,” Journal of Econometrics(Summer 1980), pp. 11-48.

Barro, Robert J. “Inflation and Growth,” this Review, forthcoming.

_______. “Inflation and Economic Growth,” Bank of EnglandQuarterly Bulletin (May 1995), pp. 166-76.

Blinder, Alan S. “The Economic Myths that the 1980s Exploded,”Business Week (November 27, 1989), p. 22.

Briault, Clive. “The Costs of Inflation,” Bank of England QuarterlyBulletin (February 1995), pp. 33-45.

Brunner, Karl, and Allan H. Meltzer. “The Uses of Money: Money in the Theory of an Exchange Economy,” American Economic Review(December 1971), pp. 784-805.

Bruno, Michael. “Does Inflation Really Lower Growth?” Finance andDevelopment (September 1995), pp. 35-38.

_______ and William Easterly. “Inflation and Growth: In Search of a Stable Relationship,” this Review, forthcoming.

Cagan, Philip. “The Monetary Dynamics of Hyperinflation,” in Studies inthe Quantity Theory of Money, Milton Friedman, ed. University ofChicago Press, 1956.

Cecchetti, Stephen G. “What Determines the Sacrifice Ratio?”Comment, in Monetary Policy, N. Gregory Mankiw, ed. University of Chicago Press, 1994, pp. 188-93.

_______.“Distinguishing Theories of the Monetary TransmissionMechanism,” this Review (May/June 1995), pp. 83-97.

Clark, Todd E. “Cross-Country Evidence on Long Run Growth andInflation,” unpublished manuscript, Federal Reserve Bank of Kansas City, 1993.

Chari, V.V.; Larry E. Jones; and Rodolfo E. Manuelli. “The Growth Effectsof Monetary Policy,” Federal Reserve Bank of Minneapolis QuarterlyReview (Fall 1995), pp. 18-32.

Cooley, Thomas F., and Gary D. Hansen. “The Inflation Tax in a RealBusiness Cycle Model,” American Economic Review (September1989), pp. 733-48.

DeLong, J. Bradford, and Lawrence H. Summers. “Macroeconomic Policyand Long-Run Growth,” Federal Reserve Bank of Kansas City EconomicReview (Fourth Quarter 1992), pp. 5-30.

Dickey, David A.; Dennis W. Jansen; and Daniel L. Thornton. “A Primeron Cointegration with an Application to Money and Income,” thisReview (March/April 1991), pp. 58-78.

Dotsey, Michael, and Peter N. Ireland. “On the Welfare Cost of Inflationin General Equilibrium,” unpublished manuscript, Federal ReserveBank of Richmond, 1993.

Ericsson, Neil R.; John S. Irons; and Ralph W. Tryon. “Output andInflation in the Long-Run,” unpublished manuscript, Board ofGovernors of the Federal Reserve, September 1993.

Fischer, Stanley. “The Role of Macroeconomic Factors in Growth,”Journal of Monetary Economics (December 1993), pp. 485-512.

_______. “Contracts, Credibility and Disinflation,” NBER workingpaper, No. 1339, April 1984.

_______. “Towards an Understanding of the Costs of Inflation: II,”in The Costs and Consequences of Inflation, Karl Brunner and Allan H.Meltzer, eds. Carnegie-Rochester Conference Series on Public Policy(No. 15, 1981), pp. 5-41.

Fortin, Bernard, and Andre Bernier. “The Welfare Cost of Unemploymentin Quebec: Harberger’s Triangle Meets Okun’s Gap,” Canadian Journalof Economics (February 1988), pp. 161-81.

Friedman, Milton. “The Optimum Quantity of Money,” in The OptimumQuantity of Money and Other Essays. Aldine Publishing Co., 1969.

Garfinkel, Michelle R. “What is an ‘Acceptable’ Rate of Inflation?–A Review of the Issues,” this Review (July/August 1989), pp. 3-15.

Golob, John E. “Inflation, Inflation Uncertainty, and Relative PriceVariability: A Survey,” unpublished manuscript, Federal Reserve Bankof Kansas City, August 1993.

Grier, Kevin B., and Gordon Tullock. “An Empirical Analysis of Cross-National Economic Growth, 1951-80,” Journal of MonetaryEconomics (September 1989), pp. 259-76.

Howitt, Peter. “Zero Inflation as a Long-Term Target for MonetaryPolicy,” in Zero Inflation: The Goal of Price Stability, Richard G. Lipsey,ed. C. D. Howe Institute, 1990.

Hurd, Michael D. “Mortality Risk and Bequests,” Econometrica (July 1989), pp. 779-813.

Ireland, Peter N. “Optimal Disinflationary Paths,” Journal of EconomicDynamics and Control (November 1995), pp. 1,429-48.

29 Marty and Thornton (1995)have argued that monetarypolicies motivated by concernsfor transitional unemploymentare likely to lead to an infla-tionary bias that makestargeting any moderateinflation rate difficult.

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34

Johansen, Soren. “Statistical Analysis of Cointegration Vectors,” Journal of Economic Dynamics and Control (June–September 1988),pp. 231-54.

Jones, Larry E., and Rodolfo E. Manuelli. “Growth and the Effects ofInflation,” Journal of Economic Dynamics and Control (November1995), pp. 1,405-28.

Keating, John W. “Structural Approaches to Vector Autoregressions ,”this Review (September/October 1992), pp. 37-57.

King, Robert G., and Sergio Rebelo. “Public Policy and EconomicGrowth: Developing Neoclassical Implications,” Journal of PoliticalEconomy (October 1990, Part 2), pp. S126-50.

Kormendi, Roger C., and Philip G. Meguire. “MacroeconomicDeterminants of Growth: Cross-Country Evidence,” Journal ofMonetary Economics (September 1985), pp. 141-63.

L a i d l e r, David. Taking Money Seriously and Other Essays. MIT press, 1990.

Levine, Ross, and David Renelt. “Cross-country Studies of Growth and Policy: Methodological, Conceptual, and Statistical Problems,”Policy, Research and External Affairs Working paper WPS 608, WorldBank, 1991.

_______ and _______. “A Sensitivity Analysis of Cross-CountryGrowth Regression,” American Economic Review ( September 1992),pp 942-63.

_______, _______, and Sara J. Zervos. “What We Have LearnedAbout Policy and Growth from Cross-Country Regressions?” AmericanEconomic Review (May 1993), pp. 426-30.

Logue, Dennis E., and Richard James Sweeney. “Inflation and RealGrowth: Some Empirical Results: A Note,” Journal of Money, Creditand Banking (November 1981), pp. 497-501.

Lucas, Robert E. Jr. “Discussion of the Fischer Paper,” in The Costs andConsequences of Inflation, Karl Brunner and Allan H. Meltzer, eds.Carnegie-Rochester Conference Series on Public Policy (No. 15,1981), pp. 43-52.

_______. “On the Welfare Cost of Inflation,” unpublishedmanuscript, February 1994.

Mankiw, N. Gregory; David Romer; and David N. Weil. “A Contributionto the Empirics of Economic Growth,” The Quarterly Journal ofEconomics (May 1992), pp. 407-37.

Marty, Alvin L., and Daniel L. Thornton. “Is There a Case for ‘Moderate’Inflation?” this Review (July/August 1995), pp. 27-37.

Okun, Arthur M. “Efficient Disinflationary Policies,” American EconomicReview (May 1978), pp. 348-52.

Orphanides, Athanasios, and Robert M. Solow. “Money, Inflation andGrowth,” in Handbook of Monetary Economics, Vol. I, B.M. Friedmanand F.H. Hahn, eds. Elsevier Science Publisher, 1990, pp. 224-61.

Phelps, Edmund S. “Disinflation without Recession: Adaptive Guidepostsand Monetary Policy,” in Studies in Macroeconomic Theory. AcademicPress, 1979.

Ramsey, Frank P. “A Mathematical Theory of Saving,” Economic Journal(December 1928), pp. 543-59.

Summers, Lawrence H. “How Should Long-Term Monetary Policy BeDetermined? Panel Discussion,” Journal of Money, Credit and Banking(August 1991, Part 2), pp. 625-31.

Tatom, John A. “The Welfare Cost of Inflation,” this Review (November1976), pp. 9-22.

Taylor, John B. “On the Relationship Between the Variability of Inflationand the Average Inflation Rate,” Carnegie-Rochester Conference Serieson Public Policy, (No. 15, 1981), pp. 57-85.

_______. “Union Wage Settlements During a Disinflation,”American Economic Review (December 1983), pp. 981-93.

Thornton, Daniel L. “Money in a Theory of Exchange,” unpublishedmanuscript, Federal Reserve Bank of St. Louis, 1995.

Wallich, Henry C. “Money and Growth: A Country Cross-SectionAnalysis,” Journal of Money, Credit and Banking (May 1969), pp. 281-302.

Zellner, Arnold. “The Cowles Commission Approach, Real Business CycleTheories, and New-Keynesian Economics: Commentary,” in TheBusiness Cycle: Theories and Evidence: Proceedings of the SixteenthAnnual Economic Policy Conference of the Federal Reserve Bank of St. Louis, Michael T. Belongia and Michelle R. Garfinkel, eds. KluwerAcademic Publishers, 1992, pp. 148-57.

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Let the level of steady-state output ateach point in time at the current “high”inflation rate be given by:

(A1) yt = eαt,

where α is the growth rate of real outputgiven the high inflation state.

When calculating the effect of apermanent reduction in the inflation rate,assume the monetary authority imple-ments the deflationary policy at time t0,when the level of output is y0, and that the new lower inflation policy is achievedat time t1. We consider the possibility that reduced inflation raises the level of output and the possibility that itpermanently raises the growth rate ofoutput, once the new lower inflation rate is achieved. For the level effect,output is unchanged until time t1 but is higher by an amount δ for all t > t1. It is convenient to express δ as a pro-portion of output at the time that thedisinflation policy is implemented, that is, δ = θy0. For the growth rate effect,output is assumed to grow at the rate αuntil time t1 and at the rate µ for all t > t1,where µ > α. Under these assumptions, the low-inflation output path is given by:

( A 2 )

If output is discounted at the rate β > µ for all t greater than t0, the differencebetween low- and high-inflation output,yields the present value of the gain,

(A3)

Dividing through by y0 and performing theintegration, yields:

(A4)

Because we have no interest in the casewhere the time horizon is finite, take thelimit of the above expression as T goes toinfinity. This yields:

(A5)

The first term on the right side ofequation A5 is the effect of a change in thegrowth rate. Hence, it is zero if µ = α (thatis, there is no growth-rate effect). Thesecond term on the right side is the effectof a shift in the level of output. This termis zero only if θ = 0. The first term is anincreasing function of µ – α and µ – β(recall that µ – β is strictly negative underthe assumptions stated) and a decreasingfunction of t1. The second term is anincreasing function of θ and µ – β and adecreasing function of t1. The importanceof the size of the discount rate relative tothe growth rate of output is clear from thisexpression. Both terms on the right side ofequation A5 approach +∞ as β approachesµ. If β is only slightly larger than µ, PVGgets very large.

T→∞lim itPVG =

e ( – )t1( – )e ( – )t1 – ( – )e ( – )t1

( – )( – )

–e ( – )t1

–

.

PVG = e ( – )(t1– t0 )

e ( – )T

( – )

–

e ( – )T

( – )

+( – )e ( – ) t1 – ( – )e ( – ) t1

( – )( – )

+e ( – )T – e ( – ) t1

–

.

pvg = ∫0

T

e t(yt*– yt )dt = y0e ( – )(t1 – t0 )

[ ∫t1

T

e ( – )tdt – ∫t1

T

e ( – )tdt] + ∫t1

T

e ( – )tdt.

yt*=

y0e t, for t0 ≤ t ≤ t1

[y0e(t1 – t0 ) + ]e t, for t > t1

.

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Appendix A

THE BENEFITS OF PRICE STABILITY

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Ericsson, Irons, and Tryon (1993)recently criticized cross-sectional studiesof inflation’s effects on growth. They arguethat the results are very sensitive to a fewvery high inflation countries and arguethat time-averaged data can give misleadingresults about the causal relationship betweeninflation and output growth. Moreover,using tests for unit roots and co-integration,they find evidence that generally does notsupport the theory that inflation has aneffect on growth. Indeed, their results forthe United States suggest that, in the longrun, inflation raises the level of outputwithout affecting the growth rate ofoutput. Taken literally, Ericsson, Irons,and Tryon’s results suggest that not only is a little inflation good for the economy,but that a lot of inflation is even better.

TESTS FOR A UNIT ROOTTheir analysis, however, depends

critically on tests for a unit root,supporting the notion that the real percapita output, X, is integrated of order 1, I(1), while the price level, P, is I(2).These tests have difficulty rejecting thenull hypothesis of a unit root when theestimated root is close to, but perhapsdifferent from, unity, as illustrated in Table B1. It shows the results fromAugmented Dickey-Fuller (ADF) tests of aunit root for X, ∆X, P, and ∆P and the cor-responding estimates of the root, λ, usingannual data for the period 1953-90. Asdid Ericsson, Irons, and Tryon, I includeda constant and a deterministic time trendin each case and used ADF tests with twolags.

The null hypothesis of a unit root isnot rejected even when the estimated rootis quite different from zero, as shown inTable B1. This is true for X and ∆P, wherethe estimated roots are 0.7006 and 0.8262,respectively. This is because of the well-known fact that such tests lack the abilityto reject the null hypothesis when it is false.1

NONSTATIONARYVARIABLES

How this problem affects the determi-nation of whether X or ∆P is nonstationaryis illustrated in Figure B1, which shows Xand ∆P over the sample period. The ADFtest cannot reject the null hypothesis of aunit root for both X and ∆P. In the case of X, Figure B1 makes it clear that thisvariable is nonstationary in the sense thatit rises fairly steadily. Hence, both the testand an inspection of the data suggest thatthe series is nonstationary.2

From a visual inspection of ∆P, it is much less clear that the inflation rate is nonstationary in the sense that it willwander off indefinitely. The problem is

Appendix B

TIME-SERIES EVIDENCE ON INFLATION’S EFFECTS ON GROWTH

1 This is referred to as the powerof the test. Hence, the Dickey-Fuller test is said to lack power.

2 Some care must be exercisedhere, however, because thedata could be stationary arounda deterministic time trend.Because our test included aconstant and a trend variable,we are more inclined to acceptthe test results in this case.

Table B1

Results of Augmented Dickey-Fuller, Unit Root Te s t s

Critical Value,Variable Estimate Root ADFt-Statistic 5% Level

X 0.7006 –2.452 –3.461

∆X –0.0548 –3.6801

P 0.9508 –2.565

∆P 0.8262 –1.496

1 Indicates the null hypothesis of a unit root is rejected at the 5 percent significance level.

Figure B1

that there is a lot of persistence in the inflation rate. When the inflation rate increases, it continues to increase for a period before it begins to fall. When falling, it continues to fall for a period before it begins to rise. It is this persistence in the inflation rate that accounts for the large estimated root. Consequently, the test results that suggest that ∆P isnonstationary must be viewed with some skepticism.3

CO-INTEGRATIONANALYSIS

Given the uncertainty of the order of integration of P, it is important toinvestigate whether Ericsson, Irons, and Tryon’s results are sensitive to theirclaim that P is I(2). To investigate this, I performed their co-integration analysiswith their specifications and using both X and ∆P and X and P.

The method of testing for co-integrationis that of Johansen (1988). This approachstarts from a very general vector auto-regression representation of the form

(B1) Yt = β(L)Yt–1 + εt,

where Y is a two-by-one vector, β(L) is apolynomial in the lag operator, L, that is,β(L) = β0 + β1 L + β2L2 + . . . + βk Lk andLz t = z t– 1. Equation B1 can be rewritten as,

(B2) ∆Yt = Γ(L)∆Yt–1 + ΠYt–k + εt.

If the elements of the Y vector, y1 andy2, are I(1), they are co-integrated, that is,

there exists a linear combination of y1 andy2 that is I(0), if the rank of the two-by-two matrix Π is 1.4 The Johansen test forco-integration is a test of the rank of Π.

Given the uncertainty about whether P is I(1) or I(2), tests for co-integrationwere performed for two specifications of Y′, namely, Y′ = (X ∆P) and Y′ = (X P). In both cases, a constant term and adeterministic time trend were includedand k was set at 2. The results are sum-marized in Table B2. Regardless of thespecification of Y, there is evidence of a single co-integrating vector.5

In the case where Y′= (X ∆P), theestimated normalized co-integrating vector was X = 1.0581∆P + 0.0179T, where T denotes the deterministic timetrend. This estimate is very close to that reported by Ericsson, Irons, andTryon. They interpret the estimated co-integrating vector as the long-runrelationship between output and inflation.The implication of this co-integratingvector—that output could be increasedsimply by increasing the inflation rate—is preposterous.

In contrast, the estimate of the co-integrating vector when Y′ = (X P) is X = –0.0695P + 0.0217T. This estimatesuggests that the level of output falls as the price level rises. Because of constantcoefficients and the log-linear specifica-tion, this result implies that the rate ofoutput growth falls as the inflation raterises. The estimate suggests that a reduc-tion of the inflation rate from 3 percent tozero would increase the per capita outputgrowth rate by 0.00209, an amount that,though small, is economically significant.

3 The same conclusion applies toaccepting the hypothesis that X is nonstationary as opposedto having a deterministic trend.The null hypothesis of a unitroot is rejected because there is considerable persistence inthe swings of X about its trendrather than getting farther andfarther from its estimatedtrend. Plotting X about itstrend over this period makesthis point clear.

4 Note that the rank of Π couldonly be two if the elements ofY are I(0). If the rank of Π iszero, there would be no linearcombination of the elements of Y that is stationary [that is,I(0)]. See Dickey, Jansen, andThornton (1991) for a moredetailed discussion of co-inte-gration and the Johansen andother tests for co-integration.

5 The level specification [Y ′ = (X P )] is preferable on theo-retical grounds becauseeconomic theory suggests thatthere should be a contempora-neous relationship between X and P, that is, output and the price level are determinedsimultaneously. Indeed, equa-tion B1 can be viewed as thereduced form of a structuralmodel of the form, AYt =B(L )Yt – 1 + vt, so that, β(L ) =A –1B(L) and et = A –1vt. Theproblem is that it is difficult toobtain estimates of the struc-tural parameters in A withoutimposing some rather severerestrictions. See Keating(1992) for a discussion of these restrictions.

MA R C H/ A P R I L 1 9 9 6

FE D E R A L RE S E RV E BA N K O F S T. LO U I S

38

Estimate Eigenvalues and Co-Integrating Vectors for Two Specifications of Equation B2

Eigenvalue X ∆P Trend Eigenvalue X P Trend

0.4399 –1.0000 1.0581 0.0179 0.4502 –1.0000 –0.0695 0.0217

0.0900 –0.3757 –1.0000 0.0075 0.1851 –0.2424 –1.0000 0.0615

Table B2